# Using Laplace transforms to solve mass-spring-damper systems

1. Apr 23, 2007

### uteng2k7

Hello everyone, I am actually a mechanical engineering major at the University of Texas, but I thought I'd ask you kind folks for help with this problem. I apologize in advance for this being difficult to read because I don't know how to type many of these symbols.

1. The problem statement, all variables and given/known data
"The response of the mass-spring-damper due to a hammerblow F at t=to is described by:

m*ydotdot + c*ydot + k*y = F(lowercase delta (t-to)).

For a particular system: ydotdot+3*ydot+2y = (lowercase delta(t-1)).

a. Using Laplace transforms, determine the mass displacement, y(t) for a system that was initially at rest. Plot y(t) for 0 <= t <= 6.

b. Next, determine and plot the force on the mass Fm(t) for 0 <= t <= 6. Neglect the gravitational force mg as it is constant and hence does not affect the dynamics of the system.

2. Relevant equations
See above--ydot is the first differentiation of y, ydotdot is the second.

3. The attempt at a solution

I took the Laplace transforms of both sides, and ended up with:

[s^2*Y(s) - s*yo - ydoto] + 3[s*Y(s)-yo]+2Y(s)=L{lowercase delta[t-1]}.

For part a, I assumed that since the system was initially at rest, y(o)=0, and ydot(o)=0.

I then solved for Y(s), and rearranging the info, I got:

Y(s) = L{lowercase delta[t-1]}/(s^2+3*s+2)+(s+1)yo/(s^2+3*s+2)+ydoto/(s^2+3s+2).

Assuming that yo=0 and ydoto=0, the last two terms should equal zero, leaving you with Y(s)= L{lowercase delta[t-1]}/(s^2+3*s+2). You could simplify the bottom term to (s+2)*(s+1), but I'm not sure how to proceed from there.

--Jeremy

2. Apr 23, 2007

### jamesrc

I'll admit I didn't read all of your work, but the Laplace transform of an impulse function is defined. Maybe you even went through that derivation in class. In any event, you'll want to use that to find Y(s), then take the inverse Laplace to find the time domain response.

ETA: more hints in case I'm not around for follow up answers:
-you'll need to do partial fractions or otherwise simplify Y(s) once you find it
-look over your shifting theorems, you may need to use one.

Last edited: Apr 23, 2007
3. Apr 23, 2007

### rbj

i have a suggestion: learn to use LaTeX in spelling out messy equations, so we can read them.