- #1
AnkleBreaker
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Homework Statement
Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'.
Spring constant, k = (7*10^4) Nm
Damping coefficient, c = (3*10^3) N/m/s
mass,m = 250kg
a) Make a Laplace Transform of system and utilize it to predict 'y' in response to various inputs (step, impulse, ramp)
Homework Equations
Sprint force, Fs = kx
Damping force, Fd = cv
F = ma
The Attempt at a Solution
In this question,
y(t) = output
x(t) = input
y(t)/x(t) = transfer function
∑F = ma (but at equillibrium, a = 0)
Therefore ∑F = 0
(ignoring force by damper for now and resolving forces)
Fs = mg
Fs = k[ (original spring length, L0) - (new spring length, X0) ] = mg
Fs = k[ L0 - ( X0 + y(t) - x(t) ) ]
After simplifying...
Fs = k( L0 - X0 ) - k( y(t) -x(t) )
∑F = -mg + k( L0 - X0 ) - k( y(t) -x(t) )
Because mg = k( L0 - X0 ) [proven above]
∑F = - k( y(t) -x(t) ) = ma
m(d^2y/dt^2) + ky(t) = kx(t)
Accounting for the damping force
Fd = -c[ (dy/dt) - (dx/dt) ]
Therefore
∑F = - k( y(t) -x(t) ) - c[ (dy/dt) - (dx/dt) ] = m(d^2y/dt^2)
m(d^2y/dt^2) + c(dy/dt) + ky(t) = kx(t) + c(dx/dt) <---- Differential equation
Applying Laplace transformation...
Y(s)ms^2 + Y(S)cs + Y(S)k = X(S)k + X(s)cs
Y(S)[ms^2 + cs + k] = X(S)[k + cs]
Therefore,
Y(S)/X(S) = (cs + k) / (ms^2 + cs + k)
I have completed the question up to that point. It would be appreciated if someone could check my working and let me know if my answers are correct or there is something wrong.
I'm also having trouble completing this part of the question:
"it to predict 'y' in response to various inputs (step, impulse, ramp)"
Am I correct in assuming, this is how you do it?
Y(S) = [ (cs + k) / (ms^2 + cs + k) ] * X(S)
(cross multiply X(S))
And then you substitute values for X(S).
Ex:
unit step, X(S) = 1/s
ramp, X(S) = 1/s^2
Is that how its done?
Any and all advice will be much appreciated. Thank you