End-to-end statistical models for predicting maximum expected
vibro-acoustic response levels under aero-acoustic loading
Paul G. Bremner1
1
AeroHydroPLUS,
PO Box 883, Del Mar, CA 92014, United States of America
Email: [email protected]
In the space industry, it is necessary to “qualify” all equipment that will fly on a launch vehicle by
vibration testing to maximum expected vibration levels. Piersol [1] has shown that there is considerable
uncertainty in flight random vibration levels. An ensemble of measurements from multiple similar launch
vehicles showed the vibration levels are approximately log normally distributed with a relative variance
of approximately unity, as summarized in Figure 1 below. MIL-SPEC 1540 and NASA HBDBK 7005
recommend the use of (Log) Normal Tolerance Limits to define the maximum expected vibration test
specification. This paper addresses the question of how to estimate maximum expected vibration levels
when using a physics-based model to design the launch vehicle.
1.E+01
Relative Variance
1.E+00
1.E-01
1.E-02
1.E-03
10
100
Frequency (Hz)
1000
Figure 1 Ensemble of flight random vibration measurements (left) and relative variance of the ensemble (right)
The simulation of linear vibration response under lift-off and ascent aeroacoustic loading, involves the
surface pressure loading spectrum and its spatial correlation, and the modal dynamics of the loaded
structure. Aeroacoustic loading is often temporally random and only partially space-correlated. Since
random loading can only be described statistically, the structure vibration response can only be predicted
statistically.
However, it is also common that the modal dynamics of the structure are quasi-random (eg. due to
uncertainties in the as-built boundary conditions) particularly at higher frequencies. In this situation,
design for the maximum expected vibration or stress or strain response requires an “end-to-end” statistical
model that correctly combines randomness in the loading with uncertainty in the structure dynamics. To
this end, it is necessary to extend existing models to predict not only the statistical mean but also the
statistical variance and probability density function for the end-to-end prediction process.
This paper will show that the total variance is a function of at least three statistically independent random
variables – the fluctuating surface pressure spectrum ps2 , the modal power acceptance of the local
structure modes jr2 Re  M s  and the modal damping Qr  1 r . Under reasonable assumptions, it will
be shown that the total relative variance of vibration response r 2 vs2  can be estimated as:
r 2 v 2 
r 2  ps2   r 2 Q  r 2  j2 Re M  
(1.1)
Sample data ensembles – available in the open literature [2][3] – are used to show that this statistical
model appears to be consistent with measured flight vibration variance, as shown in Figure 2 below.
1.E+01
2

r 2 vFlight
test 
Relative Variance
1.E+00
1.E-01
r 2 Q 
r 2  j 2 Re M  
1.E-02
r 2  ps2 
1.E-03
10
100
Frequency (Hz)
1000
Figure 2 Comparison of component vibroacoustic model relative variances with total relative variance of flight vibration
ensemble.
References
[1] Piersol, A. “Maximum Expected Environments”, Chapter 6, NASA HDBK-7005
[2] Bremner, P,, Blelloch, P., Streett, C. et al. “Validation of Methods to Predict Vibration of a Panel in
the Near Field of a Hot Supersonic Rocket Plume”, AIAA-2011-2852 Proc. AIAA Aeroacoustics
Conf., Portland OR, May 2011
[3] Cotoni, V., R.S. Langley, and M.R.F. Kidner. “Numerical and Experimental Validation of Variance
Prediction in the Statistical Energy Analysis of Built-Up Systems.” Journal of Sound and Vibration
288 (2005): 701–728.