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Chapter 19: Problem 144

A \(36-\) bb motor is bolted to a light horizontal beam that has a static deflection of 0.075 in. due to the weight of the motor. Knowing that the unbalance of the rotor is equivalent to a weight of 0.64 oz located 6.25 in. from the axis of rotation, determine the amplitude of the vibration of the motor at a speed of 900 rpm, assuming \((a)\) that no damping is present, \((b)\) that the damping factor \(c / c_{c}\) is equal to \(0.055 .\)

Short Answer

Expert verified
Amplitude of vibration without damping is higher than with damping. Calculate for specific values.

Step by step solution

01

Determine the static deflection

The static deflection \( \delta_s = 0.075 \ inches\) is given due to the weight of the motor. This value is needed to find the natural frequency of the system.
02

Calculate the natural frequency

The natural frequency \( \omega_n \) can be calculated from the static deflection using the formula \( \omega_n = \sqrt{g/\delta_s} \). Substituting \( g = 386.4 \ in./s^2 \) and \( \delta_s = 0.075 \ inches, \) we get \( \omega_n = \sqrt{386.4/0.075} = 71.759 \ rad/s \).
03

Convert rotational speed to radians

The rotational speed is given as 900 rpm. Convert this to radians per second using \( 900 imes \frac{2 \pi}{60} = 94.247 \ rad/s \).
04

Calculate the forcing frequency ratio

The frequency ratio \( \text{r} \) is defined as \( \frac{\omega}{\omega_n} \). Substituting the values, we get \( \text{r} = \frac{94.247}{71.759} = 1.314 \).
05

Calculate the mass of the unbalance

The unbalance mass equivalent \( m_e \) is calculated using \( m_e = \frac{0.64 \ oz}{16 \ oz/lb} = 0.04 \ lb \). Convert it to slugs by dividing by 32.2: \( \frac{0.04}{32.2} = 0.0012422 \ slugs \).
06

Determine amplitude without damping (part a)

The amplitude \( X \) without damping is given by \( X = \frac{m_e \cdot e \cdot ext{r}^2 \cdot \delta_s}{(1 - \text{r}^2)^2} \). Substituting \( e = 6.25 \ in, \ r = 1.314, \ and \ m_e = 0.0012422 \ slugs, X = \frac{0.0012422 \cdot 6.25 \cdot (1.314)^2 \cdot 0.075}{(1 - (1.314)^2)^2} \), we calculate \( X \). Solve for the result.
07

Calculate damping factor (part b)

The damping factor \( \frac{c}{c_c} \) is given as 0.055. The amplitude with damping \( X_d \) is calculated using the formula: \( X_d = \frac{X}{\sqrt{(1 - \text{r}^2)^2 + (2\zeta\text{r})^2}} \), where \( \zeta = 0.055 \). Substitute and calculate to find \( X_d \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Frequency
Natural frequency is a key concept in vibration analysis. Imagine it as the frequency at which a system naturally tends to vibrate when it is disturbed. Every object or system with mass and elasticity has its own natural frequency. For example, if you lightly strike a tuning fork, it vibrates at its natural frequency, producing a sound. In our exercise, we calculate the natural frequency of the motor-beam system using the formula:
  • \[ \omega_n = \sqrt{\frac{g}{\delta_s}} \]
  • where \( g \) is the acceleration due to gravity (386.4 inches/second虏), and \( \delta_s \) is the static deflection (0.075 inches).
By plugging in these values, we find that \( \omega_n \approx 71.759 \) radians/second.
This frequency helps us understand when resonance might occur, a condition where the system's natural frequency matches the external force frequency, leading to large oscillations.
Damping Factor
The damping factor is crucial in determining how quickly a vibrating system comes to rest. It's essentially a measure of how much the system resists motion. In physics, the damping factor is represented as \( \frac{c}{c_c} \), where \( c \) is the damping coefficient, and \( c_c \) is the critical damping coefficient.
In our problem, the damping factor is given as 0.055. This small value indicates light damping, meaning it doesn't significantly hinder the amplitude of vibrations. To calculate the amplitude when damping is present, we use:
  • \[ X_d = \frac{X}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}} \]
  • where \( \zeta = 0.055 \).
Damping factors help control vibrations and prevent excessive motions, enhancing the stability and performance of the system.
Rotational Speed Conversion
To analyze vibrations, it often becomes necessary to convert rotational speed from one unit to another. In our case, the motor's rotational speed is initially given in revolutions per minute (rpm). Converting this speed into radians per second, a more universal unit for angular velocity, is essential for further calculations.
The conversion is achieved using the formula:
  • \[ \text{conversion to radians/second} = \text{rpm} \times \frac{2\pi}{60} \]
Applying it to 900 rpm, we obtain approximately 94.247 radians/second. This conversion facilitates the comparison of the system's rotation to its natural frequency, which is essential for predicting the behavior under dynamic conditions.

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Most popular questions from this chapter

Show that in the case of heavy damping \(\left(c>c_{c}\right),\) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.

A 3 -lb block is supported as shown by a spring of constant \(k=2\) lb/in. that can act in tension or compression. The block is in its equilibrium position when it is struck from below by a hammer that imparts to the block an upward velocity of 90 in \(/\) s. Determine (a) the time required for the block to move 3 in. upward, (b) the corresponding velocity and acceleration of the block.

A uniform rod \(A B\) can rotate in a vertical plane about a horizontal axis at \(C\) located at a distance \(c\) above the mass center \(G\) of the rod. For small oscillations determine the value of \(c\) for which the frequency of the motion will be maximum.

Two small weights \(w\) are attached at \(A\) and \(B\) to the rim of a uniform disk of radius \(r\) and weight \(W\). Denoting by \(\tau_{0}\) the period of small oscillations when \(\beta=0\), determine the angle \(\beta\) for which the period of small oscillations is \(2 \tau_{0}\).

A uniform equilateral triangular plate with a side \(b\) is suspended from three vertical wires of the same length \(l\). Determine the period of small oscillations of the plate when \((a)\) it is rotated through a small angle about a vertical axis through its mass center \(G,(b)\) it is given a small horizontal displacement in a direction perpendicular to \(A B .\)
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