Show the variation of the transmission ratio, (X1)/(δst), with (ω)/(ω1) for an undamped dynamic

step 1 Given undamped dynamic absorber, omega 2 equals omega 1, and mu equals M2 to M1, which is 0 .25,

Show the variation of the transmission ratio, (X1)/(δst),...

Show the variation of the transmission ratio, $\frac{X_{1}}{\delta_{\mathrm{st}}}$, with $\frac{\omega}{\omega_{1}}$ for an undamped dynamic vibration absorber for $\omega_{2}=\omega_{1}$ and $m_{2}=0.25 m_{1}$.

Key Concepts

Dynamic Vibration Absorber

A dynamic vibration absorber is an auxiliary device attached to a mechanical system designed to cancel or reduce unwanted oscillations by introducing a secondary oscillating system. By tuning the absorber’s properties to specific frequencies of the primary system, it mitigates vibrations through destructive interference, improving the overall dynamic performance of the system.

Transmission Ratio

The transmission ratio in vibration analysis is defined as the ratio of the response amplitude of the primary system to a static or base excitation amplitude. It serves as a measure of how effectively vibrations are transmitted through the system, thereby being a key parameter in assessing vibration isolation or amplification effects in structural dynamics.

Frequency Ratio

The frequency ratio is a dimensionless parameter that compares the excitation frequency to a characteristic natural frequency of the system, often that of the primary structure. It is essential for understanding resonance phenomena, the tuning of dynamic absorbers, and how system responses vary over a range of operating conditions.

Undamped System

An undamped system is an idealized model where no energy dissipation occurs due to damping. Studying undamped systems provides fundamental insights into resonance and dynamic amplification without the complicating effects of energy loss, serving as a baseline for understanding real-world behaviors when damping is eventually introduced.

Mass Ratio

The mass ratio is the proportion of the absorber mass relative to the primary system’s mass. It plays a critical role in the tuning and effectiveness of the vibration absorber, influencing the system's dynamic response and the magnitude of vibration reduction achieved through proper design adjustments.

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