SOHEIL KHOSHPARVAR∗, LARS BYLANDER†, NICKOLAY IVCHENKO† AND GUNNAR TIBERT∗
∗KTH, Royal Institute of Technology
School of Engineering Sciences, Department of Mechanics SE-100 44 Stockholm, Sweden
e-mail: tibert@kth.se, web page: http://www.mech.kth.se
†KTH, Royal Institute of Technology
School of Electrical Engineering, Laboratory of Space and Plasma Physics SE-100 44 Stockholm, Sweden
e-mail: nickolay@kth.se, web page: http://www.alfvenlab.kth.se/plasma
Key words: Random vibration, Mile’s equation, root-mean square von Mises stress, finite element analysis.
Summary. Two methods for the stress analysis of a spacecraft structure under random vi-bration loading are studied and applied to a wire boom deployment structure for the future BepiColombo mission to Mercury to qualify the structure for launch.
1 INTRODUCTION
A spacecraft is typically subjected to the largest loads during launch and every part on the spacecraft must be able to sustain the random vibration launch loads without permanent deformation or failure. MEFISTO-S is a wire boom deployment system with an electric field measurement unit that will be used in one of the spacecraft of the BepiColombo mission. The qualification of each part is typically done by computing the stress factors of safety against the yield and ultimate strengths. In a structural stress analysis, the von Mises stress criterion plays a major role. In most cases, static or dynamic, when the input forces are deterministic, calculating the von Mises stress is a straightforward process. In cases when the forces are non-deterministic or, in another words, when the structure is under random or stochastic input conditions, the response parameters will also be in a statistical and probabilistic format. In random vibration analysis, the input is often given as a power spectral density (PSD). The linear method which is used to evaluate the root-mean-square (rms) acceleration, displacement and stress components cannot be directly used to derive the rms von Mises stress since it is a non-linear function of the linear stress components. The aim in this study1 is to compare the current methods available to engineers when dealing with random vibration stress levels with the MEFISTO-S structure as example.
Soheil Khoshparvar, Lars Bylander, Nickolay Ivchenko and Gunnar Tibert
2 MILE’S VERSUS RMS VON MISES STRESS
The most common method when dealing with the stress levels caused by random vibration on a spacecraft structure is to use Mile’s equation2. In that method, one chooses a single eigenfre-quency of the structure, typically the dominant one, which relates to the highest modal effective mass within the bandwidth of the imposed input frequencies. With the use of modal damping, the PSD of the input acceleration and Mile’s equation, the rms of the response acceleration is computed as
¨ xrms=
s fDπ
4ξ APSD(fD) (1)
where APSD is the PSD of the acceleration, fD is the dominant eigenfrequency and ξ is the damping ratio. This rms response acceleration is multiplied by a factor of 3 and applied to the structure as an equivalent static gravity field. Applying a field with a gravity acceleration of 3¨xrms to the structure results in a von Mises stress that is considered to cover the highest response peaks for 99.73% of the duration of the imposed random vibration disturbance.
Mile’s equation is very simple to implement, but it has some limitations that are important to consider. The equation has been derived for a single-degree-of-freedom system, so therefore it cannot consider all the eigenfrequencies within the input bandwidth for a multi-degree-of-freedom system. There is also the assumption that the input PSD is of white noise nature, so more complex inputs with changing octaves PSDs might produce errors.
The major simplification in Mile’s equation is that it is not considering the rms von Mises stress, but assumes that the rms static acceleration is causing the stresses in the structure. The most direct method to accurately estimate the rms von Mises stresses is to evaluate all the stress components in each time step and the rms value is then derived from time integration for each location in the model, but this method would computationally be very expensive to process.
Segalman et al.3 have developed an efficient method to evaluate the rms von Mises stress for structures subjected to random vibration. In this method, the rms von Mises stress is evaluated as a summation of modal stress eigenvectors and the modal cross-covariance which itself consists of the modal transfer functions and input PSDs. The rms von Mises stress is3:
σrms von Mises= s
X
i,j
ΓijTij (2)
where Tij presents the modal stress eigenvectors contribution3:
Tij = ψσTi Aψσj (3)
and A is the von Mises stress coefficients matrix. Γij in Eq. (2) is the modal cross-covariance3:
Γij =
Nf
X
a Nf
X
b
φaiφbj
1 π
Z ∞ 0
Re(Di(ω)Dj(ω)[Sf f(ω)]ab)dω
(4) where ψ is the modal stress eigenvector, φ is the displacement eigenvector, D is the modal transfer function and Sf f is the cross-spectral density matrix of the input random vibration.
Nf represents the number of forces that are imposed on the structure.
Soheil Khoshparvar, Lars Bylander, Nickolay Ivchenko and Gunnar Tibert
When implementing this method, the von Mises stress and T are to be computed for every integration point. Γ is computed only once, as it depends on the modal properties of the structure. In practice, Γ depends on the number of truncated modes, modal damping ratios and the input PSDs, so the integral in Eq. (4) can be approximated by summation over the bandwidth frequency3:
Γij =
Nf
X
a Nf
X
b
φaiφbj
Nω
X
n=1
Re(Di(ωn)Dj(ωn)[Sf f(ωn)]ab)∆ω π
!
(5)
In deriving the rms von Mises stress method3, an assumption is that all the loads are stationary with a zero mean value, which in practice might not be the case. Nevertheless, the values obtains from rms von Mises stress method are found to be conservative and accurate. The advantage of the rms von Mises stress method in comparison with Mile’s equation is that the former includes all the truncated modes and mode shapes in the bandwidth of the excitation.
3 QUALIFICATION OF THE MEFISTO-S STRUCTURE
To test the accuracy of Mile’s equation in comparison with the rms von Mises method, both methods have been applied to the MEFISTO-S structure. Figure 1 shows the maximum rms von Mises stress, with a probability of 99.73%, as a function of the number of eigenmodes included in Eq. (5). The first seven modes are the eigenfrequencies of the structure which are in the input bandwidth. The rule of thumb in calculating the rms von Mises stress is to truncate the modes of the structure with one and half times of the actual bandwidth upper bound frequency3. For each mode, the von Mises stress by Mile’s equation has been computed and the value of the von Mises stress achieved with the same probability for each mode has been recorded. As seen in Fig.
1, the Mile’s stress value for the dominant mode of the structure (fourth mode) is conservative compared to the maximum rms von Mises stress.
Figure 1: Comparison between Mile’s equation and and the rms von Mises stress method.
Soheil Khoshparvar, Lars Bylander, Nickolay Ivchenko and Gunnar Tibert
The comparison also shows that the stress by Mile’s method is only close to the rms von Mises method when the deformed shape caused by the gravity field is similar to the dominant mode shape which is used in Eq. (1). In the case of the MEFISTO-S structure, the fifth eigenmode, which is not the dominant mode, would have given a more accurate result by Mile’s equation, as the deformed shape in that mode is similar to the fifth mode shape of the structure.
4 CONCLUSIONS
The stress analysis of the MEFISTO-S structure shows that Mile’s equation presents higher stresses than the rms von Mises stress method and might therefore be a less good method from a design point of view, due to the strict mass limits in the aerospace industry. Mile’s equation is producing good results only if the deformed shape caused by the static gravity field is similar to the dominant mode shape. For other cases, Mile’s equation can give either unsafe or too conservative results.
Figure 2: Von Mises stresses (MPa) in the MEFISTO-S structure under static gravity field load (fourth eigenmode).
REFERENCES
[1] Khoshparvar, S. Random Vibration Stress Analysis of the BepiColombo Wire Boom Deploy-ment System (MSc thesis, KTH, 2010).
[2] Wijker, J. Random Vibrations in Spacecraft Structures Design: Theory and Applications (Springer, 2009).
[3] Segalman, D. J., Fulcher, C. W. G., Resse, G. M. & Field, Jr., R. V. An efficient method for calculating RMS von Mises stress in a random vibration environment. Journal of Sound and Vibration 230, 393–410 (2000).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010