Short Answer:

Harmonic excitation is a type of external periodic force that varies with time in a smooth and repetitive manner, usually in the form of a sine or cosine wave. It acts on a system and causes it to vibrate continuously with the same frequency as the excitation.

In simple words, harmonic excitation means applying a time-varying force like  to a mechanical system. This type of excitation is commonly used in vibration analysis because many real-life vibrations can be represented as a combination of harmonic forces.

Detailed Explanation :

Harmonic Excitation

Harmonic excitation refers to a time-dependent external force that acts on a mechanical or structural system and varies in a sinusoidal (harmonic) manner. It is one of the most common types of external excitations used to study the behavior of vibrating systems.

In harmonic excitation, the force applied to the system changes periodically with time according to a sine or cosine function. The mathematical form of such a force is:

Where:

•  = instantaneous value of the applied force at time
•  = amplitude of the applied force (maximum value)
•  = angular frequency of excitation (in radians per second)
•  = time

This type of excitation causes the system to vibrate continuously, and the system response depends on the amplitude, frequency, and damping of the system.

Types of Harmonic Excitation

Harmonic excitation can be classified into two main types based on the damping characteristics of the system:

1. Undamped Harmonic Excitation:
   In this case, the system has no energy loss. The equation of motion is simple and depends only on the stiffness and mass of the system. The system responds with oscillations at the same frequency as the excitation.
2. Damped Harmonic Excitation:
   When damping is present in the system, part of the vibration energy is dissipated as heat or friction. The amplitude of vibration decreases, and the phase difference between the applied force and response depends on the damping ratio.

Equation of Motion for Harmonic Excitation

Consider a single degree of freedom (SDOF) system with a mass , stiffness , and damping coefficient , subjected to a harmonic force .

The equation of motion is:

Where:

•  = mass of the vibrating system
•  = damping coefficient
•  = stiffness of the spring
•  = displacement
•  = acceleration
•  = velocity

The solution of this equation gives the total response of the system, which consists of two parts:

1. Transient Response: It occurs initially and dies out with time due to damping.
2. Steady-State Response: It remains after the transient part vanishes and has the same frequency as the excitation force.

Thus, for a harmonic excitation, the steady-state response is most important because it defines the long-term vibration behavior of the system.

Steady-State Solution

The steady-state displacement of the system under harmonic excitation is given by:

Where:

•  = amplitude of steady-state vibration
•  = phase difference between the applied force and displacement

The amplitude  can be expressed as:

Where:

•  = frequency ratio (ratio of excitation frequency to natural frequency)
•  = damping ratio

The phase angle  is given by:

This shows that the amplitude and phase of the response depend on both the frequency ratio  and the damping ratio .

Effect of Frequency on Harmonic Excitation Response

The response of a system under harmonic excitation varies with the excitation frequency ():

1. At Low Frequency ():
   The system follows the excitation almost exactly, and the amplitude is small.
2. At Resonance ():
   The amplitude reaches its maximum value. If damping is low, the amplitude becomes very large, leading to possible failure.
3. At High Frequency ():
   The mass inertia dominates, and the system cannot respond quickly. The amplitude becomes very small.

Hence, the system shows three distinct behaviors depending on the relation between the excitation frequency and natural frequency.

Phase Relationship

The phase difference () between the applied force and the resulting displacement varies with frequency:

• At low frequencies, the displacement is in phase with the applied force ().
• At resonance, the displacement lags by 90° behind the force.
• At high frequencies, the displacement lags by nearly 180° behind the force.

This phase difference is important in understanding the motion and stability of vibrating systems.

Practical Examples of Harmonic Excitation

1. Unbalanced Rotating Machinery:
   In motors, fans, or turbines, unbalanced rotation creates a harmonic force on bearings and supports.
2. Reciprocating Engines:
   The periodic motion of pistons produces harmonic forces that excite engine mounts and vehicle frames.
3. Buildings and Bridges:
   Wind or traffic loads act periodically, creating harmonic excitation in structures.
4. AC Electrical Machines:
   Alternating magnetic fields apply harmonic forces on machine components.
5. Testing Machines:
   Harmonic excitation is used in vibration testing to determine natural frequencies and damping characteristics.

Importance of Studying Harmonic Excitation

• Helps in predicting vibration response of machines and structures.
• Useful in designing isolators and dampers to reduce vibration amplitude.
• Essential for resonance analysis and avoiding harmful vibration conditions.
• Assists in diagnosing unbalance and misalignment in rotating systems.

Thus, understanding harmonic excitation allows engineers to design safer and more reliable mechanical systems.

Conclusion

In conclusion, harmonic excitation is a sinusoidal external force that acts periodically on a system and causes it to vibrate with the same frequency as the applied force. It plays a key role in studying forced vibrations and resonance phenomena. The response of a system to harmonic excitation depends on its natural frequency, damping, and excitation frequency. By analyzing harmonic excitation, engineers can predict vibration behavior, avoid resonance, and design systems with effective damping and vibration control.