1

RANDOM VIBRATION
Random vibrations are non-periodic. Knowledge of the past history of random vibration is adequate to predict the probability of occurrence of acceleration, velocity and displacement magnitudes, but it is not sufficient to predict the precise magnitude at a specific instant.
For most structural vibrations the excitation such as force or base acceleration alternates evenly about zero. Consequently, mean values characterizing the excitation as well as responses to that excitation such as displacement or stress are equal to zero.
For this reason, results of random vibrations analysis are given in the form of Root Mean Square
(RMS) values. To explain the concept of RMS value, refer to graph on the next slide which shows acceleration time history (acceleration as a function of time) of random vibration expressed in gravitational acceleration [G].
2

RANDOM VIBRATION
Acceleration [G]
Time [s]
An example of acceleration time history data collected during 1.5s
Considering the sampling rate of 5000 samples per second, this time history curve contains 7500 data. The overall mean value of acceleration time history G
RMS
2
The overall root square of overall G
RMS
2 = 0.27G
2 is G
RMS
= 0.52G is 0.27G
2 .
Random vibration is composed of a continuous spectrum of frequencies. The huge amount of time history data makes it impractical to run a dynamic time analysis.
3

RANDOM VIBRATION
Mean value of acceleration time history
G
RMS
2 = 0.27G 2
The acceleration time history has a zero mean value. However, if we multiply the function by itself we obtain a function with positive value. Its mean will not be zero and this squared function will be well suited to characterize the acceleration time history. This mean value of square acceleration time history is the mean square value and has units [G 2 ]. The square root of the mean value is the root-mean-square
(RMS) acceleration and has units of [G]. The same applies to RMS displacement, velocity, stress etc.
In random vibration, the magnitudes of acceleration, velocity, displacement etc. all follow normal distribution. The RMS value corresponds to one standard deviation σ characterizing the normal distribution. The magnitude of acceleration, as characterized by the given acceleration time history, has
68% probability of remaining below 0.52G. Consequently, it has 32% probability of being less than
-0.52G or more than 0.52G.
4

ACCELERATION POWER SPECTRAL DENSITY
Let’s assume that the acceleration time history in the previous slide is a stationary random process where probability numbers characterizing do not change with time. This acceleration time history can be used to calculate the Acceleration Power Spectral Density (PSD) * curve.
The overall G
RMS
2 of acceleration of random vibration is 0.27G
2 . However, random vibrations are composed of a large number of frequencies. Let’s say we wish to investigate G
RMS
2 individually for a number of frequencies in the range from 0 to 2000Hz. Therefore, we divide the 0-2000Hz range into
20 bins, each 100Hz wide and calculate G
RMS
2 characterizing each bin by filtering out all frequencies falling outside of the bin (next slide),
* Variation of any property with respect to frequency is called “spectrum”.
5

ACCELERATION POWER SPECTRAL DENSITY
G
RMS
2 = 0.16
G
RMS
2 = 0.30
Squared acceleration time history.
G
RMS
2 = 0.19
G
RMS
2 calculated individually for specified frequency ranges. Only three frequency ranges (bins) are illustrated here for brevity.
Having found G
RMS
2 values obtained for each frequency range we may now calculate individual “density” of
G
RMS
2 in each bin dividing G
RMS
2 in each bin by the width of the bin. Results obtained for all bins may be plotted as a function of the frequency in the center of each bin. This function is the Acceleration Power
Spectral Density (next slide).
6

ACCELERATION POWER SPECTRAL DENSITY
Constructing Acceleration Power Spectral Density function (PSD)
Three points of the Acceleration PSD curve have been calculated by dividing G
RMS
2 frequency range) by the width of the bin. in each bin (each
The Acceleration Power Spectral Density (PSD) allows compression of data and is commonly used to characterize a random process. In particular, mechanical vibrations are commonly described by the
Acceleration Power Spectral Density which is easily generated by testing equipment.
7

INTERPRETATION OF RESULTS OF RANDOM VIBRATION ANALYSIS
Example
RMS displacement is
σ = 7.5 mm.
Calculate the probability that the absolute displacement X will be over 9 mm.
1 Convert the normal distribution into standard normal distribution using the formula: z =
X- μ
σ
2 Calculate the z score: z =
X- μ
9 - 0
σ
=
7.5
= 1.2
X = 9 a score from the original normal distribution
μ = 0 the mean of the original distribution
σ = 7.5 the standard deviation of the original distribution
3 Calculate the probability | z | > 1.2
Based on Gauss distribution the probability of z staying within –1.2 < z < 1.2 equals the white area under the curve or 77%. Thus the probability of X exceeding 9 mm is 23%
White area is 0.7699
Blue area is 0..2301
–1.2 < z < 1.2
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INTERPRETATION OF RESULTS OF RANDOM VIBRATION ANALYSIS
1.0
0.8
0.5
0.3
-3.0
-2.0
-1.0
0.0
0.0
1.0
2.0
3.0
Normal standard cumulative distribution
0.4
0.3
0.2
0.1
-3.0
-2.0
-1.0
0.0
0.0
1.0
2.0
3.0
Gauss' curve
RMS displacement
Investigated displacement
Probability that absolute displacement magnitude w ill exceed the investigated displacement magnitude is
Probability that absolute displacement magnidute w ill be less than the investigated displacement magnitude is
7.5
10
18%
82%
Interpretation of results of random vibration analysis
9

Hard drive head
Model file
Model type
Material
Supports
HD HEAD.sldprt
solid
Alloy steel as shown
Objectives
Random vibration analysis with base excitation; vibration within the range of 0 – 2500Hz
Fixed restraint
Measure location
10

Hard drive head
Mesh control
Mesh control
Default mesh would be acceptable for Frequency Analysis. It would also be acceptable for subsequent Random Analysis if displacement results were our only objective. However, since we intend to analyze stresses, mesh controls are required to assure correct element shape and size in the area of stress concentrations.
11

Hard drive head
Four modes found within the requested range
12

Hard drive head
Frequency Options (left) come from already run Frequency study. In
Random Vibration options (right) specify the Upper limit 2500Hz; we investigate response to Random Vibration in the frequency range from 0 to 2500Hz.
Global damping defined as
2% of critical damping.
13

Hard drive head
Base Excitation defined as Acceleration PSD in G 2
14

Support
A4 problem4.SLDPRT
A
Hard drive head is subjected to a random base excitation in y direction. The frequency range is 0-2500Hz
The acceleration time history is in r vib1.txt. Use poweri.exe
to generate acceleration PSD
Find RMS and PSD displacement of tip A.
What is the probability that displacement exceeds the percentage of RMS displacement given in the next slide?
Deliverables:
SW model with response plot and study ready to run.
15