Centrifugal Pump Vibration Training Module

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Engineering Technical Training Modules for Nuclear

Plant Engineers

Mechanical Series: Module #14

Centrifugal Pump Vibration

1010792

Engineering Training Modules for Nuclear Plant Engineers

Mechanical Series: Module #14

Centrifugal Pump Vibration

TR-1010792

Technical Report, March 2005

EPRI Project Manager(s)

T. Eckert

M. Hooker

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EPRI Licensed Material

T

ABLE OF

C

ONTENTS

1.0 S COPE AND P URPOSE .

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2.0 S UGGESTED S KILLS AND K NOWLEDGE .

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3.0 O BJECTIVES .

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4.0 N OMENCLATURE .

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5.0 P RINCIPLES AND P ROPERTIES .

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. 12

5.1 Pump Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2 Basic Vibration Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Pump Failure Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Basic Vibration Diagnostic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.5 Acceptance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6 Pump Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 68

6.0 N UCLEAR C ONSIDERATIONS .

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7.0 U TILITY E XAMPLE E XERCISES AND S OLUTIONS .

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8.0 S OURCE D OCUMENTATION .

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9.0 I NDUSTRY O PERATING E XPERIENCE .

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10.0 P ROFICIENCY M EASURES .

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101

1


2


1.0 S

COPE AND

P

URPOSE

This module is intended to provide the utility engineer with a basic understanding of the application of vibration analysis to centrifugal pump diagnostics. Information will be provided on vibration theory, measurement, and pump troubleshooting. This module covers centrifugal pumps and does not address vibration in positive displacement pumps.

This module is limited to the development of the fundamentals of basic vibration theory, the description of the forces causing pump vibration, and the diagnostic techniques associated with vibration analysis.

This module is to provide the student with an understanding of the causes of pump vibration, and diagnosis of pump vibration situations in power plant systems. This module will also provide understanding of vibration measurement, analysis, basic critical speed calculation, and vibration response.

3


4


2.0 S

UGGESTED

S

KILLS AND

K

NOWLEDGE

It is suggested that the student review the basic engineering principles of centrifugal pump fundamentals ( Reference 1 ), and kinematics and dynamic systems.

5


6


3.0 O

BJECTIVES

Upon completion of this module, the student will be able to accomplish the following:

•

Describe types of centrifugal pumps and their typical applications.

•

Define/Describe:

Pump Curves

Best Efficiency Point

Head vs. Resistance

How to Use Performance Testing Data to Monitor Pump Internal Clearance

Degradation.

•

Present Basic Vibration Theory.

•

Present Simple Vibration Calculations.

•

Define/Describe Typical Pump Failure Modes.

•

Describe Standard Vibration Monitoring, Analysis, and Diagnostics.

7


8


4.0 N

OMENCLATURE

GPM c

GPM

T

H

H c h d

H d

H

D h s h f h v h p

H s

H% k

A

BEP

BHP c c cr

δ

=

= e =

EFF,

η pump

=

F = f =

=

=

=

=

F c

=

F k

F m ϕ

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

Generic Vibration Magnitude (displacement, velocity or acceleration)

Best Efficiency Point

Brake Horsepower

Damping Coefficient (N-sec/m, lb-sec/in)

Critical Damping Coefficient (N-sec/m, lb-sec/in)

Logarithmic Decrement

Eccentricity (inches, mm)

Pump Efficiency

Force (N, lb)

Frequency (Hz, CPM)

Dissipation Force (N, lb)

Elastic Force (N, lb)

Inertia Force (N, lb)

Phase Angle (degrees)

Speed Corrected Pump Flow (Gallons/minute)

Corrected Pump flow at Design Temperature (gpm)

Total Dynamic Head (feet)

Speed Corrected Pump Head (feet)

Static Discharge Head (feet)

Net Discharge Head (feet)

Design Pump Head

Static Suction Head (feet)

Friction Head (feet)

Suction or Discharge Velocity Head (feet)

Pressure Head (feet)

Net Suction Head (feet)

Percentage Head of the Design

Rigidity Coefficient (N/m, lb/in)

9

M, m

NPSH

NPSH

A

NPSH

R

N

N a

N cr

N d

N s

N ss

ω d

ω n

P

Q =GPM r

=

=

=

ρ

ρ ref

ρ v

Sg

Sg d

=

X x x t

T u

WHP

& x & , a

ξ

Sg

SUCTION

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=


Mass (Kg, lbm)

Net Positive Suction Head (feet)

Net positive suction head available (feet)

Net positive suction head required (feet)

Shaft Speed (revolutions/minute)

Actual Pump Speed (rpm)

Critical Speed (rpm)

Pump Design Speed (rpm)

Specific Speed

Specific Suction Speed

Natural Frequency of Damped System (rad/sec)

Natural Frequency (rad/sec)

Pressure (psi, feet)

Flow (Gallons / minute or klbs/hr)

Tuning Factor

Dynamic Amplification Factor

Density (lb/ft

3

)

Density of Water at 4 o

C (

ρ ref

=

62 .

4 lbm / ft 3 )

Specific Volume (ft

3

/lb)

Specific Gravity

Design Specific Gravity

Suction Specific Gravity

Time (sec)

Period (sec)

Unbalance

Water Horsepower

Displacement Amplitude (mm, mils)

Displacement (m or in)

Velocity (m/s or in/sec)

Acceleration (g, m/sec

2

, in/sec

2

)

Damping Factor (ratio)

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11


5.0 P

RINCIPLES AND

P

ROPERTIES

5.1 Pump Fundamentals

5.1.1 Pump Types

Pumps can be classified into two categories based on the method by which pumping energy is transmitted to the fluid. These categories are kinetic and positive displacement pumps.

This document focuses only on kinetic (centrifugal) pumps. Most kinetic pumps are centrifugal pumps, which are described in more details in Reference 1. Centrifugal pumps are designed for vertical or horizontal operation. Centrifugal pumps are divided into the following categories:

•

Single and multistage volute types with single and double suction impellers

•

Multistage diffuser types with single suction impellers

These are further divided into:

•

Radial or vertical split case types or

•

Axial or horizontal split case types

Volute type pumps are further divided into:

•

Single volute

•

Double volute

A single volute pump ( Figure 1 ) has one channel that increases in cross section thereby converting the kinetic energy into pressure. At the best efficiency point (BEP), the pressures on opposite sides of the impeller are essentially equal. However, at partial flows the pressures are not equal and could be sufficiently large to cause excessive deflection of the shaft, especially in high head pumps.

12


Figure 1: Single Volute Pump

To minimize this deflection a double volute pump ( Figure 2 ), is used. A second channel is cast into the pump casing, which equalizes the pressures across the impeller. The double volute provides similar flow channels, with outlets 180 degrees apart, which result in a considerable reduction in the radial loads on the shaft.

Figure 2: Double Volute Pump

In a diffuser casing ( Figure 3 ), the vanes are designed to form passages of gradually expanding area to ensure a uniform decrease in velocity from inlet to outlet. The multiple passages equalize the pressure at all points about the periphery of the impeller resulting in perfect radial balance.

13


Figure 3: Diffuser Pump

There are two basic elements of the centrifugal pump; the stationary element and the rotating element. The stationary element consists of the pump case, baseplate, stuffing boxes and bearings. The stationary element provides the support and enclosure for the rotating element.

The case provides the suction and discharge nozzles and directs the flow of liquid into and away from the impeller. Kinetic energy, generated by the impeller is converted into potential energy (pressure). The rotating element consists of a shaft on which one or more impellers are mounted. The rotating element provides the energy to generate the flow of liquid and the required head of the pump.

5.1.2 Pump Impeller Types

Impellers can be classified as either single or double suction ( Figure 5 & 6 ). Either type can be enclosed, semi-enclosed or open ( Figure 4 ). The terms, single and double suction pertains to the number of inlets contained in the impeller.

Enclosed Semi-Enclosed Open

Figure 4: Impeller Configurations

14


Suction Vane

Edge

Wear Ring

Hub

Suction Eye

Shroud

Discharge Vane

Edge or Tip

Figure 5: Single Suction Impeller

Discharge Vane

Edge or Tip

Wear Ring

Suction Vane

Suction

Shroud

Figure 6: Double Suction Impeller

15


A single suction impeller ( Figure 5 ) has one inlet, and a double suction impeller ( Figure 6) has two inlets.

An enclosed impeller has a side wall (shroud) on both sides of the impeller. An enclosed impeller will have a wear ring on each side of the inlet to reduce leakage from the discharge back to the suction.

A semi-enclosed impeller has a shroud on one side of the vanes while the other side is left open. These types of impellers are not furnished with wear rings, and the losses due to leakage from the discharge to the suction are higher than a closed impeller.

The open impeller has cast vanes without any shrouds. Its efficiency is low, and its use in power plants is limited.

5.1.3 Pump and Head Terminology

The term head is used as a measure of energy, and has the units of feet.

Recall from the incompressible flow module that friction head (h f

) is the energy required to overcome resistance to flow in the pipe, fittings, valves, entrances and exits.

Static suction head (h s

) is the vertical distance in feet, above the centerline of the pump inlet to the free level of the fluid source. If the free level of the fluid source is below the pump inlet, h s

will be negative and is referred to as static suction lift .

Static discharge head (h d

) is the vertical distance in feet above the pump centerline to the free level of the discharge tank.

Net suction head (H s

) is the total energy of the fluid entering the pump inlet. It includes the static suction head (h s

) , plus the pressure head (if any) in the suction tank (h p

) , plus the suction velocity head ( h v

), minus the friction head (h f

) in the suction piping ( Figure 7 ). h f h s

P, h p h

ν

Figure 7: Net Suction Head

16


Net discharge head (H d

) is the total energy of the fluid leaving the pump. It includes the static discharge head ( h d

), plus the discharge velocity head ( h v

), plus the friction head in the discharge piping ( h f

), plus the pressure head (if any) in the discharge tank ( h p

) ( Figure 8 ). h d

P, h p h f h

ν

Figure 8: Net Discharge Head

Total dynamic head ( H ) is the net discharge head minus the net suction head. It is the total amount of energy added to the fluid by the pump.

A positive head (normally atmospheric pressure) must push the liquid into the impeller of a centrifugal pump. Net positive suction head required (NPSH

R

) is the minimum fluid energy required at the inlet to the pump for satisfactory operation. NPSH

R

is specified by the pump manufacturer. It accounts for any additional frictional losses from the pump suction flange and entrance to the impellers. It is rarely, if ever, determined after installation. Net positive suction head available (NPSH

A

) is the fluid energy available at the inlet to the pump above the fluid’s vapor pressure.

If NPSH

A

is less than NPSH

R

, the fluid will cavitate. Cavitation is the vaporization of the fluid within the casing or suction line. A rule of thumb states that a margin of 2 to 3 feet is sufficient to prevent cavitation. If the fluid pressure is less than the vapor pressure, pockets of vapor will form. As vapor pockets reach the surface of the impeller, the local high fluid pressure will collapse them, causing noise, vibration, material erosion, and possible structural damage to the pump. The sudden collapse of the vapor bubbles forces the liquid at high velocity against the metal and into the pores of the metal creating surge pressures of high intensity on localized areas of the impeller. This pressure can exceed the tensile strength of the metal, which ultimately results in metal fatigue. Cavitation may also be recognized by fluctuating discharge pressure or flow and oscillating motor amps. It may sound as if the pump is “pumping rocks”. There are no known materials that are not adversely affected by cavitation. Cavitation should not be confused will air intrusion or air binding which will be discussed later.

17


5.1.4 Pump Bearings

The bearings in a centrifugal pump must perform the following functions:

• hold the rotating element in position both radially and axially, thereby eliminating rubbing

• permit the shaft to rotate with very low friction

• absorb the forces generated by the impeller

Bearings in a centrifugal pump are classified according to the direction of the loads they support. Centrifugal pump bearings may be ball or rolling element anti-friction bearings, sleeve type journal bearings, or hydrodynamic thrust bearings. Journal bearings will absorb only radial loads, hydrodynamic thrust bearings will absorb only axial loads, but anti-friction bearings will absorb both radial and axial loads.

5.1.4.1 Anti-friction Bearings

Anti-friction bearings use balls or rolling elements to support shaft and impeller load with minimal wear and friction. Since the anti-friction bearing is a precision part machined to close tolerances, cleanliness and mounting accuracy are required. In order to develop high reliability and service from anti-friction bearings, the shaft and casing must also be machined to very close tolerances and the bearing must be mounted properly. The shaft and casing bores must have almost perfect concentricity to each other so that both bearings will carry an equal share of the load. Maintenance of anti-friction bearings is relatively simple: reduce contaminants and moisture and provide adequate lubrication.

5.1.4.2 Journal or Sleeve Bearings

Journal or sleeve bearings (also called hydrodynamic bearings) ( Figure 9 ) provide support and positioning of the shaft while it is rotating. This bearing is composed of two basic parts; the rotating cylindrical journal transmits the shaft load, and the stationary bearing and housing, which support the shaft and impeller load. Journal bearings utilize the shearing action of the lubricating oil as the shaft is rotating. This lubricating oil shearing action creates an oil pressure wedge that lifts and pushes the shaft to a stable position within the bearing clearance.

18


Bearing

Oil

Shaft

Figure 9: Sleeve or Hydrodynamic Bearing

5.1.5 Pump Shaft Seals

All pumps require a means of sealing the shaft where it extends through the pump casing.

Three possible mechanisms are:

• packing

• throttle bushing with seal injection

• mechanical seals

5.1.6 Pump Application

5.1.6.1 Performance of Centrifugal Pumps

Every centrifugal pump has fixed design geometry so that flow areas are optimized to produce a given head and flow with minimal losses. This design point is called the Best

Efficiency Point ( BEP ). Operation on either side of the best efficiency point will introduce losses that will affect pump performance.

A typical centrifugal pump efficiency curve is shown in Figure 10 . This curve represents how efficiently the pump converts the mechanical energy from the driver or shaft into increased pressure energy of the fluid being pumped. Inefficiencies are caused by the fluid being recirculated within the pump casing. Inefficiencies are also caused by energy going into the fluid as increased internal energy (temperature) instead of pressure. This is caused by friction between the fluid and the impeller. Efficiencies at the BEP can be as high as 90% for many centrifugal pumps. As the figure illustrates, at no-flow conditions all of the energy is being absorbed by the fluid as an increase in its temperature and thus the pump efficiency is zero.

19


E f f i c i e n c y

80%

0%

Capacity, gpm

Figure 10: Efficiency Curve

Pump efficiency is defined as follows:

ηη

Pump

= water horsepower brake horsepower

This can also be defined as follows:

Eq. 1 total

Pump

= work by head the driver

Eq. 2

In this case, “total head” is the total useful head developed by the pump and is the sum of the changes in elevation, velocity and pressure heads through the pump. This definition does not include any provision for internal energy changes in the fluid as it passes through the pump.

This equation also does not account for inefficiencies of the pump driver (motor or turbine).

As the flow of the pump increases, the losses from the suction flange to the inlet of the impeller increase. As the losses increase, the NPSH required for the pump increases. More

NPSH is required to prevent the pump from cavitating. This curve may be provided as part of the manufacturer’s performance curves. However, occasionally it does not show the

NPSH

R

over the entire range of flow. Extrapolation of this curve must be done only after consulting with the manufacturer. A typical NPSH

R

curve is shown in Figure 11 .

20


Head

Flow

Figure 11: NPSH

R curve

For convenience, pump manufacturers display the entire set of pump performance curves on one integrated curve for a given speed and impeller diameter, as shown in Figures 12 and 13 .

NPSH

R

NPSH

R

( ft )

60

40

20

0

90

Eff. %

Head

4000

80

70

3500

3000

2500

2000

EFF %

BHP @ 1.0 Specific Gravity

0 100 200 300 400 500 600 700

FLOW ( gpm )

60

50

40

30

20

10

0

BHP

600

400

200

0

Figure 12: Typical Auxiliary Feedwater Pump Curves

21


40

BHP

5000

30

20

4000

3000

2000

1000

0

10

0

90

80

70

60

50

Head

BHP

EFF %

NPSH

R

0

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320

FLOW (1000 gpm )

Figure 13: Typical Condenser Circulating Water Pump Curves

60

50

40

30

Eff. %

90

80

70

20

10

NPSH

R

( ft )

60

40

20

0

5.1.6.2 System Resistance Curve

The required head to be generated by any pump is determined by the system resistance curve.

The system resistance curve is comprised of three basic components:

•

Static Head

•

Pressure Head

•

Friction Head

Figure 14 illustrates how these components combine to create a system resistance curve. For a given system, the pressure and static head are constant at all flow rates but, the friction head increases as flow rate increases. Once the system resistance curve has been generated, pump selection can be made that provides the flow rate necessary to meet the demands of the system.

22


Head

Friction Head

Flow

Pressure Head

Static Head

Figure 14: System Resistance Curve

It is important to remember that a pump will always operate at the intersection of the pump performance curve and the system resistance curve ( Figure 15 ).

Head

Pump Curve

Increasing system resistance

(discharge throttle valve closing)

Pump Operating

Point

System Resistance Curve

Figure 15: Pump Performance Curve

When developing the system resistance curve, all of the system’s operating conditions must be considered. For example, if a control valve is included in the system, the system curve must be evaluated with the control valve fully open. If this is not considered, the system resistance curve could intersect the pump performance curve near or beyond pump runout.

The pump could experience cavitation, due to insufficient NPSH , or the motor could be overloaded.

23


The minimum flow of the system should also be evaluated. Every pump has a minimum flow at which it can operate. As a general rule, the minimum flow for a centrifugal pump is

25% of the flow at BEP . However, some pumps can require more or less than this minimum flow. The pump manufacturer should be consulted for the exact requirement. Cavitation damage could occur if a pump is operated at less than its minimum flow requirement. This type of cavitation damage is due to the internal recirculation that occurs when impellers are operated at very low flows.

5.1.6.3 Hydraulic Performance Testing

The performance of a single constant speed pump can be determined by measuring the flow rate and the suction and discharge pressures. When periodic test results are compared to the original pump curve, performance degradation can be evaluated.

The original pump curve, as produced during shop testing by the manufacturer may use conditions different than those in the power plant. For example, the pump speed or the temperature and specific gravity of the fluid may have been different than design conditions.

A close examination of the manufacturers test conditions should be made to determine whether any corrections to the field performance data are required to obtain a direct comparison of data.

Ideally, a pump should be tested at the design flow rate or at multiple flow rates to verify pump performance along the performance curve. However, many system configurations do not allow for periodic testing at high or design flow rates. Most pumps are periodically tested at some low or minimum flow and this data is compared to previous periodic test results also at the reduced flow. An assumption is made that any pump performance degradation seen at the low flow will be similar to degradation expected at the design flow.

Although this is not exact, this assumption has been verified in the past. Full flow or design flow tests should be performed yearly or bi-yearly when system conditions allow.

When pumps operate in parallel, the flow of each pump must be measured to determine the performance of each individual pump. It is assumed that each pump is handling fluids with the same properties. If not, the appropriate corrections must be made for temperature, density, vapor pressure, etc.

When monitoring the performance of variable speed pumps, the speed of the pump must also be recorded. Once the speed is known, the pump parameters can be corrected using the pump affinity laws and comparisons can be made to the manufacturer performance test curves.

24


5.1.6.4 Estimating Degradation of Internal Clearances of Pumps

A good estimation of degradation in pump clearances can be obtained by using one or all of the three methods described below. Performance degradation is important because a cost can be associated with the current pump condition, based on lost efficiency, to provide a basis for pump maintenance to restore performance.

Performance Test Data

Calculating the % head loss at a given flow can be accomplished using pump performance data and measurements of:

•

Delta P (discharge pressure - suction pressure)

•

Temperatures

•

Flow

•

Pump speed

The following measures should be taken when measuring performance data:

•

If the pump is not directly coupled to the driver, the affinity laws should be used to correct for design speed before performing comparison of pump head.

•

All flow going through the pump must be measured (isolate recirculation lines or other pumps in parallel if flow could be bypassing the flow element or recirculating before the flow element).

•

A pump base line curve provides a reference for comparison. This curve can be corrected to include long-term degradation that is not wear related.

A utility example showing the necessary steps taken in assessing loss of performance for a variable speed pump is given in Section 7.

Balance Line Flow

Another way to estimate pump internal degradation is by a balance device leak-off flow if the pump has this option (most Ingersol Rand (IR) and Worthington style pumps have this option, Byron Jackson (BJ) pumps do not). This is accomplished using a pipe that goes back to the suction, which sometimes has an orifice that can be used to measure flow and give a good indication of internal wear.

A correlation between flow/Delta P and pump performance must be either obtained from the

OEM or developed in-house.

25


Static Lift Check

If the pump is off, a lift check also gives a very good indication of pump clearances.

A lift check is a means of checking relative ring clearance in a pump by lifting the runner/rotor/shaft in the case and measuring the relative movement of the shaft in the case, measured as close as possible to the ring fits.

5.2 Basic Vibration Theory

Vibrations are the result of dynamic forces acting on a structure. For centrifugal pumps, these forces are related to the rotation of the rotor, transported fluids, and magnetic fields.

The machine/structure dynamic relationship can be characterized by the mechanical impedance of the system. It can also be formally expressed as Newton’s second law of motion ( Equation 3 ).

∑

F

= ma Eq. 3 a

ΣΣ

F m

Figure 16: Accelerated Mass

The body acceleration a is due to the resultant force

Σ

F acting on the mass of the body m

( Figure 16 ).

26


5.2.1 Equation of Motion of a Single Degree of Freedom (SDOF) System

The vibration of many systems may be understood by simplifying their characteristics to the terms of the SDOF model. While the full study of vibration and vibrating systems is beyond the scope of this module, the general equation of motion for a single degree of freedom system with viscous damping is presented in order to show the relationship of acceleration, velocity and displacement to mass, damping and stiffness, respectively. The theoretical model is pictured below (See Figure 17 ) . k c x(t) m F(t)

Figure 17: A simple vibratory system

A vibratory system can be described by four quantities namely:

•

The mass represented by [ m ].

•

The rigidity represented by [ k ].

•

The damping represented by [ c ].

•

The excitation represented by [ F ].

F supplies dynamic energy to the system where it is stored in m as kinetic energy and in k as potential energy. Energy can only be dissipated by c . kx cx k c m x(t) m F(t) mx

F(t)

Figure 18: Free body diagram

27


Vibration of the mass due to the action of resultant forces

∑

F is designated x ( t ) . In the free body diagram ( Figure 18 ), the forces acting on the mass are the restoring spring force kx , the dissipation force c x and the excitation force F ( t ) .

The sum of these forces is

∑

F motion,

∑

F

= ma

⇒ ∑

F

=

=

F ( t )

− kx

− c & . By virtue of Newton’s second law of

F ( t )

− kx

− c x &

= m & x & Eq. 4

Which yields: m & x &

+ c x &

+ kx

=

F ( t ) Eq. 5

5.2.2 General Solution of the Equation of Motion

The solution of the equation of motion of a single degree of freedom system (SDOF) is composed of two parts: the first part is called the complementary function and is due to the homogeneous equation (with no excitation) and the second part is called the particular integral (with excitation).

( )

=

( )

+ x p

( ) Eq. 6

Where, x t Is solution of mx

+ cx

+ kx

=

0 + initial conditions x p t Is solution of mx

+ cx

+ kx

=

( )

Complementary Function – Free Oscillations

The solution of mx

+ cx

+ kx

=

0 can be proposed as:

( )

=

Ae st

Where A and s are constants that are independent of the initial conditions (I.C).

Substitution of the proposed solution in the differential equation yields:

( ms

2 + cs

+

) st =

0

We thus obtain the characteristic equation of the system: ms

2 + cs k 0

Eq. 7

Eq. 8

Eq. 9

Eq. 10

Eq. 11

The roots of this equation are:

28

EPRI Licensed Material s

1

=

1

2 m

(

− + c

2 −

4 km ) Eq. 12 s

2

=

1

2 m

(

− − c

2 −

4 km ) Eq. 13

The nature of the roots depends on the sign of

∆ = c

2 −

4 km . In particular for

∆ =

0 one can define a critical damping for the system equals to c = c cr

=

2 km .

Of importance is the case where a system damping is less than the critical damping, that is: c

< c cr

⇒ ∆ <

0 Yielding complex conjugate roots (this is the case for most mechanical systems)

The case where system damping is higher than the critical damping ( c ≥ c cr

) gives ∆ ≥

0 which yields real negative roots and does not result in any vibration.

In practice we use a non-dimensional quantity relating the actual damping of the system to the critical damping as follows:

ξ = c c cr

=

2 c km

Where:

ξ

is called the damping factor of the system.

Introducing another notation relating the system stiffness to the system mass as

Eq. 14

ω n

= k m

The homogeneous equation becomes

+

2

ξω n

+ ω 2 n x

=

0

And the roots can be written as: s

1

= − ξω n

+ ω n

ξ 2 −

1

Eq. 15

Eq. 16

Eq. 17 s

2

= − ξω n

− ω n

ξ 2 −

1

With this notation we have

∆ = 2 2 n

Thus,

∆ >

0 Corresponds to

ξ >

1

∆ =

0 Corresponds to

ξ =

1

∆ <

0 Corresponds to

ξ <

1

−

1 )

Eq. 18

29


Nature of the homogeneous solution ( ) :

Case 1 (Figure 19) :

ξ >

1

The roots are real and negative because x c

ξ 2

1

ξ

The solution can be written as:

( )

=

A e

+

2

This solution does not result in oscillations no matter what the initial conditions are: x c

( t )

→

0 when t

→ ∞ x c t = A e

1 s t

1 + A e

2 s t

2

An illustration of this type of motion is given in

Figure 19.

O Time, t

Figure 19: Response of an over-damped system


ξ =

1

The roots are equal, real and negative s

1

= s

2

= −ω n

The solution may be written as: x t

=

( A

1

+

A t e

− ω n t

This solution does not result in oscillations no matter what the initial conditions are: x c

( t )

→

0 when t

→ ∞

An illustration of this type of motion is given in Figure 20.

x c

( )

O x c t

=

( A

1

+

A t e

− ω n t

T im e , t

Figure 20: Response of a critically damped system


ξ <

1

The roots are complex conjugates. Using j

2 as: s

1

= − ξω n

+ j

ω d

= −

1 the expressions for the roots can be given

Eq. 19 s

2

= − ξω − n j

ω d

Where,

ω d

is the frequency of oscillations of the damped system.

Eq. 20

30


ω d

= ω n

1

− ξ 2

The solution of the homogeneous equation can be written using the following form:

( )

=

A e

+

2

Substituting the expressions of s

1 and s

2

yield,

( )

= e

− ξω n t

( j ω d t +

− j ω d t

)

Eq. 21

Eq. 22

Eq. 23

And using Euler’s relation e

± j θ

= cos

θ ± j

( )

= e

−ξω n t

( B

1 cos

ω d t

+

B

2 sin

ω d t )

The final form of the solution is obtained using trigonometric relations,

( )

=

Ae

−ξω n t sin(

ω d t

+ ψ

)

Eq. 24

Eq. 25

Where the amplitude A and the phase angle

ψ

are determined by the initial energy provided to the system (initial conditions) and given by:

A

=

B

2

1

+

B

2

2

Eq. 26

ψ = tan

−

1

B

2

B

1

Eq. 27

The motion described by this equation is called a harmonic motion of frequency

ω d amplitude Ae

−ξω n t

that decreases with time due to the presence of damping.

An illustration of this type is shown in Figure 21 . and

The complementary function represents the free oscillation of the system due to initial conditions. This response is transient in time since x c

( t )

→

0 when t

→ ∞

. This is the type of response that is obtained by a vibration sensor during impact testing where the hammer imparts initial condition to the system.

An example application of this type of motion is given by impact testing on a pump to determine the natural frequencies for potential resonance conditions with running frequency excitations such as vane pass. In addition to natural frequencies, vibration mode shapes and associated damping may be determined from this response.

31


Figure 21: Response of an under-damped system

Practical Application: Determination of the dynamic characteristics of a system from vibration measurements

Figure 22: Determination of system characteristics from vibration measurement

A plot of the free oscillations of a system ( Figure 22 ) provides a simple way to determine important dynamic characteristics of a pump under test conditions.

Given the time waveform ( Figure 22) as measured by a vibration transducer, the natural frequency, the damping factor, and the stiffness of the system, can be determined.

32


The ratio of two consecutive amplitudes is given by: x i x i

+

1

= e

ξω n

T

Eq. 28

Letting

δ

be the logarithmic decrement (Log-Dec),

δ = ln x i x i

+

1

= ξω n

T

=

2

π

ξ

1

− ξ 2

Eq. 29

For mechanical systems the damping ratio is a small quantity (

ξ <

0 .

1 )

⇒ δ ≈

2

πξ

Using this relation we can determine the damping ratio from the Log-Dec:

ξ ≈

2

δ

π

=

2

1

π ln x i x i

+

1

Using non-successive peaks, distant by pT sec we obtain the relation:

ξ ≈

1

2

π p ln x i x i

+ p

The natural frequency of the damped system is calculated by:

ω d

=

2

π p t i

+ p

− t i

Recalling that

ω d

= ω n

ω n

=

ω d

1

− ξ 2

, gives:

= Natural frequency in rad/sec

1

− ξ 2

The stiffness of the system can be determined from Equation 15 : k

= m

ω n

2

Particular Integral – Forced Oscillations

Eq. 30

Eq. 31

Eq. 32

Eq. 33

All rotating equipment is subject to forced vibration due to the rotation of the shaft and the other rotating elements. The residual unbalance present in the rotating parts imparts a force to the bearings as it revolves within them. This force is transmitted to the bearing supports and then to any other intermediate structures to the bearing casing or the foundation of the machine. It is the resulting vibration that we measure on the operating machine.

Vibration from sources outside the machine may also contribute to forced vibration. It is important to consider the other machines in the area, the piping system, ground vibration

33

EPRI Licensed Material transmitted through the support structure and any other potential sources of vibration when considering vibration of pumps.

The purpose of this section is to determine the solution x p

( ) called particular integral of the differential equation mx + cx

+ kx

=

( ) . For an excitation F ( t ) of a general form, various methods can be applied including the method of undetermined coefficients, Laplace and

Fourier transforms as well as numerical methods such as Range Kutta and finite difference techniques.

In the following development we will consider the solution due to a harmonic excitation.

This type of excitation is the most common in practical application. It is known as unbalance in rotating machinery. Dynamic testing of large flexible structures is usually accomplished by harmonic excitations using Electro-dynamic shakers.

Through the use of Fourier series any periodic function can be decomposed into a set of harmonic functions. The response of linear systems can be obtained by superposition.

There are several ways to represent harmonic functions. Consider the following harmonic function x ( t )

= a cos

ω t

+ b sin

ω t

=

X cos(

ω t

− ϕ

) where

X 2 = a 2 + b 2 ⇒

X

= a 2 + b 2 And the phase angle ϕ = tan

−

1 b a

Eq. 34

This function can be represented in the following ways:

Time signal representation ( Figure 23 ) x t

=

X sin

ω t Eq. 35

T period in seconds

f frequency in Hz , CPS

Or in CPM

ω frequency in rad/sec

T

=

2

ω

π

=

1 f

Eq. 36

ω = π

Eq. 37

X x(t)

T

Time t

Figure 23: Time Signal Representation

34


Vector representation ( Figure 24 )

OM vector of amplitude X and argument (

ω t

− ϕ

) x(t) is the projection of OM on the x axis

O

M

(

ω t

− ϕ

) x

( )

=

OM cos(

ω t

− ϕ

)

Figure 24: Vector Representation

Representation in the complex plan ( Figure 25 )

In complex notation z

= a

+ jb

Where j

= −

1 a b

=

=

X

X cos(

ω t sin(

ω t

−

− ϕ ϕ

)

) z

=

Xe j (

ω t

− ϕ

)

X

= ϕ a 2 + b 2

= tan

− 1 b a x(t) is the real part of z

Eq. 38

Eq. 39

Eq. 40

Eq. 41

O b

Imaginary a

M

Z = a + j b

Figure 25: Complex Representation

Real

Frequency Representation ( Figure 26 ) x ( t )

= a 2 cos(

ω t )

T period in seconds f frequency in Hz , CPS

or in CPM

ω frequency in rad/sec

T

=

2

ω

π

=

1 f and ω = π

The phase does not appear.

Eq. 42 a

O

Amplitude f

Frequency

Figure 26: Frequency Representation

35


5.2.3 Harmonic Excitations

Periodic excitations are often harmonic or sinusoidal. For linear systems ( x

∝

F ), the response of the structure is also harmonic. For example, the unbalance of shaft trains in rotating machinery produces centrifugal forces that are periodic with a frequency equal to the frequency of rotation of the machine. Vibrations resulting from such excitations are also periodic with the same frequency.

A simple mathematical method used to solve the differential equation of motion in the case of a harmonic excitation is called the mechanical impedance method. This method uses the vector representation of harmonic functions.

Introducing the notation:

F t

=

Fe Eq. 43

And x t

=

Xe Eq. 44

Where

F

=

F e j 0

Eq. 45

And

X

=

Xe Eq. 46

Are vector quantities.

Recalling the derivatives of x(t) we can write: & j x and && =

( j

ω

)

2 x

= − ω 2 x

Note: Deriving consists of a multiplication by j

ω

.

Using the notation J 0 j

ω = ω e

π j

2 ,

&

=

JXe Which gives, &

= ω

Xe j (

π

2

− ϕ

) e

&&

=

JJXe Which gives, &&

= ω 2

Xe j ( − ) e j t

Eq. 47

Eq. 48

The system forces can be represented as:

Inertia Force: mx

= m

ω 2

Xe j (

−

) e j t

Eq. 49

36


Dissipation Force: cx

= ω j (

π

2

− ϕ

) e

Elastic Force: kx

= kXe j e j t

Harmonic Excitation Force: F t

=

F e e j t

The differential equation of motion becomes:



 m

ω 2

Xe j ( − ) + j (

π

2

− ϕ )

+ kXe

− j ϕ



 e j t =

Fe e j t

Eq. 50

Eq. 51

Eq. 52

Eq. 53

This can be represented as follows:

In Figure 27:

F k represents kx

F c represents c &

F m represents m & x &

F c

O

F ϕ

ω t

F k

F m

Figure 27: Vector representation of system forces

5.2.4 Fundamental Relations

For the harmonic function x t

= a cos

ω t

+ b sin

ω t

=

X cos(

ω t

− ϕ

) Eq. 54

The derivatives are:

& ( )

= dx

= − dt

ω

X sin(

ω t

− ϕ

)

= ω

X cos(

ω t

− +

π

)

2

Eq. 55

&& ( )

=

&

= − dt

ω 2

X cos(

ω t

− ϕ

)

= − ω 2 = ω 2

X cos(

ω t

− +

) Eq. 56

These quantities can be visualized using the vector or complex representation of Figure 28 .

37

EPRI Licensed Material d represents x t displacement v represents & ( ) velocity a represents && ( ) acceleration

The physical quantities are real: v

Imaginaries d

( ω t − ϕ )

Reals O

& ( )

&& ( ) Is the real part of a a

Figure 28: Fundamental Relations

5.2.5 Polygon of Forces (see Figures 29 and 30 ):

The polygon of forces will be used primarily to visualize the effects of the various system physical quantities on the behavior of the dynamic system.

The equation of motion indicates that these forces are in dynamic equilibrium and form a polygon (sum to zero). r r r r r

F

+

F k

+

F c

+

F m

=

0 Eq. 57 ϕ

F

F k

F c

F m

O

ω t

Figure 29: Polygon of system forces

For simplicity we can set (

ω t

− ϕ

)

=

0 as axis of reference. The polygon of forces provides good insights in the characterization of a dynamic system.

Introducing the tuning factor as:

38

EPRI Licensed Material r

=

ω

ω n

Eq. 58

F k

F c

F k ϕ

F

F c

F ϕ F c

F k

F ϕ

F m O F m O

F m

O r

<

1

⇒ ϕ <

π

2 r

=

1

⇒ ϕ =

π

2 r

>

1

⇒ ϕ >

π

2

Figure 30: Polygon of system forces as a function of the tuning factor

Interpretation of Figure 30 :

•

For pump operation below resonance r

< ⇒ ϕ <

π

, the system behaves as a spring

2 because the priming force is the elastic force F k

. The elastic restoring forces stabilize the system.

•

For pump operation at resonance r

= ⇒ ϕ =

π

, the system is in resonance. In this

2 case the elastic and inertia forces cancel each other. The only parameter which acts

• against the excitation F is damping, as indicated by the dissipation force F c

π

For pump operation above resonance r > ⇒ ϕ >

2

.

, the system behaves as a mass because the priming force is the inertia force F m

. While the excitation forces tend to control vibration they have to overcome the inertia forces which tend to destabilize the system.

Particular Solution:

The forced response of the simple system due to harmonic excitation can be evaluated using the mechanical impedance method.

For a harmonic excitation

F ( t )

=

F

0 cos

ω t Eq. 59

39

EPRI Licensed Material x t

=

X cos(

ω t

− ϕ

) Eq. 60

The amplitude X and the phase angle ϕ of the forced response depend on the excitation frequency

ω

and are given by:

X

ω =

( k

− m

ω

F

0

)

+ c

ω

Eq. 61

And ϕ ω = tan

−

1 c

ω k

− m

ω 2

In terms of the damping and tuning factors the above equations can be written as:

=

( 1

− r

F

0 k

)

+

( 2

ξ r )

2

Eq. 62

Eq. 63

Eq. 66

Eq. 67

And ϕ r

= tan

− 1

2

1

−

ξ r r

2

Eq. 64

Note that

F

0

represents the static deflection X k amplification factor of the system such that:

0

and the ratio R ( r ) represents the dynamic

=

( 1

− r

1

)

+

( 2

ξ r )

2

Eq. 65

And,

X

=

RX

0

Or,

R r

=

X

X

0

A plot of the functions R ( r ) and ϕ

(r) is given in Figures 31, 32 and 33 for various values of the damping factor. The linear scale representation ( Figure 31 ) does not show the details for high values of damping, which corresponds to lower vibration amplitudes. For this reason, a logarithmic scale is often used since it provides amplification of the lower end of the graph

( Figure 32 ).

40


1 0 0 .

R

8 0 .

6 0 .

4 0 .

2 0 .

0 .

r

0 .0

1 . 0 2 .0

Figure 31: Dynamic amplification factor R r as a function of

ξ

(Linear scale)

1 00 .

0

R

ξ =

0

10 .

0

ξ = 0 1

ξ =

0 2

ξ = 0 3

1 .0

ξ =

0 6

ξ = 1

0 .1

0 1.0

2.0

r

Figure 32: Dynamic amplification factor R r

ξ

(Logarithmic scale)

41

EPRI Licensed Material ϕ

( )

1 8 0

1 6 0

ξ =

0

ξ =

0 0 1

ξ =

0 1

ξ =

0 2

1 2 0

ξ =

0 6

ξ =

1

8 0

4 0

0

0 . 0 1 . 0 2 . 0

Figure 33: Phase angle ϕ

( )

ξ r

42


Good use of data available from transient regimes (run-up and coast down) requires understanding of graphs such as the Bode Plot (vibration amplitude and phase versus speed).

The example in section 7 demonstrates construction of such graphs using vibration theory.

Practical Application: Determination of damping factor using vibration measurements

This example shows how to determine the damping factor using the method of half-power

∆ f bandwidth ( ). f

During a transient regime (run-up/coast-down) of a pump, the vibration at some location may be recorded as indicated in Figure 34 :

The half-power bandwidth uses the value of:

R

1

=

R

2

=

R max

2

Eq. 68

Based on which the associated values of r

1 and r

2 are measured giving the following simplified results:

∆ r

= r

2

− r

1

=

ω

2

ω n

−

ω

1

ω n

=

∆ f f n

=

2

ξ

Which yields an approximation of the damping factor as follows,

ξ =

1

2

∆ r

=

1

2

∆

ω

ω n

Or ξ =

1

∆ f

2 f n

Eq. 69

Eq. 70

43


R

R max

−3 dB

R max

2 ∆ r

0.

r

1 .

r

2

2.

3.

r

Figure 34: Dynamic amplification factor R r

Table 1 provides an indication of the approximate value of the damping ratio for various damping systems.

Table 1: Approximate damping ratio for various systems

Structure Damping Ratio

ξ

Car shock absorber

Plastics

Riveted Structures

Concrete

Wood

Steel

Aluminum

Bronze

0.1 to 0.5

0.04

0.03

0.02

0.003

0.0006

0.0002

0.00007

44


Practical Application: Determination of the dynamic characteristics of a system from vibration measurements using the method of amplitudes

A measurement of a coast down may not yield a good estimate of the critical speed of the system for various reasons including bad set up of the measuring instrumentation and/or bad plots of the response. The resolution of the plots may not show the existence of very close critical speeds, as is the case for dissimilar stiffness supports.

The following method can be used to calculate the critical speeds out of recorded Bode Plots.

From the amplitude measurements ( Figure 35 ), one can also determine

ξ

and

ω n

using the relation of the vibration amplitude three times (once for each selected point ( X , r ) ) then solving for

ω

. The relations obtained are complicated but can easily be programmed in a n calculator.

=

( 1

− r

X s

)

+

( 2

ξ r )

2

Eq. 71

ω 4 n

=

X X

2

2

2

1

2

2

(

2

2

− ω 2

1

)

+

2

2

(

ω 2

2

− ω 2

1

)

+

2

3

2

1

2

3

2

3

(

2

3

− ω 2

1

)

(

ω 2

3

− ω 2

1

)

+

+

2

3

2

3

2

2

2

3

(

ω 2

3

− ω 2

2

(

)

2

3

− ω 2

2

)

Eq. 72

ξ 2 =

X

2

3

(

ω 2 n

− ω 2 2

3

)

−

4

ω 2 n

( X

2

2

ω 2

2

X

2

2

(

ω 2 n

− ω 2 2

2

)

−

X

2

3

ω 2

3

)

Eq. 73

These expressions have been produced using a constant amplitude forcing function. For an excitation with variable amplitude such as the case with an unbalance ( F

0 u

2

with u

= me for an eccentric mass m), resulting amplitudes are proportional to

ω 2

. It is necessary

1 in this case to divide the reading of the amplitudes X i

by

ω i

2

.

To differentiate between any two potential consecutive critical speeds which is often the case for on-plane and off–plane critical, use three measurements below and three measurements above the critical speed.

45


X

X

3

X

2

X

1

N

1

N

2

N

3

N rpm

Figure 35: Dynamic Amplitude X

Practical Application: Determination of the dynamic characteristics of a system from vibration measurements using the method of phases

System characteristics (

ξ

and ω n

) can also be determined from a measurement of the phase angles ( Figure 36 ). This method however, does not provide accurate results due to the large uncertainty associated with the measurement of phase angles in general. The method proceeds in the same way as the previous method but only two measurement points are needed. An expression for the damping ratio is developed from the phase angle relation as follow:

In the physical vicinity of resonance, pick two pairs of measurements: ϕ

1

, N

1 and ϕ

2

, N

2

Substituting in the expression for the damping ratio and solving yields:

.

N

2 n

=

N N

2

N

1

N

2 tan tan ϕ

1 ϕ

1

−

−

N

2

N

1 tan ϕ

2 tan ϕ

2

Eq. 74

And

ξ =

N n

2 −

N

1

2 tan ϕ

1

2 N n

N

1

Eq. 75

46

EPRI Licensed Material ϕ

( )

1 8 0

1 6 0 ϕ

2

1 2 0

8 0 ϕ

1

4 0

0

0 .

N

1

N

2

Figure 36: Phase Angle ϕ

( )

N r p m

5.2.6 Analysis of Equivalent Systems

Rotating machines are multiple-degree of freedom (MDOF) systems, and may be analyzed as combinations of many SDOF sub-systems.

Systems will respond to internal or external forces, due to operation or from some other source. If the frequency of the force coincides with a natural frequency of the mechanical system, the response is said to be resonant. This resonant vibration becomes a problem because the magnitude of vibration is amplified at resonance. In systems with very little damping, this can result in violent vibration and rapid and severe failure.

Structural resonance conditions can be estimated by measuring the system support stiffness.

The following examples show the methodology used in simplifying a complicated system to a relatively simple equivalent system in order to estimate the natural frequencies for potential resonance conditions with running speed harmonics (1X, 2X, and vane pass excitations zX).

47


Considering Figure 37 , this example determines the parameters of an equivalent dynamic system.

M m

ω

Beam x

M

Support

⇒

y(t) y

Figure 37: Simplified equivalent system of a motor/pump/structure k

éq

F(t)

Unbalance is defined as u

= me

The force that acts on the beam in the vertical direction is the projection of the centrifugal force on the y-axis. Thus,

F ( t )

=

F eq sin

ω t , With F eq

= me

ω 2

Eq. 76

The equivalent stiffness of the beam is given by: k

éq

=

48 EI

L

3

An equivalent damping can be estimated and added to the equation of motion.

Eq. 77

Calculation of Critical Speeds

A rotor speed is called critical if it coincides with one of the natural frequencies of the rotor/bearing system in operation. The rotor of Figure 38 consists of a hollow shaft with a negligible mass, external diameter d e

The shaft supports a disk of mass M.

and internal diameter d i

, supported by rigid bearings.

48

y

EPRI Licensed Material x

S h a ft, E I

D isk

M z

S u p p o rt a b

Figure 38: Simple pump rotor on rigid support

The deflection at the disk position is equal to

δ r

=

( )

3

( + b

)

Using

Mg

= k eq

δ r

Yields k eq

=

Mg

δ r

⇒ k eq

=

3 EI

( a

+ b

)

( a

2 b

2

)

For a hollow shaft:

I

=

π

64

( d

4 e

− d i

4

)

The natural frequency of the equivalent system (see Figure 39 ) is

ω n

= k eq

M

=

δ g r

The critical speed is the equivalent of

ω n

in rotations per minute, thus

N cr

=

60

ω n

2

π

RPM

Eq. 78

Eq. 79

Eq. 80

Eq. 81

Eq. 82

Eq. 83

49


M g

M y(t)

L , E I a b

⇒

K eq y

Figure 39: Equivalent system of a pump rotor on rigid support

What happens to the critical speed if the supports were flexible? Let K, be the radial stiffness of the supports. This system is depicted in Figure 40 : x y

Disk, M

Shaft, EI z

K K

Fondation a b

Figure 40: Simple pump rotor on flexible support

The resulting shaft deflection is as shown in Figure 41 . The displacement of the disk alone is labeled

δ

and may be obtained geometrically by d

δ d

= δ

1

+ a a

+ b

( δ

2

− δ

1

)

Eq. 84

To this displacement the shaft deformation at the disk level

δ s

is added to calculate the equivalent stiffness.

50


L , E I

δδδδ

1 δδδδ d M g δδδδ

2

R

1

R

2 a b

Figure 41: Support displacements

The displacements

δ

1 and

δ

2

are determined directly from the support reactions R

1 and R

2

:

And

δ

1

=

R

K

1 = b a

+ b

Mg

K

Eq. 85

δ

2

=

R

K

2 = a a

+ b

Mg

K

The disk displacement is

δ = δ s

+ δ d

= δ s

+ δ

1

+ a a

+ b

( δ

2

− δ

1

)

The natural frequency is given by:

Eq. 86

Eq. 87

ω n

= k eq

M

= g

δ

Eq. 88

The critical speed is the equivalent of ω n

in rotations per minute, thus

N cr

=

60

ω n

2

π

RPM Eq. 89

Since the displacement

δ

is greater than

δ s

(larger denominator in equation 88 ), the critical speed of the system with flexible supports is less than that of the rigid supports. An increase in the flexibility of the supports generally results in a decrease of the critical speeds of rotors.

This result is generally used to de-tune resonant mechanical systems. The other way to accomplish system de-tuning is to modify the supported mass of the system or add what is called a “tuned” vibration absorber.

51


Multi-Critical Speeds Calculation

This example illustrates the behavior of a system with two critical speeds. The existence of dissimilar support stiffness in a system is one of the reasons why systems exhibit many critical speeds.

A Jeffcott rotor is a simple system consisting of a vertical shaft of negligible mass and supporting a disk in the mid-span as shown by Figure 42 .

The supports are considered rigid. The geometric position of the mass center G of the disk is described by the x and y coordinates. The disk mass is eccentric such that G is at a distance e from the geometric center P.

The shaft has a stiffness k. Energy is dissipated through a dissipation force proportional to the linear velocity of the geometric center P.

Equations of motion:

Applying Newton’s second law of motion to the mass center G in both directions x and y yields: d

2

M dt

2

( x

+ e cos

ω t

) = − kx

−

& Eq. 90 d

2

M dt

2

( y

+ e sin

ω t

)

= − ky

− & Eq. 91

52


O P

G y x y e

2 y u

P e

G ϕ

ω t

O x e

1 x z

Figure 42: Jeffcott Rotor

These equations way be written in the standard form:

M & x &

+ c &

+ kx

=

Me

ω 2 cos

ω t

=

F eq cos

ω t

M & y &

+ c &

+ ky

=

Me

ω 2 sin

ω t

=

Fe eq sin

ω t

Using the impedance method:

( k

−

M

ω 2 + j

ω c

)

X

=

F eq

( k

−

M

ω 2 + j

ω c

)

Y

=

F eq

.

e j

π

2

53

Eq. 92

Eq. 93

Eq. 94

Eq. 95

EPRI Licensed Material equal

(

π

The angle indicates that 90° separate displacements x and y. Note that the amplitudes are

2

X

=

Y

) and that the motions have the same frequency. The geometric sum of these motions gives a circle of radius: u

=

X

=

Y

=

F eq k

R ( r )

=

Me

ω 2 k

R ( r ) Eq. 96

The motion can also be normalized by the eccentricity to give: u e

=

M

ω 2 k

R

= r R Eq. 97

Where r, is the tuning factor. In terms of the non-dimensional quantities, the dynamic motion can be expressed as: u e

=

( 1

− r r

2

)

+

( 2

ξ r )

2

Eq. 98

Considering that support flexibility is generally different from x to y direction, we obtain two critical speeds. Assuming the system has an equivalent stiffness K x

≠

K y

and neglecting damping, we obtain for the disk relative motions:

X

= e r

2 x

1

− r

2 x

Eq. 99

And

Y

= e r

2 y

1

− r

2 y

Where r x

=

ω

ω nx

ω nx

=

K x

M

And r y

=

ω

ω ny

Eq. 100

Eq. 101

Eq. 102

Eq. 103

54


ω ny

=

K y

M

Eq. 104

Since X

≠

Y , the motion is an ellipse. In this case, the geometric center P describes an ellipse called orbit of vibration. In practice, this orbit can be visualized using a two-channel oscilloscope or analyzer. Two displacement probes placed 90° apart are used in practice to continuously monitor the shaft position in its bearing shell.

For an undamped system, the phase angle between the excitation and the response is 0° for speeds below the critical speed and is equal to 180° for speeds above the critical speed. The existence of two critical speeds (on plane and off plane) is easily analyzed using the following plot. Figure 43 depicts the x and y normalized displacement responses of a damped system.

X e

,

Y e

ω ϕ =

180

°

ω

ω ϕ =

360

°

1.

0.

r x

= 1 r y

=

1 r x

, r y

Figure 43: Shaft Vibration Amplitudes during Transient Regime

The amplitude peaks provide visualization of the critical speeds. The orbit plots indicate not only the shape of the shaft path but also the direction of shaft precession. The orbit describes the geometric center P of the shaft. The direction of the orbit is obtained through analysis of the phase angle between the response and the excitation.

55


5.3 Pump Failure Modes

Pump failure modes are either a hydraulic / performance type or a mechanical type.

However, in most instances these two types of pump failures are interrelated. For example, a pump may have a reduction in pump capacity that is due to increased wear at the running clearances. Care must be taken to determine which one is the cause and which one is the symptom. In this case, the performance problem is the symptom and the increased clearance

(mechanical) is the cause.

The general principle behind vibration monitoring and analysis is that some of the energy in the machine is always converted into various levels of vibration, noise, and heat. The spectral content of vibration and noise depend on the energy input and the dynamic characteristics of different parts of the machine/structure. If the machine condition changes due to wear, damage, etc., the dynamic characteristics of the machine and hence the vibration and noise will change. Vibration characteristics will also vary with changes in energy input such as speed, shaft loading, or background vibrations. Table 2 provides a breakdown of the various loading types and associated responses found in high-energy pumps.

The most common failure modes for centrifugal pumps are presented in this section including:

•

Pump runout (too little system resistance)

•

Pump deadhead (too much system resistance)

•

Air binding (entrained gases)

•

Motor overloading

•

Bearing and seal failures

56


Table 2: Loading and Associated Response of Pump and Structure

Pump

Loading

Static forces and moments on the casing:

Impeller radial thrust (primary static load of the rotor)

Weight of the rotating assembly

Static loading varying with pump flow rate

Dynamic forces and moments fixed on the rotor:

Impeller hydraulic unbalance (primary dynamic load of the rotor)

Rotor mechanical unbalance (secondary dynamic load of the rotor)

Dynamic instability mechanisms:

Pump bearing whirl

Rubs

Impeller vane pass

Rotor eccentricity

-

-

-

-

Response

Static bearing and seal forces

Static shaft deflection

Rotor axial position movement

Dynamic bearing and seal forces expressed as vibrations

Multiples and sub-multiples of running speed from analysis:

Sub-synchronous frequencies

Impact

Multiples of running speed

Structure

Loading Response

-

-

Static:

Nozzle loads

Foot loads

Dynamic:

Seismic nozzle loading

Seismic foot loading

Static structural response is not currently measured

Dynamic structural response:

Seismic response not recorded

Natural frequencies usually accounted for

Running speed components

Sub-harmonics analyzed

Bearing and seal static and dynamic

Operation

Mostly pressure, temperature and flow loading

Flow induced loading measurements and performance trending

57


5.3.1 Pump Runout

Pump runout refers to the maximum flow rate at the lowest anticipated system head. Pumps that operate in an oversized system can reach their maximum flow rates due to a system head loss that is too low. Correctly sized pumps can also reach runout due to ruptures in the system, or improper system valve lineups, that drastically reduce head loss. Runout results in:

•

Decreased Pump efficiency – the runout flow is typically much higher that the best efficiency point

•

Eventual flow loss (possible cavitation problems) – the runout flow typically has much higher NPSH requirements

•

Overheating the motor and/or pump, leading to pump failure

5.3.2 Pump Deadhead

Pump deadheading refers to a pump running with little or no flow. This is also referred to as shutoff head. This typically occurs when the pump discharge valve is closed or the system resistance at the pump discharge is higher than the pump discharge pressure. Running a pump in a deadhead condition will result in overheating the motor, bearings, and/or the pump. The energy from the impeller is still being transferred to the fluid but the energy is not leaving the pump casing. This results in the energy being converted into internal energy of the fluid instead of pressure or kinetic energy. Eventually the fluid temperature will rise causing the fluid to vaporize or cavitate and eventually will cause the temperature of the pump materials to rise and possibly expand beyond their design limits. Although each pump design determines how long it can withstand this condition, many pumps can run for 20 minutes or longer with no permanent damage. Since the deadhead flow (or no flow) is typically much lower than the best efficiency point, the pump will operate at very low efficiency causing it to vibrate more than normal and eventually degrade mechanical components. Pump deadheading can be prevented by providing a flow path that will allow the manufacturer’s suggested minimum safe flow rate. If a pump is left operating in a deadhead condition too long, it can result in its failure.

5.3.3 Air Binding

Air binding is a problem commonly caused by inadequate venting or entrained gases in the liquid caused by fittings or lines trapping gases in the piping. Entrained gases can also occur due to vortexing in a suction vessel or air leaks through packings and joints. The results can be more severe than cavitation. The air moving through the pump will not remove the energy from the pump and thus the pump will overheat and liquid flow will be greatly reduced. This problem may not be readily observable. In fact, a noisy or cavitating pump may run more quietly due to the added air space to cushion the vapor bubble collapse.

Typically, the discharge pressure and flow will be reduced or will oscillate. The motor amps

58

EPRI Licensed Material will likewise be reduced or oscillate due to the reduced or changing horsepower requirements.

5.3.4 Motor Overloading

In some cases, problems with the pump can result in motor overloading and failure. Pump discharge pressure lower than normal may indicate that the pump is operating farther out on its performance curve where a pump generally requires more horsepower. Pump internal parts may be rubbing or loose. If the pump has worn internal clearances, the efficiency of the pump will be lower which will also increase the required motor horsepower. The packing may be too tight causing the pump to draw more horsepower to develop the same performance. The fluid conditions should also be monitored for changes. More horsepower is required if the fluid specific gravity is greater than expected. If a motor is suspected of overloading, hot bearings or vibration should be investigated.

5.3.5 Bearing Failures

Water or other contamination in lube oil can cause bearing failures. A lube oil analysis program helps to detect and prevent these types of failures. Also, a proper oil seal or deflector will help prevent water and other contamination from entering the bearing.

Coupling alignment is another frequent contributor to bearing failures. One aspect of coupling alignment that should be considered is the thermal rise of the pump during operation. All pumps with higher than ambient fluid temperatures will “grow” as they heat up. This thermal growth must be considered when the pump is aligned cold so that the hot condition alignment with the driver is correct. Most manufacturers will provide the amount of thermal growth expected. However, their calculations include assumptions that may not exist in the power plant. For example, some pump manufacturers provide the amount of thermal growth based on the assumption that the pump pedestal reaches full operating temperature of the fluid. However, in the plant the pedestal may never reach this temperature since it is not insulated. In this case the thermal growth provided would be more than the pump exhibits in the plant and the pump would be mis-aligned in its operating condition.

The thermal growth should be measured in the power plant to verify the manufacturer recommendations.

High suction pressures can cause unusual axial thrust loads on a thrust bearing that may cause the bearing to fail. Also, some pumps have balancing lines installed to reduce the axial loads and the size of the thrust bearing. If this line becomes blocked, the thrust loads will increase and the bearing may fail. For this reason, valves should never be installed in a balancing line.

59


5.3.6 Mechanical Seal Failure

Mechanical seals may fail due to vibration, inadequate seal injection to cool the seals, or improper installation practices. Improper installation practices are probably the most common causes of seal failures.

5.3.7 Storage of Pump Spare Parts

Storage of pump spare parts is not normally considered a subject of pump troubleshooting, however, proper storage can and does affect the safe and reliable operation of centrifugal pumps. Every pump has spare parts that have a finite “shelf life”. For example, some pumps contain rubber o-rings. These o-rings may have a shelf life of about two years. After two years the o-ring material starts to become brittle and crack. If this o-ring were to be installed in a pump or mechanical seal, it would fail in a very short time.

Another storage requirement that should be reviewed is the storage of pump shafts. If pump shafts are stored horizontally the center of the shaft will bow due to gravity. If left in this condition long enough the bow may become permanent. A pump installed with a bowed shaft will exhibit a vibration problem. Pump shafts should be supported along the full length to prevent bowing. Ideally, the shaft should be stored vertically.

5.3.8 Lube Oil Analysis

Analyzing the lube oil in the reservoirs of the pump bearings is an excellent tool to determine bearing condition. The condition of bearings can be determined by analyzing the oil for wear metals, water content and contamination. Lube oil analysis is a tool to be used in lieu of changing the oil. It will also indicate that a bearing is failing, which will allow the bearing to be replaced prior to a catastrophic failure that could also damage the pump. The oil should be analyzed at least quarterly and the results trended.

5.4 Basic Vibration Diagnostic Techniques

Vibration analysis has a wide scope of application to high-energy pumps. Vibration sources include common structural vibrations from the interaction of rotating and non-rotating parts, as well as fluid-flow noise such as cavitation. The following sources of vibration are documented in the literature.

•

Bearing wear

•

Alignment problems

•

Unbalance problems

•

Wear of geared pump/motor couplings

•

Cracked and/or worn shaft

60


•

Mechanical looseness (bearings, pedestals, base)

•

Improper internal pump clearances

•

Impeller wear (especially that caused by off-Best Efficiency Point (BEP) operation)

•

Degradation of wear rings

•

Degradation of diffusers

•

Degradation of volutes

•

Degradation of channel rings

•

Degradation of balancing device

•

Erosion of interstage seals

•

Erosion, corrosion of pump casing and/or rotor internal flow paths

5.4.1 Vibration Measurements

Vibrations can be measured by attaching displacement, velocity, or acceleration transducers to different parts of the machine in different orientations ( Figure 44 ). Noise is measured by microphones and treated the same way as vibrations, except that noise levels are expressed in decibels (dB) and generally referenced to human ear characteristics. Vibration measurements can be divided into two major types:

•

Direct shaft motion by displacement probes often accompanied by a tachometer pulse attachment (a sensor generates a pulse with every shaft revolution used as a reference for phase measurements)

•

Casing vibration by velocity probes or accelerometers

Two standard types of vibration analysis are:

•

Traditional low-frequency vibration measurements (i.e., acceleration, velocity and displacement) which identify unbalance, misalignment, resonance, mechanical looseness and structural problems

•

Time-based enveloping or high-frequency vibration measurements, which identify problems undetectable in low-frequency vibrations due to masking by other frequencies and mechanical noise. Enveloping isolates (by filtering) very weak signals that can indicate problems with rolling elements and bearings.

61


Vertical

Displacement Probe

Horizontal

Displacement Probe

Bearing

Shaft

Bearing Support

Motor

Vertical

Accelerometer

Bearing

Casing

Axial

Accelerometer

Shaft

Figure 44: Vibration Measurements Using Displacement Probes and Accelerometers

5.4.2 Vibration Sensors

There are two fundamental types of vibration transducers: contacting and non-contacting.

Contacting sensors are physically attached to the target surface; non-contacting sensors are attached to a nearby fixed object. Non-contacting probes (usually the eddy current type) measure displacement without contacting the monitored body. Contacting probes

(transducers) are attached to the surface of the body to be monitored and they convert the mechanical motion to an electrical signal.

5.4.2.1 Non-contacting Displacement Probes

Displacement probes measure the relative motion between a stationary surface, usually the bearing casing, and a moving surface, usually the shaft ( Figure 45 ). They are typically used to monitor shaft vibration and rotor radial position for sleeve bearing supported rotors. They are also used to monitor rotor axial position and axial vibrations for thrust bearings.

62


This type of transducer requires a rigid attachment to the stationary surface. When two transducers are used at the same location (axially), they are mounted 90 degrees apart

( Figure 44 ) to provide for monitoring of shaft dynamic position with respect to the bearing.

Non-contacting probes have a frequency limitation of approximately 1 kHz, and require signal conditioning. Measurement from eddy current probes is adversely affected by shaft surface irregularities and shaft material magnetic properties.

Displacement

Probe

Fixed Object

(Bearing)

Moving Object:

(Shaft)

Figure 45: Displacement Probes

5.4.2.2 Contacting Probes

There are two fundamental types of contacting probes: velocity transducers and accelerometers ( Figure 46 ). Accelerometers are preferred in most applications over velocity transducers because of their small size, price, robustness, and wide range of frequency measurement. Accelerometers have benefited from extraordinary developments in materials sciences and electronic miniaturization. Today, a variety of accelerometers are available for low frequency, high frequency, high-temperature, triaxial, and special applications.

Accelerometers require external power, whereas velocity probes are self-generating. The accelerometer signal can generally be transmitted over longer distances. A contacting probe can be modeled by a one-degree-of-freedom dynamic system based on variations of mass, stiffness and damping elements.

The accelerometer provides a linear range of measurement below its natural frequency, whereas the velocity probe provides a linear range above its natural frequency. Different mounting practices are used, including stud mounting for high frequency and continuous monitoring, magnetic base, and a hand held “stinger” in difficult-to-reach areas for

63

EPRI Licensed Material periodic vibration monitoring. The mounting method affects the dynamic properties of the measurement.

The characteristics of the velocity probe include direct velocity measurement with integration, no external power requirements, and a good signal-to-noise ratio.

Piezoelectric Accelerometer Seismic Velocity Transducer

Transducer Case

Transducer Wiring

Transducer

Wiring

Integrated

Electronics

Vibrating Mass

Piezoelectric Element

(Spring)

Support Element

Transducer Mount

Transducer Case

Damper (oil)

Permanent Magnet

Coil

Vibrating Mass

Stiffness Element

(Spring)

Transducer Mount

Magnetic Base

Figure 46: Piezoelectric Accelerometers and Seismic Velocity Transducers

5.4.2.3 Mounting Practices

Hand-held readings provide a rapid and easy way to obtain vibration data. Repeatability and less accurate measurements are the main drawbacks. The maximum vibration frequency that can be effectively monitored is about 500 Hz using a stinger and 2000 Hz for a magnet mount.

Stud mounts provide the best mounting practice for structural vibration measurements.

The measured vibration frequency is limited by the resonant frequency of the combined accelerometer/ mounting structure assembly. Stud mounting is the preferred method for high frequency vibration measurement (usually greater than 3000 Hz).

64


5.5 Acceptance Criteria

Acceptance criteria are necessary to translate a measurement into a mechanical condition.

There are two sources of acceptance guidelines:

•

Physical constraints, such as clearances supplied by the original equipment manufacturer (OEM)

•

Established limits, determined from experience and dependent on the type of machine, measurement location, etc.

Standards for vibration limits are published by industry groups, and by national and international standards organizations including:

• the American Petroleum Institute (API),

• the American Gear Manufacturer Association (AGMA),

• the National Electrical Manufacturers Association (NEMA), the American National

Standards Institute (ANSI),

• and the International Standards Organization (ISO)

The rotating equipment OEM and vibration instrumentation vendors are also good sources for obtaining vibration limits.

5.5.1 OEM Limits

Radial and axial clearances between rotating and stationary surfaces are important physical parameters. They are usually expressed as “maximum allowable,” and should not be exceeded. Other types of OEM criteria may include limits on pressure, balance piston differential pressure, speed, temperature, etc.

5.5.2 Published Severity Criteria

Vibration limits are also established using the following information sources:

•

Independent testing organizations

•

Severity charts from PdM vendors

•

Inspection results

•

OEM recommendations

65


5.5.3 Casing Vibration Limits

Limits for casing vibration for typical machines are based on measurements made using repeatable operating conditions (measurement type, location, etc.) in a controlled environment. ISO standards 2372 and 3945 are widely used in Europe.

•

ISO 2372 is a general standard and is used primarily for shop acceptance testing

•

ISO 3945 is a more specific standard and is designed for evaluating the vibration of larger machinery in the field

Both standards contain criteria for judging machine condition from casing velocity measured at specific bearing locations. These standards apply to machines operating at 10 to 200 Hz

(600 to 12,000 rpm). Both standards require a true root mean square (RMS) amplitude measurement, make a distinction between flexibly supported and rigidly supported machines, and recognize that a support system may be rigid in one direction and flexible in the other.

Both standards are now withdrawn and replaced by a more current standard (ISO 10816).

There is generally good agreement among experts on the various limits for casing vibration.

In general, a level below 0.1 in/sec peak is considered acceptable, and a level above 0.6 in/sec peak is considered unacceptable. The advantage of velocity measurements, although limited to casings, is that frequency is included in the measurement.

Table 3 summarizes the recommended limits for overall unfiltered casing velocities.

Table 3: Recommended Limits for Overall Casing Velocity

Peak Velocity Acceptance Class

Less than 0.15 ips (3.8 mm/sec) Acceptable

0.15 to 0.25 ips (3.8 – 6.3 mm/sec) Tolerable

0.25 to 0.4 ips (6.3 – 10 mm/sec)

0.4 to 0.6 ips (10 – 15 mm/sec)

May be tolerable for moderate periods of time. Monitor closely to warn of changes

Impending failure; watch closely for changes and be prepared to shut down for repairs

Above 0.6 ips (15 mm/sec) Danger of immediate failure

Acceleration measurements are not generally used for trending, but are most useful for diagnostic work. Acceleration signals accentuate the low amplitude, high frequency signals for diagnostics.

To monitor conditions for operating gears, the AGMA specification recommends a conservative guideline limit of a constant 10 Gs of casing acceleration above 600 Hz.

66


Vibration severity criteria per ISO 2372 and 3945 are presented in Table 4 .

Table 4: Vibration severity per ISO 2372

Ranges of Radial Vibration

Severity

Quality Judgement for Separate Classes of

Machines

RMS Velocity measured in the 10-1000 Hz frequency band

Class I Class II Class III Class IV mm/sec in/sec

0.71 0.028 A

A

A

1.12 0.044 A

B

1.8 0.071

B

2.8 0.11

C B

4.5 0.18

C B

7.1 0.28

C

11.2 0.44

C

D

18 0.71 D

D

45 1.8

MACHINE CLASSES

CLASS I Small Machines to 20 HP

CLASS II Medium Machines 20 to 100 HP

CLASS III Large Machines 10-200 rev/sec, 400 HP and Larger Mounted on

Rigid Supports

CLASS IV Large Machines 10-200 rev/sec, 400 HP and Larger Mounted on

Flexible Supports

ACCEPTANCE CLASSES

A = GOOD B = SATISFACTORY C = UNSATISFACTORY D = UNACCEPTABLE

D

67


5.5.4 Shaft Vibration Limits

Shaft vibration levels are dependent on the measurement location and the shaft mode shape.

Displacement measurements are ordinarily expressed for particular operating frequencies.

Therefore, a filtered signal must be used to assess vibration severity based on displacement measurement. Tolerable displacement levels for rotating component typically decrease with frequency.

Shaft vibration levels are better indicators when expressed in terms of a displacement to diametrical clearance ratio for journal bearings. This is an effective way to monitor for potential bearing degradation.

Displacement probe measurements should always be corrected for runout due to shaft surface irregularities and material magnetic properties. Running speed shaft motion (1X) is generally the predominant contributor to shaft vibration amplitude and is indicative of unbalance. If another condition exists, such as misalignment or instability, running speed vibration may not be the only determinant of tolerable vibration levels. Higher and even lower order harmonics of the frequency of rotation may also be important since additional information about the shaft behavior may be found at these frequencies. One practice is to electronically differentiate the signal to velocity and use the limits set for overall velocity measurement.

API and AGMA specify that, for the purpose of acceptance, maximum shaft amplitude peak-

12 , 000 to-peak expressed in mils shall not exceed or 2 mils whichever is less. In this

RPM criterion, shaft motion includes runout, which can be no more than 25% of the allowable displacement. This guideline is established for new machinery; operating machinery can tolerate higher levels.

5.6 Pump Diagnostics

In addition to the use of Predictive Maintenance (PdM) technologies, walk-down checks, and visual inspections are also useful methods of monitoring machine health. A change in sound can also signal potential problems.

Visual observations provide evidence of the following:

•

Foundation and grout degradation

•

Hold-down bolt integrity

•

Cracks in welded areas

•

Oil leaks

•

Process fluid leaks

68


•

Dirt buildup

•

Other degradation

5.6.1 Interpretation of Vibration Spectra

Due to the cyclic nature of the dynamic forces present in rotating machinery, vibration signatures contain frequency information on individual pump components. Spectrum frequencies can be divided into harmonic, sub-harmonic and super-harmonic frequencies of the machine running speed.

The sub-harmonic parts of the spectrum may contain components that indicate oil whirl in journal bearings, or impeller/casing rubs in pumps. Instability mechanisms and structural natural frequencies often show up as sub-harmonic spectrum components.

All mechanical structures have natural frequencies and characteristic modes of vibration.

When structural natural frequencies are excited by forces due to operation of the machine or motion of its support, they are called resonance conditions.

The harmonic part of the spectrum may contain multiples of the frequency of rotation.

Mechanical and hydraulic unbalance will appear exactly at these frequencies. Excitations due to hydraulic forces around the impeller will be expressed as vane pass frequencies in the spectrum that are determined by the running speed times the number of impeller vanes or blades.

Mechanical forces due to torque transmission through gears will develop gear mesh frequencies in the spectrum. The gear mesh frequency is the shaft rotation frequency multiplied by the number of gear teeth. Solid disc resonant frequencies such as those in a flywheel are independent of running speed.

Rolling element bearings can have several frequencies that appear in the spectrum if there are bearing defects. Bearing fault frequencies can be calculated using values of bearing geometry and shaft running speed. All modern PdM software provides a calculation routine for bearing fault frequencies, along with a comprehensive bearing library.

Different faults may be expressed by the same frequency component. A mechanical unbalance, bent shaft, electrical problem, misalignment, and a host of other conditions may show up at once-per-revolution frequency. Other diagnostic techniques are used to differentiate among these conditions.

Unbalanced mechanical and hydraulic radial forces cause excessive vibration, unnecessary wear, and noise. Pressure differentials between stages create axial loads. Such unbalanced axial forces must be controlled to prevent damage to bearings and pump internals. Devices to balance the impeller thrust loads include a balance drum, a balance disk, or some combination of the two.

69


5.6.2 Mechanical Unbalance

Unbalance, the most common cause of vibration in rotating machinery, occurs when the actual center of the rotating mass is not exactly at its geometric center. This eccentricity causes a heavy side of the rotating component, creating a synchronous rotating force vector.

For linear systems, unbalance produces a vibration directly proportional to the unbalance amount, and has a frequency equal to the running speed of the machine (Figure 47).

Figure 47: Vibration Spectrum Indicating a Mechanical Unbalance Condition

Mechanical unbalance in rotating machinery is usually caused by errors in design, manufacture, assembly, initial balancing, or impeller damage. Unbalance can lead to excessive vibration, bearing wear, and seal leakage. Impellers should be statically and dynamically balanced so that the maximum residual unbalance is less than W/N, where W = impeller weight and N is the pump speed in rpm.

An unbalance condition may involve different modes of the rotor bearing system. For rigid rotor modes, the unbalance condition may be static, couple, or dynamic. The balancing technique to use depends on the unbalance condition:

•

A static unbalance condition requires balancing using only one rotor plane

70


•

A couple-unbalance condition requires two rotor planes for the correction weights placed 180 degrees apart but is treated as a one plane balancing technique

•

A dynamic unbalance condition requires a two plane balancing technique based on influence coefficients.

Flexible rotors are more complex; unbalance correction requires several balancing planes in addition to knowledge of the vibration mode shapes involved.

5.6.3 Hydraulic Unbalance

Hydraulic unbalance is induced by fluid flow and is usually caused by poor suction piping arrangement and design. A flow restriction or an elbow too close to the pump suction causes fluid to assume different velocities within the pipe. If these velocities do not equalize before reaching the impeller, the hydraulic unbalance will impose a high radial vibration at the running frequency of the machine.

5.6.4 Bent Shaft

A bent shaft will appear as an unbalance condition, accompanied by high axial vibration due to contortion of the rotor configuration. A twice-per-revolution harmonic will also appear in the spectrum. Phase readings are usually taken to differentiate between an unbalanced rotor and a bent shaft. Phase readings at both bearings in the same direction indicate an in-phase relationship, whereas readings in the axial direction indicate an out-of-phase condition.

5.6.5 Misalignment

Misalignment is considered the second most prevalent vibration source. It is caused by the non-coincidence of rotating axes of coupled components. Misalignment results in a high axial vibration reading, in addition to multiples of running speed, mainly the two-times (2X) component ( Figure 48 ). The axial reading may be as high as twice the vertical reading.

Misalignment causes pump vibrations resulting in seal leakage, overheated bearings, and coupling wear. Soft foot and pipe stresses contribute a great deal to misalignment.

5.6.6 Cavitation

Cavitation occurs at low pressure regions within the pump. The presence of vapor cavities occurs whenever the pumping medium pressure drops to the vapor pressure of that medium.

When these vapor cavities reach a higher pressure region of the pump, they quickly collapse or implode, causing noise, vibration and pitting of the impeller/diffuser. Figure 49 shows the effects of entrained air or gases on centrifugal pump performance .

Cavitation can be avoided by maintaining a sufficient NPSH and by maintaining sub-cooled fluid at the impeller suction.

71


5.6.7 Recirculation

Hydraulic excitation forces and pressure pulsations created by excessive flow deceleration at partial load can result in pump sub-component failures. These forces and pulsations are the result of excessive flow recirculation in the impeller inlet, diffuser, or volute sections of the pump. In multi-stage pumps, radial and axial balance is necessary.

Recirculation is caused by high suction pressure pulling fluid back through seals and impeller clearances thus reducing pump output flow. Recirculation causes excessive vibration, noise and wear in pump internals. Recirculation can be reduced by increasing discharge flow or by installing a bypass line from the discharge to the suction.

Figure 48: Vibration Spectrum Indicating a Misalignment

5.6.8 Impeller/Diffuser Interaction

Turbulence occurs when the fluid interacts with the diffuser at less than the rated flow rate.

Improper clearance between the impeller and casing is the main cause of turbulence and the resulting sub-synchronous and random vibrations. Further evidence of impeller/diffuser interaction is acoustic resonance in the discharge piping caused by pressure pulsations. The piping resonates at the vane-pass frequency ( Figure 50 ), regardless of the number of fixed vanes.

72


5.6.9 Temperature

Fluid viscosity is a function of temperature. A change in viscosity alters liquid handling characteristics in a system. Increased fluid temperature can cause a pump to work harder, lose efficiency and cavitate. Flow restrictions may have similar effects on the pump.

5.6.10 Pressure

Piping transients such as water hammer can occur when pressure in certain regions of a piping system is reduced to the vapor pressure of the fluid. The subsequent collapse of a large volume of vapor results in the pressure transient. Pressure transients in piping systems are also caused by rapid changes in pump operation and valve position.

LIQUID WITH NO GAS

H

E

A

D

4000

3500

3000 feet 2500

2000

MIN FLOW 0%

5 % MORE GAS IN LIQUID

0 100 200 300 400 500 600 700

FLOW ( gpm )

Figure 49: Effects of Entrained Air or Gas on Centrifugal Pump Performance

73


Figure 50: Vibration Spectrum Indicating a Vane-Pass Excitation (5X) with Sum and

Difference Frequencies

5.6.11 Flow

Flow rates vary depending on variations in static head, valve alignment, system loads, and flow control valve operations and machinery wear. Measurement of balance drum flow

(typically 27-30 gpm) is a good indicator of wear on the balance drum, sleeve and ring.

When flow increases to approximately 40-45 gpm, this indicates excessive wear of internal pump components.

5.6.12 Anti-Friction Bearings

In anti-friction bearings, structural deformation due to metal-to-metal contact of the balls or rollers and/or races produces a high frequency, low amplitude vibration ( Figure 51 ).

The useful life of a rolling element bearing depends on two basic sets of variables:

•

The application set of variables, such as load, speed, temperature, mounting, lubrication, etc.

•

The configuration of the bearing itself, including the design, material and method of fabrication

74


Review of literature has shown that it is possible for a superior quality bearing to fail to achieve rated life predictions. Many such bearing failures are caused by improper mounting and lubrication, as well as contamination, high temperature and load. Manufacturing defects also play an important role in bearing failures. As many as 10 % of new bearings are reported to have manufacturing defects. Improper storage practices may also reduce the life of a bearing.

Typical defects include those found in the inner or outer race, the balls or rollers, and the retainers (sometimes referred to as ball separators or cages). Other failure causes include improper internal clearances and the imposition of either thrust or radial loads.

Most anti-friction bearing failures have characteristic frequencies that are readily discernible from the fundamental running frequency of the rotating equipment.

Anti-friction Bearing Frequencies:

The predominant frequencies generated by anti-friction type bearings can be classified as:

•

Outer race frequency (BPFO)

•

Inner race frequency (BPFI)

•

Ball spin frequency (BSF)

•

Train frequency (FTF)

Race Frequencies (BPFO and BPFI) are produced as the balls or rollers pass over a defect in the raceway. As a ball or roller strikes the raceway, it produces a particular response at BSF.

The defect can impact both races during each revolution; thus, the response can be two times the operating speed. Rotation of the cage and ball or roller assembly or train produces FTF.

Also, when particular faults occur, harmonics are generated with unique characteristic frequencies.

75


Figure 51: Vibration Spectrum Showing a Bearing Fault (BSF)

Spectrum Analysis of Rolling Element Bearings:

Formulas have been developed to calculate bearing characteristic frequencies associated with a ball or roller bearing given the following:

•

Rotating speed

•

Number of balls or rollers

•

Diameter of ball or roller

•

Pitch diameter

•

Contact angle

In general, there is no set rule to determine when a critical condition has been reached.

However, experience has shown that trends in the spectrum, such as the following, may indicate a developing bearing fault:

•

Shifting from single peaks to a broad band of energy with the running speed superimposed

76


•

An increase in amplitude

•

The presence of any bearing frequency related peaks

•

The appearance of single peaks at characteristic frequencies that were not present during start-up

•

A combination frequency caused by the sum and/or difference of several characteristic frequencies

5.6.13 Sleeve Bearings

Excessive clearances in sleeve bearings can cause vibration problems at rotating frequency

(1X). This condition can be determined by taking both shaft and bearing casing displacement readings in the same direction.

Frequency analysis of the response of fluid film bearings can be categorized as follows:

•

Running speed harmonics

•

Oil whirl sub-harmonics

•

Natural frequencies of the rotating elements (attached to the shaft)

•

A multiple of the running speed times any combination of the pads (tilt pad bearings)

The one-time frequency of rotation is usually associated with unbalance excitations and can also be generated by the bearing itself. In fluid film bearings, the synchronous frequency can be caused by a single wipe of the bearing inner surface, a shaft offset, or a flat spot on the shaft.

As the shaft rotates in its bearings with a slight offset, misalignment or thermal growth, a portion of the rotating element can contact the portion of the stationary housing, ring seals, etc. This can produce a response by any part attached to the journal, or by the journal itself.

These responses will be the resonant frequencies associated with these parts.

5.6.14 Oil Whip

Oil whip is another condition that is exhibited by rotating equipment with sleeve bearings.

Due to oil wedge buildup and breakdown, this condition occurs normally at slightly less than half of shaft rotating frequency. The primary cause of an oil whip condition is the unloading of the bearing, allowing the shaft to ride too high in its bearing. However, other oil parameters, such as viscosity, temperature, pressure, and the location of the oil injection in the bearing, can all be factors. Oil whip occurs when the whirl frequency locks on to a rotor natural frequency.

77


5.6.15 Looseness

Mechanical looseness produces running speed harmonics at generally decreasing magnitudes

( Figure 52 ), depending upon the degree of looseness and the machine design. In horizontal machines, rotating forces impact twice in each revolution due to excessive clearances or lack of tightness. The vibration response reflects this double impact. A strobe light attachment synchronized to the frequency of rotation is typically used to investigate looseness of assembled parts.

Figure 52: Vibration Spectrum Indicating Looseness

5.6.16 Gear Defects

Gear defects generally produce a low amplitude vibration at gear mesh frequencies, and harmonics with side bands at multiples of running speed ( Figure 53 ).

5.6.17 Foundation/Structural Problems

Foundation failure can cause erratic changes in vibration amplitude and phase, especially during transient regime operations of high-speed rotating machinery. Bases, bedplates or baseplates, machine feet and mounting bolts are often overlooked as potential sources of vibration problems.

Soft foot is a term loosely applied to several mounting related faulty conditions. Soft foot can include structural looseness, deformation of the machine feet due to static or dynamic loading, weakness of the baseplate and a degraded foundation. Soft foot is a structural problem, although it is frequently considered an alignment-related problem.

Attached structural components such as piping, seismic restraints and hangers might be poorly installed or adjusted, resulting in unwanted loads on the pump casing that can cause distortions resulting in excessive vibrations.

78


Figure 53: Vibration Spectrum Showing a Gear-mesh Excitation

5.6.18 Resonance

Resonance occurs when a forcing frequency falls within the range of the natural frequency of the excited system ( Figure 54 ). Resonance conditions can amplify the vibration to dangerous levels, depending largely on the amount of damping present in the excited mode of vibration.

79


Figure 54: Vibration Spectrum Indicating a Resonance Condition Excited by Fan 9X Blade

Pass In the Presence of Misalignment

Rotating machinery critical speeds present a special case of resonance conditions during runup or coast-down operations. They involve the synchronizing of the rotating frequency with the rotor natural frequency.

5.6.19 Seal leakage

Mechanical seals are considered to have failed when leakage becomes excessive or when excessive reduction in pressure in the sealing system is experienced. Mechanical seals do not provide zero seal leakage.

80


81


6.0 N

UCLEAR

C

ONSIDERATIONS

Both nuclear and non-nuclear applications are based on the same engineering principles. The difference for nuclear applications is primarily in Quality Assurance, Quality Control, and documentation of the design, installation, and inspection results. Also, pumps that are considered to be safety related must be able to withstand worst case conditions with respect to flow rates, running duration and environmental conditions (temperature, humidity, etc.), for the duration of the postulated design basis event. When considering the service life of safety related pumps, these issues must be factored in addition to “normal” operating condition factors over the life of the plant.

Vibration characteristics of pumps in nuclear power plants are especially important due to the relative inaccessibility of much of the equipment. Safety requirements often allow only limited opportunities and/or time for testing and these in-service tests must often be conducted at offdesign conditions for the particular pump

83


84


7.0 U

TILITY

E

XAMPLE EXERCISES AND

S

OLUTIONS

7.1 Exercise No. 1 (Pump Internal Degradation Assessment)

Example Performance Calculation

The following example demonstrates calculations involved in quantifying loss of performance for a variable speed pump (AFWP). The design speed ( N d

) for this pump is

3370 rpm. Two identical pumps are tested for internal degradation assessment.

Given the data in the Table 5 , calculate the head loss of each pump and assess whether maintenance is needed.

Solution:

1 st -

Calculate the Specific Volume

ρ

: v

The specific volume is defined as the volume per unit mass. It is the reciprocal of density. Given the temperature and the pressure, a look-up in the steam tables provides the specific volume.

2 nd

- Calculate the Pump Head H :

H

=

144 *

[ (

ρ v

* P

)

DISCHARGE

−

(

ρ v

* P

)

SUCTION

]

Eq. 105

Where, P is the pressure in psi from Table 5 .

3 rd

- Calculate Corrected Pump Head H for actual speed c

N using affinity law: a

H c

=

H *





N d

N a





2

Where, N d is design speed from above and N a is actual speed from Table 5 .

Eq. 106

4 th

- Measure the pump flow Q then calculate flow in GPM :

GPM

=

7 .

481 * 1000 * Q *

ρ

V

60

SUCTION

Eq. 107

85


5 th

- Calculate Corrected Pump flow GPM c for actual speed N a

using affinity law:

GPM c

=

GPM

N

*

N a d

Eq. 108

6 th

- Calculate Corrected Pump flow GPM

T at design temperature

The density is calculated as the reciprocal of the specific volume:

ρ =

1 /

ρ v

Eq. 109

The specific gravity is the ratio of the density of the substance to that of a reference substance. Water at 4 o

C is usually used as a reference for liquids.

Sg

=

ρ

ρ ref

Eq. 110

The design specific gravity is 0.887 as given by the pump curve (at 350 o

F)

GPM

T

=

GPM c

*

Sg d

Sg

SUCTION

Eq. 111

Where, Sg d is the design specific gravity and Sg

SUCTION is the suction specific gravity.

7 th

- Obtain the Design Pump Head H

D in feet:

Read the pump curve based on corrected flow GPM

T at design temperature and record the design pump head H

D

in feet.

8 th

- Calculate the Percentage Head Loss H% due to degradation

H %

=

( 1

−





H

H

D c





) * 100 Eq. 112

86


Table 5: Performance Testing Based Pump Internal Degradation Assessment

Design Data

Water density at 4 o

C, ρ ref

(lbm/ft

3

)

Design specific gravity

Design speed, (rpm)

Test Data and Calculations

Testing speed N d

, (rpm)

Suction temperature, ( o

F)

Suction pressure, (psia)

Suction specific volume, (ft

3

/lb)

Discharge temperature, ( o

F)

Discharge pressure, (psia)

Discharge specific volume, (ft

3

/lb)

3370

Pump A Pump B

3360 3334 Measured

345

230.32

0.0179

Notes

62.40

Used to calculate the specific gravity

0.887 Water property at 350 o

F

345 Measured

229.42 Measured

0.0179

Reference

Lookup in steam table given temperature & pressure

348.67 348.74 Measured

2602.77 2605.97 Measured

0.0177 0.0177

Lookup in steam table given temperature & pressure

6629.8

6669.3

998

2227.4

6638.0 Eq.105

6782.1 Eq.106

850 Measured

1897.1 Eq.107

Pump Head, (ft)

Speed Corrected Pump Head, (ft)

Flow, (Klbs/hr)

Flow, (gpm)

Speed Corrected Pump Flow,

(gpm)

Suction density, (lb/ft

3

)

Suction specific gravity

Corrected pump flow at design temp

2234.0

55.87

0.895

2213.3

1917.5 Eq.108

55.87 Eq.109

0.895 Eq.110

1899.8 Eq.111

Design pump head, (ft)

Head loss

6900

3.3%

7400

Read from pump curve given flow at design T

8.3% Eq. 112

The expected head is 100 % but due to aging one might expect less than 100 % of design head. Any head loss more than what is expected due to aging will be considered due to pump internal clearance degradation. A general rule would be to use a delta H% of 5 to 7 depending on the application.

Conclusion: Based on loss of performance (Head Loss > 7%), pump B is degraded enough to warrant maintenance to rebuild pump internals.

87


7.2 Exercise No. 2 (Estimation of Pump Structural Dynamic Characteristics by Bump Test)

A numerical example demonstrating the use of a recorded time waveform to extract system dynamic characteristics is presented below.

Vibration measurements on a pump of mass m = 100 lbm, have produced the plot shown in

Figure 55 . At the fifth cycle time indication was 0.5 seconds vibration was 4 mils p-p (2 mils peak).

8

2

T i m e , s e c

Figure 55: Example of Free Oscillations Recording

Determine the pump structural dynamic characteristics.

Solution:

The vibration maximums may be truncated in some areas as shown in Figure 55 .

Pertinent data has been summarized in Table 6 :

Cycles

Time (sec)

Table 6: Free Oscillations Measurement

Vibration Maximums

0

0

8

1

0.1

-

2

0.2

-

3

0.3

-

( x Mils) i

4

0.4

-

5

0.5

2

The log-dec can be calculated by ( Eq. 29 )

88


δ = ln x i x i + 1

Considering p

=

5 cycles the above equation gives ( Eq.30

),

ξ ≈

δ

2

π p

=

1

2

π p ln x i x i + p

=

1

2

π

.

5 ln

8

2

=

1

31 .

4 ln 4

=

0 .

044

Calculate the period

T

= t i

+ p p

− t i =

0 .

5

−

0

5

=

0 .

1 sec

Calculate the frequency f

=

1

T

=

1

0 .

1

=

10 Hz

Calculate the damped natural frequency in rad/sec ( Eq.32

)

ω d

=

2

π

T

=

6 .

28

0 .

1

=

62 .

8 rad / sec

Calculate the undamped natural frequency ( Eq.33

)

ω n

=

ω d =

62 .

8

1

− ξ 2

1

−

0 .

044

2

=

62 .

9 rad / sec

Estimate the system stiffness ( Eq.15

) k

= m

ω n

2 =

100 x 62 .

9

2 =

395641 lbs / in

89


7.3 Exercise No. 3 (Pump Coast-Down Data - Bode Plot)

Problem Statement:

It was determined that an unbalance u = 1 oz-in exists in the pump of the previous example.

Plot the dynamic amplitude X r ϕ r with r

= ω

/

ω n

varying from 0 to 2.

Solution:

For each pump operating speed there is a corresponding tuning factor r where

ω n

=

62 .

9 rad / sec

= ω ω n

(see previous example). Given r, first calculate

ω

, then the corresponding pump speed in RPM, N

=

60

ω

.

2

π

Using the unbalance and the stiffness we evaluate (from Eq.63

) X

0

=

F

0 k

= u

ω 2 k

The calculation of ( Eq.65

) R r

=

1

is next.

( 1

− r )

+

( 2

ξ r )

2

Then, the vibration amplitude ( Eq.66

) X

=

RX

0 and phase angle ( Eq.64

) ϕ r

= tan

−

1

2

1

−

ξ r r

2 are tabulated (see Table 7 ). A plot of the vibration amplitude and phase is produced in Figure 56 . This is a typical pump coast down synchronous response plot (1X).

Table 7: Example Calculation of Vibration Response due to Harmonic Excitation r =

ω

ω n

ω = rw n

Rad/sec

Pump

Speed

RPM

X

0

= u

ω k

Mils

2

R ( r )

X

=

RX

0

Mils ϕ

( )

Degrees

0.00 0.00

0.25 15.73

0.50 31.45

0.75 47.18

0.85 53.47

1.00 62.90

1.10 69.19

1.25 78.63

1.50 94.35

750

900

1.75 110.08 1050

2.00 125.80 1200

0.

150

300

450

510

600

660

0.0 1.000

0.04 1.066

0.156 1.331

0.350 2.260

0.450 3.479

0.625 11.364

0.756 4.325

0.977 1.745

1.406 0.796

1.914 0.484

2.500 0.333

0.00

0.04

0.21

0.80

0.0

1.3

3.3

8.5

1.57 15.1

7.1 90.0

3.27 155.3

1.71 169.0

1.12 174.0

0.93 175.8

0.83 176.7

90


8

7

6

5

4

1

0

3

2

200

150

100

50

0 200 400 600 800 1000 1200

0

Pump Speed - RPM

Amplitude - Mils Peak Phase - Degrees

Figure 56: Plot of Vibration Response – Bode Plot

91


7.4 Exercise No. 4 (Vibration Isolation)

Problem Statement:

A pump weighing 1200 pounds is to be supported on four springs, each of stiffness k (lb/in.).

If the unit operates at 3200 rpm, what value of K=4k would be used if only 10% of the shaking force of the pump is to be transmitted to the supporting structure? Neglect damping and assume only vertical motion.

Solution:

To obtain 90% isolation r

= ω

/

ω n

>

1

Therefore, ( Eq.113

comes directly from Eq.63

)

F

T

F

0

= r

2

1

−

1

Eq.113

0 .

1

= r

2

1

−

1 r

2 r

=

11

=

ω

ω n

=

3 .

32

Using Eq.15 with M=W/g

ω 2 n

=

ω 2

11

=

K

⋅ g / W

ω =

3200 rpm

=

334 .

9 rad / sec

K

= W

ω 2

11 g

=

( 1200 )

(

334 .

9

)

2

/ 11 ( 386 )

=

31 , 704 lb / in

Each spring requires a stiffness of k=K/4. k

= K

4

=

31 , 704

4

=

7 , 926 lb / in

92


7.5 Exercise No. 5 (Mass Addition)

Problem Statement:

A pump weighing 5000 pounds is rigidly attached to a foundation of 15,000 lb, which has a vibration at 900 cpm (pump speed) at 12 mils peak to peak. How much mass needs to be added to the foundation to obtain an acceptable level of vibration at 0.35 in/sec- peak?

Solution:

The present magnitude of vibration velocity is v

=

2

π

900

0 .

006 in

=

60

0 .

56 in / sec

Since ( Eq.3

)

F

= ma

=

W g a

=

W g

( 2

π fv )

=

20 , 000

386

2

π

900

0 .

56

60

=

2 , 732 .

6 lb

Total required weight for the specified vibration limit of 0.35 in/sec is

W

=

Fg v 2

π f

=

2 , 732 .

6 x

.

35

386

2

π ⋅

15

=

32 , 000 lb

Added weight needed is total weight minus (pump + current foundation)

W

=

32 , 000

−

20 , 000

=

12 , 000 lb

93


94


8.0 S

OURCE

D

OCUMENTATION

1.

Engineering Technical Training Modules for Nuclear Plant Engineers , EPRI TR-109623,

Mechanical Series, Module #4, Centrifugal Pumps, April 1999.

2.

DeLaval Engineering Handbook, 3 rd

edition , Hans Gartmann, editor, McGraw-Hill, New

York, 1970.

3.

Fluid Mechanics and Hydraulics, 2 nd

editio n, Ranald V. Giles, McGraw-Hill, New York,

1962.

4.

Fluid Mechanics With Engineering Applications, 7 th

edition , Robert L. Daugherty and Joseph

B. Franzini, McGraw-Hill, New York, 1977.

5.

Pump Handbook, 2 nd

edition , Igor J. Karassik, et. al., McGraw-Hill, New York, 1986.

6.

Machinery Vibration: Measurement and Analysis , Victor Wowk, McGraw-Hill, New York,

1991.

7.

Machinery Vibration: Theory and Applications, 2 nd

edition , Francis S. Tse at al, Allyn and

Bacon, Inc., Boston, 1978.

8.

Shock & Vibration Handbook, 3 rd

edition , Cyril M. Harris, editor, McGraw-Hill, New York,

1988.

9.

Theory of Vibration with Applications , William M. Thompson, Prentice-Hall, Inc.,

Englewood Cliffs, NJ, 1972.

10.

Flow-Induced Vibration, 2 nd

edition , Robert D. Blevins, Van Nostrand Reinhold, New York,

1990.

11.

Marks’ Standard Handbook for Mechanical Engineers , T. Baumeister, et. al., McGraw-Hill,

New York, 1996.

12.

Root Cause Analysis and Corrective Actions for Power Plants , Chong Chiu, Failure

Prevention, Inc., San Clemente, CA, 1992.

13.

Shaft Alignment Handbook, 2 nd

edition , expanded, John Piotrowski, Marcel Dekker, Inc.,

New York, 1995.

95


96


9.0 I

NDUSTRY

O

PERATING

E

XPERIENCE

Diagnostic Table Using Vibration Analysis

Cause Frequency Amplitude

Unbalance 1 x RPM

Misalignment

(1, 2, 3, …) x

RPM

Proportional to unbalance Radial

- steady

Axial – high

Eccentricity 1 x RPM Varies

Bent shaft (1 to 2) x

RPM

Thermal bow

1 x RPM

Looseness (1, 1.5, 2, 2.5,

3, …) x RPM

Axial - high

Varies

Proportional to load

Soft foot 1 to 2 x RPM Proportional to load

Large Electrical 1 x RPM or 1 to 2 x line frequency

Sleeve bearings wear and clearance

(1, 2, 3, 4, …) x RPM

Oil whip .5 x RPM

May be higher in

Vertical than

Horizontal

Oil whirl

Anti-friction bearings

(.42 to .48) x

RPM

Radial – unsteady, excessive

Radial – unsteady, sometimes severe

Radial - low

Rubbing

BPFI, BPFO,

BSF, FTF and

Harmonics

(0-0.5)x, 1x, and higher harmonics

Erratic

Phase

1 Reference mark - steady

1, 2 or 3 reference marks

0 or 180 o between

Horizontal and Vertical

180 o

out of phase axially

1 Reference mark - steady

2 reference marks, slightly erratic

Erratic

Erratic

Erratic

Erratic

Erratic

Erratic

Notes

Most common cause of vibration, no phase change

Second most common cause of vibration.

Axial amplitude may be twice the vertical or horizontal.

Balancing may reduce vibration in one direction but increase it in the other

Same radial phase on both bearings Orbit and phase are good parameters to monitor

Increasing vibration during load variations and startup from a cold condition

Frequently coupled with misalignment

Strobe may help. Amplitude depends on load

Check mountings for variations in amplitude

When power is turned off vibrations disappears instantly

Compare shaft to bearing displacement readings. Oil analysis best monitor for wear

Frequency is near one-half running speed

(machine speed is nearly 2x critical speed)

Oil temperature is a good indicator

Caused by unloading of bearing. Tangential destabilizing force due to lube film in the direction of rotation adds energy to vibration

Use velocity, acceleration or spike energy

Similar to impact, may excite many system frequencies

97


Gears

Diagnostic Table Using Vibration Analysis (Continued)

GMF=Z x

RPM

Foundation Unsteady

Radial - low

Erratic

Resonance System specific

Cracks 1x, 2x RPM,

High

Variable during transients. Drop in higher harmonics

High radial and axial

Hydraulic

Forces

Vane Pass =

Z x RPM and harmonics

Cavitation Random high frequency +

Vane Pass

High radial and axial

Erratic

Unstable reference

Erratic

Phase shift

NA

NA

Use velocity or acceleration. Tooth wear is better indicated by side-bands around GMF and excitation of tooth natural frequency.

Higher tooth load will increase amplitude at

GMF. Backlash is characterized by decreasing amplitude at GMF when load is increased. Gear misalignment shows with higher 2x and 3x GMF. A cracked or broken tooth is best seen on the time signal. A hunting tooth problem shows at very low frequencies

Strobe may help

Increased levels at resonant frequency.

Often appears on old machines pedestals

Increased levels at resonant frequency

Phase is a good indicator. 2x RPM excitation of critical speed during coast down.

Use velocity or acceleration. Due to uneven internal gap between rotating vanes and diffuser. May excite natural frequencies.

Flow obstructions are common causes.

Due mainly to insufficient suction pressure and the presence of vapor and air in the liquid.

98


99


10.0 P

ROFICIENCY

M

EASURES

10.1 Proficiency Measures (Questions):

1.

Select the statement that describes pump cavitation. a Vapor bubbles are formed when the enthalpy difference between pump discharge and pump suction exceeds the latent heat of vaporization. b Vapor bubbles are formed and enter a high-pressure region where they collapse. c Vapor bubbles are formed when the localized pressure exceeds the vapor pressure at the existing temperature. d Vapor bubbles are discharged from the pump where they impinge on downstream piping and cause a water hammer.

2.

The presence of air in the casing may result in __________ when the pump is started. a vortexing b pump runout c head loss d gas binding

3.

Pump head is a measure of a Energy expressed in MW b Fluid dynamic pressure expressed in PSI c Energy expressed in feet d Kinetic energy of the fluid leaving the pump expressed in feet

4.

Pump internal clearance degradation can best be estimated using a Performance test data and evaluating the pump head loss at operating speed b Vibration analysis c A static rotor lift and measuring the axial end-play d The balance line differential pressure at BEP

5.

The general solution of the equation of motion of a SDOF system is a The forced vibration response given the initial conditions b The sum of the free vibration response and the forced vibration response c The free vibration response of the damped system with initial conditions d The mechanical impedance method given a harmonic excitation

6.

The natural frequency of a system depends on a The external and internal excitations of the dynamic system b The initial conditions

101

EPRI Licensed Material c The physical properties of the mechanical system d Amount of residual unbalance

7.

The natural frequency of a pump/structure system can be altered by a Changing the mass, stiffness, or damping of the system b Modifying the support of the system c Performing a balancing of the pump rotor d Aligning the pump shaft to specifications

8.

A critical speed can be defined as a The pump speed that is critical to operations b The pump design speed c A natural frequency of the rotor that lies within the operating speed range d The natural frequency of the rotor/bearing system

9.

A pump critical speed can be determined by a A Bode plot b A bump test while the pump is in operation c Performance testing d The method of the half-power bandwidth

10.

An acceptable definition of unbalance is a The amount of rotor eccentricity b The residual vibration amplitude c The centrifugal force at synchronous frequency d A weight times eccentricity

11.

Alignment is a condition that can be created by a Thermal growth b An unbalance condition c Improper lubrication d Off BEP operation

12.

The most commonly diagnosed mechanical conditions in a pump using vibration are a Thermal growth and leakage b Unbalance and misalignment c Cavitation and recirculation d Bent shaft and vane pass excitation

13.

The most common type of vibration measurement is usually made by: a A displacement probe b An accelerometer c A velocity transducer d A strobe light

102


11.2 Proficiency Measures (Solutions)

1) b Vapor bubbles are formed and enter a high pressure region where they collapse

2) d gas binding

3) d Kinetic energy of the fluid leaving the pump expressed in feet

4) a Performance test data and evaluating the pump head loss at operating speed

5) b The sum of the free vibration response and the forced vibration response

6) c The physical properties of the mechanical system

7) a Changing the mass, stiffness, or damping of the system

8) c A natural frequency of the rotor that lies within the operating speed range

9) a A Bode plot

10) d A weight times eccentricity

11) a Thermal growth

12) b Unbalance and misalignment

13) b An accelerometer

103


104

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1

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