How Is Work Calculated in a Forced Oscillation with Resistance During Resonance?

[bour1992]

Homework Statement

How can I calculate the work which is produced by the resistance force in a forced oscillation in one period?
The only forces on the oscillatory body are the resistance force and the external force.
The oscillatory body is in resonance.

Homework Equations

resistance force: $$F_{res}=-bv$$ (b is the damping constant)
external force:$$F_{ext}= F_{max} \cbullet \cos\omega t$$
$$x=Asin\omega t$$
$$u=u_{max}cos\omega t$$Thanks in advance

 

[ehild]

How can you calculate work in general?

ehild

 

[bour1992]

the work in a constant force is: W=F*d.
Moreover the work can be calculated from the area from graph F-d.

I can't find the work with either ways.

 

[jhooper3581]

A definition of work can be found in this http://en.wikipedia.org/wiki/Work_(physics)" .

 

[ehild]

the work in a constant force is: W=F*d.
Moreover the work can be calculated from the area from graph F-d.

I can't find the work with either ways.

The work is not constant here, and the area can be calculated as integral of force with respect to the displacement.

There is an other way to get work, by integrating power with respect to time for a given time period. And power (P) is the scalar product of force (F) and velocity (v). In case of one-dimensional motion,

$$P=Fv$$, and work done in one period is

$$W=\int_0^T{F(t)v(t)dt}$$

You know that $$F =-bv$$, and $$v(t)=v_{max}cos(\omega t)$$. Write the product of them and integrate.

ehild