An engine flywheel has an eccentricity of 1.2 cm and mass of 40 kg. Assuming a damping ratio o

step 1 The model here is a Jeffcott rotor on a flexible, simply supported shaft with a disc at midspan.

An engine flywheel has an eccentricity of 1.2 cm and mass...

An engine flywheel has an eccentricity of $1.2 \mathrm{~cm}$ and mass of $40 \mathrm{~kg}$. Assuming a damping ratio of 0.05 , what is the necessary stiffness of its bearings to limit its whirl amplitude to $1.2 \mathrm{~mm}$ at all speeds between 1000 and $2000 \mathrm{r} / \mathrm{min}$ ?

Key Concepts

Stiffness in Dynamic Systems

Stiffness is a property that measures a system’s resistance to deformation under applied forces. In dynamic systems, sufficient stiffness is critical to limit displacements and vibrations induced by dynamic loading. It directly influences the system's natural frequency and the amplitude response, making it a key parameter in preventing excessive vibrations in rotating machinery.

Mass-Spring-Damper System

A mass-spring-damper system is a fundamental model used to analyze the response of mechanical systems to dynamic loads. In this model, the mass represents inertia, the spring symbolizes the system's stiffness, and the damper reflects energy dissipation. It provides a framework to predict how a system will behave when subjected to forces, helping engineers design systems that can handle vibrations effectively.

Natural Frequency

Natural frequency is the inherent oscillation rate of a system without any external forcing or damping, determined by the system’s mass and stiffness. It is critical in design because if a system is excited near its natural frequency, resonance can occur, leading to significantly amplified vibrations which can compromise stability and safety.

Damping Ratio

Damping ratio is a dimensionless measure that indicates how quickly oscillations in a system decay after a disturbance. It quantifies the level of damping relative to critical damping. In practical applications, an appropriate damping ratio is essential to control vibration amplitudes and prevent sustained oscillations that could lead to mechanical failure.

Unbalance Forces in Rotating Machinery

Unbalance forces arise when the mass distribution in a rotating system is not perfectly symmetrical. These forces, which can be caused by even small eccentricities, generate periodic excitation that leads to vibrations. Understanding these forces is important for designing supports like bearings to minimize vibration and ensure smooth operation.

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