A 150-kg sewing machine operates at 1200 r / min and has a rotating unbalance of 0.45 kg-m. Wha

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A 150-kg sewing machine operates at 1200 r / min and has a...

A $150-\mathrm{kg}$ sewing machine operates at $1200 \mathrm{r} / \mathrm{min}$ and has a rotating unbalance of $0.45 \mathrm{~kg}-\mathrm{m}$. What is the maximum stiffness of an undamped isolator such that the force transmitted to the machine's foundation is less than 2000 N?

Key Concepts

Vibration Isolation

Vibration isolation involves reducing the transmission of vibrations from a vibrating machine (or structure) to its supporting foundation or surrounding environment. It typically relies on compliant elements, such as springs or elastomeric pads, to absorb and attenuate the energy from dynamic loads, thereby preventing excessive dynamic forces from being transmitted to the structure.

Rotating Unbalance

Rotating unbalance refers to the uneven distribution of mass in a rotating system, where the product of an imbalance mass and its offset (eccentricity) generates centrifugal forces when the system rotates. This imbalance leads to periodic dynamic forces that can excite vibrations and cause oscillating loads on both the machine structure and its supports.

Dynamic Force Calculation

Calculating the dynamic force generated by a rotating unbalance typically involves converting the rotational speed into angular velocity and then using the relationship between centrifugal force and angular velocity. The force due to an unbalanced rotating mass is proportional to the product of the unbalance (mass times offset) and the square of the angular velocity, highlighting the importance of both the speed and magnitude of the unbalance in determining the excitation force.

Stiffness in Vibration Isolation Systems

The stiffness of an isolator (often modeled as a spring) determines how much displacement occurs for a given applied force according to Hooke’s law. In vibration isolation, the isolator’s stiffness controls the transmission of vibrational energy: a lower stiffness generally allows more relative displacement, thereby reducing the force transmitted to the support structure, while a higher stiffness limits displacement but can lead to higher transmitted forces.

Natural Frequency and Frequency Ratio

The natural frequency of a mass-spring system, determined by both the mass and the stiffness of the isolator, is a critical parameter in vibration control. The effectiveness of isolation depends on the ratio of the excitation (forcing) frequency to the system’s natural frequency. When the excitation frequency is much higher than the natural frequency, the isolation system can more effectively attenuate vibrations, resulting in lower transmitted forces to the foundation.

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