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**1.**

A mass of 0.5 kg hangs on a spring. The stiffness of the spring is 100 N/m, and the mechanical resistance is 1.4 kg/s. The force (N) driving the system is f=2cost5t.

(a) What will be the steady-state values of the displacement amplitude, speed amplitude, and average power dissipation?

(b) What is the phase angle between speed and force?

(c) What is the resonance frequency and what would be the displacement amplitude, speed amplitude, and average power dissipation at this frequency and for the same force magnitude as in (a)?

(d) What is the Q of the system, and over what range of frequencies will the power loss be at least 50% of its resonance value?

**2.**

eq. 1

eq. 2

eq. 3

Z

_{m}

^{2}=R

_{m}

^{2}+ (omega*m-s/omega)^2

## The Attempt at a Solution

m= 0.5 kg

stiffness=k=s(in my book)=100 N/m

mechanical resistance=c=R

_{m}(in my book)=1.4 kg/s

f=2cost5t

(a) I used eq. 1 and found the displacement amplitude to be .023 m or 2.3 cm.

To get the speed amplitude, I differentiated the displacement equation. The only thing I differentiated was the forcing function since that is the only thing that's dependent on time. So basically I multiplied the derivative of f by .023. This resulted in -.115 m/s. So the velocity amplitude is .115 m/s or 11.5 cm/s.

For the average power dissipation I used eq. 2

And using eq. 3 I found what Z

_{m}was.

power dissipation=.009 W

(b) Here I'm not so sure I'm right. I said that it was pi/2. F=ma, and the acceleration response is a cosine function (differentiated the velocity response from (a)). The speed response is a sine function. Cosine and sine are out of phase by pi/2 radians. Does my logic make sense? Or is there something I messed up.

(c) Here I'm also not sure I did this right. In my book it says the following:

The resonance angular frequency omega_o is defined as that at which the mechanical reactance X_m vanishes and the mechanical impedance is pure real with its minimum value, Z_m=R_m.

So in order for Z_m to equal R_m in eq. 3, the term in (...) must be zero.

omega*m-s/omega=0

omega*m=s/omega

omega^2=s/m

omega=(s/m)^1/2

I got 14.142 rad/s. This does not seem right because it is more than the natural frequency of the driving force which is 5 rad/s. Using this number in the displacement amplitude I came up with .101 m or 10.1 cm. This made me question my result for the resonance frequency.

(d) I didn't bother doing yet because of the questionable answer I got above.

Any help would be greatly appreciated. I've spent the past 2 days trying to figure this out.

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