### A4 handout

```ASSIGNMENT 4 Problem 1
Find displacement and stresses in the crankshaft when engine runs
at the first natural frequency of the crankshaft
ASSIGNMENT 4 PROBLEM 2 35%
Support
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 rvib1.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.
ASSIGNMENT 4 PROBLEM 2 35%
Jon
Richard
110%
90%
ASSIGNMENT 4 PROBLEM 2 35%
2.0
1.5
1.0
0.5
0.0
0
0.5
1
-0.5
1.5
2
time [s]
-1.0
-1.5
-2.0
Acceleration time history in r_vib1.txt
This acceleration time history contains 7680 data obtained during 1.5 s with 5120 samples per second. Overall G RMS = 0.44.
The amount of data makes it impractical to run a dynamic time analysis. Assuming that this is a stationary random process we will use
this acceleration time history to calculate the Acceleration Power Spectral Density curve.
Acceleration time history is stored in r_vib1.txt
ASSIGNMENT 4 PROBLEM 2 35%
Required input:
Enter input filename:
r_vib1.txt
Specify Input file type:
acceleration
Select Input Unit
G
Select Output Band Type
constant spectral bandwidth
Select the number of samples per segment
64
Mean removal
yes
Window type
Hanning (see below for explanation)
2.0
1.5
1.0
0.5
0.0
0
0.5
1
1.5
2
-0.5
-1.0
-1.5
-2.0
Rectangular windows works well with nonstationary data, ideally with quiet periods
before and after the main events
Hanning window is
recommended for stationary
data
Input to program poweri.exe required to generate Acceleration Power Spectral Density curve from acceleration time
ASSIGNMENT 2 PROBLEM 4 35%
Input to program poweri.exe required to generate acceleration power spectral density curve from acceleration time history.
ASSIGNMENT 4 PROBLEM 2 35%
Output files from program poweri.exe. File A_G2Hz.psd must now be processed in Excel.
Once parsed into columns, paste it into SW
ASSIGNMENT 4 PROBLEM 2 35%
30000
25000
20000
15000
10000
5000
0
0
500
1000
1500
2000
2500
frequency [Hz]
Acceleration PSD curve we’ll use as input to FEA. Plot has been created in Excel using
“smooth curve” option.
ASSIGNMENT 4 PROBLEM 2 35%
G RMS
Hz
2 0.0016
Δf = 2.5Hz
0.0014
0.0012
Number of samples
per segment: 2048
0.0010
0.0008
0.0006
0.0004
frequency [Hz]
0.0002
0.0000
G RMS
Hz
2.0
1.5
2 0.0016
0
500
1000
1500
0.0012
0.5
0.0008
0.5
1
-0.5
1.5
0.0006
2
time [s]
0.0004
-1.0
0.0002
-1.5
0.0000
frequency [Hz]
0
-2.0
Time history
G RMS
Hz
2
500
1000
1500
0.0016
2000
2500
3000
Δf = 40Hz
0.0014
0.0012
Number of
samples per
segment: 128
0.0010
Acceleration PSD curves produced using different numbers of
bins (frequency ranges). The area under the curve is
approximately the same for all curves.
3000
Number of
samples per
segment: 512
0.0010
0
2500
Δf = 10Hz
0.0014
1.0
0.0
2000
0.0008
0.0006
0.0004
0.0002
frequency [Hz]
0.0000
0
500
1000
1500
2000
2500
3000
ASSIGNMENT 4 PROBLEM 2 35%
Acceleration time
PSD generator
history data
poweri.exe
Excel
SW
Random Vibration
2.0
1.5
1.0
0.5
0.0
0
0.5
1
1.5
2
-0.5
-1.0
-1.5
-2.0
Acceleration PSD
r_vib1.txt
Time domain
Frequency domain
Summary of PSD curve generation
ASSIGNMENT 4 PROBLEM 2 35%
Mode 1
390Hz
Mode 3
1272Hz
Mode 2
1182Hz
Mode 4
2005Hz
Results of modal analysis
ASSIGNMENT 4 PROBLEM 2 35%
RMS displacement result
RMS displacement value
investigated displacement value
probability that absolute displacement will exceed that displacement is
probability that absolute displacement will be less than that displacement
8.80E-02
1.00E-01
26%
74%
RESPONSE SPECTRUM METHOD
14
SEISMIC RECORDS
The ground displacements recorded by a seismometer located directly above a fault that
ruptured during the 1985 Mw = 8.1, Michaocan, Mexico earthquake.
Ground motion histories (North South directions) for El Centro (1940)
“How to do seismic analysis using finite elements”
Phil Cooper, Philip Hoby, Nawal Prinja
15
SEISMIC RECORDS
CALIFORNIA ZONE 2
G
The earthquake is a non-stationary process and the
methods of Random Vibrations can not be used.
Here, each graph contains 6145 data points
Dynamic Time analysis is impractical.
time [s]
Special analysis method called Response Spectrum
Method has been developed to analyze long
duration non stationary processes like the
CALIFORNIA ZONE 4
earthquake or pyrotechnic shock events.
G
time [s]
16
RESPONSE SPECTRA
Each data point on the response spectrum curve represents
the peak response from a time history analysis of the
earthquake applied to 1-DOF oscillator system. The ordinate
defines the natural period of the oscillator, and the abscissa
the peak response. Repeating the procedure for a great many
frequencies defines a continuous curve for an assumed level
of damping. The figure shows the response of 2% damped
system tuned to 1Hz and 2Hz and the transfer of the peak
calculated responses to the response spectrum graph.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
17
RESPONSE SPECTRA
Response spectra are typically presented for a damping ratio of 5% that is considered to be typical for buildings. This does not mean that a
damping ration of 5% is appropriate for any given analysis.
When response spectra are used as an input, a single smoothed spectrum derived from several events or several response spectra from
different events are used. The use of smoothed response spectra implies the use of several earthquake records.
A given shock response spectrum does not have a unique corresponding time history.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
18
RESPONSE SPECTRA
There are a number of approximate methods for scaling 5% damped spectra for other damping levels.
Given two consistent spectral of different damping values, the spectrum for an intermediate damping values can
be calculated based on interpolation between spectral amplitude and natural logarithm of damping ratio.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
19
CONCEPT OF THE RESPONSE SPECTRUM
The concept of the Response Spectrum Method is based on the following observations:
1
A system vibrating in resonance can be described as a single degree of freedom harmonic oscillator system with certain equivalent mass,
equivalent stiffness and damping.
2
A response of a system with more that one resonant frequency can be represented as a combination of responses of harmonic
oscillators, each harmonic oscillator corresponding to particular resonant frequency. This is the basis of the modal superposition method.
3
If the excitation frequency is equal to one of structure resonance frequencies, then the system response to that excitation is controlled
only by system damping. Mass does not matter, stiffness does not matter, they have no impact on system’s response. The only thing
controlling your system response is its damping!
Imagine two vastly different systems: A harmonic oscillator and a bridge. Those two systems have two important things in common: the
natural frequency and damping. Now imagine that both the harmonic oscillator and the bridge are excited by the same excitation that
happens to have the same frequency as the resonance frequency of both the harmonic oscillator and the bridge. The response (e.g. the
maximum displacement) will be the same for the harmonic oscillator and the bridge!
4
Say you agree that structure dynamic response can be adequately described by using the modal superposition method based on four
modes. Then rather than studying the response of the actual structure, you can study the response of four harmonic oscillators with
natural frequencies corresponding to those four modes and damping the same as damping of the structure you study. In particular, if
you study the structure response to seismic excitation, then rather than testing the actual structure, you can just subject those four
oscillators to the earthquake acceleration time history and record e.g. the maximum acceleration of each oscillator as a function of
oscillator’s frequency. This way you will build the response spectrum curve.
5
If the natural frequencies of the examined structure fall “in-between” frequencies of your oscillators, the structure response can be
interpolated. So if you know structure natural frequencies and you also know the acceleration response spectrum curve you have just
constructed by examining the response on harmonic oscillators, you can find out by interpolation, the maximum acceleration of the
structure corresponding to each mode. Double integration will give you the maximum displacement.
6
Notice that the information of interactions between modes has been lost in the above process. Therefore, it must be re-built using the
SRSS or the Absolute Sum method, as described further.
20
GENERATING RESPONSE SPECTRUM
The Response Spectrum model is a series of Single Degree of Freedom (SDOF) harmonic oscillators with
different natural frequencies subjected to a particular earthquake ground motion. Each SDOF system
has a unique time history response to a given base input. The shock response spectrum is the peak
absolute acceleration of each SDOF oscillator to the history base input.
The SDOF oscillators are arranged in order of ascending natural frequency. The oscillator on the far left is
the most compliant one; and the oscillator on the far right is the most stiff system.
We subjected the oscillators to a half sine shock. Note that:
1.
The most compliant system is "isolated" from the shock pulse.
2.
The stiffer systems almost follow the input pulse exactly. They have very little reverberation.
3.
Some of the middle systems actually reach the highest amplitude due to a transient resonance effect.
21
Base excitation
GENERATING RESPONSE SPECTRUM
1
0
0
50
100
150
200
250
300
350
400
time
Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata
22
2.0
vertegi4
yg
GENERATING RESPONSE SPECTRUM
1.5
G [ m / s2 ]
1.0
Base input
0.5
0.0
yg
-0.5
-1.0
-1.5
-2.0
0
10
20
time [ s ]
30
40
time
Step 1 A synthesized waveform; typical
acceleration time history
Step 2 A series of SDOF harmonic oscillators is subjected to the excitation as
defined by the waveform from step 1.
10
Maximum
acceleration
1
0.1
0.01
0.001
0.1
1
10
100
Frequency [Hz]
Step 3
Response spectrum curve ready to be used as input in the Dynamic
Shock analysis. It is usually plotted in log scale.
23
USING RESPONSE SPECTRUM FOR FEA
Maximum
acceleration
10
1
0.1
0.01
0.001
0.1
1
10
100
Frequency [Hz]
Response spectrum methods is ubiquitous in earthquake engineering, with very widespread support in general purpose FE
systems, and almost universal application. It is important to remember that it is a response spectrum: i.e. once you know the
natural frequencies of the structure, in simple cases domminated by a single mode, the respoinse can be simply read from a
graph. FE implementation of the procedure merely automate the interpolation, and add post-processing to combine effects of
mutliple modes and shaking directions.
24
USING RESPONSE SPECTRUM FOR FEA
Step 1 Modal analysis of the analyzed structure determines modes, frequencies and mass participation
factors. Structure is thus represented by a series of SDOF oscillators.
Step 2 The acceleration response for each mode (for each SDOF oscillator) is interpolated using the
response spectrum curve (at the resonant frequency of corresponding mode).
Step 3 The complete response of each mode is found based on the acceleration response found in step 2
Step 8 An assumption is made to find the combined response of all SDOF oscillators because the time at
which oscillator (each mode) reached its peak value was discarded when the response was calculated.
There is no precise way of combining the modes to find the total response.
25
USING RESPONSE SPECTRUM FOR FEA
Full model shows 40.6% total mass participation
factor based on 20 modes.
Truncated model shows 56.5% total mass participation
factor based on the same 20 modes.
Note on mass participation:
Most analysis standards call for the total mass participation factor of 80% or more.
However, this requirement is often difficult to satisfy especially if, due to the
nature of supports, only a small portion of structure participates in vibrations.
26
USING RESPONSE SPECTRUM FOR FEA
Modal combination methods
An assumption has to be made to find the resultant structural response since the time at which the mode reached its
peak value was discarded when the response was calculated. There is no precise way of combining the modes to find
the total response.
The Square Root of the Sum of the Squares
Tends to underestimate the response if modes are spaced out closely. Closely spaced modes are typical for structures
with high torsional stiffness and in structures with identical members. Modes are closely spaced if, for example, there
are repetitive elements in the structure. Modes are considered closely spaced if:
CF 
fi  fi 1
 0.1
fi 1
Absolute Sum The sum of the absolute values of the contributions of each mode
Tends to overestimate the response and can be very conservative.
27
GENERATING RESPONSE SPECTRUM CURVE
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
-0.5
-1
-1.5
-2
vertegi4.txt
Acceleration time history has 6145 data points
This is input to Shock Response curve generator.
28
GENERATING RESPONSE SPECTRUM CURVE
vertegi4 mm.txt
Input file in ASCII
format containing
acceleration time
history.
The starting frequency may be
defined by a testing standard or
some other convention.
Consideration should also be
given to the natural frequency of
the structure. The starting
frequency defines the frequency
of the lowest frequency
oscillator.
The amplification
factor relates to 
using the formula
below.
 Is % of critical
damping:
Q
1
2 ζ
The amplification
factor 25 corresponds
to
= 2 %
The amplification
factor 10 corresponds
to
= 5 %
Octave spacing is similar to the frequency spacing on a piano keyboard. Each successive
key is 1/12 octave higher than the previous key. The frequency spacing is sometimes
chosen by an established convention, such as some testing standard or by experience.
The frequency spacing effectively defines the number of oscillators.
Primary shock is the response
that occurs during the shock
pulse. Residual shock pulse is the
response that occurs after the
shock pulse. These terms mainly
apply to a well-defined shock
pulse such as a half-sine pulse.
The earthquake shock, however,
may taper off in a very gradual
manner. Thus a somewhat
arbitrary point must be chosen as
the "end" of the shock.
Response spectrum generator; Courtesy of Vibrationdata
29
GENERATING RESPONSE SPECTRUM CURVE
seconds
16.93
16.935
16.94
16.945
16.95
16.955
16.96
16.965
16.97
16.975
16.98
16.985
16.99
16.995
17
17.005
17.01
17.015
17.02
17.025
17.03
17.035
17.04
17.045
17.05
17.055
17.06
17.065
17.07
17.075
m/s2
-825.69
-530.89
-248.48
-1.18
244.61
477.66
613.73
775.75
980.39
1151.4
1334.03
1536.91
1740.5
1970.92
2206.33
2355.73
2508.93
2662.86
2734.67
2756.32
2739.77
2688.63
2619.15
2552.68
2540.24
2545.58
2595.85
2786.82
3067.25
3432.76
m/s2
-0.08257
-0.05309
-0.02485
-0.00012
0.024461
0.047766
0.061373
0.077575
0.098039
0.11514
0.133403
0.153691
0.17405
0.197092
0.220633
0.235573
0.250893
0.266286
0.273467
0.275632
0.273977
0.268863
0.261915
0.255268
0.254024
0.254558
0.259585
0.278682
0.306725
0.343276
Input
Time history of an earthquake
Maximum positive and maximum negative acceleration
(6145 data points) as function of time
frequency
mm/s2
0.1000
94.52
0.1122
111.40
0.1260
131.80
0.1414
156.80
0.1587
187.50
0.1782
225.90
0.2000
273.90
0.2245
334.70
0.2520
411.20
0.2828
506.90
0.3175
630.80
0.3564
799.00
0.4000
1013.00
0.4490
1294.00
0.5040
1713.00
0.5657
2244.00
0.6350
3149.00
0.7127
4458.00
0.8000
7013.00
0.8980
13120.00
1.0080
20710.00
1.1310
22560.00
Output from SRS generator (after processing in Excel)
Max. absolute acceleration (49 data points) as function of
frequency for 5% damping
30
GENERATING RESPONSE SPECTRUM CURVE
60000
Maximum absolute
acceleration mm/s2
50000
40000
30000
20000
10000
0
0
1
10
100
frequency
Generated Shock Response Spectrum curve
31
GENERATING RESPONSE SPECTRUM CURVE
Acceleration time
SRS generator
history data
SRS.exe
Excel
SolidWorks
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
-0.5
-1
-1.5
-2
SRS.txt
vertegi4.txt
Frequency domain
Time domain
Summary of SRS curve generation
32
SEISMIC ANALYSIS
Example of a model
33
```