Harmonic Excitation-Basic Questions

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In summary, the solution to the governing differential equation of a single degree of freedom system subjected to harmonic excitation is obtained by adding the complimentary solution and the particular solution. The complimentary solution represents the free vibration response, which decays with time due to damping and is dependent on initial conditions. The particular solution is the result of the applied force and is influenced by damping, with the amplitude of the steady state vibration being greater in an undamped system and lower in a damped system. For a very long time, the amplitude of the motion will stabilize to a value affected by damping, but the rate of change will trend towards zero. Lighter damping leads to a longer transient response, meaning that it takes more cycles for the amplitude of the deformation
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jrm2002
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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?
 
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  • #2
jrm2002 said:
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
 

Related to Harmonic Excitation-Basic Questions

1. What is harmonic excitation?

Harmonic excitation is a type of external force or vibration that is applied to a system or structure at a constant frequency. It can cause the system to resonate and produce vibrations at the same frequency as the force.

2. How is harmonic excitation different from random excitation?

Harmonic excitation is a regular and predictable force that is applied at a constant frequency, while random excitation is an unpredictable force that varies in frequency and amplitude.

3. What are some common examples of harmonic excitation?

Some common examples of harmonic excitation include musical instruments, wind turbines, and car suspensions. In these cases, the constant vibrations are desirable and serve a specific purpose.

4. How does harmonic excitation affect structures and machines?

Harmonic excitation can cause structures and machines to vibrate and potentially lead to structural damage or malfunction if the frequency of the force matches the system's natural frequency. On the other hand, it can also be harnessed to enhance the performance of some structures and machines.

5. How do engineers account for harmonic excitation in their designs?

Engineers use various techniques such as frequency analysis and damping methods to account for harmonic excitation in their designs. They also consider the natural frequency of the system and ensure that it does not coincide with the frequency of the excitation to prevent resonance.

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