The mass–spring–damper system of Figure 1.6 models the suspension system of a quarter-car. Obtain a state-space model in which the output represents the body mass vertical movement y and the input represents the tyre vertical movement u(t) due to the road surface. All displacements are measured from equilibrium positions.

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The equations of motion, using Newton’s second law, may be written down for the body mass and axle/wheel mass from which a state-space model can be deduced.
Alternatively a block diagram for the system, which is more informative for modelling purposes, may be drawn up as followswhere s denotes the Laplace ‘s’ and upper case variables X, Y, Y1 denote the corresponding Laplace transforms of the corresponding lower case time domain variables x(t), [latex]y(t), y_{1}(t) ,y_{1}(t)[/latex] is the vertical displacement of the axle/wheel mass. Using basic block diagram rules this block diagram may be reduced to the input/output transfer function model
[latex]\overset{X}{ ightarrow} \frac{K_{1}(K+B_{s})}{(M_{1}s^{2}+K_{1})(M_{s}^{2}+B_{s}+K)+M_{s}^{2}(K+B_{s})}\overset{Y}{ ightarrow}[/latex]

or the time domain differential equation model

[latex]M_{1}M\frac{d^{4}y}{dt^{4}}+B(M_{1}+M)\frac{d^{3}y}{dt^{3}}(K_{1}M+KM_{1}+KM)\frac{d^{2}y}{dt^{2}}+K_{1}B\frac{dy}{dt}+K_{1}Ky=K_{1}K_{2}x+K_{1}B\frac{dx}{dt}[/latex]

A possible state space model is

[latex]\begin{bmatrix}z_{1}^{\cdot }\ z_{2}^{\cdot }\ z_{3}^{\cdot }\ z_{4}^{\cdot }\end{bmatrix}=\begin{bmatrix}-B(M_{1}+M) & 1 & 0 &0 \ \frac{-(K_{1}M+KM_{1}+KM)}{MM_{1}} & 0& 1 & 0\ \frac{-K_{1}B}{M_{1}M}& 0& 0 & 1\ \frac{-K_{1}K}{M_{1}M} & 0& 0 & 0\end{bmatrix}\begin{bmatrix}z_{1}\ z_{2}\ z_{3}\ z_{4}\end{bmatrix}+\begin{bmatrix}0\ 0\ \frac{K_{1}B}{M_{1}M}\ \frac{K_{1}K}{MM_{1}}\end{bmatrix}x(t)[/latex]

[latex]y=\begin{bmatrix}1 & 0 & 0 & 0\end{bmatrix}z(t) , z=\begin{bmatrix}z1 & z2 & z3 &z4 \end{bmatrix}^{T}[/latex]

Clearly alternative forms may be written down, such as, for example, the companion form of equation in the text. Disadvantage is that its output y is not one of the state variables.

Question : The mass–spring–damper system of Figure 1.6 models the suspe...

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