appendix a random response sdof system

Appendix ARandom Response SDOF System

In this appendix, the random response of a single degree of freedom (SDOF) system,excited by a random enforced acceleration, is explained. The spectral approach isused. This appendix is added to this book because in a number of chapters thismethodis applied to compute the power spectral density (PSD) and the root mean square(r.m.s.) value of the response of the mass of the SDOF system. The SDOF system isshown in Fig.A.1.

The equation of motion of the SDOF system is given by

mX (t) + cX (t) + kX (t) = cU (t) + kU (t), (A.1)

where all variables are denoted in Fig.A.1. When the variables in the time domainare transformed into the frequency domain, Eq. (A.1) can be written as follows

X (ω) + 2jζωnX (ω) + ω2nX (ω) =

[−2jζ

ωn

ω−

(ωn

ω

)2]U (ω), (A.2)

where ζ is the damping ratio,ωn the natural frequency, andω the excitation frequency.The acceleration X (ω) can be expressed in the enforced acceleration U (ω).

X (ω) =[

2jζ(

ωnω

) + 1

1 − (ωnω

)2 + 2jζ(

ωnω

)]U (ω) = H (jω)U (ω), (A.3)

where H (jω) is the frequency transfer function (FRF).The PSD of the acceleration WX can be expressed in the PSD of the enforced

acceleration WUWX (f ) = |H (f )|2WU (f ), (A.4)

© Springer International Publishing AG 2018J. Wijker, Miles’ Equation in Random Vibrations, Solid Mechanicsand Its Applications 248, https://doi.org/10.1007/978-3-319-73114-8

169

170 Appendix A: Random Response SDOF System

Fig. A.1 SDOF system,enforced acceleration

m

k c

moving base

U(t)

X(t)

and

H (jf ) =⎡⎢⎣ 2jζ

(fnf

)+ 1

1 −(fnf

)2 + 2jζ(fnf

)⎤⎥⎦ , (A.5)

where the rad/s are replaced by cycles/s (Hz).The r.m.s. value of the acceleration X can be calculated using

xrms =∫ ∞

0WX (f )df . (A.6)

The integral in Eq. (A.6) can be approximated (numerical quadrature) by thetrapezoidal method to calculate xrms numerically [1].

xrms(fn) =√√√√ N∑

k=1

|H (fn, fk)|2WX (fk)Δfk , (A.7)

where Δfk is the step size (increment) between two subsequent frequency steps.

Example

Given the random acceleration specification in TableA.1 [2]. Calculate the randomresponse spectrum (RRS) of that acceleration specification for an instrument withmass m = 10kg and an amplification factor Q = 10. The RRS is the r.m.s. response

Table A.1 Specifications for random testing [2]

Frequency (Hz) PSD (g2/Hz)

20–100 +3dB/oct.

100–400 0.05 × m+20m+1

400–2000 −3dB/oct.

m (kg) Mass equipment with unknown location

Appendix A: Random Response SDOF System 171

Fig. A.2 RSS of randomacceleration specificationTableA.1

Hz

101 102 103 104

g

100

10 1

102Random Response Spectrum, RRS, Q=10

of a SDOF system to the random excitation as function of the natural frequency andfor a given damping ratio ζ . The RRS is presented in Fig.A.2. The RRS may beconsidered to reflect the damaging effect of the environment.

The PSD of the random acceleration spectrum can be reconstructed from the RRSusing Miles’ equation

WU (fn) = RRS2(fn)π2 fnQ

. (A.8)

Problems

A.1 An given RRS has a logarithmic varying acceleration of 10g at 20Hz and 300gat 2000Hz. The frequency varies linearly. Calculate the corresponding PSD of theacceleration specification with an amplification factor Q = 10.

A.2 Calculate the RSS from the PSD of the acceleration spectrum computed inProblem A.1 with a damping ratio ζ = 0.05.

Table A.2 Random acceleration specificationWU

Spectrum 1 Spectrum 2

Frequency (Hz) (g2/Hz) Frequency (Hz) (g2/Hz)

20–100 3dB/oct

100–500 0.04 20–2000 0.01

500–2000 −6dB/oct

172 Appendix A: Random Response SDOF System

Table A.3 Random vibration specificationWU

Frequency (Hz) PSD random acceleration (g2/Hz)

20–80 3dB/oct.

80–500 0.04

500–2000 −6dB/oct.

A.3 In TableA.2 two spectra of the enforced random accelerationWU are specified.Evaluate the potential damaging characteristics of both spectra using the RSS (Q =10).

A.4 Calculate the extreme response spectrum ERS(f ) of the random enforced ac-celeration specification given in TableA.3 using Miles’ equation. The ERS is anequivalent of the shock response spectrum (SRS).

Calculate Grms and perform the following steps:

• CalculateERS(f ) = n√

π2 fQWU (f ), with f the frequency between 20 and 2000Hz

and Q = 10.• n = √

2 ln(fT ) is a (peak) factor on the standard deviation to yield maximumresponse. The test duration is T = 120 s.

Answer: Grms = 6.78g

References

1. Schwarz HR (1989) Numerical analysis. Wiley, New York. ISBN 0-471-92065-72. GirardA, Imbert JF,MoreauD (1989) Derivation of European satellite equipment

test specification from vibro-acoustic test data. Acta Astronaut 10(10):797–803

Appendix BQuasi-static, Random, and Acoustic Loads

B.1 Introduction

During flight, the spacecraft is subjected to static and dynamic loads. Within theframe of this book about Miles’ equation, only the following static and dynamicloads are discussed:

• Quasi-static load.• Acoustic loads.• Random vibrations.

The specifications about the quasi-static design loads, the manner how the acous-tics loads and the random vibrations are prescribed need more explanation, whichwill be given in the following sections.

B.2 Quasi-static Load Specifications

FigureB.1 shows a typical longitudinal static acceleration time history for the Soyuzlaunch vehicle during its ascent flight. The highest longitudinal acceleration occursjust before the first stage cutoff and does not exceed 4.3g. The highest lateral staticacceleration may be up to 0.4g at maximum dynamic pressure and takes into accountthe effect of wind and gust encountered in this phase [1]. The very low frequencydynamic loads, which are expected to have no influence on the dynamic behavior ofthe spacecraft, are added to the static acceleration. The static acceleration increasedby the low frequency dynamic loads is called the quasi-static loads (QSL).

The quasi-static design loads are multiplied by the factors of safety to constructdesign limit loads to design the spacecraft structure and to analyze strength andstiffness characteristics of the spacecraft structure such as: permanent deformation,ultimate strength, buckling.


173

174 Appendix B: Quasi-static, Random, and Acoustic Loads

Fig. B.1 Typical longitudinal static acceleration [1]

B.3 Acoustic and Random Load Specification

B.3.1 Introduction

The acoustic noise inside the fairing (shroud) of the launch vehicle is assumed to bereverberant (diffuse); however, this is amathematical representation of the real soundfield. The reverberant sound field representation is well accepted in the spacecraftstructures design community. This sound field under the fairing will excite the outerstructure of the spacecraft as well as the subsystems mounted to the outside of thespacecraft, such as solar array wing, antennae dishes. The random sound pressuresexciting the outer structure of the spacecraft are transferred into random structuralvibrations and acoustic pressures of inside cavities. The interior acoustic noise willexcite the internal spacecraft structure and is transferred to random structural vibra-tions too. This process is illustrated in Fig.B.2. The random structural vibrationswill excite panel mounted equipment and components at the base. In general, smallequipment is not sensitive to acoustic noise.

The typical representation of acoustic load and random acceleration specificationsis discussed in subsequent sections [2, 3].

B.3.2 Acoustic Loads

Acoustic loads appear as design specifications for spacecraft and spacecraft attach-ments such as solar arrays and antennae. Acoustic loads are generated during launch,

Appendix B: Quasi-static, Random, and Acoustic Loads 175

Acoustic noise

Outer structure spacecraft

Internal structure(s)

Equipment & Components

Propagation

Sound pressures

Interior acoustic noise Mechanical random vibrations

Fig. B.2 Mechanism to produce random vibrations

or in acoustic facilities for test purposes, e.g., reverberation chamber.1 It is very com-mon to specify a reverberant sound field [1], which means that the intensity of thesound is the same for all directions. More about the fundamentals of acoustics orsound can be read in [4–6].

In general, the acoustic loads are described as sound pressure levels (SPL) andspecified in decibels (dB) [1, 7]. The SPL is defined by

SPL = 10log

(p

pref

)2

, (B.1)

where p is the rms pressure in a certain frequency bandwith frequency bandwidthΔf ,mostly one octave band or one-third octave band, and pref is the reference pressure2 × 10−5 Pa.

The xth octave band of two sequential frequencies f1 and f2 is given by

f2f1

= 2x, (B.2)

where x = 1 for the octave, and x = 13 when the one-third octave band; then

f2f1

= 2

1

3 = 1.260.

1A reverberation chamber is an enclosure with thick, rigid walls and smooth interior surfaces thatstrongly reflect sound waves.


The center frequency fc (Hz) is defined by

fc = √fminfmax, (B.3)

where fmin (fmax) is the minimum (maximum) frequency (Hz).The frequency bandwidth Δf (Hz) is given by

Δf = fmax − fmin. (B.4)

With fmaxfmin

= 2x, the bandwidthΔf canbe expressed in termsof the center frequencyfc as follows

Δf =(2

x2 − 2

−x2

)fc. (B.5)

When

• x = 1, the one octave band width is Δf = 0.7071fc.• x = 1

3 , the one-third octave band width is Δf = 0.2316fc.

The PSD of the pressure field Wp(fc) (Pa2/Hz) in the frequency band with centerfrequency fc, bandwidth Δf , and r.m.s. pressure p(fc) is defined as

Wp(fc) = p2(fc)

Δf (fc). (B.6)

The mean square value p2rms of the sound pressure level is given by

p2rms =∫ ∞

0W (f )df =

k∑i=1

Wi(fc)Δf =k∑

i=1

p2i (fc), (B.7)

where k is the number of one octave or one-third octave bands.The overall sound pressure level (OASPL) in dB is defined as

OASPL = 10log

(p2rmsp2ref

). (B.8)

The following relation determines the conversion of a one-third octave band to aone octave band;

SPL1−octave = 10 log

[3∑

k=1

10SPL 1

3 −octave

10

], (B.9)

and the following relation determines the conversion of a one octave band to a one-third octave band


SPL 13−octave = SPL1−octave + 10 log

[Δf 1

3−octave

Δf1−octave

]. (B.10)

B.3.2.1 Reverberant Sound Field

The cross-power spectral density of the pressure Wp(f , x)(Pa2/Hz), in a reverberant(diffuse) sound field (acoustic chamber), at position x and frequency f (Hz), is givenby [8]

Wp(f , x) = Wp(f )sin k|Δx|k|Δx| , (B.11)

whereWp(f )(Pa2/Hz) is the auto-power spectral density of the (reference) pressure p(Pa), k = 2π f /c is the wave number, c(m/s) is the speed of sound in air, |Δx|(m) thedistance between the positions x1 and x2. It is stated in [9] that the spectral densitybetween two distinct points x1 and x2 in space is:

• dependent on the relative distance |Δx| between two points only, i.e., it is inde-pendent of the absolute coordinates of any of the two points. Hence, the soundfield is homogenous.

• independent of the direction of the vector Δx between the two points, i.e., it is thesame in all directions. Hence, the sound field is isotropic.

FigureB.3 shows a plot of the normalized diffuse (reverberant) field spatial cross-spectral density in air (c = 340 m/s) for frequencies f = 50, 150, and 500Hz asa function of the distance Δx. This Fig.B.3 is created using the computer algebrasystem wxMaxima®, version 11.08.0.

Fig. B.3 Normalized diffuse pressure field cross-spectral density, y = sin(kΔx)/(kΔx), f = 50Hz(blue), f = 150Hz (red), f = 500 Hz (green), c = 340 m/s


A reverberant sound field can be simulated by a combination of (at least 26) planewaves caused by point sources placed equally distributed on a sphere with a largeradius [10].

B.3.3 Random Enforced Acceleration

In most cases, the random mechanical loads for spacecraft and subsystems of space-craft are specified in a very special manner. The power spectral density (PSD) valuesof the acceleration depend on the frequency (Hz). In general, the frequency range isbetween 20 and 2000Hz. The specificationmust be accompanied by theGrms value ofthe random acceleration in the frequency range. An example of a typical accelerationspecification is given below.

• 20–150Hz 6dB/oct• 150–700Hz Wu = 0.04 g2/Hz• 700–2000Hz −3dB/oct• Grms = 7.3g.

The graphical representation of the random acceleration specification is shown inFig.B.4.

The octave band between f2 and f1 is defined by

101

102

103

104

10−4

10−3

10−2

10−1

Frequency (Hz)

PS

D A

ccel

erat

ion

g2 /Hz

Random Vibration Specification

Fig. B.4 Specification PSD acceleration Grms = 7.3g


f2f1

= 21. (B.12)

The power of 1 denotes the octave band. The number x is the number of octavesbetween two frequencies f , and the reference frequency fref can be obtained using

f

fref= 2x, (B.13)

this yields

x =ln

(ffref

)ln 2

=log

(f

fref .

)log 2

. (B.14)

The relation between the PSD values depends on the number of dBs per octave n(dB/oct) and the number of octaves between two frequencies f and fref . The relationin dB between Wu(f ) and Wu(fref ) is given by

10 log

{Wu(f )

Wu(fref )

}= nx =

n log(

ffref

)log 2

, (B.15)

or {Wu(f )

Wu(fref )

}=

(f

fref

) n10 log 2

≈(

f

fref

) n3

. (B.16)

If both the frequency f axis and the axis of the PSD function W (f ) have a logscale then the angle m (dB/freq) can be obtained by

m = logWu(f ) − logWu(fref )

log f − log fref=

log{

Wu(f )Wu(fref )

}

log(

ffref

) = n

3. (B.17)

Finally the derivation of the following expression is obtained, a relation betweenthe PSD functions and the frequencies

Wu(f ) = Wu(fref )

(f

fref

) n3

= Wu(fref )

(f

fref

)m

. (B.18)

The total root mean square (r.m.s.) value (magnitude) of u(t) is equal to the squareroot of the area bounded by the PSD function between frequency limits f1 and f2.This can be written as

urms =√E{u2(t)

} =∫ f2

f1

Wu(f )df (B.19)


Substituting (B.18) into (B.19) we obtain the following expression for urms

urms =√E{u2(t)

}

=√∫ f2

f1

Wu(f1)

(f

f1

)m

df =√√√√Wu(f1)f1

m + 1

[(f2f1

)m+1

− 1

], f1 < f2

=√

−∫ f1

f2

Wu(f2)

(f

f2

)m

df =√√√√Wu(f2)f2

m + 1

[1 −

(f1f2

)m+1], f1 < f2.

(B.20)The parameters needed to calculate the Grms value of the random acceleration

spectrum are illustrated in Fig.B.5.The specification of the PSD (sometimes called acceleration spectral density (AS-

D)) of the enforced acceleration or base excitation can be divided into three regions:

• Spectrum with a positive slope n1 (rising).• Flat spectrum (slope is zero).• Spectrum with negative slope n2 (falling).

B.3.3.1 Spectrum with a Positive Slope

FigureB.5 shows a rising spectrum with a constant slope n1 > 0 between f1 and f2.The constant slope is expressed in decibels per octave. The area A1 can be calculatedas follows

A1 ={W (f2)f2m1 + 1

[1 −

(f1f2

)m1+1]}

, (B.21)

wherem1 = n1/3, and n1 > 0 is the increase of the PSD value in decibels per octave.

Fig. B.5 Calculation of Grms


B.3.3.2 Flat Spectrum

For a flat spectrum with a zero slope between f2 and f3 with m1 = 0 in (B.21), thearea A2 becomes

A2 = W (f2)[f3 − f2

], (B.22)

as shown in Fig.B.5.

B.3.3.3 Spectrum with a Negative Slope

For a falling spectrum of a constant slope n2 < 0 between f3 and f4, the constantslope is expressed in decibels per octave. The area A3 can be calculated as follows

A3 ={W (f3)f3m2 + 1

[(f4f3

)m2+1

− 1

]}, m2 �= −1, (B.23)

wherem2 = n2/3, andn2(< 0) is the decrease of the PSDvalue in decibels per octave.Equation (B.23) is not applicable if m2 = −1. In that case we have to calculate thevalue of A3 when limm2→−1. This limit can be found using L’Hôpital’s Rule2 [11]. If

u(m2) =(f4f3

)m2+1 − 1 and v(m2) = m2 + 1, then L’Hôpital’s rule gives

A3 = W (f3)f3 ln

(f4f3

)= 2.30W (f3)f3 log

(f4f3

), m2 = −1. (B.24)

The Grms of the enforced random acceleration specification can be obtained, asillustrated in Fig.B.5, by the following expression

Grms = √A1 + A2 + A3. (B.25)

B.4 Random Vibration Test Tolerances

At the end of this appendix, something is said about tolerance on power spectraldensities and r.m.s. values of the random enforced accelerations during the randomvibration test (TableB.1).

2Let u(a) = v(a) = 0. If there exists a neighborhood of x = a such that (1) v(x) �= 0, except forx = a, and (2) u′(x) and v′(x) exist and do not vanish simultaneously, then

limx→a

u(x)

v(x)= lim

x→a

u′(x)v′(x)

whenever the limit on the right exists.


In the ECSS, testing standard [12] are the allowable tolerances for several kindsof tests specified.

Problems

B.1 Given the following random acceleration specification in TableB.2. Calculatethe slopes and Grms.Answers: 3dB/oct., −3dB/oct, Grms = 6.06g.

B.2 Presented inTableB.3 the ‘Vega-Users-Manual-issue-04-April-2014’SPLqual-ification acoustic vibration levels.

Table B.1 Random vibration test tolerances [12]

Frequency range (Hz) Amplitude PSD/rms

20–1000 −1/ + 3dB

1000–2000 ±3dB

Overall Grms ±10 % g

Table B.2 Random acceleration spectrum

Frequency (Hz) ASD g2/Hz Slopes

20 0.01005 ? dB/oct

80 0.04

350 0.04

2000 0.00704 ? dB/oct

Grms ?

Table B.3 Acoustic vibration test levels

Octave center frequency (Hz) SPL qualification levels (dB)pref = 2.0 × 10−5 Pa

31.5 123

63 126

125 129

250 135

500 138

1000 130

2000 123

OASPL (20–2828Hz) 140.8

Test duration 120s


Perform the following assignments:

• Set up the series of frequencies in the one-third octave band.• Convert the one octave band SPLs in TableB.3 into the one-third octave band.

References

1. Arianespace (2012)Soyuzuser’smanual, Issue2,Rev0edition.www.arianespace.com

2. Wijker JJ (2008) Spacecraft structures. Springer, Berlin. ISBN 978-3-540-755524

3. Wijker JJ (2009) Random vibrations in spacecraft structures design, theoryand applications. Number SMIA 165 in solid mechanics and its applications.Springer, Berlin. ISBN 978-90-481-2727-6

4. Fahy F (1985) Sound and structural vibration, radiation, transmission and re-sponse. Academic Press Limited, London. ISBN 0-12-247671-9

5. Norton M, Karczub D (1990) Fundamentals of noise and vibration analysisfor engineers, 2nd edn. Cambridge University Press, Cambridge. ISBN 0-251-49913-5

6. Smith PW Jr, Lyon RH (1965) Sound and structural vibration. Technical reportNASA CR-160, NASA

7. Arianespace (2011) ARIANE 5 user’s Manual, Issue 5, Revision 1 edition8. Waterhouse RV, Cook RK, Berendt RD, Edelman S, Thompson MC (1955)

Measurement of correlation coefficients in reverberant sound fields. J AcoustSoc Am 27(6):1072–1077

9. WittingM (1986) Modelling of diffuse sound field and dynamic response analy-sis of lightweight structures. PhD thesis, Technische Universität München, HertUtz Verlag. ISBN 3-89675-678-8

10. Hamdi MA, Gardner B (2007) Vibro-acoustic analysis of space structures: stateof the art and analysis guidelines. ESA Contract 20371/NL/SFe VASI-ESI-TN-010, ESA

11. Korn GA, Korn TM (1961)Mathematical handbook for scientists and engineers.McGraw-Hill Book Company, New York

12. European Cooperation of Space Engineering, Noordwijk, The Netherlands. S-pace engineering testing, ECSS-E-ST-10-03C, 3rd edn, 1st June 2012

www.arianespace.com

www.arianespace.com

Appendix CSimulation of the Random Time Series

C.1 Introduction

In this appendix, the transformation of a PSD function in a random time series withthe FFT is discussed. The back transformation of random time series into PSD usingIFFT is discussed too.

C.2 Random Time Series

The simulation of random time series is discussed in [1–3]. In this appendix, wediscuss the efficiency of fast computation of the random time series. The analysis isbased on [2]. If Φ1

x (ω) is the one-sided PSD of the desired signal x(t), then x(t) mayapproximated by

x(t) = √2N−1∑n=0

[Φ1x (ωn)Δω] 1

2 cos(ωnt − φn), (C.1)

where φn are uniformly distributed random numbers on the interval (0 − 2π) andωn = nΔω and Δω = ωmax/N . ωmax is the maximum frequency in the powerspectrum Φ1

x (ω), and N is the total number of terms in the summation.A considerable improvement in the computational effort can be obtained by re-

casting (C.1) to allow the use of the fast Fourier transform (FFT). To accomplishthis, (C.1) may be written as

x(t) = �[√

2N−1∑n=0

[Φ1x (ωn)Δω] 1

2 ej(ωnt−φn)

]. (C.2)


185

186 Appendix C: Simulation of the Random Time Series

If the simulated time series x(t) is needed only at discrete values of time t, letxn = x(tk) = x(kΔt), where the time duration between the equally spaced samplesis Δt. Evaluation of (C.2) at time t = tk gives

x(tk) = x(kΔt) = �[√

2N−1∑n=0

[Φ1x (ωn)Δω] 1

2 ej(ωnkΔt−φn)

]. (C.3)

To satisfy the Nyquist sampling criterion, the time series, x(t), must be sampled ata high enough rate to obtain two samples during one period of the highest frequencycomponent ωmax of interest in the original PSD Φ1(ω). Hence, Δt is chosen to be

Δt ≤ π

ωmax. (C.4)

The term ωnkΔt in (C.3) can be rewritten as follows

ωnkΔt = nΔωkΔt = nωmax

Nk

π

ωmax= nk2π

2N. (C.5)

Thus, (C.3) can expressed in the following form

x(kΔt) = �[√

2N−1∑n=0

[Φ1x (ωn)Δω] 1

2 e−jφnejnk2π2N

]. (C.6)

Equation (C.6) may be evaluated using the FFT algorithm. Given a discretesequence an, the FFT provides an efficient means of computing Ak , where

Ak =N−1∑n=0

anejnk2π2N , k = 0, 1, 2, . . . ,N − 1. (C.7)

Equation (C.6) may be evaluated using the FFT by defining a sequence

an = [Φ1x (ω)Δω] 1

2 e−jφn n ≤ N − 1,

= 0, n ≥ N .(C.8)

Equation (C.6) may then be written as

xk = x(kΔt) = �[√

22N−1∑n=0

anejnk2π2N

]k = 0, 1, 2, . . . , 2N − 1. (C.9)

Appendix C: Simulation of the Random Time Series 187

Table C.1 Random vibration specification [4]

Frequency (Hz) W(f) g2/Hz

20–100 3dB/oct

100 0.2

600 0.2

600–2000 −3dB/oct

Overall 15.94Grms

In finding the real part of (C.9), we may use the complex conjugate3 of the right sideof (C.9) to give

xk = x(kΔt) = �[√

22N−1∑n=0

ane−jnk2π2N

]k = 0, 1, 2, . . . , 2N − 1. (C.10)

This is equivalent toxk = √

2�[FFT (an)]. (C.11)

Note that the length of the sequence an is 2N .The PSD function Φ1

x (ω) can be reconstructed using the following expression

Φ1x,n = 2

|IFFT (xk)|2Δω

k = 0, 1, 2, . . . , 2N − 1, n = 0, 1, 2, . . . ,N − 1, (C.12)

where IFFT is the inverse fast Fourier transform.If the productΦ1

x (ω)Δω is replaced byW 1x (f )Δf , (C.13) can bewritten as follows

W 1x,n = 2

|IFFT (xk)|2Δf

k = 0, 1, 2, . . . , 2N − 1, n = 0, 1, 2, . . . ,N − 1, (C.13)

where W (f ) is the PSD function in the cyclic frequency domain and f is the cyclicfrequency (Hz).

Example

The enforced random acceleration spectrum is specified in TableC.1. This randomacceleration spectrum will be simulated by random time series. The number of termsin time series isN = 500, and the maximum frequency of interest is fmax = 2000Hz.The simulated random time series is shown in Fig.C.1. The reconstructed PSD func-tionW (f ) from the random time series in Fig.C.1 is shown in Fig.C.2. The artificialvalue of the PSD below 20Hz isW = 1.0 × 10−6 g2/Hz.

3cos u = �[eju] = �[e−ju].

188 Appendix C: Simulation of the Random Time Series

Time (s)0 0.05 0.1 0.15 0.2 0.25

(g)

-50

-40

-30

-20

-10

0

10

20

30

40

50Random time series enforced acceleration (g), N=500, f

max=2000 Hz

Fig. C.1 Simulated random time series from TableC.1, 1σ = 15.95g, Umax = 43.66g

Frequency (Hz)

100 101 102 103 104

(g2/H

z)

10-6

10-5

10-4

10-3

10-2

10-1

100

Reconstructed PSD enforced acceleration W(f), N=500, fmax

=2000 Hz

Fig. C.2 Reconstructed PSD W (f ) from Fig.C.1

Appendix C: Simulation of the Random Time Series 189

Problems

C.1 This problem is taken from [3]. The PSD function is given by

W (f ) = W0α

α2 + f 2, 0 ≤ f ≤ fu,

where

W0 =[arctan

fuα

]−1

,

where fu = 40Hz and α = 4Hz.Perform the following assignments:

• Simulate the random time series.• Reconstruct the original PSD function from the random time series.

References

1. Cacko J, Bily M, Burkoveecky J (1988) Random processes: measurement, anal-ysis and simulation, vol 8. Fundamental studies in engineering. Elsevier, Ams-terdam. ISBN 0-444-98942

2. Miles RN (1992) Effect of spectral shape on acoustic fatigue life estimation. JSound Vib 153(2):376–386

3. Shinozuka M, Jan CM (1972) Digital simulation of random processes and itsapplications. J Sound Vib 25(1):111–128

4. Chung YT, Krebs DJ, Peebles JH (2001) Estimation of payload random vi-bration loads for proper structure design. In: AIAA-2001-1667, Seattle, 42ndAIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materialsconference 16–19 April 2001

Appendix DComputation of SRS

D.1 Introduction

The principle of the calculation of the shock response spectrum (SRS) is illustrated inFig.D.1. For quite a number of SDOF elements, the transient responses are computedwhen they are excited at the base by a transient acceleration. The maximum valuesof the transient response Gk , k = 1, 2, . . . , n of the SDOF element k with naturalfrequency fk and amplification factorQ are plotted, successively. The horizontal axis(ordinate) represents the frequency (Hz) and the vertical axis (abscise) the maximumaccelerations (g). The figure represents the SRS. The SRS was already mentioned byM.A. Biot in 1933 [1] and later in 1941 [2]. The theoretical description of the SRSwas done within the frame of earthquake engineering.

D.2 Single Degree of Freedom System (SDOF)

The equation of motion of the SDOF element (see Fig.D.2) excited at the base byan enforced acceleration u(t) (m/s2)

mx = −c(x − u) − k(x − u), (D.1)

where x (m/s2) is the absolute acceleration, k(N/m) the stiffness of the spring, and c(Ns/m) the damping of the damper. The transfer function H (s) of this SDOF systemin the complex Laplace domain assuming zero initial conditions of the velocity x anddisplacement x is given by

H (s) = X (s)

U (s)= 2ζωnωns + ω2

n

s2 + 2ζωns + ω2n

, (D.2)

where the damping ratio is ζ = c/2√km and the natural frequency is ωn = √

k/m.


191

192 Appendix D: Computation of SRS

Fig. D.1 Development of a shock response spectrum

Fig. D.2 Base-acceleration-excited SDOFelement m

k c

moving base

u(t)

x(t)

For a given amplification factor Q = 1/2ζ , natural frequency ωn, and base exci-tation u(t), it is theoretically possible to calculate the acceleration response x(t) inthe form of the time function as well and consequently to determine the peak valueof this time function. The input acceleration signal is in the form of a sampled signalwith a sample interval Ts = 1/fs, not in the form of a continuous signal. The transferfunction Eq. (D.2) has to be approximated by the Z-transform, i.e., to be transformedinto a discrete system (digital filter).

Appendix D: Computation of SRS 193

D.3 Discrete Approximation of Continuous TransferFunction

The complex function in the s-plane is approximated by a discrete function in thez-plane (z-Transform, [3]), i.e., the approximation of the differential equation ofmotion by a difference equation, which is a problem of numerical integration [4].The continuous function between two adjacent samples may be approximated by aconstant value or by a ramp or by any suited function. A mapping of the s-planeon the z-plane has to be find saving the important properties of continuous systemafter transformation into a discrete system of the same order. The transfer functionH (z) can be approximated by the First-order hold theorem [3]) using the followingexpression

H (z) = (z − 1)2

TzZ

{L−1

[H (s)

s2

]}(D.3)

The digital filter corresponding to the SDOF system responses is the second-orderfilter infinite impulse response (IIR), with the general z-transform expression [4]:

H (z) = X (z)

U (z)= β0 + β1z−1 + β2z−2

1 + α1z−1 + α2z−1, (D.4)

where β0, β1, β2, α1, and α2 are the filter parameters.The transfer function H (z) corresponds to the difference equation, which is en-

abling to calculate the response function in the time domain.

x(k) = β0u(k) + β1u(k − 1) + β2u(k − 2) − α1x(k − 1) − α2x(k − 2). (D.5)

Equation (D.3) is completely worked out by Irvine in [5]. The digital filter parametersare calculated by equations given below:

β0 = 1 − exp(−A) sin(B)/B,

β1 = 2 exp(−A)[sin(B)/B − cos(B)],β2 = exp(−2A) − exp(−A) sin(B)/B,

α1 = −2 exp(−A) cos(B),

α2 = exp(−2A),

A = ζωnTs = ωnTs2Q

,

B = ωnTs√1 − ζ 2 = ωnTs

√1 − 1

4Q2.

(D.6)

To compute the SRS the following MATLABTM function can be applied,


function [y,f]=srs(x,fs,fmin,fmax,fno,Q)

% SRS calculation using Smallwood improved method

%

% [y,f]=srs(x,fs,fmin,fmax,fno,Q)

% y maximum accelerations calculated

% f frequency vector (Hz)

% x data vector (enforced acceleration)

% fs samping frequency (Hz)

% fmin lowest frequency of interest (Hz)

% fmax highest frequency of interest (Hz)

% fno number of frequencies

% Q Amplification factor

%

y=zeros(fno,1);

f=zeros(fno,1);

%

%

k1=log(fmax/fmin)/(fno-1);

k2=pi/Q/fs;

k3=2*pi/fs*sqrt(1-1/(4*Qˆ2));

%

%

for n=1:fno;

f0=fmin*exp(k1*(n-1));

A=k2*f0;

B=k3*f0;

a=[1,-2*cos(B)*exp(-A),exp(-2*A)];

b=[1-exp(-A)*sin(B)/B,2*exp(-A)*(sin(B)/B-cos(B)),

exp(-2*A)-exp(-A)*sin(B)/B];

z=filter(b,a,x);

y(n,1)=max(z);

f(n,1)=f0;

end

%endfunction

Example

The random response spectrum shown in Fig.D.3 is measured on a solar array wingplaced into a reverent acoustic chamber. It is asked to compute a SRS of this giv-en random acceleration spectrum. Before the SRS can be generated, the randomacceleration spectrum is transformed into the time domain applying the transforma-tion method discussed in Appendix C. The number of frequency steps is N = 500.The SRS associated with the accelerations in the time domain is shown in Fig.D.5.


Hz

101 102 103 104

g2/H

z

10-5

10-4

10-3

10-2

10-1

100

101

Fig. D.3 Measured random acceleration spectrum, Grms = 24.98g

Time (s)

0 0.1 0.2 0.3 0.4 0.5 0.6

(g)

-100

-80

-60

-40

-20

0

20

40

60

80Random time series enforced acceleration (g), N=500, f

max=2000 Hz

Fig. D.4 Measured random acceleration in time domain, Grms = 24.98g

Because the SRS is also based on peak responses, the equivalent random accelerationvibration specification can be obtained using Eq. (10.11) (Fig.D.5).

http://dx.doi.org/10.1007/978-3-319-73114-8_10


(Hz)

101 102 103 104

(g)

100

101

102

103SRS, Q=10, T=0.5s

Fig. D.5 Shock response spectrum

g

f(Hz)

SRS

m

k

fn = 12π

km

fn

G

G

Fig. D.6 Application SRS

Example

In this example, an application of the SRS is presented. In Fig.D.6, the dynamicsystem (SDOF element) has to be designed to withstand the acceleration G at thenatural frequency fn. The inertia load is mG, and the stress distribution inside thestructure can be calculated.

Problems

D.1 This problem is taken from [6]. Find the ramp invariant simulation of


t0 1 2 3 4 5 6 7 8

g

0

0.1

0.2

0.3

0.4

0.5

0.6Continuous verse discrete solution

TheoreticalDiscrete

Fig. D.7 Continuous verse discrete solution

H (s) = 1

1 + s,

and illustrate its response g(t) to f (t) = 2t exp(−t), 0 ≤ t ≤ 8s. H (z) is found asfollows:

H (z) = (z − 1)2

TzZ

{L−1

[H (s)

s2

]}

Define the response function in the time domain. Perform the response analysis inthe time domain with T = 0.6 s, and compare the numerical results with analyticalsolution of g(t) = t2 exp(−t).Answers:

H (z) = (T − 1 + exp(−T ))z − T exp(−T ) − 1 − exp(−T )

T (z − exp(−T )),

g(m) = 1

T

[(T − 1 + exp(−T ))f (m) − (T exp(−T ) − 1 − exp(−T )f (m − 1)

]+ exp(T )g(m − 1).

FigureD.7.


g101 102 103

100

101

102 SRS Half-Sine Pulse, A=10g,11ms

Fig. D.8 SRS half-sine pulse, A = 10g, duration 11ms

D.2 The input acceleration time history is a half-sine pulse of 11 ms is sampled at2000Hz [4]

x = A sin(ωt), 0 ≤ t ≤ 0.011,

where A = 10 g.

• Calculate ω.• Compute the corresponding SRS of x with Q = 10.

Answer: ω = 1000π/11 rad/s, SRS see Fig.D.8.

D.3 The input acceleration is time history sampled at 2000Hz which is given by [7]

x = u(t)e−ηωt sin(ωt) + u(t + τ)Ae−νt sin(ν(t + τ)), 0 ≤ t ≤ 0.25,

where u(t) is the Heaviside step function, A = −0.1995, η = 0.05, ω = 2π(100),ν = 2π(10), and τ = −0.015757.

• Compute the corresponding SRS of x with Q = 10.

References

1. Biot MA (1933) Theory of elastic systems vibrating under transient impulse withan application to earthquake-proof buildings. Proc Natl Acad Sci 19(2):261–268


2. Biot MA (1941) A mechanical analyzer for the prediction of earthquake stresses.Bull Seismol Soc Am 31(2):150–171

3. Stearns SD (2003) Digital signal processing, with examples in matlab. The elec-trical engineering and applied signal processing series. CRC Press, Boca Raton.ISBN 0-8493-1091-1

4. Tuma J, Koci P (2009) Calculation of shock response spectrum. Colloqium, 2–4Feb 2009

5. Irvine T (2013) Derivation of the filter coefficients for the ramp invariant methodas applied to the base excitation of a single-degree-of-freedom system. www.vibrationdata.com, revision b edition, 3 April 2013

6. Stearns SD (1975) Digital signal analysis. Hayden Book Company, Inc. ISBN0-8104-5828-4

7. Smallwood DO (1981) An improved recursive formula for calculating shockresponse spectra. Shock Vib Bull 51:4–10

www.vibrationdata.com

www.vibrationdata.com

Appendix EApplication Rayleigh’s Quotient

E.1 Introduction

In this appendix, the application Rayleigh’s quotient to estimate quickly the natu-ral frequency associated with a particular vibration mode of a dynamic system isdiscussed.

E.2 Rayleigh’s Quotient

The theory behind Rayleigh’s quotient or Rayleigh’s method can be found in maytextbooks, but will be briefly recapitulated in this Appendix. Rayleigh’s quotient isdefined by the following equation

R(u) =12 (u)

T [K](u))12 (u)

T [M ](u)) = U

T ∗ , (E.1)

where U is the strain energy, T ∗ the ‘kinetic energy’, [K] the stiffness matrix, [M ]the mass matrix, and (u) the assumed mode. For continuous dynamic systems, thestrain energy and ‘kinetic energy’ can be written as

U = 1

2

∫V

σεdV , (E.2)

T ∗ = 1

2

∫V

ρudV , (E.3)

with σ the stress and ε the strain distribution associated with the assumed displace-ment u and ρ the density. V is the volume of the continuous system.

Rayleigh’s quotient is very useful:


201

202 Appendix E: Application Rayleigh’s Quotient

• It is insensitive to errors in the choice of the assumed mode.• It is often possible to guess at the shape of an assumed mode.

Rayleigh’s quotient has a stationary value when the assumed mode (u) is in thevicinity of any vibration mode (φi), with the following well-known orthogonalityproperties

(φi)T [M ](φj) = δij

(φi)T [K](φj) = ω2

i δij,(E.4)

where δij is Kronecker’s delta, δii = 1, δij = 0, i �= j.The assumed mode (u) is depicted on an independent set of eigenvectors or vi-

bration modes, such that

(u) =n∑

k=1

ck(φk), (E.5)

where ck , k = 1, 2, . . . , n are the weighting factors. Equation (E.5) is substituted inEq. (E.1), then

(u)T [K](u)) =n∑

k=1

c2k(φk)T [K](φk), (E.6)

(u)T [M ](u)) =n∑

k=1

c2k(φk)T [M ](φk), (E.7)

due to the orthogonality relations Eq. (E.4).When (φk)T [K](φk) = ω2

k (φk)T [M ](φk),

Rayleigh’s quotient can be written

R(u) = ω2 =∑n

k=1 c2kω

2k∑n

k=1 c2k

. (E.8)

If the assumed mode (u) ≈ (φr), then cr � ck , (r �= k). The Rayleigh quotient canbe further evaluated

R(u) =c2rω

2r + c2r

∑k=1,2,...,

k �=r

(c2kc2r

)2ω2k

c2r + c2r∑

k=1,2,...,k �=r

(c2kc2r

)2 =ω2r + ∑

k=1,2,...,k �=r

(c2kc2r

)2ω2k

1 + 1∑

k=1,2,...,k �=r

(c2kc2r

)2 . (E.9)

The fraction∣∣∣ c2kc2r

∣∣∣ = εk 1. In case r = 1

Appendix E: Application Rayleigh’s Quotient 203

R(u) =ω21 + ∑

k=2,3,...

(c2kc21

)2ω2k

1 + 1∑

k=2,3,...

(c2kc21

)2

≈ ω21 +

n∑k=2,3,...

ε2kω2k − ω2

1

n∑k=2,3,...

ε2k ,

= ω21 +

n∑k=2,3,...

(ω2k − ω2

1)ε2k .

(E.10)

The stationary value of R(u) is

dR

dε= 0, if εk = 0, k = 2, 3, . . . (E.11)

General conclusions about Eq. (E.10) can now be drawn:

• Rayleigh’s quotient is never lower than the first eigenvalue, R(u) ≥ ω21, and• similarly it can be shown that R(u) ≤ ω2

n. Rayleigh’s quotient is never higher thanthe highest eigenvalue.

• In general, Rayleigh’s quotientω21 ≤ R(u) ≤ ω2

n. For continuous dynamic systems,ω2n → ∞.

Example

A tapered bar is shown in Fig.E.1, with length L, and a varying cross section A(x) =Ao(1−x/2L)2,A(0) = Ao, andA(L) = Ao/4.Young’smodulus of the bar isE, and thedensity is ρ. Calculate Rayleigh’s quotient when the assumed mode or displacementfunction is u(x) = (x/L)α .

The strain energy U and kinetic energy T ∗ of the tapered bar are

U = 1

2

∫V

σεdV = 1

2

∫ L

0EA(x)

(du

dx

)2

dx,

T ∗ = 1

2

∫V

ρu(x)2dV = 1

2

∫ L

0ρA(x)u(x)2dx.

(E.12)

Fig. E.1 A tapered beam

x

L

Ao

ρ, E

A(x) = 1 − x2L

2


Rayleigh’s quotient can now be obtained

R(u) = U

T ∗ = 35E

8ρL2, (E.13)

and the lowest approximation of the natural frequency ω is

ωn =2.0917

√Eρ

L> ωtheor =

2.029√

Eρ

L. (E.14)

Problems

E.1 A dynamic system consists of three coupled SDOF elements as shown inFig.E.2. The mass m = 100kg and the spring stiffness k = 1.0 × 109 N/m.

Perform the following assignments:

1. Calculate the static displacement vector (u) = (x1, x2, x3)T under an 1g gravi-tational field (1g = 9.81m/s2).

2. Calculate Rayleigh’s quotient R(u) (Rad/s)2, and associated natural frequencyfn(u) (Hz).

3. Solve undamped eigenvalue problem [K] − ω2i [M ](φi) = 0, and subsequently

the natural frequencies fn,i = ωi/2π, i = 1, 2, 3.

Answers: (u) = 1.×10−4(0.0589, 0.1079, 0.1373)T , R(u) = 8.0831×105(Rad/s)2,fn(u) = 143.0903Hz, (fn) = (142.7539, 469.1769, 777.0698)T (Hz).

E.2 Acantilevered beam is also supported at the tip by a rotational and a translationalspring, kφ and kw respectively [1]. The system is illustrated in Fig.E.3. Calculate the

Fig. E.2 Three SDOFdynamic system m

k

x1

2m

2k

x2

3m

3k

x3

1g

Appendix E: Application Rayleigh’s Quotient 205

Fig. E.3 Cantilevered beam

EI, mkφ = EI

L

kw = 3EIL3

L

Φ(x)x

Fig. E.4 S/C modeled as acantilevered beam

L

L2

M2

M3

Φ(x) = 1 − cos πx2L

x

EIm

EIm

Φ(0) = 0

Φ (0) = 0

Φ (L) = 0

eigenvalue ω2n with the aid of Rayleigh’s quotient with the following assumed mode

[2, 3]:Φ(x) = c1x

2(6L2 − 4Lx + x2).

Answer: ω2n = 31.0673 EI

L4m (Rad/s)

E.3 A spacecraft is mathematically modeled as a cantilevered beam with bendingstiffness EI , distributed mass m, and two discrete masses M1,M2. The dynamicsystem is shown in Fig.E.4.

The assumed mode isΦ(x) = 1 − cos

(πx

2L

).

Calculate using Rayleigh’s principle the natural frequency ωn corresponding to theassumed mode Φ(x).

Answer: ωn =√

3.0440EI(0..2268mL+0.858M1+M2)L

E.4 Investigate and report about the Ritz method [1, 4, 5] to obtain naturalfrequencies.


References

1. Rao SS (2011)Mechanical vibrations 5th edn. Prentice Hall, Upper Saddle River,Number ISBN 978-0-13-21-2819-3

2. Ludolph G, Potma AP, Legger RJ (1963) Sterkteleer. Number Tweede deel inLeerboek der mechanica. Wolters, Groningen, achttiende druk edition

3. Temple G, BickleyWG (1956) Rayleigh’s principle and applications to engineer-ing. Dover Publications, New York

4. Leissa AW (2005) The historical bases of the Rayleigh and Ritz method. J SoundVib 287:961–978

5. Michlin SG (1962) Variationsmethoden der Mathematische Physik. Akademie-Verlag, Berlin

Appendix FRandom Fatigue Estimation

F.1 Introduction

In this appendix, three methods to predict the fatigue damage and fatigue life of partsof structures exposed to random vibrations are presented. Among many methods,three fatigue damage estimation methods are discussed:

• The narrowband method [1, 2].• The Dirlik methods, [3].• The Steinberg’s three band method [4].

Examples are provided and problems given.

F.2 Statistical Characteristics of Random Processes

In the frequency domain, the random process X is defined by the one-sided PSDWx(f ),with f the frequency (Hz). The statistical characteristics of a stationary processcan be described by the moment of the PSD [5].

The nth spectral moment mn of the PSD Wx(f ) at frequency f is given by

mn =∫ ∞

0f nWx(f )df . (F.1)

For fatigue prediction, analyses up to m4 are normally used. The even spectral mo-ments represent the standard deviation of the random process X , and its time deriva-tive X are

σx = √m0, σx = √

m2. (F.2)

The spread of the random process or spectral width is estimated using the param-eter αi, which has the general form


207

208 Appendix F: Random Fatigue Estimation

αi = mi√m0m2×i

, (F.3)

where α2 is most commonly used and is the negative of the correlation between Xand X . It takes values from 0 to 1. The higher the value to more narrow is the randomprocess in the frequency domain. It approaches 0 as the random process is a whitenoise process.

Another frequently used spectral parameter is δ (Vanmarke’s bandwidthparameter)

δ =√1 − α2

1 . (F.4)

The expected peak occurrence frequency νp and the expected positive zero-crossing rate (characteristic frequency) ν0 are defined as

νp =√m4

m2, ν0 =

√m2

m0. (F.5)

Example

The response of Problem 11.5, Chap. 11 will be used to calculate the statisticalcharacteristics of the PSD of the bending stress. The PSD of the bending stressWσb(f ) is illustrated in Fig.F.1.

The statistical properties of the PSD function of the bending stress Wσb(f ) arepresented in TableF.1.

Fig. F.1 PSD bending stress

http://dx.doi.org/10.1007/978-3-319-73114-8_11

Appendix F: Random Fatigue Estimation 209

Table F.1 Statistical properties Wσb (f )√m0 (Pa) α2 ν0 (Hz) νp (Hz)

4.7434 × 107 0.0000 57.3858 58.1468

The PSD response is narrowbanded because the spread factor α2 = 0.0000. Thepositive zero crossings ν0 and the averaged number of peaks νp are about equal tothe fundamental frequency fn = 56.2745Hz.

The foundation for the frequency domain approximation of a cycle distributionwas set by Rice in his famous paper “Mathematical Analysis of random Noise” in[6] where he analytically defined the probability density function of the peaks pp(a)based on the PSD

pp(a) =√1 − α2

2√2πσx

exp

(− a2

2σ 2x (1 − α2

2)

)+ α2a

σ 2x

exp

(− a2

2σ 2x

)Φ

⎛⎝ α2a

σx

√1 − α2

2

⎞⎠ ,

(F.6)where a is the peak amplitude. Φ(z) is the standard normal cumulative distributionfunction

Φ(z) = 1√2π

∫ z

−∞exp

(−t2

2

)dt. (F.7)

F.3 Fatigue Damage and Fatigue life

The cumulated fatigue damage D is based the Palgren–Miner rule

D =n∑

i=1

niNi

, (F.8)

where Ni indicates number of cycles to failure when the specimen or component issubjected to stress with an amplitude si and ni is the applied number of stress cyclesof si. Eq. (F.8) indicates that n various stress amplitudes are applied to the componentfor numbers of cycle. When the damage ratio D = 1, the component will fail.

The relation between the occurring stress s and the corresponding allowable num-ber of cycles N can be described by the s-N (Wöhler) curve or Basquin equation

skN = C, (F.9)

where k and C are material constants. The Basquin equation is illustrated in Fig.F.2.The expected damage D per second is based on the Palgren–Miner rule can be

estimated by the following expression


Fig. F.2 s − N curve

s1

s2

N1 N2

skN = c

Typical s-N (or Wohler) curve

Failure cycle (N)

Stress

D = νp

C

∫ ∞

0skpa(s)ds. (F.10)

The fatigue life estimate is given by

T = 1

D. (F.11)

Among many methods in the frequency domain, three methods to estimate thefatigue damage D per second will be discussed in this appendix:

• Narrowband method to estimate DNB [1, 2].• Dirlik method to estimate DDK [3].• Steinberg’s three stage method to estimate DSB [4].

F.4 Narrow-Band Method

For a narrowband random process, it is reasonable to assume that every peak iscoincident with a cycle and that the cycle amplitudes are Rayleigh distributed (seeEq. (F.6) with α2 = 0). The narrowband expression was originally presented byMiles [2] and is here defined for stress amplitudes

DNB = ν0C−1(

√2m0)

kΓ

(1 + k

2

), (F.12)

where ν0 is the expected zero-crossing frequency with positive slope, which is veryclosed to the peak intensity νp for the narrowband process. C and k are materialfatigue constants, m0 is the zero-order spectral moment. For a narrowband process,the parameter α2 ≈ 0. The gamma function is defined as


Γ (x) =∫ ∞

0tx−1 exp(−t)dt. (F.13)

F.5 Dirlik Method

The Dirlik method [3, 7] approximates the cycle-amplitude distribution by using acombination of one exponential and two Rayleigh probability densities. It is basedon numerical simulations of the time histories for two different groups of spectra.This method is considered as one of the best to estimate fatigue damage. The rain-flow-cycle amplitude probability density estimate is given by

pa(s) = 1√m0

[D1

Qexp

(− Z

Q

)+ D2Z

R2exp

(− Z2

2R2

)+ D3Z exp

(−Z2

2

)],

(F.14)whereZ is the normalized amplitude and xm is themean frequency as definedhereafter

Z = s√m0

, xm = m1

m0

√m2

m4, (F.15)

and the parameters D1, D2, D3, R, and Q are defined as

D1 = 2(xm − α22)

1 + α22

, D2 = 1 − α2 − D1 + D21

1 − R,

D3 = 1 − D1 − D2, R = α2 − xm − D21

1 − α2 − D1 + D21

,

Q = 1.25(α2 − D3 − D2R)

D1,

(F.16)

while α2 is already defined in Eq. (F.3). The rain-flow damage intensity under thePalgren–Miner rule is calculated by substituting pa(s) Eq. (F.14) in Eq. (F.10)

DDK = νpmk/20

C

[D1Q

kΓ (1 + k) + √2kΓ

(1 + k

2

) (D2|R|k + D3

)]. (F.17)

The parameters C and k are defined in Eq. (F.9).

F.6 Steinberg’s Three Band Method

The assumption of the Steinberg’s method is that the probability density functionof the random stress response follows a Gaussian distribution, and therefore the


expected values of the stress response amplitudes are bound by certain probabilitylevels:

68.27% Chance that the amplitude of the stress cycles will not exceed the peak ofone times the r.m.s. of the stress response signal.

27.1% Chance that the amplitude of the stress cycles will not exceed the peak oftwo times the r.m.s. of the stress response signal.

4.3% That the stress cycles will not exceed the peak of three times the r.m.s. ofthe stress response signal.

There are no stress cycles occurring with peaks greater than three times the r.m.s.value of the stress.

The expected fatigue damage per second is given by:

DSB = νo

C

[0.683

(√mo

)k + 0.271(2√mo

)k + 0.043(3√mo

)k]. (F.18)

Example

The 1σ shear stress τp = 0.07637 MPa of Problem 2.10 is used to calculate thefatigue life time of the honeycomb core. The natural frequency fn = 75Hz. Thefollowing (artificial) s-N curve is applied in the fatigue damage calculations

s6N = 1.9847 × 1035 cycles Pascal6. (F.19)

The narrowband method and the Steinberg’s method are applied to estimate thefatigue life T of the honeycomb core.

Narrow-band method DNB = fnc−1(√2τp)kΓ

(1 + k

2

) = 0.0036,TNB = 1/DNB = 278 s,

Steinberg’s method DSB = fnτ kp c

−1[0.683 + 0.271 × 2k + 0.043 × 3k

] =0.0037,TSB = 1/DSB = 270 s.

Problems

F.1 This problem is a continuation of Problem 11.5, in particular the FEA part,which will be more or less recapitulated. The natural frequency of the dynamicsystem illustrated in Fig. 11.14 will be calculated using the finite element analysis(FEA) method. The beam will be modeled with one Bernoulli beam element withthe following mass and stiffness matrices [8].

http://dx.doi.org/10.1007/978-3-319-73114-8_2

http://dx.doi.org/10.1007/978-3-319-73114-8_11

http://dx.doi.org/10.1007/978-3-319-73114-8_11


Fig. F.3 Finite elementmodel beam 1 2

w1 w2

φ2φ1

E, I, m, L

[M ] = mL

420

⎡⎢⎢⎣

156 22L 54 −13L22L 4L2 13L −3L2

54 13L 156 −22L−13L −3L2 −22L 4L2

⎤⎥⎥⎦ [K] = EI

L

⎡⎢⎢⎣

12 6L −12 6L6L 4L2 −6L 2L2

−12 −6L 12 −6L6L 2L2 −6L 4L2

⎤⎥⎥⎦(F.20)

The finite element model (FEM) is illustrated in Fig.F.3.To introduce the fixation, the large mass approach will be applied (massMl = 109

kg and second moment of mass Il = 109 kgm2) in the direction of w1 and φ1. Theupdated mass matrix is

[M ] = mL

420

⎡⎢⎢⎣156 + Ml

420mL 22L 54 −13L

22L 4L2 + Il420mL 13L −3L2

54 13L 156 + M 420mL −22L

−13L −3L2 −22L 4L2

⎤⎥⎥⎦ , (F.21)

where the mass of the box M is introduced too. To calculate the unitary frequencyresponse functionsH (f ) of the accelerations at node 1, degrees of freedom (DOF)w1

and w2 a constant enforced acceleration U = Ml will be applied to node 1, DOF w1.That means that the resulting enforced acceleration is equal to one. This is principlethe large mass approach. Later on the PSD of the accelerations will be computed.

The natural frequencies fn and corresponding vibration [Φ]modes are given here-after

(fn) =

⎛⎜⎜⎝

0.00000.000056.27451503.18

⎞⎟⎟⎠ , [Φ] =

⎡⎢⎢⎣0.0000 −0.0000 −0.0000 0.00000.0000 0.0000 −0.0000 0.00000.0000 −0.0000 2.1820 0.38990.0000 0.0000 21.7353 340.4000

⎤⎥⎥⎦ (F.22)

The orthogonal damping matrix [C] can be obtained using Eq. (11.10), with ζ =0.05.

The PSD of the accelerations of DOFs w1 and w2 as well as the bending momentin node 1, φ1 direction are shown Fig.F.4.

The r.m.s. acceleration of the discrete mass M is w2,rms = 20.9057 g, and ther.m.s. of the bending moment in node 1 is Mb = 5.8107Nm, and the r.m.s. of thebending stress is σb = 47.4341 MPa. Use this stress to calculate the fatigue lifetimewith the following methods in conjunction with the s-N curve Eq. (11.44):

http://dx.doi.org/10.1007/978-3-319-73114-8_11

http://dx.doi.org/10.1007/978-3-319-73114-8_11


(a) Accelerations (b) Bending moment

Fig. F.4 PSD accelerations and bending moment

• Narrowband method, TNB = 1/DNB.• Dirlik method, TDK = 1/DDK .• Steinberg method, TSB = 1/DSB.

Answers: TNB = 4.0360 × 104 s, TDK = 4.0624 × 104 s, TSB = 3.9839 × 104 s.Repeat the FE analysis with your own favorite FE package and perform again the

fatigue life analyses.

References

1. Crandall SH, Mark WD (1973) Random vibration in mechanical systems. Aca-demic Press

2. Miles JW (1954) On structural fatigue under random loading. J Aeronaut Sci21(11):753–762

3. Dirlik T (1985) Application of computers in fatigue Analysis. PhD thesis, Uni-versity of Warwick, Coventry, England

4. Steinberg DS (2000) Vibration analysis for electronic equipment, 3rd edn. Wiley,New York. ISBN 0-471-37685-X

5. MrsnikM,Slavic J,BoltezarM(2013)Frequency-domainmethods for a vibration-fatigue-life estimation-application real data. Int J Fatigue 47:8–17. https://doi.org/10.1016/j.ijfatigue.2012.07.005

6. Wax N (1954) Selected papers on noise and stochastic processes. Dover publica-tions. ISBN 0-486-60262

7. Benasciutti D (2004) Fatigue analysis of random loadings. PhD thesis, Universtyof Ferrara, Department of Engineering, December 2004

8. Kwon YW, Bang H (2000) The finite element method, using MATLAB 2nd edn.CRC Press, Boca Raton. ISBN 0-8493-0096-7

https://doi.org/10.1016/j.ijfatigue.2012.07.005

https://doi.org/10.1016/j.ijfatigue.2012.07.005

Appendix GJohn Wilder Miles (1920–2008)

G.1 Obituary Notice

John W. Miles, renowned scientist and research professor emeritus of applied me-chanics and geophysics at Scripps Institution of Oceanography, UC, San Diego, diedOctober 20, 2008, in Santa Barbara, Calif., following a stroke. He was 87years old[1]. Miles had been at Scripps since 1964 as a researcher and professor in the CecilH. and Ida M. Green Institute of Geophysics and Planetary Physics (IGPP) and alsoserved as vice chancellor for academic affairs at UC San Diego from 1980 to 1983.

Miles was well regarded for his pioneering work in theoretical fluid mechanicsand in 1957 proposed a wind-wave growth model, a theory he continued to refineuntil late in his career. His theoretical model is considered as one of the corner-stones for the current generation of numerical wave prediction models, and a majorcontribution to the field of oceanweather forecasting andwave dynamics. In 2003, re-searchers at Johns Hopkins University and UC Irvine provided the first experimentalmeasurements supporting important aspects of Miles’ theory.

Miles devoted the first 20years of his research to electrical and aeronautical engi-neering. When he joined Scripps, he turned his mathematical abilities to geophysicalfluid dynamics and made numerous contributions to all aspects of fluid dynamics,including supersonic flow, ocean tides, the stability of currents and water waves andtheir nonlinear interactions, along with extensive work in the application of mathe-matical methodology.

‘John was a very generous colleague with a great love for research and an im-mense appetite for work’, said Rick Salmon, Scripps professor of oceanography andcolleague of Miles.

Miles was born in Cincinnati, Ohio, in 1920, and received his B.S. in 1942, M.S.in 1943, and A.E. and Ph.D. in 1944, all from the California Institute of Technolo-gy. During World War II, he worked at the Massachusetts Institute of Technology?sRadiation Laboratory and for Lockheed Aircraft Corporation, followed by an ap-pointment as professor of engineering and geophysics at UCLA from 1945 to 1961(Fig.G.1).


215

216 Appendix G: John Wilder Miles (1920–2008)

Fig. G.1 John W. Miles1920–2008

He has received several prestigious awards, including the Timoshenko Medal ofthe American Society of Mechanical Engineers in 1982, and was designated theOtto Laporte Lecturer by the American Physical Society in 1983. He was elected amember of the National Academy of Sciences in 1979.

Miles’ research productivity was legendary, with more than 400 publicationsover his working 60-year career. He served as associate editor of the Journal of FluidMechanics and editor/coeditor of several other scientific journals, covering U.S. andforeign research in fluid mechanics, applied mechanics and mathematical analyses.Within the frame of this book the following publication is of great interest: ‘Onstructural fatigue under random loading’, J. Aero. Sci. 21, 753-762 (Nov., 1954).

He was a longtime resident of La Jolla, Calif., and is survived by three daughters,Patsy Fiske of Santa Barbara, Calif., Ann Leslie Albanese of Solvang, Calif., andDiana Jose of San Diego, Calif. At his request there will be no memorial services.His ashes were scattered at sea.

Reference

1. Reisewitz A (2008) Obituary notice distinguished scientist and professor: Johnw.miles. UC San Diego, News Center. http://ucsdnews.ucsd.edu/archive/newsrel/science/10-08JohnWMiles.asp

http://ucsdnews.ucsd.edu/archive/newsrel/science/10-08JohnWMiles.asp

http://ucsdnews.ucsd.edu/archive/newsrel/science/10-08JohnWMiles.asp

Index

AAcceleration

spectral density, 180Acceptance

test, 107Acoustic

pressure field, 91Acronyms

list, xviiAssumed

mode, 11, 42

BBasquin

equation, 209Beam element

load vector, 32mass matrix, 28stiffness matrix, 28

CCenter

of mass, 146CoM, 146

DDamping

matrix, 147Dirlik

method, 4, 207Displacement

mode, 11Dunkerley’s

method, 52

Dwellresponse, 106

EEquivalent

static acceleration field, 25static finite element analysis, 37static force field, 25

ERS, 172European

Space Agency, 79Space Technology Center, 79

Extremeresponse spectrum, 112, 172

FFast

Fourier Transform, 187Fatigue

damage, 209damage spectrum, 107life, 209

FFT, 187First

moment of area, 40passage failure, 46

First-OrderHold Theorem, 193

Fokker-Planck-Kolmogorov equation, 5FPKE, 7Fundamental

vibration mode, 91


217

218 Index

GGeneral

environmental verification standard, 63Generalised

coordinate, 91mass, 56

GEVS, 63

HHeaviside

step function, 198Homogenous

sound field, 177

IIFFT, 187IIR, 193Infinite

Impulse Response, 193Inverse

Fast Fourier Transform, 187Isotropic

sound field, 177

JJacobian

matrix, 14John

Wilder Miles, 215Joint

acceptance, 100

KKinetic

energy, 201

LL’Hôpital’s rule, 181Large

European Acoustic Facility, 111mass approach, 162, 213

LEAF, 111

MMAC, 57Mass

acceleration curve, 57matrix, 147

participation, 57participation approach, 48

MilesJohn Wilder, 215

Miles’equation, 1, 8, 57

Miner’scumulative damage ratio, 160

Modaleffective mass, 25participation factor, 25, 56power, 59static deflection, 33

Myosotisequations, 53, 145

NNarrow-band

method, 4, 207Notch, 61

OOASPL, 41, 176Obituary

notice, 4One

octave band, 175third octave band, 175

Orbitalreplacement unit, 62

Orthogonaldamping matrix, 29

ORU, 62Overall

sound pressure level, 176

PPalgren–Miner, 209Poisson’s

ratio, 83Power

spectral density, 4PSD, 4

QQSL, 4, 57Qualification

test, 107Quasi

static load, 4

Index 219

RR.m.s., 1Random

response spectrum, 171time series, 185vibration load factor, 46vibration test, 107

Rayleigh’sprinciple, 201quotient, 4, 93, 201

Referencepressure, 175

Residualmass, 60

Rigidbody mode, 154

Ritzmethod, 205

Rootmean square, 1

RRS, 171

SS-N curve, 209SDOF

response curve, 84Second

moment of area, 40Shape

factor, 81Shell

structure, 91Shock

response spectrum, 4, 191Sine

sweep response, 106Sinusoidal

response, 106vibration test, 107

Sinusoidal--random equivalence, 106

Soundpressure level, 32, 175

Spacetransportation system, 62

Spann’s

component predictor, 84Spatial

distribution, 91Spectral

moment, 207SPL, 32, 175SRS, 4, 191Static

load factor, 63Steinberg’s

three band method, 4, 207Stiffness

matrix, 147Strain

energy, 201Structural

damping coefficient, 83STS, 62Sweep

rate, 106Symbols

list, xvii

TThin

walled shell, 91Three

band technique, 160Three-sigma

design approach, 45Time

duration, 107Trapezoidal

method, 170

VVibration

response spectrum, 112VRS, 112

ZZ

transform, 192

appendix a random response sdof system

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