Quarter-car suspension models are used to study the dynamics of a vehicle’s suspension. The model consists of: the wheel, the suspension system (damper and coil) and a quarter of the vehicle’s body mass.

The modeling approach is to use lumped parameters, which means that a quarter of the vehicle body and the wheel are concentrated in distinct single mass parameters. The stiffness and the damping of the suspension and of the wheel are also concentrated in lumped parameters.

where:

m_{1} [kg] – the mass of a quarter of the vehicle body

m_{2} [kg] – the mass of the wheel and suspension

k_{1} [N/m] – spring constant (stiffness) of the suspension system

c_{1} [Ns/m] – damping constant of the suspension system

k_{2} [N/m] – spring constant (stiffness) of the wheel and tire

c_{2} [Ns/m] – damping constant of the wheel and tire

z_{1} [m] – displacement of the vehicle body (output)

z_{2} [m] – displacement of the wheel (output)

u [m] – road profile change (input)

The road profile is considered to be the input into the system. The purpose of the study is to analyze the system’s response (outputs) for a step input of the road profile, which can be regarded as the wheel going above a steep rigid object on the road (rock, brick, etc.)

Before writing down the equations, we need to draw the free body diagram (FBD) of the system. From this we will deduce the force equilibrium equations for both masses. Since there are two coupled masses, the dynamic system has **two degrees of freedom**.

where:

F_{e1} [N] – the elastic force of the suspension system

F_{d1} [N] – the damping force of the suspension system

F_{i1} [N] – the inertial force of the vehicle body

F_{e1} [N] – the elastic force of the wheel and tire

F_{d1} [N] – the damping force of the wheel and tire

F_{i1} [N] – the inertial force of the wheel and tire

The directions of the forces are set according to the following reasoning. Imagine that the road has a bump and the wheel rolls over it. The profile change of the road will cause the tire to compress. This will generate an elastic force *F _{e2}* and a damping force

*F*which will push the wheel up. Due to inertia, the wheel will try to resist the position change through the inertial force

_{d2}*F*. The elastic and damping forces are also acting on the road but this is not in the interest of our study.

_{i2}The suspension system will oppose the upward movement of the wheel and it will push on it with the elastic force *F _{e1}* and a damping force

*F*. According to Newton’s third law of motion, for each action there is a reaction. Therefore the elastic force

_{d1}*F*and the damping force

_{e1}*F*will push the vehicle body upwards. Due to inertia, the body mass will try to resist the position change through the inertial force

_{d1}*F*.

_{i1}The equation of force equilibrium for the **body mass** is:

The inertial force of the body, and the elastic and damping forces of the suspension system are expressed as:

\[ \begin{split}F_{e1} &= k_1 (z_2 – z_1)\\

F_{d1} &= c_1 \left ( \frac{dz_2}{dt} -\frac{dz_1}{dt} \right )\\

F_{i1} &= m_1 \frac{d^2 z_1}{dt^2}

\end{split} \]

Replacing the expressions of the forces in equation (1), we get:

\[\bbox[#FFFF9D]{m_1 \frac{d^2 z_1}{dt^2} =k_1 (z_2 – z_1) +c_1 \left ( \frac{dz_2}{dt} -\frac{dz_1}{dt} \right )} \tag{2}\]The equation of force equilibrium for the **wheel mass** is:

The inertial force of the wheel, and the elastic and damping forces of the wheel system are expressed as:

\[ \begin{split}F_{e2} &= k_2 (u – z_2)\\

F_{d2} &= c_2 \left ( \frac{du}{dt} -\frac{dz_2}{dt} \right )\\

F_{i2} &= m_2 \frac{d^2 z_2}{dt^2}

\end{split} \]

Replacing the expressions of the forces in equation (3), we get:

\[\bbox[#FFFF9D]{m_2 \frac{d^2 z_2}{dt^2} =k_2 (u – z_2) +c_2 \left ( \frac{du}{dt} -\frac{dz_2}{dt} \right ) – k_1 (z_2 – z_1) – c_1 \left ( \frac{dz_2}{dt} -\frac{dz_1}{dt} \right )} \tag{4}\]The **ordinary differential equations (ODE)** (2) and (4) describe the dynamic behavior of the quarter-car model.

Before integration in Xcos, we need to keep on the left-hand side of the equal sign only the second derivatives of the displacements. This will give:

\[ \begin{split}\frac{d^2 z_1}{dt^2} &= \frac{1}{m_1} \left ( k_1 (z_2 – z_1) +c_1 \left ( \frac{dz_2}{dt} -\frac{dz_1}{dt} \right ) \right )\\

\frac{d^2 z_2}{dt^2} &= \frac{1}{m_2} \left ( k_2 (u – z_2) +c_2 \left ( \frac{du}{dt} -\frac{dz_2}{dt} \right ) – k_1 (z_2 – z_1) – c_1 \left ( \frac{dz_2}{dt} -\frac{dz_1}{dt} \right ) \right )

\end{split} \]

Before building the Xcos block diagram model, we need to load the parameters of the system in the Scilab workspace or in the `Set Context`

environment from Xcos.

`m1 = 290;`

`m2 = 60;`

`k1 = 16200;`

`k2 = 191000;`

`c1 = 1000;`

`c2 = 2500;`

Based on the differential equations above we build the Xcos block diagram model:

The road input `u`

is modeled as a step input, with a rising edge and falling edge. `v1`

is the vertical translational speed of the body of the vehicle and `v2`

is the vertical translational speed of the wheel.

All the initial conditions of the integrators are set to zero. The simulation is set to run for `8`

seconds.

The results of the simulations are saved in the Scilab workspace, under the variable `simOut`

. To plot the outputs we use the Scilab script:

// Read simulation outputs t = simOut.time; u = simOut.values(:,1); z1 = simOut.values(:,2); z2 = simOut.values(:,3); v1 = simOut.values(:,4); v2 = simOut.values(:,5); // Plot results subplot(2,1,1) plot(t,u,'g') plot(t,z1,'b') plot(t,z2,'r') xgrid() ylabel('Displacement [m]') legend('u','z1','z2',1) subplot(2,1,2) plot(t,v1,'b') plot(t,v2,'r') xgrid() xlabel('Time [s]') ylabel('Speed [m/s]') legend('v1','v2',1)

After running the post-processing Scilab script we get the following graphical window:

At time `t = 1 s`

the road profile `u`

(input) has a step change (rising edge) of `0.1 m`

. The wheel will be the first to move, with high speed and significant damping. The vehicle body will follow, with smaller speed but higher amplitude and several oscillations. After `4 s`

there is another step input (falling edge) of the road profile.

The results show a good approximation of the behavior of a real suspension system. By changing the road input signal and wheel and suspension parameters we can evaluate the dynamic behavior of the system for different road disturbances and suspension settings.

For any questions, observations and queries regarding this article, use the comment form below.

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## Farhan

can i get that scilab file?

## roynugrah

can i get that scilab file?? i have make the same scilab likes you for my practice, but at the end i cant get my graph, and the graph that appear is an empty graph

## Anthony Stark

Make sure that the simulation time is at least 8s. You can also check if you have the variables saved in the workspace. Double-click on them and you should see some values.