# Response of a Single Degree of Freedom System to HArmonic Excitation

• jrm2002
In summary, The book "Dynamics of Structures" by Anil K. Chopra discusses the response of a single degree of freedom system to an undamped system. The author mentions that an example of a harmonic excitation is the force due to unbalanced rotating machinery, which is when a solid body is rotating around an axis that doesn't go through its center of gravity. The author also explains that the displacement is said to be in phase with the applied force if the ratio of the forcing frequency to the natural frequency is greater than 1 and out of phase if the ratio is less than 1. The physical significance of being in or out of phase is a matter of personal interpretation.
jrm2002
I have been reading "Response of a Single Degree of Freedom System to Undamped system" and have been referring the book on Dynamics of Structures by Anil K. Chopra.I have got the following questions:

1) One , basic question I have is that the author says that an example of a harmonic excitation is the force due to unbalanced rotating machinery. What is the meaning of an unbalanced rotating machinery(am not from a mechanical background!) . Can anyone give some more (simple) examples of harmonic excitation?

2) On solving the governing differential equation of a single degree of freedom system subjected to a harmonic excitation, gives the author mentions that the displacement is said to be in phase with the applied force ( that is in the same direction as the applied force) if the ratio of the forcing frequency to the natural frequency is greater than 1 and the displacement is said to be out of phase with the applied force(that is the displacement is in opposite direction of the applied force) if the ratio of the forcing frequency to the natural frequency is less than 1.

Now, my question is , that does this out of phase or in phase have any physical significance whatsoever?

jrm2002 said:
1) One , basic question I have is that the author says that an example of a harmonic excitation is the force due to unbalanced rotating machinery. What is the meaning of an unbalanced rotating machinery(am not from a mechanical background!) . Can anyone give some more (simple) examples of harmonic excitation?

A thing (solid body) which is rotating around an axis which doesn't go through its center of gravity.

2) On solving the governing differential equation of a single degree of freedom system subjected to a harmonic excitation, gives the author mentions that the displacement is said to be in phase with the applied force ( that is in the same direction as the applied force) if the ratio of the forcing frequency to the natural frequency is greater than 1 and the displacement is said to be out of phase with the applied force(that is the displacement is in opposite direction of the applied force) if the ratio of the forcing frequency to the natural frequency is less than 1.

Now, my question is , that does this out of phase or in phase have any physical significance whatsoever?

First of all, the ratios mentioned above should be MUCH greater or MUCH smaller than 1.

Being in or out of phase is just a property of the motion at hand. Whether you consider that of physical significance or not, is a matter of taste. It just tells you whether the system you study "moves with" or "moves against" the movement that makes the forcing.

1) An unbalanced rotating machinery refers to a machine or equipment that has uneven distribution of mass or weight, causing it to vibrate or oscillate when it rotates. Some examples of unbalanced rotating machinery are washing machines, air conditioning units, and car engines.

Other examples of harmonic excitation could include a pendulum, a swing, or a tuning fork. These all have a natural frequency at which they oscillate, and if a force is applied at that same frequency, it will cause the system to vibrate or oscillate with larger amplitudes.

2) The concept of in-phase and out-of-phase responses in a single degree of freedom system has physical significance in understanding the dynamic behavior of the system. When the forcing frequency is greater than the natural frequency, the system is being forced to vibrate at a frequency that is faster than its natural frequency. In this case, the displacement will be in the same direction as the applied force, which means the system is amplifying the force and responding with larger amplitudes.

On the other hand, when the forcing frequency is lower than the natural frequency, the system is being forced to vibrate at a frequency that is slower than its natural frequency. In this case, the displacement will be in the opposite direction of the applied force, which means the system is resisting the force and responding with smaller amplitudes. This is important to consider in designing structures and machines to ensure they can withstand and dampen these types of harmonic excitations.

## 1. What is a single degree of freedom system?

A single degree of freedom system is a simplified model used to study the dynamic behavior of a physical system. It consists of a single mass connected to a spring and damper, and is commonly used to analyze the response of structures to external forces.

## 2. What is harmonic excitation?

Harmonic excitation refers to a type of external force that varies sinusoidally with time. In other words, it is a repeating force that has a constant amplitude and frequency.

## 3. How does a single degree of freedom system respond to harmonic excitation?

The response of a single degree of freedom system to harmonic excitation depends on the frequency of the excitation and the characteristics of the system (e.g. mass, spring stiffness, damping ratio). At certain frequencies, the system may experience resonance, resulting in large amplitude vibrations. At other frequencies, the system may experience a smaller response or even no response at all.

## 4. What is the equation of motion for a single degree of freedom system?

The equation of motion for a single degree of freedom system is typically written as: mX''(t) + cX'(t) + kX(t) = F(t)where m is the mass, c is the damping coefficient, k is the spring stiffness, and F(t) is the applied force at time t. This equation describes how the mass responds to the external forces acting on it.

## 5. How is the response of a single degree of freedom system to harmonic excitation calculated?

The response of a single degree of freedom system to harmonic excitation can be calculated using the equation of motion and solving for the displacement (X(t)) at each time step. This can be done analytically or numerically using numerical methods such as the Euler method or the Runge-Kutta method. Additionally, software programs such as MATLAB or SIMULINK can be used to simulate and analyze the response of a single degree of freedom system to harmonic excitation.

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