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EQUIVALENT STATIC LOADS FOR RANDOM VIBRATION

Revision M

By Tom Irvine

Email: tomirvine@aol.com

October 8, 2010

_____________________________________________________________________________________________

The following approach in the main text is intended primarily for single-degree-of-freedom

systems. Some consideration is also given for multi-degree-of-freedom systems.

Introduction

A particular engineering design problem is to determine the equivalent static load for equipment

subjected to base excitation random vibration. The goal is to determine peak response values.

The resulting peak values may be used in a quasi-static analysis, or perhaps in a fatigue

calculation. The response levels could be used to analyze the stress in brackets and mounting

hardware, for example.

Limitations

Limitations of this approach are discussed in Appendices F through K.

A particular concern for either a multi-degree-of-freedom system or a continuous system is that

the static deflection shape may not properly simulate the predominant dynamic mode shape. In

this case, the equivalent static load may be as much as one order of magnitude more conservative

than the true dynamic load in terms of the resulting stress levels.

Load Specification

Ideally, the dynamics engineer and the static stress engineer would mutually understand, agree

upon, and document the following parameters for the given component.

1. Mass, center-of-gravity, and inertia properties

2. Effective modal mass and participation factors

3. Stiffness

4. Damping

5. Natural frequencies

6. Dynamic mode shapes

7. Static deflection shape

8. Response acceleration

9. Modal velocity

10. Relative displacement

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11. Transmitted force from the base to the component in each

of three axes

12. Bending moment at the base interface about each of three

axes

13. The manner in which the equivalent static loads and

moments will be applied to the component, such as point

load, body load, distributed load, etc.

14. Dynamic stress and strain at critical locations if the

component is best represented as a continuous system

15. Response limit criteria, such as yield stress, ultimate stress,

fatigue, or loss of clearance

Each of the response parameters should be given in terms of frequency response function, power

spectral density, and an overall response level.

Furthermore, assumptions must be documented, including a discussion of conservatism.

Again, this list is very idealistic.

Importance of Modal Velocity

Bateman wrote in Reference 24:

Of the three motion parameters (displacement, velocity, and acceleration) describing a

shock spectrum, velocity is the parameter of greatest interest from the viewpoint of

damage potential. This is because the maximum stresses in a structure subjected to a

dynamic load typically are due to the responses of the normal modes of the structure,

that is, the responses at natural frequencies. At any given natural frequency, stress is

proportional to the modal (relative) response velocity. Specifically,

EVC maxmax

(1)

where

max

= Maximum modal stress in the structure

max

V

= Maximum modal velocity of the structural response

E

= Elastic modulus

= Mass density of the structural material

= Constant of proportionality dependent upon the geometry of

the structure (often assumed for complex equipment to be

4 < C < 8 )

C

Some additional research is needed to further develop equation 1 so that it can be used for

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equivalent quasi-static loads for random vibration. Its fundamental principle is valid, however.

Further information on the relationship between stress and velocity is given in Reference 25.

Importance of Relative Displacement

Relative displacement is needed for the spring force calculation. Note that the transmitted force

for an SDOF system is simply the mass times the response acceleration.

Specifying the relative displacement for an SDOF system may seem redundant because the

relative displacement can be calculated from the response acceleration and the natural frequency

per equation (7) given later in this paper.

But specifying the relative displacement for an SDOF system is a good habit.

The reason is that the relationship between the relative displacement and the response

acceleration for a multi-degree-of-freedom (MDOF) or continuous system is complex. Any

offset of the component’s center-of-gravity (CG) further complicates the calculation due to

coupling between translational and rotational motion in the modal responses.

The relative displacement calculation for an MDOF system is beyond the scope of a hand

calculation, but the calculation can be made via a suitable Matlab script. A dynamic model is

required as shown in Appendices H and I.

Furthermore, examples of continuous structures are shown in Appendices J & K. The structures

are beams. The bending stress for the equivalent static analysis of each beam correlates better

with relative displacement than with response acceleration.

Model

The first step is to determine the acceleration response of the component.

Model the component as an SDOF system, if appropriate, as shown in Figure 1.

Figure 1.

m

k

c

x

y

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where

M

C

K

X

Y

is the mass

is the viscous damping coefficient

is the stiffness

is the absolute displacement of the mass

is the base input displacement

Furthermore, the relative displacement z is

z = x – y (2)

The natural frequency of the system fn is

1

k

m

fn

2

(3)

Acceleration Response

The Miles’ equation is a simplified method of calculating the response of a single-degree-of-

freedom system to a random vibration base input, where the input is in the form of a power

spectral density.

The overall acceleration response

GRMS

x

is

fn

xf ,

GRMSn

22

where

Fn is the natural frequency

P is the base input acceleration power spectral density at the natural frequency

is the damping ratio

P

(4)

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Note that the damping is often represented in terms of the quality factor Q.

1

Q

2

Equation (4), or an equivalent form, is given in numerous references, including those listed in

Table 1.

Table 1. Miles’ equation References

Reference Author Equation

1 Himelblau (10.3)

2 Fackler (4-7)

3 Steinberg (8-36)

4 Luhrs -

5 Mil-Std-810G -

6 Caruso (1)

Furthermore, the Miles’ equation is an approximate formula that assumes a flat power spectral

density from zero to infinity Hz. As a rule-of-thumb, it may be used if the power spectral density

is flat over at least two octaves centered at the natural frequency.

An alternate response equation that allows for a shaped power spectral density input is given in

Appendix A.

Relative Displacement & Spring Force

Consider a single-degree-of-freedom (SDOF) system subject to a white noise base input and with

constant damping. The Miles’ equation set shows the following with respect to the natural

frequency fn:

(5)

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246

76

225

59

516.6-12

28

Response Acceleration

n

f

(6)

Relative Displacement

5 . 1

n

f /1

(7)

Relative Displacement

= Response Acceleration

2

n

/

(8)

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where

Equation (8) is derived in Reference 18.

Consider that the stress is proportional to the force transmitted through the mounting spring. The

spring force F is equal to the stiffness k times the relative displacement z.

F = k z (9)

RMS and Standard Deviation

The RMS value is related the mean and standard deviation values as follows:

RMS2 = mean2 + 2 (10)

Note that the RMS value is equal to the 1 value assuming a zero mean.

A 3 value is thus three times the RMS value for a zero mean.

Peak Acceleration

There is no method to predict the exact peak acceleration value for a random time history.

An instantaneous peak value of 3 is often taken as the peak equivalent static acceleration. A

higher or lower value may be appropriate for given situation.

Some sample guidelines for peak acceleration are given in Table 2. Some of the authors have

intended their respective equations for design purposes. Others have intended their equations for

“Test Damage Potential.”

Table 2.

Sample Design Guidelines for Peak Response Acceleration or Transmitted Force

Design or Test

Equation

nn

f2

Refer. Author Page Qualifying Statements

1

Himelblau,

et al

3

190

However, the response may

be non-linear and

non-Gaussian

3 is the usual assumption

for the equivalent peak

sinusoidal level.

Theoretically, any large

acceleration may occur.

2 Fackler

3 76

4 Luhrs

3 59

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Table 2.

Sample Design Guidelines for Peak Response Acceleration or Transmitted Force

(continued)

Design or Test

Equation

3 for

STS Payloads

2 for

ELV Payloads

McDonnell

Douglas

Scharton &

Pankow

Refer. Author Page Qualifying Statements

7 NASA 2.4-3

Minimum Probability Level

Requirements

8

4

4-16 Equivalent Static Load

10

5

- See Appendix C.

11

DiMaggio,

Sako, Rubin

n

Eq (22)

See Appendices B and D for

the equation to calculate n via

the Rayleigh distribution.

See Appendix E for equation

to calculate Cn.

12 Ahlin Cn -

Furthermore, some references are concerned with fatigue rather than peak acceleration, as shown

in Table 3.

Table 3.

Design Guidelines for Fatigue based on

Miner’s Cumulative Damage Index

Reference

3

6

Author

Steinberg

Caruso

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Note that the Miner’s Index considers the number of stress cycles at the 1, 2, and 3levels.

Modal Transient Analysis

The input acceleration may be available as a measured time history. If so, a modal transient

analysis can be performed. The numerical engine may be the same as that used in the shock

response spectrum calculation. The advantage of this approach is that it accounts for the

response peaks that are potentially above 3It is also useful when the base input is non-

stationary or when its histogram deviates from the normal ideal.

The modal transient approach can still be used if a power spectral density function is given

without a corresponding time history. In this case a time history can be synthesized to meet the

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power spectral density, as shown in Appendix B. This approach effectively requires the time

history to be stationary with a normal distribution.

Furthermore, a time domain analysis would be useful if fatigue is a concern. In this case, the

rainflow cycle counting method could be used.

Special Case

Consider a system that has a natural frequency that is much higher than the maximum base input

frequency. An example would be a very stiff bar that was subjected to a low frequency base

excitation in the bar’s longitudinal axis.

This case is beyond the scope of Miles’ equation, since the Miles’ equation takes the input power

spectral density at the natural frequency. The formula in Appendix A can handle this case,

however.

As the natural frequency becomes increasingly higher than the maximum frequency of the input

acceleration, the following responses occur:

1. The response acceleration converges to the input acceleration.

2. The relative displacement approaches zero.

Furthermore, the following rule-of-thumb is given in Reference 24:

Quasi-static acceleration includes pure static acceleration as well as low-frequency

excitations. The range of frequencies that can be considered quasi-static is a function

of the first normal mode of vibration of the equipment. Any dynamic excitation at a

frequency less than about 20 percent of the lowest normal mode (natural) frequency of

the equipment can be considered quasi-static. For example, an earthquake excitation

that could cause severe damage to a building could be considered quasi-static to an

automobile radio.

Case History

A case history for random load factor derivation for a NASA programs is given in Reference 22.

Error Source Summary

Here is a list of error sources discussed in this paper, including the appendices.

1. An SDOF system may be an inadequate model for a component or

structure.

2. An SDOF model cannot account for spatial variation in either the input or

the response.

3. A CG offset leads to coupling between translational and rotational modes,

thus causing the transmitted forces to vary between the mounting springs.

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4. Instantaneous peak values can occur in the time domain as high as

5depending on the duration and natural frequency.

5. The static deflection shape is not the same as the dynamic mode shape, thus

affecting the strain calculations.

Base Input & Component Response Concerns

The derivation of the base input level is beyond the scope of this paper, but a few points

are mentioned here as an aside.

1. The base input time history may have a histogram which departs from the

Gaussian ideal, with a kurtosis value > 3. A solution for this problem is

given in Reference 19.

2. Consider a component in its field or flight environment. The base excitation

at the component’s respective input points may vary by location in terms of

amplitude and phase. As a first approximation, the field response of the

component would be less than if the loads were uniform and in phase at the

input points, which would be the case during a shaker table test. On the other

hand, consider a beam simply-supported at each end. A uniform base input

would not excite the beam’s second bending mode. However, this mode

could be excited in a field environment where the inputs were non-uniform.

3. The base input level might not account for any force-limiting or mass-loading

effects from the component.

4. A structure or component may have a nonlinear response. Consider a

component mounted to a plate or shell, where the mounting structure is

excited by acoustical energy on the opposite side. At higher acoustic levels,

the structure will undergo membrane effects which limit its vibration

response, thus limiting the base input to the attached component.

5. Component damping tends to be non-linear. The damping tends to increase

as the input level increases. This increase can be due to joint slipping for

example. This should be considered in the context of adding margin to the

input levels.

6. Conservative enveloping may have been used to derive the component base

input level. In some case, the input level may be the maximum of all three

axes.

Conclusion

The task of deriving an accurate equivalent static load for a component or secondary structure is

very challenging.

There are numerous error sources. Some of the sources in this paper could lead to an under-

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prediction of the load, such as omitting potential peaks > 3. Other sources could result in an

over-prediction, as shown for the cantilever beam example in Appendix J.

Ideally, these issues could be resolved by thorough testing and analysis.

Components could be instrumented with both accelerometers and strain gages and then exposed

to shaker table testing. This would allow a correlation between strain and acceleration response.

The input level should be varied to evaluate potential non-linearity. The resulting stress can then

be calculated from the strain.

Component modal testing would also useful to identify natural frequencies, mode shapes, and

modal damping values. This can be achieved to some extent by taking transmissibility

measurements during a shaker table test.

The test results could then be used to calibrate a finite element model. The calibration could be

as simple as a uniform scaling of the stiffness so that the model fundamental frequency matches

the measured natural frequency.

The test results would also provide the needed modal damping. Note that damping cannot be

calculated from theory. It can only be measured.

Cost and schedule often limit the amount of analysis which can be performed. But ideally, the

calibrated finite element model could be used for the dynamic stress calculation via a modal

transient or frequency response function approach. Note that the analyst may choose to perform

the post-processing via Matlab scripts using the frequency response functions from the finite

element analysis.

Otherwise, the calibrated finite element model could be used for a static analysis.

The proper approach for a given component must be considered on a case-by-case basis.

Engineering judgment is required.

Future Research

Further research is needed in terms of base input derivation, response analysis, and testing.

Another concern is material response. There are some references that report that steel and other

materials are able to withstand higher stresses than their respective ultimate limits if the time

history peak duration is of the order of 1 millisecond or less. See Appendix K.

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Appendices

Table 4. Appendix Organization

Appendix Title

A SDOF Acceleration Response

B Normal Probability Values & Rayleigh Distribution

C Excerpt from Reference 10

D Excerpt from Reference 11

E Excerpt from Reference 12

F Excerpt from Reference 14

G Excerpts from References 15 & 16

H Two-degree-of-freedom System, Example 1

I Two-degree-of-freedom System, Example 2

J Cantilever Beam Example

K Beam Simply-supported at each End Example

L Material Stress Limits

References

1. H. Himelblau et al, NASA-HDBK-7005, Dynamic Environmental Criteria, Jet Propulsion

Laboratory, California Institute of Technology, 2001.

2. W. Fackler, Equivalence Techniques for Vibration Testing, SVM-9, The Shock and

Vibration Information Center, Naval Research Laboratory, United States Department of

Defense, Washington D.C., 1972.

3. Dave Steinberg, Vibration Analysis for Electronic Equipment, Wiley-Interscience, New

York, 1988.

4. H. Luhrs, Random Vibration Effects on Piece Part Applications, Proceedings of the

Institute of Environmental Sciences, Los Angeles, California, 1982.

5. MIL-STD-810G, “Environmental Test Methods and Engineering Guidelines,” United

States Department of Defense, Washington D.C., October 2008.

6. H. Caruso and E. Szymkowiak, A Clarification of the Shock/Vibration Equivalence in

Mil-Std-180D/E, Journal of Environmental Sciences, 1989.

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7. General Environmental Verification Specification for STS & ELV Payloads, Subsystems,

and Components, NASA Goddard Space Flight Center, 1996.

8. Vibration, Shock, and Acoustics; McDonnell Douglas Astronautics Company, Western

Division, 1971.

9. W. Thomson, Theory of Vibration with Applications, Second Edition, Prentice- Hall,

New Jersey, 1981.

10. T. Scharton & D. Pankow, Extreme Peaks in Random Vibration Testing, Spacecraft and

Launch Vehicle Dynamic Environments Workshop, Aerospace/JPL, Hawthorne,

California, 2006.

11. DiMaggio, S. J., Sako, B. H., and Rubin, S., Analysis of Nonstationary Vibroacoustic

Flight Data Using a Damage-Potential Basis, AIAA Dynamic Specialists Conference,

2003 (also, Aerospace Report No. TOR-2002(1413)-1838, 1 August 2002).

12. K. Ahlin, Comparison of Test Specifications and Measured Field Data, Sound &

Vibration, September 2006.

13. T. Irvine, An Introduction to the Vibration Response Spectrum, Vibrationdata

Publications, 2000.

14. R. Simmons, S. Gordon, B. Coladonato; Miles’ Equation, NASA Goddard Space Flight

Center, May 2001.

15. H. Lee, Testing for Random Limit Load Versus Static Limit Load, NASA TM-108542,

1997.

16. H. Lee, A Simplistic Look at Limit Stresses from Random Loading, NASA TM-108427,

1993.

17. T. Irvine, The Steady-State Frequency Response Function of a Multi-degree-of-freedom

System to Harmonic Base Excitation, Rev C, Vibrationdata, 2010.

18. T. Irvine, Derivation of Miles’ equation for Relative Displacement Response to Base

Excitation, Vibrationdata, 2008.

19. T. Irvine, Deriving a Random Vibration Maximum Expected Level with Consideration

for Kurtosis, Vibrationdata, 2010.

20. T. Irvine, Steady-State Vibration Response of a Cantilever Beam Subjected to Base

Excitation, Rev A, 2009.

21. W. Young, Roark's Formulas for Stress & Strain, 6th edition, McGraw-Hill, New York,

1989.

22. Robert L. Towner, Practical Approaches for Random Load Factors, PowerPoint

Presentation, Mentors-Peers, Jacobs ESTS Group, November 21, 2008.

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23. C. Lalanne, Sinusoidal Vibration (Mechanical Vibration and Shock), Taylor & Francis,

New York, 1999.

24. A. Piersol and T. Paez, eds., Harris’ Shock and Vibration Handbook, 6th Edition, NY,

McGraw-Hill, 2010.

25. T. Irvine, Shock & Vibration Stress as a Function of Velocity, Vibrationdata, 2010.

26. R. Huston and H. Josephs, Practical Stress Analysis in Engineering Design, Dekker,

CRC Press, 2008. See Table 13.1.

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APPENDIX A

SDOF Acceleration Response

The acceleration response x GRMS of a single-degree-of-freedom system to a base input power

spectral density is

i

12

i 1

2

i

2

2

2

i

N

1 (2

)

ˆ

x f ,

n

Y (f ) f ,

f /f

i GRMS APSD iiin

(A-1)

where

fn

is the natural frequency

ˆY (f )

APSD n is the base input acceleration power spectral density

The corresponding relative displacement is

2

1

ZX

RMSGRMS

2 fn

(A-2)

Equation (A-1) allows for a shaped base input power spectral density, defined over a finite

frequency domain. It is thus less restrictive than the Miles’ equation. Equation (A-1) is derived

in Reference 13.

Note that equation (A-1) is the usual method for the vibration response spectrum calculation,

where the natural frequency is an independent variable.

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APPENDIX B

Normal Probability Values

Note that the RMS value is equal to the 1 value assuming a zero mean. The 1 value is the

standard deviation.

Consider a broadband random vibration time history x(t), which has a normal distribution.

The precise amplitude x(t) cannot be calculated for a given time. Nevertheless, the probability

that x(t) is inside or outside of certain limits can be expressed in terms of statistical theory.

The probability values for the instantaneous amplitude are given in Tables B-1 and B-2 for

selected levels in terms of the standard deviation or value.

Table B-1.

Probability for a Random Signal with Normal Distribution and Zero Mean

Statement Probability Ratio Percent Probability

- < x < +

0.6827 68.27%

-2 < x < +2

-3 < x < +3

-4 < x < +4

0.9545 95.45%

0.9973 99.73%

0.99994 99.994%

Table B-2.

Probability for a Random Signal with Normal Distribution and Zero Mean

Statement Probability Ratio Percent Probability

| x | >

| x | > 2

| x | > 3

| x | > 4

0.3173 31.73%

0.0455 4.55%

0.0027 0.27%

6e-005 0.006%

Furthermore, the probability that an instantaneous amplitude is less than +3is P = 0.99865.

This is equivalent to saying that 1 out of every 741 points will exceed +3

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Rayleigh Distribution

The following section is based on Reference 9.

Consider the response of single-degree-of-freedom distribution to a broadband time history. The

response is approximately a constant frequency oscillation with a slowly varying amplitude and

phase.

The probability distribution of the instantaneous acceleration is the same as that for the

broadband random function.

The absolute values of the response peaks, however, will have a Rayleigh distribution, as shown

in Table B-3.

Table B-3. Rayleigh Distribution Probability

Prob [ A > ]

0.5 88.25 %

1.0 60.65 %

1.5 32.47 %

2.0 13.53 %

2.5 4.39 %

3.0 1.11 %

3.5 0.22 %

4.0 0.034 %

4.5 4.0e-03 %

5.0 3.7e-04 %

5.5 2.7e-05 %

6.0 1.5e-06 %

The values in Table B-3 are calculated from

2

2

1

expAP

(B-1)

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Determine the value for which exactly one peak is expected to occur for a natural frequency fn

and a duration T.

1T fn

2

2

1

exp

(B-2)

T fn

1

2

2

1

exp

(B-3)

T fn

1

ln

2

2

1

(B-4)

T fn

1

ln2

2

(B-5)

T fnln2

2

(B-6)

)T fn( ln2

(B-7a)

Thus a 4.33 peak response is expected for a system with a natural frequency of 200 Hz exposed

a random vibration with a normal distribution over a 60 second duration.

60 )( Hz200( ln2)Tfn( ln2

Example

An experimental verification of equation (B-7a) for a particular case is given in the following

example.

33 . 4sec)

(B-7b)

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0.001

0.01

0.1

1001000202000

Synthesis, 6.06 GRMS

Specification, 6.06 GRMS

FREQUENCY (Hz)

ACCEL (G2/Hz)

POWER SPECTRAL DENSITY

Figure B-1.

The synthesis curve agrees well with the specification, although it has a drop-out near 20 Hz.

The actual synthesized time history is shown in Figure B-2.

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-100

-80

-60

-40

-20

0

20

40

60

80

100

010 2030405060

TIME (SEC)

ACCEL (G)

SYNTHESIZED BASE INPUT

Figure B-2.

The sample rate is 20,000 samples per second.

Amplitude Stats

mean = -0.0001157 std = 6.058 rms = 6.058

max = 29.82 at = 28.2 sec

min = -29.45 at = 53.14 sec

crest factor = 4.923 kurtosis = 3.034

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Figure B-3.

The histogram of the base input has a normal distribution.

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-200

-150

-100

-50

0

50

100

150

200

0102030405060

TIME (SEC)

ACCEL (G)

RESPONSE TIME HISTORY

Figure B-4.

The response of single-degree-of-freedom is shown in Figure B-4. The system has a natural

frequency of 200 Hz and an amplification factor Q=10. The base input is the time history in

Figure B-2.

Amplitude Stats

mean = -2.133e-005 std = 11.19 rms = 11.19

max = 44.57 at = 1.44 sec

min = -47.18 at = 49.07 sec

crest factor = 4.218 kurtosis = 2.93

The theoretical crest factor is

)(Hz200(ln2)Tfn(ln2

33 . 4sec)60

(B-8)

The theoretical value is 2.7% higher than the experiment value.

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Figure B-5.

The histogram follows a normal distribution.

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Figure B-6.

The histogram of the absolute peak from the response time history follows a Rayleigh

distribution.

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APPENDIX C

Excerpt from Reference 10

Test data shows that occurrence of extreme peaks exceeds that predicted by Rayleigh

distribution.

For a Gaussian probability distribution, the probability of | x | > 5 is 6E-7.

If one has 60 seconds of white noise digitized at 20,000 points per seconds, the probability of a

peak exceeding 5 is 60 *20,000*6E-7=0.73

This may explain why one sees more extreme peaks than predicted by a Rayleigh distribution.

Extreme single peaks in the input acceleration are less of a concern because they do not appear to

produce near-resonant amplification of the response.

Extreme peaks in the base force and acceleration responses are a very real threat.

A simple experiment indicted that the strength of hard steel and carbon rods, and presumably of

other brittle materials, does not increase with frequencies up to 500 Hz.

Data in the literature indicate that the increase in strength of aluminum and many other materials

is small even at considerably higher frequencies.

Given the frequent observance of five sigma test peaks in time histories of responses in random

vibration tests, three sigma design strength requirements, such as those in NASA-STD-5002,

appear inconsistent.

The options are to increase mission limit loads, or to decrease test margins.

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APPENDIX D

Excerpt from Reference 11

Reference 11 uses the Rayleigh distribution.

Equation (22) from this reference is

2

max

A

) fnT(ln2

2

This equation is equivalent to equations (B-6) and (B-7) in Appendix B of this paper.

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APPENDIX E

Excerpt from Reference 12

The following formula is intended for comparing random vibration to shock. It is not a design

level per se, but may be used as such.

Consider a single-degree-of-freedom system with the index n. The maximum response

n

max

can be estimated by the following equations.

T fn ln2

n

c

(E-1)

n

c

5772. 0

n

c

n

C

(E-2)

nn

C

n

max

(E-3)

where

fn is the natural frequency

T is the duration

ln

is the natural logarithm function

n

is the standard deviation of the oscillator response

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APPENDIX F

Excerpt from Reference 14

The authors of Reference 14 give the following warning:

MILES’ EQUATION DOES NOT GIVE AN EQUIVALENT STATIC LOAD –

Calculating the GRMS value at a resonant peak after a random vibration test and

multiplying it by the test article mass does not mean that the test article was subjected to

that same, equivalent static load. It simply provides a statistical calculation of the peak

load for a SDOF system. The actual loading on a multiple DOF system due to random

input depends on the response of multiple modes, the mode shapes and the amount of

effective mass participating in each mode. Static testing must still be done.

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APPENDIX G

Excerpt from References 15 & 16

NASA engineers performed experimental static and vibration testing an “AEPI fiberglass

pedestal” structure.

The following conclusions were made:

1. Strain, in general, is lower during random testing than during an equivalent static

loading as predicted by Miles’ equation.

2. The Miles’ equation equivalent static loading clearly develops stresses an order of

magnitude above those created by the random environments.

The reference also noted:

A study completed in 1993 by the Marshall Space Flight Center (MSFC) Random

Loads/Criteria Issues Team concluded, after an extensive literature search, that almost

no analytical or empirical documentation exists on the subject of the relationship

between random limit load (stress) and static limit load (stress). The consensus of the

team was that it is a complex subject and requires a carefully planned effort to produce

and effective, yet practical solution.

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APPENDIX H

Two-degree-of-Freedom-System, Example 1

Figure H-1.

The variables are

y is the base displacement

xi

is the absolute displacement of mass i

zi

is the relative displacement of mass i

j

is the modal damping for mode j

j

is the natural frequency (rad/sec) for mode j

is the base excitation frequency

A is the base acceleration amplitude

ki

is the stiffness for spring i

mi

is mass i

The mass-normalized eigenvectors in column format are

m1

k 1

m2

k 2

x1

x2

y

Page 30

30

2221

12 11

q ˆ

q ˆ

q ˆ

q ˆ

The equation of motion from Reference 17 is

zm0

22

Define participation factors 1

211111

mq ˆ

mq ˆ

221122

q ˆ

mq ˆ

The relative displacement transfer functions are

k

y

m

y

m

z

z

k

kkkz0m

2

1

2

1

22

22111

(H-1)

and 2

.

2

(H-2)

2

m

(H-3)

22

2

2

2

2

j

12

]

11

2

2

1

1

j

11

]

1

2[

q ˆ

2[

q ˆ

A/ (f)Zˆ

(H-4)

22

2

2

2

2

2j

22

]

11

2

2

1

1

j

21

]

2

[

q ˆ

2[

q ˆ

A/ (f)Zˆ

(H-5)

Example

Consider the system in Figure H-1. Assign the following values.

Table H-1. Parameters

Variable Value

1

m

20.0 lbm

2

m

10.0 lbm

1

k

60,000 lbf/in

2

k

40,000 lbf/in

Page 31

31

Furthermore, assume that each mode has a damping value of 5%.

The following parameters were calculated for the sample system via a Matlab script.

Natural Frequencies =

126.2 Hz

268.5 Hz

Modes Shapes (column format) =

-2.823 -3.366

-4.76 3.993

Participation Factors =

-0.2696

-0.07097

Effective Modal Mass =

28.06 lbm

1.944 lbm

Total Modal Mass = 30.0000 lbm

Page 32

32

0.01

0.1

1

10

100

10 1001000

Mass 2

Mass 1

FREQUENCY (Hz)

TRANS (G/G)

ACCELERATION RESPONSE TRANSFER FUNCTION MAGNITUDE

Figure H-1.

The resulting transfer functions were also calculated via a Matlab script.

Page 33

33

0.00001

0.0001

0.001

0.01

0.1

110 1001000

Mass 1 - Mass 2

Mass 2

Mass 1

FREQUENCY (Hz)

TRANS (inch/G)

RELATIVE DISPLACEMENT TRANSFER FUNCTION MAGNITUDE

Figure H-2.

Table H-2.

Magnitude Values at Fundamental Mode, Frequency = 126.2 Hz

Magnitude Parameter Mass 1 Mass 2

Acceleration Response (G/G) 7.7 12.8

Relative Displacement (inch/G) 0.0047 0.0079

Note that:

Relative Displacement = Acceleration Response /

2

1

This equation is true for the transfer function values, at least for a two-degree-of-freedom system

with well-separated modal frequencies. It will have some error, however, for the response

acceleration and relative displacement overall RMS values, as shown later in this appendix.

Page 34

34

0.001

0.01

0.1

1

10

100

10 1001000

Mass 2 18.9 GRMS

Mass 1 11.9 GRMS

FREQUENCY (Hz)

ACCEL (G2/Hz)

ACCELERATION POWER SPECTRAL DENSITY

Figure H-3.

Table H-3. PSD, Base Input, 14.1 GRMS

Freq (Hz) Accel (G^2/Hz)

10 0.1

2000 0.1

The PSD in Table H-3 is applied to the system in Figure H-1. The acceleration response is

shown in Figure H-3.

Page 35

35

10-10

10-9

10-8

10-7

10-6

10-5

10 100 1000

Spring 1 0.0066 inch RMS

FREQUENCY (Hz)

REL DISP (inch2/Hz)

RELATIVE DISPLACEMENT POWER SPECTRAL DENSITY

Figure H-4.

Table H-4. Overall Levels

Element Parameter RMS Value

3 Value

Mass 1 Acceleration Response 11.9 G 35.7 G

Spring 1

Note that Spring 1 is the base spring. The force transmitted through Spring 1 is 1188 lbf 3

Also note that for the RMS values

Hz2 .1262

The reason for this difference is that the second modal peak has a higher contribution in the

acceleration transfer function than in the relative displacement transfer function. This difference

is amplified when each of the transfer functions is squared for the PSD response calculations.

Relative Displacement 0.0066 inch 0.0198 inch

inch0073 . 0

G/ sec/in386G 9 .11

2

2

> 0.0066 inch

Page 36

36

Equivalent SDOF System

How would the two-degree-of-freedom system be modeled as an SDOF system?

A conservative approach would be to taken the highest acceleration response from Figure H-3

and apply it to the total mass.

The maximum response acceleration was 18.9 GRMS, or 56.7 G 3

The combined mass is 30 lbm.

The force transmitted via the base spring to the boundary interface is thus 1701 lbf 3

The base spring has a stiffness of 60,000 lbf/in

Thus the base spring relative displacement would be 0.0284 inch 3.

Table H-5. Spring 1 Results, Model Comparison

Transmitted Force

(lbf 3 (inch 3

SDOF 1701 0.0284

Model

Relative Displacement

2-DOF 1188 0.0198

The SDOF result is 3.1 dB higher for each parameter.

Page 37

37

APPENDIX I

Two-degree-of-Freedom-System, Example 2

Figure I-1.

The system has a CG offset if

The system is statically coupled if

The rotation is positive in the clockwise direction.

21

LL

.

2211

Lk L k

.

k 1

k 2

L1

y

L2

x

k 1 ( y - x - L1 )

)

k 2 ( y - x + L2 )

)

Page 38

38

The variables are

y is the base displacement

x is the translation of the CG

is the rotation about the CG

m is the mass

J is the polar mass moment of inertia

k i

is the stiffness for spring i

z i

is the relative displacement for spring i

j

is the modal damping for mode j

j

is the natural frequency (rad/sec) for mode j

is the base excitation frequency

A is the base acceleration amplitude

The mass-normalized eigenvectors in column format are

2221

q ˆ

q ˆ

The equation of motion are taken from Reference 17.

12 11

q ˆ

q ˆ

y

L k Lk

k k

x

L k L k Lk L k

Lk Lk k k

x

J0

0m

2211

21

2

22

2

112211

221121

(I-1)

The relative displacement for spring 1 is

z1 = z + L1 (I-2)

Page 39

39

The Fourier transform for the relative displacement in spring 1 is

2

1

[

The relative displacement for spring 2 is

z2 = z - L2 (I-4)

2

22

2

2

22

2

1 12

21

11

2

21

2

1 11

111

j][

q ˆ

L q ˆ

q ˆ

j]

q ˆ

L q ˆ

q ˆ

AmZ

(I-3)

The Fourier transform for the relative displacement in spring 2 is

2

22

2

2

22

2

2 12

21

11

2

2

1

21

2

211

112

j][

q ˆ

L q ˆ

q ˆ

j][

q ˆ

L q ˆ

q ˆ

AmZ

(I-5)

The Fourier transform for the translational acceleration at the CG is

1

[

The Fourier transform for the rotational acceleration is

22

2

2

2

21

j

12

11

2

2

2

11

]

2

a

2][

q ˆ

q ˆ

2j

q ˆ

m1AX

(I-6)

22

2

2

2

21

j

22

]

11

2

2

1

21

j

11

]

1

2[

q ˆ

q ˆ

2[

q ˆ

q ˆ

Am

(I-7)

Page 40

40

Example

Consider the system in Figure I-1. Assign the following values. The values are based on a

slender rod, aluminum, diameter =1 inch, total length=24 inch.

Table H-1. Parameters

Variable Value

m

18.9 lbm

J 907 lbm in^2

1

k

20,000 lbf/in

2

k

20,000 lbf/in

1

L

8 in

2

L

16 in

Furthermore, assume that each mode has a damping value of 5%.

The following parameters were calculated for the sample system via a Matlab script.

The mass matrix is

m =

0.0490 0

0 2.3497

The stiffness matrix is

k =

40000 160000

160000 6400000

Natural Frequencies =

133.8 Hz

267.9 Hz

Modes Shapes (column format) =

-4.4 1.029

0.1486 0.6352

Participation Factors =

0.1336

1.543

Page 41

41

0.01

0.1

1

10

100

101001000

FREQUENCY (Hz)

TRANS (G/G)

ACCELERATION TRANSFER FUNCTION FOR CG

Figure I-2.

Page 42

42

10-10

10-9

10-8

10-7

10-6

10-5

101001000

Spring 2 0.0095 inch RMS

Spring 1 0.0035 inch RMS

FREQUENCY (Hz)

REL DISP (inch2/Hz)

RELATIVE DISPLACEMENT POWER SPECTRAL DENSITY

Figure I-3.

Page 43

43

0.001

0.01

0.1

1

10

101001000

FREQUENCY (Hz)

ACCEL (G2/Hz)

ACCELERATION POWER SPECTRAL DENSITY

AT SYSTEM CG 13.8 GRMS OVERALL

Figure I-4.

Table I-2. PSD, Base Input, 14.1 GRMS

Freq (Hz) Accel (G^2/Hz)

10 0.1

2000 0.1

The PSD in Table I-2 is applied to the system in Figure I-1. The acceleration response is shown

in Figure I-4.

Page 44

44

0.00001

0.0001

0.001

0.01

10 1001000

Spring 2

Spring 1

FREQUENCY (Hz)

TRANS (inch/G)

RELATIVE DISPLACEMENT TRANSFER FUNCTIONS

Figure I-5.

Table I-3. Overall Levels, Two-degree-of-freedom Model

Element Parameter RMS Value

3 Value

Mass Acceleration Response 13.8 G 41.4 G

Spring 1 Relative Displacement 0.0035 inch 0.0105 inch

Spring 2

The results in Table I-3 show an error that would occur if the system had been modeled as an

SDOF system with translation only. The spring displacement in this case would have been

0.0065 inches. This would be 3.3 dB less that the Spring 1 displacement in Table I-3.

Relative Displacement 0.0095 inch 0.0285 inch

Page 45

45

APPENDIX J

Cantilever Beam

Figure J-1.

The following equations are taken from Reference 20.

The relative displacement response

, xY

to base acceleration is

, x

x

n

2

m

1

nn

2

n

nn

2 j

Y

W

Y

(J-1)

, x

x

n

x

m

1

nn

2

2

n

nn

2 j

Y

dx

d

W

Y

(J-2)

y(x, t)

w(t)

Page 46

46

, x

x

n

m

1

nn

2

2

n

n

2

2

n

2

2

2 j

Y

dx

d

W

Y

x

(J-3)

The bending moment is

, x

, x

Y

x

EIM

2

2

(J-4)

The bending stress is

I

Mc

(J-5)

, x

, x

Y

x

cE

2

2

(J-6)

, x

x

n

m

1

nn

2

2

n

n

2

2

n

2 j

Y

dx

d

W

cE

(J-7)

f , x

f

x

n

m

1

nn

2

2

n

n

2

2

n

2

ff 2 jff

Y

dx

d

4

1

W

2cE

(J-8)

Page 47

47

f , x

f

x

n

m

1

nn

2

2

n

n

2

2

n

ff 2 jff

Y

dx

d

W

cE

2

1

(J-9)

The bending stress transfer function is

f , x

f

x

n

W

m

1

nn

2

2

n

n

2

2

n

s

ff 2 jff

Y

dx

d

cE

2

1

) f , x (H

(J-10)

The eigenvalue roots for the cantilever beam are

Table J-1. Roots

Index n

1

2

3

4

The first mode shape and its derivatives are

L

L

n

1.875104

4.694091

7.854575

10.99554

x sinxsinh73410 . 0

x cosxcosh

1

) x (

1

Y

1111

x

1

(J-12)

(J-11)

cosxcosh73410. 0

x sinxsinh

L

) x (

1

Y

dx

d

111

1

x sinxsinh73410 . 0

x cosxcosh

L

) x (

1

Y

dx

d

1111

2

1

2

2

(J-13)

Page 48

48

The second mode shape and its derivatives are

1

) x (Y

2

x sinxsinh 1.01847x cosx cosh

L

2222

(J-14)

x cosxcosh1.01847x sinxsinh

L

) x (

2

Y

dx

d

2222

2

(J-15)

x sinxsinh1.01847x cosxcosh

L

) x (

2

Y

dx

d

2222

2

2

2

2

(J-16)

The third mode shape and its derivatives are

1

) x (Y

3

x sinxsinh0.99922x cosxcosh

L

3333

x

(J-18)

(J-17)

cosxcosh 0.99922x sinxsinh

L

) x (

3

Y

dx

d

3333

3

x sinxsinh0.99922x cosxcosh

L

) x (

3

Y

dx

d

3333

2

3

2

2

(J-19)

The fourth mode shape and its derivatives are

1

) x (Y

4

x sinxsinh1.00003x cosxcosh

L

4444

x

(J-21)

(J-20)

cosxcosh1.00003x sinxsinh

L

) x (

4

Y

dx

d

4444

4

x sinxsinh1.00003x cosxcosh

L

) x (

4

Y

dx

d

4444

2

4

2

2

(J-22)

Page 49

49

Example

Consider a beam with the following properties:

Cross-Section

Boundary Conditions

Material

Diameter

Circular

Fixed-Free

Aluminum

D = 0.5 inch

Cross-Section Area A = 0.1963 in^2

Length L = 24 inch

Area Moment of Inertia I = 0.003068 in^4

Elastic Modulus E = 1.0e+07 lbf/in^2

Stiffness EI = 30680 lbf in^2

Mass per Volume

v

= 0.1 lbm / in^3 ( 0.000259 lbf sec^2/in^4 )

Mass per Length

= 0.01963 lbm/in (5.08e-05 lbf sec^2/in^2)

Mass

L

= 0.471 lbm (1.22E-03 lbf sec^2/in)

Viscous Damping Ratio

The normal modes and frequency response function analysis are performed via a Matlab script.

The normal modes results are:

Table J-2. Natural Frequency Results, Cantilever Beam

Mode fn (Hz) Factor Modal Mass

( lbf sec^2/in )

1 23.86 0.02736 0.000748

= 0.05

Participation Effective Effective

Modal Mass

(lbm)

0.289

2 149.53 0.01516 0.00023 0.089

3 418.69 0.00889 7.90E-05 0.031

4 820.47 0.00635 4.04E-05 0.016

Note that the mode shape and participation factors are considered as dimensionless, but they

must be consistent with respect to one another.

Page 50

50

0.1

1

10

100

110 10010002000

FREQUENCY (Hz)

[ RESPONSE ACCEL / BASE ACCEL ] ( G / G )

ACCELERATION AT FREE END TRANSFER FUNCTION

Figure J-2.