The armature of a variable-speed electric motor, of mass 200 kg, has an unbalance due to manufa

step 1 We are given a mass of 200 kilograms. K is 10 kilonitons per meter, so that would be 10 ,000 new

The armature of a variable-speed electric motor, of mass 200...

The armature of a variable-speed electric motor, of mass $200 \mathrm{~kg}$, has an unbalance due to manufacturing errors. The motor is mounted on an isolator having a stiffness of $10 \mathrm{kN} / \mathrm{m}$ and a dashpot having a damping ratio of 0.15 . (a) Find the speed range over which the amplitude of the fluctuating force transmitted to the foundation will be larger than the exciting force. (b) Find the speed range over which the transmitted force amplitude will be less than $10 \%$ of the exciting force amplitude.

Key Concepts

Dynamic Amplification Factor (Transmissibility)

The dynamic amplification factor, often referred to as transmissibility, quantifies the ratio between the amplitude of the system’s response and the amplitude of the applied excitation force. This factor depends on both the frequency ratio, which is the ratio of the excitation frequency to the natural frequency, and the damping ratio. It is pivotal in assessing the vibration isolation performance of a system, determining conditions where the transmitted force is amplified or attenuated relative to the excitatory force.

Damping Ratio

The damping ratio is a dimensionless measure of the damping in a system relative to critical damping. It indicates how quickly the oscillations decay over time. In forced vibration analysis, the damping ratio influences the peak dynamic response, determines the sharpness of resonance, and affects the system’s ability to isolate or transmit vibratory energy.

Single-Degree-of-Freedom (SDOF) System

A single-degree-of-freedom system is a fundamental model used to analyze dynamic behavior in mechanical systems. It consists of a mass, a spring, and a damping element, which together describe how the system responds to external forces. This model is widely used in vibration analysis and helps in understanding the resonance phenomenon and the effect of damping on system response.

Natural Frequency

The natural frequency is the frequency at which a system oscillates when it is disturbed from its equilibrium position and then left to vibrate freely without external forcing or damping effects. It is determined by the system's mass and stiffness, and plays a critical role in vibration analysis, particularly in predicting resonance, which occurs when the system is excited at or near this frequency.

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