Question 5.21: 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 5.102 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 \text{sin}(ωt) , where Z_0 = 0.01 \text{m} and ω = 3.5 \text{rad/s} . The mathematical model of the system is given by an ordinary differential equation
m\ddot{x} + b\dot{x} + kx = b\dot{z} + kz
where m = 3000 \text{kg}, b = 2000 \text{N ⋅ s/m}, and k = 50 \text{kN/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} (kz + b\dot{z} – kx – b\dot{x})
The corresponding Simulink block diagram is shown in Figure 5.103. 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 the block and type 0.01 for the Amplitude and 3.5 for the Frequency to define the input z(t) = 0.01 \text{sin}(3.5t).
b. The transfer function relating the input z(t) to the output x(t) is
\frac{X(s)}{Z(s)} = \frac{bs + k}{ms^2 + bs + k}
The Simulink block diagram built based on the transfer function is shown in Figure 5.104, where a Transfer Fcn block is used to represent the vehicle system. Double-click the block and type [b k] for the Numerator coefficient and [m b k] 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 5.105, which can be created by following these steps:
