## Related questions with answers

A spring-mass-damper system, as shown in Figure , is used as a shock absorber for a large high-performance motorcycle. The original parameters selected are $m=1 \mathrm{~kg}, b=9 \mathrm{~N} \mathrm{~s} / \mathrm{m}$, and $k=20 \mathrm{~N} / \mathrm{m}$. (a) Determine the system matrix, the characteristic roots, and the transition matrix $\boldsymbol{\Phi}(t)$. The harsh initial conditions are assumed to be $y(0)=1$ and $d y /\left.d t\right|_{t=0}=2$. (b) Plot the response of $y(t)$ and $y(t)$ for the first two seconds. (c) Redesign the shock absorber by changing the spring constant and the damping constant in order to reduce the effect of a high rate of acceleration force $\dot{y}(t)$ on the rider. The mass must remain constant at $1 \mathrm{~kg}$.

Solution

VerifiedThree transfer functions are given to us under **a)**, **b)** and **c)**. We are instructed to compute a state variable representation of the transfer function using the *m-file* $ss$ function.

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