Short Answer:

A forced response is the behavior or motion of a system when it is continuously acted upon by an external periodic or non-periodic force. Unlike free response, which occurs without external excitation, the forced response depends directly on the nature and frequency of the applied force.

In mechanical systems, the forced response represents how the system vibrates under external influences such as rotating unbalanced masses, engine vibrations, or fluctuating loads. It helps engineers understand how external forces affect the amplitude, phase, and frequency of vibrations in machines and structures.

Detailed Explanation :

Forced Response

The forced response of a system refers to the vibration or motion that occurs due to the continuous application of an external force. This force can be periodic (repeating in time, like a sinusoidal force) or non-periodic (such as a random or transient force).

When a mechanical system is subjected to such a force, it vibrates not only at its natural frequency but also at the frequency of the external excitation. The resulting motion is a combination of two parts: the transient response (which dies out with time) and the steady-state response (which remains as long as the external force acts).

In real-life engineering systems, almost all vibrations are forced vibrations, since machines and structures are continuously influenced by external forces such as engine torque, unbalanced rotation, or environmental disturbances like wind or earthquakes.

1. Equation of Forced Response

For a single degree of freedom (SDOF) system consisting of a mass , damping , and stiffness , subjected to an external force , the governing differential equation is:

where,

•  = mass of the system
•  = damping coefficient
•  = stiffness of the spring
•  = external force applied
•  = displacement response

If the external force is harmonic, such as , then the system vibrates under steady-state forced vibration conditions.

2. Types of Forced Response

The forced response of a system can be classified based on the nature of the external force:

1. Periodic Forced Response:
   The excitation is repetitive in nature (e.g., sinusoidal force). Examples include unbalanced rotating machinery and reciprocating engines.
2. Non-Periodic Forced Response:
   The excitation is random or irregular (e.g., road roughness on vehicles or earthquake ground motion).
3. Transient Forced Response:
   The external force acts for a short duration, such as a shock load or impact.
4. Steady-State Forced Response:
   Occurs when a continuous external force acts for a long time, leading to a stable vibration pattern at the excitation frequency.

3. Solution of Forced Response

The general solution of the forced vibration equation is given by:

where,

•  = homogeneous solution or transient response (depends on initial conditions and system properties, decays with time).
•  = particular solution or steady-state response (depends on the external force and continues as long as the force acts).

After a long time, the transient part  vanishes due to damping, and the steady-state response  dominates.

4. Steady-State Forced Response for Harmonic Force

When the applied force is harmonic, i.e., , the steady-state response is also sinusoidal and can be expressed as:

where,

•  = amplitude of forced vibration
•  = phase angle between applied force and displacement
•  = frequency of excitation

The amplitude  of vibration is given by:

where,

•  is the frequency ratio
•  is the natural frequency
•  is the damping ratio

This expression shows that the amplitude of forced vibration depends on the ratio of excitation frequency to natural frequency and on the damping present in the system.

5. Resonance in Forced Response

Resonance occurs when the frequency of the external force equals the natural frequency of the system ().

At resonance, the amplitude of vibration becomes extremely large if damping is small, which can lead to failure or damage in mechanical and structural systems.

• In an undamped system, theoretically, the amplitude becomes infinite at resonance.
• In a damped system, the amplitude remains finite, but still very high.

Engineers design systems to avoid resonance by altering mass, stiffness, or damping characteristics.

6. Effect of Damping on Forced Response

Damping plays a vital role in reducing vibration amplitude and controlling resonance.

• Low damping: High amplitude near resonance.
• High damping: Smaller amplitude but slower response to force.

The phase angle between the applied force and the system’s motion also depends on damping.

• At low frequency (), displacement is almost in phase with force.
• At resonance, displacement lags force by 90°.
• At high frequency (), displacement lags force by almost 180°.

Thus, damping ensures stable and safe operation under forced vibration conditions.

7. Physical Examples of Forced Response

1. Unbalanced Rotating Machines:
   Shafts and rotors experience periodic external forces due to imbalance, producing forced vibrations.
2. Vehicle Suspensions:
   When a car moves over rough terrain, the road irregularities act as external excitations.
3. Engine Mounts:
   Engine vibrations are transmitted to the vehicle frame, representing forced response behavior.
4. Building Structures:
   Structures experience forced vibrations during earthquakes or wind gusts.
5. Industrial Equipment:
   Machines operating with fluctuating loads or cutting tools experience forced vibrations.

8. Importance of Studying Forced Response

• To avoid resonance: Helps design systems that do not vibrate excessively at certain frequencies.
• To determine stability: Shows how systems behave under constant external loading.
• For design optimization: Guides engineers in selecting suitable stiffness and damping values.
• For improving comfort and durability: Reduces vibration levels in vehicles and machinery.
• For dynamic analysis: Essential in designing vibration isolators and absorbers.

Understanding the forced response allows engineers to ensure that mechanical systems operate safely, efficiently, and without excessive wear.

Conclusion:

The forced response is the motion of a system caused by a continuous external force acting on it. It consists of a transient part that decays over time and a steady-state part that remains as long as the excitation persists. The forced response depends on excitation frequency, damping, and natural frequency. When the excitation frequency equals the natural frequency, resonance occurs, leading to large vibrations. Studying the forced response is vital for designing stable, efficient, and vibration-free mechanical systems.