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Chapter 2: Problem 27

A \(12 \mathrm{~kg}\) spool that is \(1 \mathrm{~m}\) in radius is pinned to a viscoelastic rod of negligible mass with effective properties \(k=10 \mathrm{~N} / \mathrm{m}\) and \(c=8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). The end of the rod is attached to a rigid support as shown. Determine the natural frequency of the system if the spool rolls without slipping.

Short Answer

Expert verified
Answer: The natural frequency of the system is approximately 0.91 rad/s.

Step by step solution

01

Define the variables and parameters

Let's define the variables and parameters given in the problem: - \(m\): mass of the spool (\(m = 12\,\text{kg}\)) - \(R\): radius of the spool (\(R = 1\,\text{m}\)) - \(k\): spring constant of the viscoelastic rod (\(k = 10\,\text{N/m}\)) - \(c\): damping coefficient of the viscoelastic rod (\(c = 8\,\text{N·s/m}\)) - \(x\): extension of the viscoelastic rod from the equilibrium position
02

Determine the rolling condition and force acting on the spool

As the spool rolls without slipping, its angular velocity \(\omega\) is related to its linear velocity \(v\) through the radius \(R\) as follows: $$ v = \omega R $$ The force acting on the spool is the force exerted by the viscoelastic rod, which is a combination of the elastic (spring) force and the damping force: $$ F(x) = -kx - cx' $$ where \(x'\) denotes the time derivative of \(x\), i.e., the velocity of the spool.
03

Determine the equation of motion for the spool using Newton's laws

The net force acting on the spool will cause it to accelerate, which can be described by Newton's second law as \(F = ma\), where \(a = x''\) is the acceleration of the spool. So we have: $$ F(x) = m x'' $$ Substituting the force expression from Step 2, we get the equation of motion for the spool: $$ -kx - cx' = m x'' $$
04

Find the natural frequency of the system

The natural frequency (\(\omega_n\)) of a system is the frequency of oscillation in the absence of damping. For an undamped system (\(c = 0\)), the equation of motion can be written as: $$ -kx = m x'' $$ Dividing both sides by \(m\) and rearranging the terms, we get: $$ x'' + \frac{k}{m}x = 0 $$ This is a standard second-order ordinary differential equation (ODE) for simple harmonic motion. The natural frequency \(\omega_n\) can be extracted from the ODE as follows: $$ \omega_n = \sqrt{\frac{k}{m}} $$ Substituting the given values of \(k\) and \(m\), we calculate the natural frequency: $$ \omega_n = \sqrt{\frac{10}{12}} \approx 0.91\,\text{rad/s} $$
05

Write down the final answer

The natural frequency of the system when the spool rolls without slipping is approximately \(0.91\,\text{rad/s}\).

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Most popular questions from this chapter

A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(1 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\). Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(5 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (c) Determine the response and plot its history if the damping coefficient is \(10 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\). Determine the response of the horizontally configured system if the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\). What is the amplitude, period and phase lag for the motion? Sketch and label the response history of the system.

Determine the natural period of a typical ice cube floating in water. Measure the dimensions of a typical cube from your refrigerator and calculate its natural frequency in water. (The dimensions may vary depending on your particular ice tray.) Confirm your "experiment." Place an ice cube in water, displace it slightly and release it. Make an approximate measure of the period of an oscillation with your wrist watch, or a stop watch if available. Repeat this operation several times and compare the average measured value with the calculated value.

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). Determine the response of the vertically configured system if the mass is displaced 1 meter downward and released from rest. What is the amplitude, period and phase lag for the motion? Sketch and label the response history of the system.

Two packages are placed on a spring scale whose plate weighs \(10 \mathrm{lb}\) and whose stiffness is \(50 \mathrm{lb} / \mathrm{in}\). When one package is accidentally knocked off the scale the remaining package is observed to oscillate through 3 cycles per second. What is the weight of the remaining package?
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