Damped Harmonic Motion with a Sinusoidal Driving Force

[roldy]

1. An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the force and speed as a function of the driving frequency and find the frequencies for which the phase angle is 45°.



2. m\ddot{x} +R_m\dot{x}+kx=F_osin\omegat

see attachment for rest of equations





3. m= 0.5 kg, s=100N/m, R_m=1.4 kg/s, F_o=2N

So my first question is this, is omega the independent variable in this case? Meaning, I solve everything that I am able to and leave omega alone. Also, is the differential equation in #2 the right form?

I am confused at how I obtain the equation of motion so that I can plot this.

 

[vela]

Yes, ω is the driving frequency, so it's the independent variable for what's being asked in the problem. I'm not sure what you mean by "leaving ω alone." You want to express the speed amplitude and the phase angle as a function of ω and the constant parameters of the system.

Yes, your differential equation is correct. It is the equation of motion for the system. Note that k is the spring constant, which you called also called s in the other equations.

I think you have a typo in your equation for the phase angle. Also, what does c represent in that formula?

 

[roldy]

The equation for the phase angle is wrong it should be tan^{-1}(H)
where H=\frac{\omega*m-k/\omega}{R_{m}}.

I'm still a little confused about what they are asking for in regards to plotting.

 

[vela]

By "speed amplitude," I assume the problem is asking for the amplitude of v(t). It will depend on ω.