Question

A system with m = 0.010 kg , k = 36 N/m and b = 0.5 kg/s is driven by a harmonically varying
force of amplitude 3.6 N. Find the amplitude A and the phase constant δ of the steady-state motion
when the angular frequency of the driving force is :
a) 8.0 rad/s,
b) 80 rad/s
c) 800 rad/s.

*Be careful, tan(δ) is a periodic function and δ should be between 0 and 180°.

Answer 1

Let the driving force on the oscillator is

The differential equation of motion with damping becomes

[ k is the spring constant of the oscillator, m is the mass of the oscillator, b is damping coefficient]

Or

[ Where ]

The above equation has two solutions (a) Complementary Function (CF) and (b) Particular Integral(PI).

CF will give us the transient solution and PI will give us the steady state response.

To find CF, we set  ..................(i)

let The trial solution of the above equation is so putting this the trial solution in equation(i) we get

So the roots are

From the given data we find

So the roots are imaginary and hence its solution can be written as

[ where ]

To find PI we set ........(ii)

Let the  trial solution of the equation(ii) is .............(iii)

putting (iii) in (ii) we get

Or

Compairing the coefficients of we get

From the above two equations , we get amplitude of PI is

and phase constant  

So, the general solution becomes

At steady state [ when ] CF vanishes and PI exists.

So at steady state the solution becomes

So amplitude at steady state is

Now when

We get

Now when

We get

Now when

We get ​​​​​​​

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