Question

Figure 1 shows a system with two masses of which the origins are set up for the springs of \(k_{1}, k_{2}\), and \(k_{3}\) to be unstretched. The system is excited by the base motion of \(x_{b}(t)\).

(a) By drawing the FBDs of the two masses and applying the Newton's \(2^{n d}\) Law of Motion, find a matrix equation of motion.

(b) Find the undamped, natural frequencies and the corresponding mode shapes of the system for the given system parameters of \(k_{1}=k_{2}=k_{3}=100 \mathrm{kN} / \mathrm{m}\) and \(\mathrm{m}_{1}=3 m_{2}=12 \mathrm{~kg}\).

(c) Normalize the mode shapes with respect to the mass matrix.

(d) Using the light damping approximation, state the modal differential equations (including the definition of modal forces). The damping coefficients are given as \(c_{1}=c_{3}=20 \mathrm{~N} \mathrm{sec} / \mathrm{m}\) and \(c_{2}=25 \mathrm{~N} \mathrm{sec} / \mathrm{m}\)

(e) Using the modal solution approach and the light damping approximation, determine the steadystate vibration responses of the two bodies with the base motion of \(x_{b}(t)=A \sin (\omega t)\) where \(A=0.1 \mathrm{~m}\) and \(\omega=120 \mathrm{rad} / \mathrm{sec} .\) Plot the responses from 0 to \(0.2 \mathrm{sec}\)

Answer 1

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