Harmonic Excitation-Basic Questions

[jrm2002]

The solution of the governing differential equation of a single degree of freedom system subjected to harmonic excitation is obtained as the sum of "Complimentary Solution" and "Particular Solution".

The complimentary solution correspomds to the free vibration response and is
dependent on initial conditions.It decays with time as a consequence of damping.Right?

Now, the particular solution is the outcome of the applied force which is influenced by damping and the amplitude of the steady state vibration is more in an undamped system and less in a damped system.Right?

My questions:

1)Consider the particular solution which exists as a result of the applied force.Suppose if we continue to apply the force for a very lng time "t", will damping continue causing the lowering the amplitude of the response upto this time "t" at the same rate?Why?

2) Also, it has been observed that lighter the damping , more is the number of cycles required to achieve a steadt state response, i.e. the amplitude of the deformation being constant.What does this signify?

 

[OlderDan]

The solution of the governing differential equation of a single degree of freedom system subjected to harmonic excitation is obtained as the sum of "Complimentary Solution" and "Particular Solution".

The complimentary solution correspomds to the free vibration response and is
dependent on initial conditions.It decays with time as a consequence of damping.Right?

Now, the particular solution is the outcome of the applied force which is influenced by damping and the amplitude of the steady state vibration is more in an undamped system and less in a damped system.Right?

My questions:

1)Consider the particular solution which exists as a result of the applied force.Suppose if we continue to apply the force for a very lng time "t", will damping continue causing the lowering the amplitude of the response upto this time "t" at the same rate?Why?

2) Also, it has been observed that lighter the damping , more is the number of cycles required to achieve a steadt state response, i.e. the amplitude of the deformation being constant.What does this signify?

I do not quite understand what you are asking in #1. After a very long time, the amplitude of the motion will stabilize to a value that is affected by the damping, but the rate of lowering the amplitude trends to zero as this happens. There is a gradual reduction in the amplitude and the transient part has a decaying exponential factor, so the rate of change drops as the amplitude drops.

The transient response is just the response of the undriven oscillator to a set of initial conditions. It has the underdamped//critically damped//overdamped cases. For light damping it takes more cycles to die out. What dies it signify? I don't know what that means exactly. Maybe it just means that the tranient part of the driven solution is the undriven solution.

Here is a nice little paper on the subject

http://www.scar.utoronto.ca/~pat/fun/NEWT1D/PDF/OSCDAMP.PDF