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For a spring-mass system with base excitation, derive the following expression for the relative motion z(t) = x(t) - y(t), for an undamped system:
x(t) = 1/ωn∫ot ysinωn(t-τ)dτ
Now, assume that the base motion is specified by a velocity pulse of the base given by:
v(t) = (vo - vo(t/to)) = vo(U(t) - t/to) where U(t) is the Unit Step Function
Consider that the velocity of the base has a jump from zero to v0 instantly at t=0.
Derive an expression for the acceleration for the base motion from the velocity equation above
Derive an expression for z(t) using equation and the acceleration term you have calculated for the base
If the peak amplitude of z occurs at a t < t1, derive an expression for the maximum amplitude z.
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Consider just the vertical (y) response of the system when the beam is being stretched. Show that the spring constant, ka, is approximately 70.6 mN/m. What is the steady state amplitude of the vertical response? See Figure A below for an engine..
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