Equivalence Random and Sinusoidal Vibration

Abstract

In general, the sinusoidal vibration (sweep) test is in the frequency range between 5 and 100 Hz and the random vibration test has a frequency range between 20–2000 Hz. Therefore, it is not so straightforward to replace a random vibration test by an equivalent sinusoidal vibration test or vice versa. Damaging and fatigue aspects have to be considered transferring a specific sinusoidal specification into equivalent random vibration specification. Examples and problems are provided.

Problems

9.1

Prove that $W_{\ddot{U}}=\frac{A_{sine}^2}{\pi f_n}$.

Hint: Assume $Q=1$.

Table 9.1 Random vibration specification

9.2

This problem is based on an early version of the Soyuz L/V manual, Issue 1, revision 0, June 2006, Sect. 4.3.3.3, [6]. The verification of the S/C structure compliance with the random vibration environment in the 20–100 Hz frequency range shall be performed with one of the three following methodologies:

• 1. Perform a dedicated random vibration test.

• 2. Conduct the sine vibration qualification test up to 100 Hz and apply input levels high enough to cover the random vibration environment (equivalency obtained with Miles’ equation)

  $$\begin{aligned} G_{rms}=\sqrt{\frac{\pi }{2} f_n Q W_{\ddot{U}}(f_n)}. \end{aligned}$$

• 3. Conduct the sine vibration qualification test up to 100 Hz so as to restitute the structural transfer function and demonstrate the compliance of the S/C secondary structure with the random vibration environment by analysis.

Above 100 Hz, S/C qualification with respect to random vibration environment is obtained through the acoustic vibration test.

In Fig. 9.2 is shown that an scientific instrument with mass $m_s$ and stiffness $k_s$ is mounted on top of the S/C with masses $m_1,m_2$ (kg) and spring stiffnesses $k_1, k_2$ (N/m). The instrument mass $m_s=20$ kg and the lowest natural frequency is $f_s=125$ Hz. Define the spring stiffness $k_s$.

Fig. 9.2

Instrument mounted on top of S/C

Fig. 9.3

Frequency response (transfer) functions S/C

The spring stiffness $k_1=10^7$ N/m and spring stiffness $k_2=2k_1$ N/m. The mass $m_1=100$ kg and mass $m_2=2m_1$ kg and the damping ratio $\zeta =5$%.

• 1. Perform the random response analysis with the random enforced vibration levels (20–2000 Hz) as provided in Table 9.1 using your favorite FEA software package.

  • Calculate the undamped natural frequencies $f_n$ (Hz) and associated modal effective masses $M_{eff}$ (kg).

  • Calculate r.m.s. acceleration of the masses $m_s, m_1, m_2$, $\ddot{x}_{s,rms},\ddot{x}_{1,rms}, \ddot{x}_{2,rms}$, Number of frequency steps $N=2000$.

  • Discuss why the first vibration mode is most important when a base excitation response analysis is performed?

• 2. Calculate the equivalent sine excitation amplitude $A_{sine}$ (g) (5–100Hz) using the following expression:

  $$ A_{sine}=\sqrt{2} \frac{G_{rms}}{Q}~ \text {g}. $$

  Derive this equation and why do we use the $G_{rms}=\ddot{x}_{s,rms}$.

  Perform a sinusoidal enforced acceleration with $A_{sine}$ as amplitude and define maximum acceleration response $\ddot{x}_{s,rms}$ of mass $m_s$ and associated excitation frequency. Discuss if we meet the requirement of the sinusoidal responses cover the random responses.

• 3. Calculate the frequency response (transfer) functions of the S/C between 5–100 Hz. (Fig. 9.3), Number of frequency steps $N=200$. After that compute with the aid of Miles’equation the r.m.s. acceleration of the instrument $m_s$, using the natural frequency $f_n$ of the fundamental mode and associated frequency transfer function $|H(f_n)|$ and PSD $W_{\ddot{U}}(f_n)$. Discuss this result compared to the methodologies 1 and 2.

Answers: $k_s=1.2337e+07$ N/m.

• 1. $f_n= (33.24, 68.63, 138.81)$ Hz, $M_{eff}=277.56, 42.4, 0.01$ kg. $(\ddot{x}_{s,rms},\ddot{x}_{1,rms}, \ddot{x}_{2,rms})=3.68, 3.27, 2.16$ g.

• 2. $A_{sine}=0.52$ g. $\ddot{x}_{s,max}=7.07$ g, $f_{max}=32.87$ Hz.

• 3. $\ddot{x}_{s,rms}=4.81$ g.