MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
FACULTY OF MECHANICAL ENGINEERING
UNIVERSITI TEKNOLOGI MARA
PULAU PINANG
PROGRAMME :
Bachelor of Mechanical Engineering (Hons)-EM 241
COURSE
Applied Mechanics Lab (MEC 424)
:
LAB SHEET NO :
Title:
D1 ( Dynamics and Vibrations 1)
Free Undamped and Damped Mechanical Vibration of a Single Degree-of-Freedom System.
Objectives:
1. To determine the period and natural frequency of vibrating system.
2. Identify the effects of damper in a Single Degree-of-Freedom vibrating system.
Apparatus:
GUNTT TM 105 Universal Vibration System
1
2
k
3
4
Legend
1. Adjustment knob ;2. Spring (k=0.84 N/mm); 3. Chart plotter
4. Beam (size: 700 x 25 x 12mm, weight: 1.68 kg); 4. Damper
a= 650mm, b=200mm; Chart speed= 20 mm/s
5
Fig. 1 : The Universal vibration system and the control unit
Introduction:
A. Undamped free vibration
The system shown in fig. 1 can be classified as a single degree of freedom (SDOF) system, since the
vertical direction is sufficient to be used as generalized coordinate that describes the motion. The
undamped motion of SDOF system is simple harmonic motion. Method in establishing the equation of
motion follows the fundamental steps in analyzing and solving dynamic system problems by first
drawing the free body diagram and subsequently applying the Newton’s laws of motion. Thus, the
equation of motion for the system shown in fig. 1 is obtained as:
2
𝑘𝑎
𝜃̈ + 𝐼 𝜃 = 0
𝑜
Prepared By: Bakri Bin Ali, FKMPP
(1.1)
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Where k = spring constant, a = the length of the beam, 𝐼𝑜 = mass moment of inertia
Equation 1.1 can be further simplified into:
𝜃̈ + 𝜔𝑛 2 𝜃 = 0
(1.2)
Where 𝜔𝑛 is the natural frequency of the system.
𝑘𝑎2
𝐼𝑜
𝜔𝑛 = √
Therefore,
(1.3)
B. Damped free vibration
The damper is a device or system that dissipates the mechanical energy. Viscous damper is the most
common damping element used to retard the vibration. Using similar approach, the equation of motion
for the damped vibration with damping constant, c, is obtained as:
2
2
𝑐𝑏
𝑘𝑎
𝜃̈ + 𝐼 𝜃̇ + 𝐼 𝜃 = 0
𝑜
𝑜
(1.4)
Equation 1.4 can be further simplified into:
𝜃̈ + 2𝜔𝑛 𝜃̇ + 𝜔𝑛 2 𝜃 = 0
Where the damping ratio,  =
𝑐
2𝑚𝜔𝑛
(1.5)
, and the damped oscillation frequency can be determined from
𝜔𝑑 = 𝜔𝑛 √1 − 2.
The solution for equation 1.5 leads to a characteristic equation which contains the characteristic roots.
Depending on the damping constant value, c, the solution for the roots reveal three cases of vibrating
mode:
a). overdamped – the value of the damping constant large enough to prevent the mass from vibrating
b). underdamped – the value of the damping constant is small enough to allow the mass to oscillate.
c). critically damped – the borderline case between case a and b
The free response of vibrating system as shown in fig. 3 illustrates the three modes of vibration.
Fig. 2 : Free response of vibrating system
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Experimental Guide and expected result
The undamped free vibration experiment of SDOF can be executed by simply giving the initial
displacement of the beam (without damper). The beam will oscillate and the traces of the oscillation can
be recorded via the graph paper attached on the chart plotter. Analyzing the chart would reveal the
period and frequency of the oscillation. As a comparison, theoretical natural frequency of the system
can be estimated from equation 1.1.
The damped free vibration experiment can be done by fixing the damper in place as shown in fig. 1. The
value for damping constant is adjustable via the adjusting screw (fully closed, c = 15 N.s/m) as shown in
fig. 2. In order to analyze the behavior of damped vibrating system, several sets of experiment can be
executed by varying the damping constant. The traces of beam oscillations for different values of c,
could be established following the similar steps above (undamped).
Result:
cc =
Needle Valve
position
kgm2.
, IO =
Theoretical
f (Hz)
T(s)
c(N/s)
Experimental

f (Hz)
T(s)
c(N/s)

Fully closed
Slightly open
Medium
Maximum open
No damper
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
FACULTY OF MECHANICAL ENGINEERING
UNIVERSITI TEKNOLOGI MARA
PULAU PINANG
PROGRAMME :
Bachelor of Mechanical Engineering (Manufacturing)-EM 241
COURSE
Applied Mechanics Lab (MEC 424)
:
LAB SHEET NO :
Title:
D2 (Dynamics and Vibrations 2)
Forced Undamped and Damped Vibration of a Single Degree-of-Freedom System.
Objectives:
1. Identify the modes of vibration for undamped forced vibration systems at different frequency ratios.
2. Estimates the damping ratio using Frequency Response Function.
3. Identify the behavior of mechanical system at resonance.
Apparatus:
GUNTT TM 105 Universal Vibration System
1
1. Adjustment knob.
2. Spring (k=3.00 N/mm)
k
k
3. Chart plotter
2
4
3
6
4. Beam (700x25x12mm, 1.68kg)
5. Damper
5
6. Exciter
b= 200mm, a=650mm
Exciter shaft
Unbalance
mass (100g)
10mm
Prepared By: Bakri Bin Ali, FKMPP
Chart speed= 20 mm/s
If the exciter is constrained to move in vertical direction, then this
force will fluctuate with magnitude F(t)= Fo sin t, where Fo is a
constant centripetal force due to rotation of unbalanced mass and
given by;
Fo= mω2r ;
m= unbalanced mass= 100g
ω= angular speed of the rotor (rad/s) = 2πf
r = position of the unbalanced mass from the
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rotor centre=10mm.
f= speed of the rotor (Hz)
MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Introduction:
A. Undamped forced vibration
Recall the single-degree-of-freedom mass-spring system from experiment 1. Consider the effects of a
time-varying force, f (t), applied to a mass, starting from rest. Application of Newton’s second law to the
free-body diagram following similar approach described in experiment 1 gives the equation of motion,
i.e
2
𝑘𝑎
𝜃̈ + 𝐼 𝜃 = 𝐹(𝑡)
𝑜
(2.1)
The solution for equation 2.1 consists of the particular solution (xp) and complimentary solution (xc). The
𝐹
static deflection of the mass due to the force F is given by 𝛿𝑠𝑡 = 𝑘 , and the ratio of the amplitude of
vibration to the amplitude of zero frequency deflection is called the magnification factor (M).
𝑋
1
Therefore,
𝑀=
=
(2.2)
𝜔 2
𝛿𝑠𝑡
|1−(
𝜔𝑛
) |
The system response at different frequency ratio range is shown in fig. 2. The value of the frequency
ratio, r, equal to one has a special significance; it is termed resonance: the condition where the driving
frequency, is equal to the system’s natural frequency and it is marked by large response amplitudes.
Fig. 2 The time response of the system at different frequency ratio range, a) when r > 1, b) when 0 < r
< 1, c) when r  1 (beating), and d) when r = 1 (resonance)
B. Damped forced vibration
Recall the single-degree-of-freedom mass-spring-damper system from experiment 1. Consider the
effects of a time-varying force, f (t) ), applied to a mass, starting from rest. The equation of motion is
obtained as:
2
2
𝑐𝑏
𝑘𝑎
𝜃̈ + 𝐼 𝜃̇ + 𝐼 𝜃 = 𝐹(𝑡)
𝑜
Prepared By: Bakri Bin Ali, FKMPP
𝑜
(2.3)
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MEC 424: APPLIED MECHANICS LAB
The natural frequency for the system can be obtained from, 𝜔2 =
Dynamics and Vibrations
𝑘𝑎 2
,
𝐼𝑜
and the frequency of the
𝑐
oscillation is given by, 𝜔𝑑 = 𝜔√1 − 2 . Where the damping ratio,  = 𝑐 . In similar approach, the ratio
𝑐
of the amplitude of vibration to the amplitude of zero frequency deflection gives the magnification
factor (M).
𝑀=
1
√(1−𝑟 2 )2 +(2𝑟)2
(2.4)
Fig. 3 illustrates the correlation between the magnification factor and frequency ratio at different values
of damping ratio. It can also be observed that at damping ratio equals unity, the resonance effect takes
place. The effect of resonance, in some cases may be beneficial in engineering design, such as in
industrial shaker design, but in most cases resonance is undesirable, such as in automotive suspension
or machine tool designs.
Fig. 3 Relationship between the magnification factor and frequency ratio at different value of damping
ratio
Experimental Guides and expected result
Undamped forced vibration: This experiment can be implemented by installing the exciter motor on the
arm of the beam. The motor will induce external excitation to the vibrating beam system and its
oscillation can be recorded using graph paper attached on the chart plotter. Analyzing the chart would
reveal the period and frequency of the oscillation. The frequency at which the resonance occurred can
be determined by analyzing the trends of the plotted graphs. Therefore several sets of experiment could
be done at varying frequencies (motor rotation speeds). As a comparison, theoretical values of
magnification factor, amplitude of oscillation can be determined from equations 2.1 and 2.2.
Damped forced vibration: This experiment can be done by repeating the undamped forced vibration but
with the damper fixed in place as shown in fig. 1. The frequency at which the resonance occurred can be
determined by analyzing the trends of the plotted graph. Therefore several sets of experiment could be
done at varying frequencies (motor rotation speed) as well as different values of damping constant (fully
closed valve, c = 15 N.s/m). As a comparison, theoretical values of magnification factor, amplitude of
oscillation can be determined from equations 2.4.
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Result:
cc =
kgm2,
, IO =
Needle Valve
Fully closed
f (Hz)
Xp(mm)
(rad/s)
Fo=__________kgm/s2
/n
MF

1
2
3
4
5
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
FACULTY OF MECHANICAL ENGINEERING
UNIVERSITI TEKNOLOGI MARA
PULAU PINANG
PROGRAMME :
Bachelor of Mechanical Engineering (Hons)-EM 241
COURSE
Applied Mechanics Lab (MEC 424)
:
LAB SHEET NO :
Title:
D3 (Dynamics and Vibrations 3)
Determine of critical speed of whirling machinery.
Objective:
To determine the critical speed of whirling machinery to avoid resonance condition.
Apparatus:
GUNTT TM 105 Universal Vibration System
6. Secure nut.
7. Rubber pressure plate.
8. Circular Chart paper
9. Disc.
5
10. Plotter pen
Exciter shaft
Unbalance
mass (100g)
Prepared By: Bakri Bin Ali, FKMPP
10mm
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Introduction:
Theory:
If the unbalance rotating machinery is constraint to move in vertical direction, then the excitation force
will fluctuate with magnitude F(t)= Fo sin t, where Fo is a constant centripetal force due to rotation of
unbalanced mass and given by;
Fo= mω2r ;
m= unbalanced mass
ω= angular speed of the rotor (rad/s) = 2πf
r = position of the unbalanced mass from the rotor centre.
f= speed of the rotor (Hz)
The equation for forced vibration written as;
x + ωn2x = Fo sin t
for undamped vibration and
x + 2ζωn x + ωn2x = Fo sin t
for damped vibration.
The steady state solution of this problem can be written as:
x(t)= Xp sin (t +)
The result states that the mass will oscillate at the same frequency, f, of the applied force, but with a
phase shift . The phase shift, is defined by the following formula.
= tan-1 [(2ζ r)/(1-r2)]
The amplitude of the vibration “Xp ” is defined by the following formula.
Xp= (Fo/k) 1-( ω/ωn)22+[2ζ ω/ωn]2
Or
Xp= (Fo/k) 1- r22+[2ζ r]2
Where r = ω/ωn or f/fn and defined as the ratio of the harmonic force frequency over the undamped
natural frequency of the mass–spring–damper model. The plot of these functions, called "the frequency
response of the system", presents one of the most important features in forced vibration.
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Procedure:
Set the equipment as shown in the diagram. Put a circular chart on the disc, secure with rubber pressure
plate and tighten the nut. Set plotter pen to be vertically inline with the center of the chart on the 30m
radius circle (middle ring). Mark the position of unbalance mass on chart paper (inline with the locking
screw).Tighten the needle valve for a maximum damping coefficient. lift the plotter pen and slowly
Increase the excitation frequency knob for 1 revolution. Release the pen to plot the graph. Again, lift the
plotter, increase the speed and release pen. Repeat up to 5 different frequency. Change the chart paper.
Loose the needle valve for about one revolution and repeat the above procedure for 4 position of
needle valve. Lastly, detach the damper from the bar and repeat the procedure. Find the data required
as in the table shown.
Result:
ωn =
rad/s (from second Experiment,D2)
x1, x2,x3= amplitude for
different frequency
Unbalance position mark
1, 2, 3= phase shift for
different frequency
1. Needle Valve Fully closed
Speed Controller
ω(rad/s)
ω/ωn
Xp(mm)
( ̊ )
f(Hz)
5.0
8.5
12.0
15.5
19.0
Repeat table for different position of needle valve: Slightly open, Medium, Maximum open and No
damper.
Prepared By: Bakri Bin Ali, FKMPP
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MEC 424: APPLIED MECHANICS LAB
Dynamics and Vibrations
Discussion:
Conclusion:
References:
Prepared By: Bakri Bin Ali, FKMPP
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