Notching and Mass Participation

Abstract

The mass participation is one of the approaches for notching analysis, in which the modal effective mass and apparent mass in conjunction with Miles’ equation are the basic elements to determine notched random acceleration input comparing $3 \sigma$ reaction loads with the reaction loads caused by the quasi-static design loads (QSL). The quasi-static design limit load applied for the design of equipment and instruments is mostly based on experience from previous spacecraft projects and is defined, in general, using the mass acceleration curve (MAC). Procedures to calculate random load factors and the estimation of the depth of the notch (modification of input spectrum) using the mass participation are discussed. Examples and problems with answers are given.

Problems

5.1

A massless simply supported circular plate with bending stiffness D supports at the center a mass-spring system with mass M and spring stiffness k. Poisson’s ratio is $\nu =0.3$. The radius of the circular plate is $r=a$. The system is shown in Fig. 5.5. The simplified system is illustrated in Fig. 5.6. The deflection in the center of the circular plate due to an unit load $F=1$ is given by [7]

$$\begin{aligned} \delta =\frac{(3+\nu ) a^2}{16 \pi (1+ \nu ) D}. \end{aligned}$$

(5.22)

• Calculate the lowest natural frequency $f_n$ (Hz) of the complete system, when $k=1/\delta$.

• The simply supported plate is excited by a random enforced acceleration $\ddot{U}$ (PSD $W_{\ddot{U}}$). The SDOF system had been qualified against $\ddot{\gamma }$ QSL. Describe the procedure to define a notch.

Answers: $f_n=\frac{1}{2 \pi } \sqrt{\frac{1}{2 \delta M}}$ Hz.

Fig. 5.5

Simply supported circular plate and SDOF element

Fig. 5.6

Simplified system

5.2

This problem is the continuation of Problem 5.1. The simply supported plate has a mass per unit of area $m_p$. Use Dunkerley’s method to calculate the lowest natural frequency $f_n$ of the dynamic system presented in Fig. 5.5. The Dunkerley’s systems are shown in Fig. 5.7. The deflection w(r) can be written as [8]

$$\begin{aligned} w(r,t)=\eta (t) \varPhi (r)=\eta (t) \left[ (4+\nu ) a^3-3(2+\nu ) a r^2 +2(1+\nu )r^2 \right] \end{aligned}$$

(5.23)

The natural frequency $\omega _{11}^2$ can be obtained using Rayleigh’s quotient. The natural frequency $\omega _{21}^2$ can calculated with the displacement method with $F=1$ ($k=1/\delta$). Describe the procedure to define a notch when the panel will be overloaded at $r=0$.

Answers: $1/\omega ^2_n=1/\omega ^2_{11}+1/\omega ^2_{21}$, $\omega ^2_{11}=24.4491D/a^4m_p$, $\omega ^2_{21}=9.9008D/a^2M$.

Fig. 5.7

Dunkerley systems

Fig. 5.8

3 SDOF system

5.3

A 3 DOF system is shown in Fig. 5.8. This dynamical system is excited by a random enforced acceleration $\ddot{U}$. The spectrum is given in Table 5.1. The discrete masses are $m_1=200, m_2=100, m_3=50$ kg, and the stiffness of the springs are $k_1=3\times 10^8,k_2=2\times 10^8,k_3=1\times 10^8$ N/m, respectively. The modal damping ratio $\zeta =0.05$. The static design load factor $\ddot{\gamma }_{stat}=25$ g. Investigate if a modification (notches) of the random input spectrum is needed.

• Calculate the natural frequencies $f_k$ and associated modal effective masses $M_{eff}$.

• Calculate random load factor $\ddot{\gamma }=3F_{rms}/M_{tot}$(g) (Eq. 5.3).

• Define $W_{\ddot{U}}(f_k)$ (g).

• Calculate the factor $\bar{A}$.

• Calculate $W^*_{\ddot{U}}(f_k)\le W_{\ddot{U}}(f_k)$ (g).

• Define $A_{notch}$ (g$^2$).

• Calculate $G_{rms}$ of modified input spectrum.

• Calculate new random load factor $\ddot{\gamma }$.

Answers: $f_k=119.95, 239.34, 343.95$ Hz, $M_{eff}=297.56, 47.30, 5.14$ kg, $\ddot{\gamma }=48.16$ g, $W_{\ddot{U}}(f_k)=0.18, 0.18, 0.18$ g, $\bar{A}=616.77$, $W^*_{\ddot{U}}(f_k)=0.0373, 0.18, 0.18$ g, $A_{notch}=17.61, 0, 0$ g$^2$, $G_{rms}=12.33$ g, $\ddot{\gamma } =23.88$ g.