A 100-kg turbine operates at 2000 r / min. What percent isolation is achieved if the turbine is

step 1 This question covered the concept of hoax law and from the hoax law the spring constant k is giv

A 100-kg turbine operates at 2000 r / min. What percent...

A $100-\mathrm{kg}$ turbine operates at $2000 \mathrm{r} / \mathrm{min}$. What percent isolation is achieved if the turbine is mounted on four identical springs in parallel, each of stiffness $3 \times 10^5 \mathrm{~N} / \mathrm{m}$ ?

Key Concepts

Frequency Ratio and Vibration Isolation

The ratio of the operating (or excitation) frequency to the natural frequency of the system is central to evaluating vibration isolation performance. A higher frequency ratio typically results in better isolation, as the transmitted vibrations are reduced when the forcing frequency is much higher than the natural frequency. This concept helps in understanding how effectively the system can isolate vibrations.

Transmissibility and Percent Isolation

Transmissibility is a measure of how much vibration is transmitted through a system relative to the input vibration. It is calculated based on the frequency ratio and, in damped systems, the damping ratio as well. Percent isolation is derived from the transmissibility, indicating the percentage reduction in the vibration amplitude. This concept is critical in designing systems that minimize the transmission of unwanted vibrations.

Effective Spring Constant

When multiple springs support a load in parallel, their stiffnesses add up to produce an effective spring constant. This concept is crucial because it determines the overall stiffness of the support system for vibration isolation. In problems like these, calculating the combined stiffness allows one to determine how the system will respond to dynamic loads.

Natural Frequency of a Mass?Spring System

The natural frequency is the frequency at which a system tends to oscillate in the absence of damping or external forces. It is determined by the mass and the effective spring constant of the system. This frequency is a key parameter because it sets the baseline response of the system to dynamic loading and is used to compare against the operating or excitation frequency.

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