The center of gravity of a 12-kg rotor is 1.2 cm from its geometric center. The rotor is mounte

step 1 Given K is 3 .75 mega -newons per meter. That's equal to 3 .75 times 10 to the 6 newtons per met

The center of gravity of a 12-kg rotor is 1.2 cm from its...

The center of gravity of a $12-\mathrm{kg}$ rotor is $1.2 \mathrm{~cm}$ from its geometric center. The rotor is mounted on a shaft and spring-loaded bearings of stiffness $1.4 \times 10^5 \mathrm{~N} / \mathrm{m}$. Assuming a damping ratio of 0.05 , what is the amplitude of whirling when the rotor operates at $1500 \mathrm{r} / \mathrm{min}$ ?

Key Concepts

Rotor Imbalance

Rotor imbalance occurs when the center of mass does not coincide with the geometric center of rotation, causing eccentric forces during rotation. This unbalance creates periodic forces that excite the system, leading to vibrations that need to be analyzed in rotor dynamics.

Spring-Mass-Damper System

The rotor and its bearings can be modeled as a spring-mass-damper system, where the mass represents the rotor mass, the stiffness of the bearings serves as the spring constant, and damping represents energy dissipation in the system. This fundamental model is essential for understanding and predicting vibrational responses.

Natural Frequency

Natural frequency is the frequency at which a system tends to oscillate in the absence of external forces. In a spring-mass model, it is determined by the stiffness and the mass. Knowing the natural frequency is critical for evaluating potential resonance when the excitation frequency (due to imbalance) is near this value.

Damping Ratio

The damping ratio is a dimensionless measure that indicates how oscillations in a system decay after a disturbance. It quantifies the level of energy dissipation relative to critical damping. A low damping ratio, such as 0.05, indicates that the system is underdamped and will exhibit oscillatory behavior, which influences the amplitude of vibration.

Forced Vibration and Resonance

Forced vibration occurs when an external periodic force, such as that generated by rotor imbalance, acts on the system. When the rotational forcing frequency comes close to the natural frequency, resonance can occur, significantly amplifying the vibratory response. Analyzing this interaction is key to predicting whirling amplitude.

Whirling Motion

Whirling motion refers to the orbital vibration of the rotor resulting from the combination of rotor imbalance and the dynamic response of the support structure. It encapsulates the amplitude and pattern of the rotor’s movement under the influence of cyclic unbalanced forces and is crucial for assessing machine stability and performance.

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