A 200-kg machine is to be mounted on an elastic surface of equivalent stiffness 2 ×10^5 N/m with no damping. During operation, the machine is subject to a constant force of 2000 N for 3 s. Can vibrations be eliminated without adding damping? If so, what is the maximum deflection of the machine?

Question

A 200-kg machine is to be mounted on an elastic surface of equivalent stiffness [latex]2 imes 10^{5} N/m[/latex] with no damping. During operation, the machine is subject to a constant force of 2000 N for 3 s. Can vibrations be eliminated without adding damping? If so, what is the maximum deflection of the machine?

Accepted Answer

The differential equation governing motion of the machine is

[latex] \ddot{x}+x^{2}_{n} x =F{0}[u(t)-y(t-3)][/latex] (a)

Where [latex] F_{0}=2000 N[/latex] and [latex] \omega _{n}=31.63 rad/s[/latex]. The Laplace transform of F(t) is obtained by using the second shifting theorem

[latex] L\left\{{F_{0}[y(t)-u(t-3)]}\right\} =\frac{F_{0}}{s}(1-e^{-3s})[/latex] (b)

Then from Equation (5.43) with [latex]x(0) = 0 [/latex] and [latex]\dot{x}(0)=0,[/latex]

[latex] X(s)=\frac{F_{0}}{m}L^{-1}\left\{\frac{1-e^{3_{s}}}{s(s^{2+\omega ^{2}_{n}})}\right\}[/latex] (c)

[latex] x(t)=\frac{1}{m_{eq}}L^{-1\left\{\frac{F(s)}{s^{2}+2\zeta \omega _{n}s+ \omega ^{2}_{n}} \right\} }L^{-1}\left\{\frac{(s+2\zeta \omega _{n})x(0)+\dot{x}(0)}{s^{2}+2\zeta \omega _{n}s}+\omega ^{2}_{n} \right\} [/latex] (5.43)

Partial fraction decomposition yields

[latex] X(s)=\frac{F_{0}}{m\omega ^{2}_{n}}\left(\frac{1}{s}- \frac{s}{s^{2}+\omega ^{2}_{n}} \right) (1-2^{-3t})[/latex] (d)

The second shifting theorem is used to help invert the transform

[latex] x(t)=\frac{F_{0}}{m\omega ^{2}_{n}}[1-\cos\omega _{n}t-u(t-3)(1-cos\omega _{n}(t-3))][/latex] (e)

The solution for [latex] t \gt 3[/latex] is

[latex] x(t)=\frac{F_{0}}{m\omega ^{2}_{n}}[\cos\omega _{n}(t-3)-\cos\omega _{n}t] t\gt 3s[/latex] (f)

For no steady-state motion

[latex] \cos\omega_{n}t =\cos\omega _{n}(t-3)[/latex] (g)

which is satisfied if [latex] 3\omega_{n}=2n\pi[/latex] for any positive integer n. Thus steady-state vibrations are eliminated by requiring

[latex] \omega _{n}=\frac{2n\pi}{3}= 2.09 n rad/s[/latex] (h)

For n = 15, [latex]\omega-{n} = 31.35 rad/s[/latex], which is attained if m = 203.5 kg. Thus steady-state vibrations are eliminated if 3.5 kg is rigidly added to the machine. The machine undergoes 15 cycles while the force is applied, and motion ceases when the force is removed. The maximum displacement during operation is

[latex] x_{\max}=\frac{2F_{0}}{m\omega ^{2}_{n}}=\frac{2F_{0}}{k} =0.02m[/latex] (i)

Question 5.9: A 200-kg machine is to be mounted on an elastic surface of e...

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