A sensitive instrument of mass 100 kg is installed at a location that is subjected to harmonic

step 1 We're given the instrument mass, M is 100 kilograms. The base frequency F is 20 hertz. The base

A sensitive instrument of mass 100 kg is installed at a...

A sensitive instrument of mass $100 \mathrm{~kg}$ is installed at a location that is subjected to harmonic motion with frequency $20 \mathrm{~Hz}$ and acceleration $0.5 \mathrm{~m} / \mathrm{s}^{2}$. If the instrument is supported on an isolator having a stiffness $k=25 \times 10^{4} \mathrm{~N} / \mathrm{m}$ and a damping ratio $\zeta=0.05,$ determine the maximum acceleration experienced by the instrument.

Key Concepts

Harmonic Excitation

Harmonic excitation refers to an external force or motion applied to a system that varies sinusoidally with time. In dynamic systems, it is critical for analyzing the response of structures or instruments subjected to periodic disturbances, where the frequency and amplitude of the applied force determine the response magnitude.

Vibration Isolation System

A vibration isolation system is designed to reduce the transmission of vibrations from a source to a sensitive instrument. It typically comprises components like springs and dampers that absorb and dissipate energy, thereby minimizing the impact of external vibrations on the system's performance.

Damped Harmonic Oscillator

A damped harmonic oscillator is a standard model in dynamics that represents a mass-spring-damper system. The system's equation of motion includes inertial, damping, and stiffness components, which together describe how the system oscillates and eventually settles under the influence of a driving force.

Frequency Response Function

The frequency response function (FRF) quantifies how a system responds to harmonic excitation at different frequencies. It relates the amplitude and phase of the system’s output (such as acceleration or displacement) to the input excitation, and is essential for predicting resonance and vibration amplification effects.

Resonance

Resonance occurs when the excitation frequency approaches the system's natural frequency, leading to a significant increase in the amplitude of response. In vibration isolation applications, resonance can result in undesirable motion or excessive forces being transmitted to a sensitive component.

Damping Ratio

The damping ratio is a dimensionless parameter that characterizes the level of damping in a system relative to critical damping. It influences the peak amplitude of the response near resonance and affects how quickly oscillations decay over time.

Stiffness

Stiffness, typically represented by a spring constant in mechanical systems, is a measure of a system's resistance to deformation. In vibration isolation and dynamic analysis, stiffness is crucial for determining the natural frequency of the system and, consequently, its dynamic response to external excitations.

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