Question 5.18: A Single-Degree-of-Freedom Vehicle Model The mass–spring–dam...
A Single-Degree-of-Freedom Vehicle Model
The mass-spring-damper system shown in Figure 5.101 represents a vehicle traveling on a rough road. Assume that the surface of the road can be approximated as a sine wave z=Z_{0} \sin (\omega t), where Z_{0}=0.01 \mathrm{~m} and \omega=3.5 \mathrm{rad} / \mathrm{s}. The mathematical model of the system is given by an ordinary differential equation
m \ddot{x}+b \dot{x}+k x=b \dot{z}+k z
where m=3000 \mathrm{~kg}, b=2000 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}, and k=50 \mathrm{kN} / \mathrm{m}.
a. Build a Simulink model of the system based on the mathematical representation and find the displacement output x(t).
b. Convert the ordinary differential equation to a transfer function and repeat Part (a). Assume zero initial conditions.
c. Build a Simscape model of the physical system and find the displacement output x(t).
a. Solving for the highest derivative of the output x gives
\ddot{x}=\frac{1}{m}(k z+b \dot{z}-k x-b \dot{x}) .
The corresponding Simulink block diagram is shown in Figure 5.102. Note that the displacement input z(t) is a sine function, which can be defined using a Sine Wave block available in the library of Sources. Double-click on the block and type 0.01 for the Amplitude and 3.5 for the Frequency to define the input z(t)=0.01 \sin (3.5 t).
b. The transfer function relating the input z(t) to the output x(t) is
\frac{X(s)}{Z(s)}=\frac{b s+k}{m s^{2}+b s+k} \text {. }
The Simulink block diagram built based on the transfer function is shown in Figure 5.103, where a Transfer Fcn block is used to represent the vehicle system. Double-click on the block and type [ \left[\begin{array}{ll}\mathrm{b} & \mathrm{k}\end{array}\right] for the Numerator coefficient and \left[\begin{array}{lll}m & \mathrm{~b} & \mathrm{k}\end{array}\right] for the Denominator coefficient to define the transfer function X(s) / Z(s).
c. The Simscape block diagram corresponding to the physical system is shown in Figure 5.104, which can be created by following these steps:
1. Type ssc_new at the MATLAB Command window to open the main Simscape library and create a new model.
2. Open the library of Simscape/Foundation Library/Mechanical/Translational Elements and drag the Mass, Translational Damper, and Translational spring into the model window. Double-click on these blocks to define the parameters Mass, Damping coefficient, and spring rate as \mathrm{m}, \mathrm{b}, and \mathrm{k}.
3. To add the representation of the displacement input, open the library of simscape/ Foundation Library/Mechanical/Mechanical Sources and drag the Ideal Translational Velocity Source into the model window. Note that two types of inputs are available in Simscape for translational mechanical systems, and they are used to define either a force or a velocity input. Therefore, the displacement input function in this example must be converted to the velocity by taking the time derivative.
4. To add the sensor to measure the displacement of the mass, open the library of Simscape/Foundation Library/Mechanical/Mechanical Sensors and drag the Ideal Translational Motion Sensor into the model window.
5. Now open Simulink libraries to add the source and the scope. As mentioned in Step 3, taking the time derivative of z(t) gives the velocity input function \dot{z}(t)=0.035 \cos (3.5 t). Open the library of Simulink/Sources to drag the Clock block and open the library of Simulink/User-Defined Functions to drag the Fun block. Double-click on the Fun block and type 0.035^{*} \cos (3.5 * u) for the Expression, where u is the default name of the input to the Fun block, and here it represents the time t. Note that the Simulink-PS Converter and PS-Simulink Converter blocks are used to convert Simulink signals into physical signals or vice versa.
6. Orient the blocks and connect them as shown in Figure 5.104.
Define the values of the parameters m, b, and k in the MATLAB Command window. Run all simulations and the same curve as shown in Figure 5.105 can be obtained, which is the resulting displacement output x(t) of the vehicle due to the roughness of the road.
