Lab 3: Model-based design

Part 1: Quarter-car suspension model

The vertical movement of a car's suspension system can be modelled by a so-called quarter-car model. The car is modeled with a mass $m_c$ which follows the ground through a wheel with a shock absorber. The shock absorber consists of a spring and a damper. The wheel follows the surface of the ground, which can be modelled as an external disturbance.

In this part you will first identify the system and implement a Simulink model that represents the dynamics of (a quarter of) the car. The figures below show the mass riding over the ground (left) and a model of the car with its mass, spring and damper (right).

Assume that the empty car weighs ($1000 + 100\cdot p_1$) kg and is at equilibrium when $x=0$, and that the damper and spring of the shock absorber are linear. The force of the spring is then $F_s = K_s \cdot x(t)$, with $K_s=2500$ N/m and the force of the damper is $F_d= C_s \cdot v(t)$, with initially $C_s=250$ Ns/m.

Remember that $\dot x(t) = v(t)$, $\dot v(t) = a(t)$. Using Newton's second law of motion you get that at equilibrium the following condition: \begin{equation} F_{gravity} + F_s + F_d = \frac{m_{car}}{4} \cdot a(t) + K_s \cdot x(t) + C_s \cdot v(t) = 0 \end{equation}

You can rewrite this equation so that it is easily defined in Matlab Simulink: \begin{equation} a(t) = \frac{-K_s \cdot x(t) -C_s \cdot v(t)}{m_{car}/4} \end{equation}

This system can be implemented by creating the following Simulink system:

The right-most integrator has an initial condition of $x=0.2$ which models an external excitation. The other integrators have initial condition $v=a=0$.

You will now first investigate a system and corresponding Simulink model that describes the motion of the body of the car. Run the model by opening mbd_exercise1_sl_passive.slx in Matlab, and clicking the 'play' icon (green circle with a triangle inside).

Open the scope by double clicking on it. You should see a waveform that starts at a displacement of 0.2 meter.

The parameters of this model are determined in the Model Workspace. You can access this through View\Model Explorer\Model Explorer (or press CTRL+H); then select the Model Workspace in the Model Hierarchy. In this view, you can change the parameter values for $c,k,m_{car}$ and the initial displacement.

Question: What is the maximum absolute velocity for the body of the car when the displacement is increased to 0.3 meters? At (roughly) what time does the maximum velocity occur?

You will now tune the value of the damper such that the car stops moving as soon as possible, without showing any overshoot and oscillatory behaviour. The passive system can be either underdamped, critically damped, or overdamped. The type of the response depends on the roots of the following auxiliary equation: \[ \frac{m_{car}}{4}\cdot r^2 + C_s \cdot r + K_s = 0 \]

Question: Critical damping is achieved when there is exactly one real-valued root for this equation. Derive the expression for critical damping of this system. (Hint: use the quadratic formula.)

Question: Calculate the value of the damper $C_s$ for which the car is critically damped.

Modelling sprung and unsprung mass of a quarter-car model

In this exercise you will use Matlab Simscape to model the mechanical system of the previous part in a different way.

In Simscape, the mechanical elements are modelled more directly, and the differential equations are automatically derived. The Simscape signals have a physical interpretation associated with them, unlike the Simulink signals. Signals with translational movements are drawn with green lines, quantities such as position and velocity are drawn with brown lines, and electrical quantities are drawn with blue lines. Normal scopes can be used by converting the physical signals back to Simulink signals. The model of part 1 for use in Simscape can be found in mbd_exercise2a_ss_sprung_mass.slx:

Question: Compare the signal of the scope with the results of the previous question (using the same $k$, $c$, $m_{car}$ as before). What differences do you see?

Question: Where is the 0.2 meter initial excitation modelled in the given Simscape model?

The figure belows shows a mechanical diagram for the sprung and unsprung mass of a quarter car.

Question: Modify the Simscape model to include the mass of one wheel (7 kg), and its deformation (a spring of $K_t$=150 kN between the wheel mass and the road surface). The system should correspond to the diagram in figure shown above.

Question: Measure the position of the wheel mass and plot it in the same scope as the position of the car's body with respect to the road.

Question: Is the impact of the unsprung mass significant?

Instead of using an excitation, apply an external force upon the body of the car. The force should be equivalent to that of a person of $10\cdot (p_2+1)$ kg entering and exiting the car. Assume the weight of the person is distributed evenly over all four wheels.

Question: Remove the excitation of 0.2 meters from the Simscapes model. Add a pulse generator that applies the force of the entering/exiting mass car's body every 30 seconds. Check the position of the wheel and body of the car in the modified model.

Question

Part 3: Measuring with a strain gauge

In this exercise, you will investigate a model of a strain gauge circuit; the sensor that was assumed in Simulink was ideal.

You will combine analysis with modelling to gain understanding in some properties of the strain gauge circuit. You will use Simscape to simulate a strain gauge circuit; such a circuit can be used to connect to a Simscape model that captures the mechanical aspect of the system. A basic setup of the Simscape model has been provided in mbd_exercise3_ss_strain.slx.

The strain gauge is combined with three fixed resistors in a bridge circuit (see figure below) with a voltage supply $V_r$. The fixed resistors $R_3$ and $R_4$ have a resistance of $R_0=1000~\Ohm$, and the resistor $R_2$ has a resistance of $k\cdot R_0$. The maximal current through the strain gauge should be limited to 10 mA. The electrical equivalent circuit of this sensor is shown in the figure below.

Question: Show that the output voltage $v_s$ of the sensor circuit is equal to: \[ v_s = \frac{k-x-1}{(2+x)(1+k)}V_r \]

Question: Finish the model from mbd_exercise3_ss_strain.slx to incorporate the whole bridge and so that you are able to view the resulting output voltage. If the input signal (which would normally break the strain gauge) is amplified, you can show the non-linear behaviour of the strain gauge circuit. Excite the strain gauge with a sinusoidal source with amplitude 10, then plot and check the resulting waveform. Use that $k=p_2$.

Question: Show that the theoretical maximum strain $x$ that can be represented by $v_s$ is in the range $[-1,+\inf)$, and find the associated $v_s$. (Hint: the resistance of the strain gauge cannot become negative)