The suspension system of a car is modelled by the following differential equation:
M*((d^2*y)/(d*t^2) + D*(dy/dt) + Ky( t ) = x( t )
where y(t) is the vertical position of the car body, x(t) is the external force coming from the ground acting on the car body, M is the mass of the car body, and D and K are the damping coefficient and spring coefficient of the suspension system, respectively. For the given mass M = 1000kgs, we need to design the suspension system by choosing D and K such that the car is responsive but comfortable. Being responsive means that the step response should be reasonably quick, and being comfortable requires the oscillating frequency not to be too fast. More specifically, we are required to find D and K such that the step response is slightly oscillatory with oscillating period of 0.5 second and overshoot of 10%. Overshoot is defined to be the ratio of the maximum peak and the steady-state value of the step response.
1.1 Determine the values of D and K and plot the resulting step response. Verify the oscillating period and overshoot. (Hint: Express the transfer function of the system in the form of
H(s)= k/(s^2 +2ςω s+ω^2)

and use the fact that the oscillating frequency and overshoot are controlled by
the natural frequency and damping ratio. You may tune D and K while simulating the step response until you get the desired results.)
To determine the values of D and K and plot the resulting step response, we need to follow these steps:
Step 1: Express the differential equation in the standard form of a second-order linear homogeneous differential equation:
M * (d^2y/dt^2) + D * (dy/dt) + Ky(t) = x(t)
Step 2: Take the Laplace transform of both sides of the equation to convert the differential equation into an algebraic equation:
M * (s^2Y(s) - s * y(0) - y'(0)) + D * (sY(s) - y(0)) + KY(s) = X(s)
Step 3: Rearrange the equation to solve for Y(s) (the Laplace transform of y(t)):
Y(s) = (s^2 * M * y(0) + s * M * y'(0) + D * y(0) + X(s)) / (M * s^2 + D * s + K)
Step 4: Express the transfer function of the system in the form of a second-order transfer function:
H(s) = Y(s) / X(s) = K / (M * s^2 + D * s + K)
Step 5: Compare the transfer function to the standard form of a second-order transfer function and identify the values of ω (the natural frequency) and ζ (the damping ratio):
H(s) = K / (M * s^2 + D * s + K) = ω^2 / (s^2 + 2ζωs + ω^2)
From this comparison, we can see that ω^2 = K / M and 2ζω = D / M.
Step 6: Choose values for the oscillating period (T) and overshoot (OS) to find the desired natural frequency (ω) and damping ratio (ζ):
Given oscillating period (T) = 0.5 seconds and overshoot (OS) = 10%,
We know that T = 2π / ω and OS = exp(-(π * ζ) / √(1 - ζ^2)) * 100%
Solving for ω and ζ:
T = 2π / ω
0.5 = 2π / ω
ω = 2π / 0.5
ω = 4π rad/s
OS = exp(-(π * ζ) / √(1 - ζ^2)) * 100%
10 = exp(-πζ / √(1- ζ^2)) * 100
0.1 = exp(-πζ / √(1- ζ^2))
-πζ / √(1- ζ^2) = ln(0.1)
π^2ζ^2 / (1- ζ^2) = ln(10)^2
π^2ζ^2 = (1- ζ^2) * ln(10)^2
π^2ζ^2 + ζ^2 * ln(10)^2 = ln(10)^2
ζ^2 * (π^2 + ln(10)^2) = ln(10)^2
ζ = sqrt(ln(10)^2 / (π^2 + ln(10)^2))
ζ ≈ 0.264
Step 7: Substitute the given values of M, ω, and ζ into the expressions for D and K:
For ω = 4π rad/s and ζ ≈ 0.264, we can substitute these values into the equations:
ω^2 = K / M
(4π)^2 = K / 1000
K = (4π)^2 * 1000
2ζω = D / M
2 * 0.264 * 4π = D / 1000
D = 2 * 0.264 * 4π * 1000
Step 8: Evaluate the expressions for D and K to find their values:
K ≈ (4π)^2 * 1000
D ≈ 2 * 0.264 * 4π * 1000
Step 9: Simulate the step response of the system using the values of D and K determined in Step 8. Plot the resulting step response to verify the oscillating period and overshoot.
By following these steps, you can determine the values of D and K and plot the resulting step response.
To determine the values of D and K and plot the resulting step response, we need to follow these steps:
Step 1: Express the differential equation in the form of a transfer function.
To do this, we first need to take the Laplace transform of the differential equation. The Laplace transform of the equation is:
M * (s^2 * Y(s) + s * D * Y(s) + K * Y(s)) = X(s)
Dividing both sides by Y(s) and rearranging the equation, we get:
Y(s) / X(s) = 1 / (M * s^2 + s * D + K)
This equation represents the transfer function of the system, which we can further manipulate to the form given in the hint:
H(s) = k / (s^2 + 2ζωs + ω^2)
Comparing the two equations, we can see that:
k = 1 / M
2ζω = D / M
ω^2 = K / M
Step 2: Determine the values of k, ζ, and ω.
Using the given values of M = 1000 kg and the desired specifications of oscillating period and overshoot, we can determine the values of k, ζ, and ω.
The oscillating period T = 0.5 seconds, so ω = 2π / T = 2π / 0.5 = 4π rad/s.
The overshoot is 10%, so we have the relation: overshoot = exp(-πζ/√(1 - ζ^2)) * 100
Plugging in the values, 10 = exp(-πζ/√(1 - ζ^2)) * 100
Solving this equation numerically, we find that ζ ≈ 0.516.
Using the value of ζ, we can solve for D and K:
2 * ζ * ω = D / M = (2 * 0.516 * 4π) * 1000 ≈ 3223.43 Ns/m
ω^2 = K / M = (4π)^2 * 1000 ≈ 39,478.42 N/m
Therefore, D ≈ 3223.43 Ns/m and K ≈ 39,478.42 N/m.
Step 3: Simulate and plot the step response.
Using these values of D and K, we can now simulate and plot the step response of the system.
Unfortunately, as an AI text-based bot, I am unable to simulate and plot the step response in real-time. However, you can use software tools like MATLAB or Python with the control system libraries to simulate and plot the step response using the transfer function obtained in Step 1. Just substitute the values of D and K into the transfer function and plot the step response.