Free Forced Vibration

Summary

This video lecture explores the concept of force vibration, where vibrations occur continuously due to external forces. Using an example of a pump on a bearing, the instructor discusses the difference between free vibrations and forced vibrations. Key concepts like amplitude, frequency, and resonance are covered, with mathematical equations explaining the motion and forces involved. The impact of frequency ratios between external force and the system's natural frequency on vibration behavior is also analyzed, along with practical considerations for preventing excessive vibration in mechanical systems.

Takeaways

• 😀 Force vibration refers to continuous vibrations caused by an external force acting on an object, such as a pump on a bearing.
• 😀 Free vibration occurs when an object vibrates after being displaced once and released, while force vibration involves a constant external force causing continuous movement.
• 😀 In force vibration, there are two frequencies: the frequency of the point mass and the frequency of the applied external force.
• 😀 The amplitude of vibration can be described by the equation X = A * sin(ωt), where A represents the constant amplitude and X represents the changing displacement.
• 😀 The natural frequency of an object, denoted ωn, is defined by the equation ωn = √(k/m), where k is the spring constant and m is the mass of the object.
• 😀 The equation of motion for force vibration includes both the mass and the external force, leading to a comparison between the force and the displacement amplitude.
• 😀 Resonance occurs when the frequency of the external force matches or is close to the natural frequency of the vibrating system, resulting in large amplitudes.
• 😀 If the frequency of the external force is lower than the natural frequency, the vibration amplitude remains relatively small.
• 😀 If the frequency of the external force is much higher than the natural frequency, the amplitude decreases and the vibration becomes less significant.
• 😀 When designing machinery or structures prone to vibration, it's crucial to calculate the natural frequency and select external forces with frequencies that won't cause excessive vibrations or resonance.

Q & A

What is force vibration, and how does it occur?

Force vibration refers to continuous oscillations caused by an external force applied to an object, which results in it vibrating. The example given in the transcript is a pump placed on bearings, where the pump acts as the external force causing the bearing to vibrate.

What is the difference between free vibration and force vibration?

In free vibration, the system oscillates after being displaced from its equilibrium position and then left alone, where the motion eventually ceases. Force vibration, on the other hand, involves an ongoing external force causing continuous oscillation of the system.

What is the importance of the two frequencies mentioned in force vibration?

In force vibration, two frequencies are crucial: the frequency of the external force (force frequency) and the natural frequency of the vibrating object (natural frequency). Both affect the behavior and amplitude of the vibration.

How is the amplitude in force vibration determined?

The amplitude in force vibration is influenced by the relationship between the force frequency and the natural frequency. The formula for amplitude is X = A * sin(ωt), where X represents the displacement and ω is the angular frequency. The amplitude can vary depending on the interaction between the two frequencies.

What happens when the external force frequency matches the natural frequency?

When the external force frequency (ω_force) matches the natural frequency (ω_n), resonance occurs, and the amplitude of vibration becomes very large. This can cause excessive vibrations and potential damage to the system.

What are the three conditions for the interaction between the force frequency and natural frequency?

The three conditions discussed are: 1) When the external force frequency is smaller than the natural frequency, the amplitude remains small. 2) When the frequencies match, resonance occurs, causing a large amplitude. 3) When the external force frequency is greater than the natural frequency, the amplitude decreases over time.

What is the formula used to determine the relationship between force and displacement in force vibration?

The relationship between force (F) and displacement (X) is given by the formula X/F = 1 / (k - ω_n^2), where k is the spring constant, and ω_n is the natural frequency. This formula helps determine how the amplitude varies with the applied force.

Why is it important to control the frequency of external forces in vibrating systems?

Controlling the frequency of external forces is important because if the external force frequency matches or is close to the natural frequency of the system, it can lead to resonance, causing excessive vibrations. Properly managing the external force frequency can prevent damage to machinery and structures.

How do changes in mass or spring constant affect the natural frequency?

The natural frequency (ω_n) is determined by the formula ω_n = √(k/m), where k is the spring constant, and m is the mass of the object. Adjusting either mass or the spring constant can change the natural frequency of the system.

What practical example is given to explain the effects of resonance in force vibration?

A practical example given is the case of a pump placed on bearings. If the frequency of the pump’s vibrations matches the natural frequency of the bearings, resonance occurs, leading to amplified vibrations, which could cause the pump and bearings to wear out or fail.