Problem 2: The simplified quarter-car model of a vehicle suspension is given in Figure 3. In this simplified model, the masses if the wheel, tire, and axle are neglected, and the mass m represents the vehicle mass. The spring constant models the elasticity of both the tire and the suspension spring. The damping constant c models the shock absorber. The equilibrium position when y=0 is x=0. The road surface displacement can be derived from the road profile and the car's speed. (Note that the input has derivative) Body m x k c Suspension Road y Datum level m k(y - x) c(? - ?) Figure 2. Model for Problem 2. m = 300 kg, c = 100 N ? s / m, k = 1.5 × 10? N / m a. Obtain the differential equation model using symbols. b. Obtain the transfer function X(s) / Y(s). c. Solve and plot for the step response using MATLAB residue() function with y = 0.2u(t) meters, that is, Y(s) = 0.2 / s d. Use MATLAB function step() to solve and plot the response. e. Establish the state-variable model, i.e., find the matrices: A, B, C, D (Note: use Equations 2-33 to 2-38 in your textbook since the input contains its derivative) f. use MATLAB to obtain the transfer function: X(s) / Y(s), and compare your result in (b). g. use MATLAB/Simulink to simulate the response.
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Differential Equation Model: The differential equation model can be obtained by applying Newton's second law (F=ma) to the mass m. The forces acting on the mass are the spring force (-kx) and the damping force (-cx'), where x' is the derivative of x with respect Show more…
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The elements of the "swing-axle" type of independent rear suspension for automobiles are depicted in the figure. The differential $D$ is rigidly attached to the car frame. The half-axles are pivoted at their inboard ends (point $O$ for the half-axle shown) and are rigidly attached to the wheels. Suspension elements not shown constrain the wheel motion to the plane of the figure. The weight of the wheeltire assembly is $W=100 \mathrm{lb}$, and its mass moment of inertia about a diametral axis passing through its mass center $G$ is 1 lb-ft-sec $^{2}$. The weight of the half-axle is negligible. The spring rate and shockabsorber damping coefficient are $k=50 \mathrm{lb} / \mathrm{in.}$ and $c=200$ lb-sec/ft, respectively. If a static tire imbalance is present, as represented by the additional concentrated weight $w=0.5 \mathrm{lb}$ as shown, determine the angular velocity $\omega$ which results in the suspension system being driven at its undamped natural frequency. What would be the corresponding vehicle speed $v ?$ Determine the damping ratio $\zeta$ Assume small angular deflections and neglect gyroscopic effects and any car frame vibration. In order to avoid the complications associated with the varying normal force exerted by the road on the tire, treat the vehicle as being on a lift with the wheels hanging free.
We consider a car driving at velocity V over a road. The road surface is approximated as sinusoidal in cross section, providing a base motion of y(t) = Y sin(Wbt), where Wb = 2π/T. Y = 1 cm and A = 6 m. This is used to simulate a 1-cm bump. Mass of car Velocity of car Suspension system Neglected unsprung mass 0.02 Road surface Figure: Actual physical system and simplified model We assume that the suspension system provides an equivalent stiffness k of 2 × 10^4 N/m and damping coefficient c of 10 N.s/m. The mass of the car (empty) is 1200 kg. We assume that up to 5 passengers, each with a mass of 75 kg, are riding in the car. Write the equation of motion and derive the expression of the amplitude of the steady-state displacement of the car and the transmitted force. Show all steps.
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