Figure $8$: Tire Deflection with Softer Suspension Spring

Figure $9$: Suspension Deflection ComparisonI. QUARTER CAR SUSPENSION MODEL:

Figure $1$ shows the conventional quarter car

suspension model where all coefficients are

assumed $tobe$ linear. Equations $of$ motion for

the conventional quarter car suspension model

can $be$ written $as$:

$m_z{z}_z=-k_z(z_s-z_u)-c_p(z_z^{}-z_u^{})+f_a-f_d$

$m_u{z}_u^{}=k_z(z_s-z_u)+c_p(z_s^{}-z_u^{})+k_t(z_u-z_r)-f_a$

where the state variables are defined $as$:

$x_1=z_s-z_u$

$x_2=z_s^{}$

$x_3=z_u-z_r$

$x_4=z_u^{}$

Figure $1$ shows the conventional quarter car

suspension model where all coefficients are

assumed to be linear. Equations of motion for

the conventional quarter car suspension model

can be written as:

$m_s{z}_z^{}=-k_s(z_z-z_w)-c_p(z_z^{}-z_u^{})+f_a-f_d$

$m_u{z}_u^{}=k_z(z_s-z_w)+c_p(z_s^{}-z_u^{})+k_t(z_w-z_q)-f_a$

where the state variables are defined as:

$x_1=z_s-z_u$

$x_2=z_s^{}$

$x_3=z_u-z_r$

$x_4=z_u$

And the state equation becomes:

$x^{}=Ax+B_1{f}_a+B_2{z}_r+B_3{f}_d$

The state matrices are given as:

$A=\left[\begin{array}{cccc}0& 1& 0& -1\\ -\frac{k_z}{m_z}& -\frac{c_p}{m_z}& 0& \frac{c_p}{m_z}\\ 0& 0& 0& 1\\ \frac{k_z}{m_z}& \frac{c_p}{m_u}& -\frac{k_t}{m_u}& -\frac{c_p}{m_u}\end{array}\right]$

$fd$ is a disturbance force acting on the sprung mass and will be taken as zero.

The quarter car suspension is modeled in Simulink using the above state space equations. The

Simulink model is shown in Figure $3.$