# Homework Help: How to get transfer function (freq. domain) from Newton's 2nd Law Eqn

1. Apr 15, 2014

### annas425

1. The problem statement, all variables and given/known data

I am given the following dynamic system. I want to find the transfer function, θ1(s)/T(s).

The main body of the satellite, J1, acts like a large mass flying through space; space is essentially without friction and therefore no damping is imparted on the main body. The solar collector array acts like a second order underdamped system attached to the satellite, adding second order underdamped modes to the system. The satellite has thrusters that can impart a torque T(t) on the satellite main body; attitude θ1(t) can be measured; gravity from nearby planets is negligible about the axis of rotation; the structure attaching the main body to the solar collector acts as a torsional spring k and torsional damper b.

T(s) is the plant input, and θ1(s) is the plant output.

2. Relevant equations

Newton's 2nd Law (F=ma)

Laplace Transform?

3. The attempt at a solution

I came up with the following two equations (which I believe describe each body in the above dynamic system):

J1*θ1''(t) = -b*[θ1'(t) - θ2'(t)] - k*[θ1(t) - θ2(t)] + T(t)

J2*θ2''(t) = -b*[θ2'(t) - θ1'(t)] - k*[θ2(t) - θ1(t)]

As you can see, these are in the time domain (where θ''(t) = a(t), the second derivative of position; θ'(t) = v(t), the first derivative of position). I am unsure how to now get the transfer function from these equations (and in turn get these into the frequency domain). The transfer function is θ1(s)/T(s).

Should I take the Laplace Transform of the two equations? I am sure there is something else I am missing. Any light shed on this is GREATLY appreciated! Have a nice day :)

2. Apr 16, 2014

### donpacino

Taking the laplace transform of the two equations is a good place to start

3. Apr 16, 2014

### rude man

And the problem statement is ... ?