Effect of Damping on Steady State Vibration:Harmonic Excitation

[jrm2002]

Considering the response of a single degree of freedom system to harmonic excitation with viscous damping , following conclusions can be drawn:

Now,

The response of a single degree of freedom system to harmonic excitation can be split into:

a) Steady Sate response (or vibration) which is a result of the applied force.
b)Transient vibration which is the the result of the free vibration and is dependent on the initial conditions.Right?

Now, the transient vibration decays with time as a consequence of damping.Right?

But, it is also observed that the amplitude of the steady state response incraeses with time.This can be mathematically be proved easily as a consequence of the solution of the differential equation.

My question is:

1)What is the physical reasoning for the amplitude of the steady state response increasing with time?

Besides,
It is also found that the amplitude of the steady stae response is more in systems without damping (a theoretical case though) and is less in systems with damping.Right?

My question is:

2)That means, damping palys a role in reducing the amplitude of both steady state response as well as transient response?But, the transient (free vibration) response eventually decays completely as a consequence of dapming.Right?
Can anyone throw some more light on this?

 

[AndyW]

I can confirm that the conclusions drawn are correct. The steady state response is a result of the applied force and it depends on the frequency and amplitude of the excitation. The transient response, on the other hand, is a result of the initial conditions of the system and it decays with time due to damping.

The physical reasoning for the amplitude of the steady state response increasing with time is due to the phenomenon known as resonance. When the frequency of the excitation matches the natural frequency of the system, the amplitude of the steady state response increases because the system is able to absorb and store more energy from the applied force. This can be seen mathematically through the solution of the differential equation, where the amplitude of the steady state response is directly proportional to the frequency of excitation.

In systems without damping, the amplitude of the steady state response can theoretically become infinite at resonance. However, in real systems with damping, the amplitude is limited and can never reach infinity. This is because damping plays a crucial role in reducing the amplitude of both the steady state and transient responses. Damping dissipates energy from the system, thus reducing the amplitude of the response.

As for the transient response eventually decaying completely, this is also due to damping. Damping causes the energy of the system to dissipate over time, resulting in a decrease in the amplitude of the transient response until it eventually reaches zero. This is why systems with high amounts of damping will have a faster decay of the transient response compared to systems with low damping.

Overall, damping plays a critical role in controlling the amplitude of both the steady state and transient responses in a single degree of freedom system. It is important to consider the level of damping in a system to ensure that the response remains within safe and manageable limits.

 

[sir-pinski]

1) The physical reasoning for the amplitude of the steady state response increasing with time is due to the energy dissipation caused by damping. As the system vibrates, the damping force converts the kinetic energy into heat, causing a decrease in the amplitude of the vibration. However, the applied force continues to provide energy to the system, resulting in an increase in the amplitude of the steady state response over time. This can also be seen mathematically through the solution of the differential equation, where the amplitude of the steady state response is directly proportional to the forcing frequency and inversely proportional to the damping coefficient.

2) Yes, damping plays a role in reducing the amplitude of both the steady state response and transient response. The transient response eventually decays completely due to the energy dissipation caused by damping. This is because the damping force acts in the opposite direction of the motion, gradually reducing the amplitude of the vibration until it reaches zero. In systems without damping, the transient response would continue indefinitely, leading to larger amplitudes in the steady state response. Thus, damping plays a crucial role in controlling the amplitude of vibrations in a system.