Evaluation of Vibration Test Severity by FDS and ERS
D.H.Cho1
1
Korea Aerospace Industries, Ltd., Airframe Design Section
802,Yucheon-Ri, Sanam-Myun, Sacheon-City, Gyeongnam, Korea
email: dh0101@koreaaero.com
Abstract
The Fatigue Damage Spectrum(FDS) and Extreme Response Spectrum(ERS) were proposed by Lalanne.
These spectra can be used as the severity comparison tools between different types of environmental
vibration test specifications. In this paper, a little different ways will be proposed to calculate them. For
the wide-band random vibration, instead of Probability Density Function(PDF) of peaks, Dirlik's empirical
PDF equation of ranges will be used to calculate FDS. Rayleigh distribution of peaks will be assumed to
calculate ERS. For the Sine-on-Random(SOR) vibration, the sinusoidal component will be converted to
equivalent narrow-band Power Spectral Density(PSD) and added to background wide-band random
component. The rest procedures are the same as random vibration. These proposed procedures will be
verified by comparing with Rain-Flow Counting method. Finally, two case study results will be presented
on vibration test specifications of trainer aircraft to show the usefulness of proposed procedure.
1
Introduction
It is inevitable that the equipment installed on the supersonic jet aircraft are exposed to severe vibration
environment introduced by a variety of sources such as aerodynamic turbulence and jet engine operation.
In order to verify the performance and fatigue life of these equipments under the vibration environment,
qualification tests should be performed at the aircraft development phase. The standard vibration test
procedures approved by internationally authoritative organizations are adopted usually as the aircraft
development specifications. These standards have been continuously revised with the progresses of testing
facilities and technology. The aircraft developers want to use Off-the-Shelf(OTS) equipment as many as
possible to cut down the test expenditure. However, the old OTS items were verified, in general, with old
version of the standards written in different test method or spectra. Even the recently developed OTS
items for another aircraft were often verified with different vibration specification. In this case, the
developers should retest the equipment with considerable cost and time even though the aircraft
specification is actually less severe than that of equipment. To eliminate this unnecessary cost increase, it
is recommended that the test severities of two specifications be compared before the test.
Lalanne proposed FDS and ERS as vibration severity evaluation tools[1]. FDS is for vibration fatigue life
and ERS is for the maximum instantaneous response stress generated in the structure. The unknown
internal component is modeled as single degree of freedom(SDOF) vibration system as shown in Figure
1(a). The fatigue damage(FD) and extreme response(ER) are calculated with assumed natural
frequency(fo) and input excitation spectrum(U). The stress(σ) generated in the structure is assumed to be
proportional to the displacement difference between the excitation(U) and mass(X), i.e. the deflection(Z)
of the spring, as equation(1). Then, the frequency response function(H) between the deflection of the
spring and excitation displacement is given as equation (2).
σ(f o , f ) = qZ(f o , f ) = q ( X (f o , f ) − U (f ))
(1)
H (f o , f ) = Z(f o , f ) / U (f ) = f 2 /((f o2 − f 2 ) + j2ςf o f )
(2)
where, f is excitation frequency, q is unknown proportional factor, ζ is damping ratio.
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f1(t), x1(t)
z(t)=x(t)-u(t)
x(t)
m1
f2(t), x2(t)
m
c3
k3
m2
c
u(t), excitation
k
c1
u(t)
c2
(a) Single DOF
k1
k2
(b) Multi DOF
Figure 1 : Simplified Model of Unknown Internal Structure
1.1
FD
The accumulated fatigue damage can be calculated by Baskin’s Stress-Life relation(3) and Miner’s
Rule(4).
σib N i = c
FD(f o ) =
(3)
∑ D =∑ ( n
i
i
i
i
/ Ni ) =
∑ (q
b
/ c) * Ti * n i * Z ib
(4)
i
where, b, c are unknown material properties. Ni, Ti, ni, Zi are allowable cycle number, excitation duration,
cycle number of the stress during unit time, and the deflection level at proper accumulation index, i,
respectively. The index may be frequency or deflection range. The excitation frequency is convenient for
sinusoidal vibration and deflection range for random, because the random FD is accumulated from PDF
not PSD. The cycle number calculation procedures vary according to the type of vibration.
1.2
ER
The extreme response can be given as equation(5).
{
}
ER (f o ) = Z max = f 2 /((f o2 − f 2 ) + j2ςf o f ) ⋅ U (f )
max
(5)
ER varies according to unknown damping ratio and independent of fatigue characteristics of material.
1.3
FDS & ERS
The FDS and ERS are completed for all fo in the frequency range of interest [1-3]. It is unlikely that the
absolute value of those spectra are available because of the unknown information of the internal
components, i.e., the values of q, b, c and ζ. They can be used, however, as the relative vibration test
severity comparison tool between different test procedures for the same unknown equipment with
assumed material properties.
FDS is calculated assuming the unknown internal component as SDOF system, but the actual internal
components are usually consisted of multi degree of freedom system(MDOF) as shown in Figure 1(b). If
the internal component is assumed as linear system, all deflections and fatigue damages of the internal
spring components can be obtained by superposition of each vibration mode. If the fatigue damage of a
test specification of all modes at any frequency is greater or less than another one, then the superposition
results will be the same. Therefore the assumption of SDOF is reasonable.
DYNAMIC TESTING : METHODS AND INSTRUMENTATION
2
2.1
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Classified Severity Evaluation by Types of Vibration
Sinusoidal Vibration
For sine sweeping excitation, approximating the deflection amplitude and excitation frequency as constant
for very short time interval, the FD and ER can be directly calculated by equations (4),(5) with the
deflection of spring. That is, for short time interval, Di is calculated and then, accumulated for total
sweeping time to calculate FD(fo). Finally the FDS is completed for all fo in the frequency range of interest.
The sweeping rate is reflected as duration at the constant amplitude. The ERS is completed by finding
maximum response at each assumed natural frequency.
2.2
Random Vibration
2.2.1
FDS
The PDF of peak values was proposed as equation(6) to calculate FDS for random vibration [1].
−v2
1 − r 2 2 (1− r 2 ) vr
p(z) =
e
+ e
2π
2
−v2
2



vr
1 + erf 


 2(1 − r 2 )  



(6)
where, v = z / z rms , r = n 0+ n p+ : irregularity factor .
In this paper, the PDF of range, i.e., the difference between successive peak and valley is used to calculate
FDS. Rice proposed a series of moment equations(7) of the random vibration PSD. Bendat used only mo to
estimate the PDF, p(S), of ranges as equation(8). Bendat’s procedure is proper estimation for narrow-band
random data, but somewhat conservative for the wide-band random data[5]. Dirlik proposed equation(9)
to estimate PDF of ranges of wide-band random data properly, as well as narrow-band.
∫
m i = f i G (f )df
(7)
S2
S − 8mo
p(S) =
e
4m o
−Y2
 A − Y A Y − Y 22

Q
1
2
2
R
p(S) =
e +
e
+ A 3Ye 2
Q
W2

n (S) = E (P) ⋅ T ⋅ p(S)
(8)

/2 m
o


(9)
(10)
where, range S = ∆z peak , Y = S / 2 m o and A, Q, W are the functions of ranges and moments, mi. n(S)
is expected number of ranges during Time T. E(P) = m 4 m 2 is the expected number of peaks during
unit time[5].
The deflection PSD of spring shown in Figure 1(a) is presented in Figure 2 assuming natural frequency as
500Hz, for example. Both models(Bendat and Dirlik) will be useful for the single degree of freedom
system as the response shows narrow-band distribution near the natural frequency. For comparison, the
deflection PSD was inverse fourier transformed into time domain with random phase angle, and then the
PDFs of all data, peaks, ranges were calculated and presented in Figure 3. The result of Rain-flow
Counting[6] is also shown, which is generally adopted in time domain fatigue life estimation. As shown in
Figure 3, while Lalanne’s equation follows the distribution of peaks, Bendat’s Narrow-band and Dirlik’s
equations are similar to the results of Rain-flow Counting, as expected. It is unclear that which equation is
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more reliable, but some commercial fatigue analysis program, such as MSC.Fatigue[4], also adopt the
PDF of ranges. Figure 4 shows the FDS calculated with Dirlik’s equation.
1.0E-01
(2)
(3)
1.0E-06
2
2
PSD of Displacement (m /Hz)
1.0E-04
PSD of Acceleration (g /Hz)
1.0E-02
1.0E-08
1.0E-02
(1)
1.0E-10
1.0E-12
(1) Deflection (z=u-x), fn=500Hz
(2) Input Displacement (u, m)
(3) Input Acceleration (g)
1.0E-14
1.0E-16
10
100
1000
1.0E-03
10000
Frequency (Hz)
Figure 2 : Power Spectral Density of Random Vibration
2.2.2
ERS
Lalanne proposed several procedures to calculate ERS of random vibration. One procedure was to
calculate the number of up-crossings of threshold value with positive slope assuming Gaussian
distribution of all data. The ER was defined as the level of peak which exceeded only once. The other one
was to calculate the probability distribution with the proposed PDF equation(6) and find the maximum
peak level in the same way as previous one.
Steinberg[7] proposed that the PDF of the peak accelerations showed the Rayleigh distribution like
equation(11) in random vibration environment. In this paper, Rayleigh distribution was assumed, and ER
was decided through equation(12) to (14) in the similar manner as lalanne.
p(z peak ) =
z peak
2
z rms
− z 2peak
e
2 z 2rms
(11)
n p = Ep ⋅ T = m4 m2 ⋅ T
(12)
n p ⋅ 1 −

(13)
∫
z max
0
p( z peak )dz peak  = 1

( )
ER (f o ) = z max = 2z 2rms log e n p
(14)
where, T is test duration, Ep number of peaks per second, np number of total peaks. This result(14) is the
same as Lalanne’s ERS of random vibration. Figure 4 shows the ERS.
DYNAMIC TESTING : METHODS AND INSTRUMENTATION
1729
Probability Density Function (p(z),p(∆z))
1500
(T1)
(T2)
(T3)
(T1)
1200
(T2)
(P2) p(z_peak) _Empirical_Lalanne
(P3a) p(∆z_peak)_Empirical_Dirlik
(P3b) p(∆z_peak)_Empirical_Narrowband
900
(P2)
(T3)
600
p(z)
p(z_peak)
p(∆z_peak)_rain flow counting
300
(P3b)
(P3a)
0
0
0.001
0.002
0.003
0.004
0.005
Relative Displacement (z=x-u)
0.006
0.007
Figure 3 : Probability Density Function of Random Vibration
1.0E+08
1.0E+03
(1) FDS
(2) ERS
b=6
ζ=0.05
1.0E+04
1.0E+00
1.0E+02
1.0E+01
1.0E-04
1.0E+00
1.0E-08
1.0E-01
E.R.S.
F.D.S.
(1)
(2)
1.0E-12
1.0E-02
1.0E-16
1.0E-03
1.0E-20
1.0E-04
10000
1
10
100
Natural Frequency (Hz)
1000
Figure 4 : FDS and ERS of Random Vibration
2.3
Sine-on-Random Vibration
SOR is swept sine superimposed on a wide-band random vibration. Some vibration test specifications,
such as Gun-firing vibration environment, were defined as SOR test in the old versions of military
specifications. Therefore, the FDS and ERS for SOR vibration test specifications are also required.
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2.3.1
P ROCEEDINGS OF ISMA2010 INCLUDING USD2010
FDS
Lalanne proposed PDF of peaks of SOR vibration data as equation(15)
p(z) =
v
z rms
e
−
v2 +a 2
2 I
o (av)
(15)
2i
1
 av 
where, I o (av) =  
, v = z / z a ,rms , a = z s / z a ,rms . za,rms is root-mean-square(RMS) value of
2
i  2  (i!)
random component, and zs is the amplitude of sinusoidal component.
∑
In this paper, the sinusoidal component was converted to narrow-band PSD as equation(16) in order to
apply the PDF equation of stress ranges. Figure 5 shows the PDFs of SOR vibration data, for example.
100Hz sinusoidal component was superimposed on wide-band random PSD shown in Figure 2. The
Dirlik’s equation and Rain-flow Counting results show good agreement. The Bendat’s narrow-band
equation shifted a little to higher frequency, which means some conservatism for wide-band vibration as
mentioned earlier. By these results, it was thought acceptable to convert the sinusoidal component to
narrow-band PSD.
G SOR (f s ) = G R (f s ) + U (f s ) 2 / 2∆f
(16)
where, fs, U(fs) are frequency and amplitude of sinusoidal component. GSOR, GR are PSDs of SOR and
background random vibration, and ∆f is the frequency resolution of PSD.
Probability Density Function (p(z),p(∆z))
800
(T3)
p(∆z_peak)_rain flow counting
(P3a) p(∆z_peak)_Empirical_Dirlik
(P3b) p(∆z_peak)_Empirical_Narrowband
600
(P3a)
400
(T3)
200
(P3b)
0
0
0.001
0.002
0.003
0.004
0.005
Relative Displacement (z=x-u)
0.006
0.007
Figure 5 : Probability Density Function of Sine on Random Vibration
2.3.2
ERS
Lalanne calculated the probability distribution with the proposed PDF equation(15) for SOR, and find the
level of peak which exceeded only once. This level was defined as ER(fo) of SOR vibration.
In this paper, the sinusoidal component was converted to narrow-band random PSD and ER(fo) was
defined assuming the Rayleigh distribution of peaks like random data.
DYNAMIC TESTING : METHODS AND INSTRUMENTATION
3
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Application
The value of nonlinear characteristics, b, was assumed as 6 according to the military specification[9],
which showed that the values of b ranged, typically, between 6 and 6.5 for most materials used in the
aircraft. Damping ratio, ζ, was assumed as 0.05. The other unknown values, q and c, were assumed as 1.
3.1
Random .vs. Sinusoidal Vibration
The proposed procedures were applied to the environmental vibration specification of the military aircraft.
The test specification for newly developed aircraft was defined as random vibration as shown in Figure 6
and the sinusoidal vibration was also allowed only for the OTS equipments. The test specification was
developed without considering the FDS and ERS. Only the military specification and that of similar
aircraft developed earlier were referenced.
For the sinusoidal sweeping test, ten minutes sine dwell test were added at four natural frequencies
identified during one hour sine sweeping test. These natural frequencies may be different from those of
actual internal components because the natural frequencies can be identified only at the outside of the
equipment. FDS and ERS for these two test specifications are compared in Figure 7. ‘Sine_min’
(‘Sine_max’) is for the case where dwell frequency is far from (coincide with by chance) actual natural
frequency of internal structure. The actual FDS will be between these two lines according to the dwell
frequencies.
Both FDS and ERS of sinusoidal specification are higher than those of random specification at most of
frequency ranges. Now, it can be considered appropriate to accept those OTS equipment without
additional random test.
1.E-01
2
PSD of Random Acceleration (g /Hz)
Sinusoidal Accelecration (G peak)
1.E+03
1.E+02
(2)
1.E+01
1.E-02
(1)
1.E+00
(1) Sinusoidal
(2) Random
1.E-01
1
10
100
Frequency (Hz)
1000
Figure 6 : Environmental Vibration Test Specification
1.E-03
10000
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1.0E+04
(2)
1.0E+06
(3)
1.0E+02
F.D.S.
1.0E-02
(1)
1.0E-06
(1) FDS _ Random
(2) FDS _ Sine_max
(3) FDS _ Sine_min
(4) ERS _ Random
(5) ERS _ Sine
1.0E+02
1.0E+01
1.0E+00
(5)
1.0E-10
1.0E+03
1.0E-01
(4)
1.0E-14
E.R.S.
1.0E+10
1.0E-02
1.0E-18
1.0E-03
b=6
ζ=0.05
1.0E-22
1.0E-04
1.0E-26
1
10
100
Natural Frequency (Hz)
1000
1.0E-05
10000
Figure 7 : FDS and ERS Comparison between Random and Sinusoidal Excitation
3.2
SOR .vs. Random
Gun-firing vibration specification was defined as SOR test in an old version of military standard[8] as
shown in Figure 8(a). SOR test was adopted as environmental vibration test specification for several
military aircraft developed in 1970s. Constant pure random PSD test was adopted, however, in the late
version[9], and some new equipment were verified by pure random test. These equipment should be
retested by SOR specification to be installed on the aircrafts developed according to old version of
standards. In order to exempt SOR test, the test severity needs to be evaluated.
Figure 8(b) shows two pure random PSDs. One is the sweeping narrow-band PSD converted from
sinusoidal added on the constant background random PSD. This is regarded as equivalent to SOR. The
other is constant wide-band PSD. The FDS and ERS are shown in Figure 9. ‘RND_1’ means that the
constant PSD levels were obtained merely by dividing the sinusoidal peak level by sweeping band width,
usually ±10% of center frequency corresponding to gun-firing and higher harmonics. In this case, the
FDS and ERS of pure random are lower than SOR in most frequency range. ‘RND_2’ means that the
constant PSD level is double of ‘RND_1’. In this case, both of FDS and ERS of pure random are severe
than SOR in all frequency ranges and it seems a little conservative. The 1.5 times of ‘RND_1’ may be
proposed as proper level of equivalent severity.
4
Conclusions
The FDS and ERS were adopted as tools to compare the severity of several environmental test
specifications defined in different types of vibration. The PDF of stress range was proposed instead of that
of peaks to develop FDS of random vibration. The Rayleigh distribution was assumed for the peaks of
random vibration to develop ERS. A simple idea was proposed to evaluate the severity of SOR test. It is
converting the sinusoidal peaks to narrow-band PSD and treating as if the resulting specification is pure
wide-band random vibration.
These proposed procedures were verified by comparing with Rain-Flow Counting method, and finally
applied to two cases of specification comparison. The first case is between random and sinusoidal
DYNAMIC TESTING : METHODS AND INSTRUMENTATION
1733
sweeping, and the second is between SOR and wide-band random PSD. These results show the usefulness
of FDS and ERS as vibration severity comparison tools.
1.E+02
1.E+00
(1)
1.E-01
(2)
1.E-01
Distance = 20"
from Gun Muzzle
(1) Sinusoidal
(2) Background Random
1.E-02
10
1.E-02
1000
100
Frequency (Hz)
(a) Swept Sine on Random
1.E+04
(1) Gpeak → Instantaneous PSD
(2) Gpeak → Band Average PSD
1.E+03
(1)
Narrow band sweeping
(2)
Constant
2
PSD (g /Hz)
1.E+02
1.E+01
1.E+00
1.E-01
Distance = 20"
from Gun Muzzle
1.E-02
10
100
Frequency (Hz)
(b) Random Only
Figure 8 : Gun Vibration Test Specification
1000
2
1.E+00
PSD (g /Hz)
Gpeak
1.E+01
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1.E+08
(1)
(1) FDS_SOR
(2) FDS_RND_1
(3) FDS_RND_2
1.E+06
(3)
1.E+04
FDS
1.E+02
1.E+00
(2)
1.E-02
1.E-04
1.E-06
b=6
ζ=0.05
1.E-08
1.E-10
10
100
1000
Natural Frequency (Hz)
(a) FDS
1.E+01
(1)
1.E+00
(3)
(1) ERS_SOR
(2) ERS_RND_1
(3) ERS_RND_2
ERS
(2)
1.E-01
1.E-02
1.E-03
10
100
Natural Frequency (Hz)
(b) ERS
Figure 9 : FDS and ERS of Gun Vibration Test Specification
1000
DYNAMIC TESTING : METHODS AND INSTRUMENTATION
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References
[1] C.Lalanne, Specification Development, Vol.5, 2nd Edition, ISTE Ltd, (2009)
[2] C.Lalanne, Maximum Response and Fatigue Damage Spectra – Part I, The journal of environmental
sciences, July/Augst, (1984), pp.35-40
[3] C.Lalanne, Maximum Response and Fatigue Damage Spectra – Part II, The journal of environmental
sciences, September/October, (1984), pp.40-44
[4] Bishop NWM, Finite Element Based Fatigue Analysis, Americas User Conference, Oct 5-9, Sheraton
Universal Hotel, Universal City, California
[5] A.Halfpenny, Frequency Domain Approach for Fatigue Life Estimation from Finite Element
Analysis, International Conference on Damage Assessment of Structures, (1999)
[6] ASTM, Standard Practices for Cycle Counting in Fatigue Analysis, E1049-85, (Reapproved 2005)
[7] D.S.Steinberg, Vibration Analysis for Electronic Equipment, 3rd Edition, John Wiley & Sons, Inc.,
(2000)
[8] Military Standard, Environmental Test Methods, MIL-STD-810C, (1975)
[9] Military Standard, Environmental Test Methods and Engineering Guidelines, MIL-STD-810E,
(1989)
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