Damped and Forced Oscillations
Equations of motion for damped and forced Oscillations
Damped Oscillations.Consider the motion of a body in a viscous fluid in which the resistance to motion is proportional to the velocity. Suppose that the body has a weight W and that it is acted upon by a force acting towards x=0 by a constant force F and by a resistance Then applying Newtons Second Law the equation of motion is :
By putting x=b+y
Statical equilibrium occurs when x=b, and y is the displacement from the equilibrium position. The Auxiliary equation is :
- If k < n there is slight damping, so
- If k=n the motion is critically damped and there will be no oscillations. Both roots of the auxiliary equation are equal to -k.
- If k>n the result is a high degree of damping.
This can be written as equal to and , hence
Forced OscillationsIf in addition to the force which causes Simple Harmonic Motion and the force which causes damping, the body is acted upon by a force the equation of Motion now becomes:
and k and n have the same meaning as before. To solve this equation let x=u be a solution of the above equation in which case: 12) from (13) 11) is obtained by adding together u which is any particular solution and naturally depends upon f(t) and z which is the general solution for free oscillations. i.e. when there is no applied force f(t). We have already seen that the latter is damped out as t increases and hence the solution approaches x=u as t tends towards infinity. For this reason u is called the Steady State and the part z which dies away is called the transient. In most cases the applied force is either a constant one or it s periodic of period . In which case f(t) is of the form and the steady state can be found by assuming a trial solution and finding A and B by making the resulting equation identically true, i.e. by making the coefficients of and the same on the left hand side of the equation as on the right. This method of finding A and B is shown in the following example.
Example - Damped Oscillation
A body which weighs 8 lbs. is acted upon by three forces
- 6.25x lb.wt acting towards x = 0.
- A constant force of 2.5 lbs.wt.
- A resistance of .
- r = 0.7
- r = 2.5
- r = 6.5
By Newton's Second Law the equation of motion is:
- In the first case r = 0.7 so the above equation becomes:
The Period of the oscillations is:Had there been no damping the Period would have been :
- In the second case r = 2.5 and the equation of motion is :
- In the third case r = 6.5 and the equations become:
When r=0.7 When r=2.5