Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 5

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.

Short Answer

Expert verified
Based on the given problem and solution, the amplitude of the motion is 1 m, the period is \(\frac{2 \pi}{\sqrt{2}} \mathrm{~s}\), and the phase lag is 0. The response history is represented by a cosine wave oscillating between +1 m and -1 m with an oscillation frequency of \(\sqrt{2} \mathrm{~rad/s}\).

Step by step solution

01

Calculate the natural frequency

The natural frequency of the system can be found using the formula: $$\omega_n = \sqrt{\frac{k}{m}}$$Where \(\omega_n\) is the natural frequency, \(k = 4 \mathrm{~N} / \mathrm{m}\) is the stiffness of the spring, and \(m = 2 \mathrm{~kg}\) is the mass. Using the given values, the natural frequency can be calculated as:$$\omega_n = \sqrt{\frac{4}{2}} = \sqrt{2} \mathrm{~rad/s}$$
02

Determine the amplitude and phase

Since the system is initially displaced 1 meter downward and released from rest, the initial displacement \(x(0) = 1 \mathrm{~m}\) and initial velocity \(\dot{x}(0) = 0 \mathrm{~m/s}\). The general solution for the displacement of a single degree of freedom system can be written as:$$x(t) = A \cos(\omega_n t + \phi)$$Where \(A\) is the amplitude of the motion, \(\omega_n\) is the natural frequency, \(t\) is the time, and \(\phi\) is the phase angle. Using the initial conditions, we get:$$1 = A \cos(\phi)$$And:$$0 = -A \sqrt{2} \sin(\phi)$$The first equation suggests that the amplitude \(A=1 \mathrm{~m}\), and since the system is released from rest, the phase angle \(\phi=0\).
03

Calculate the period of the motion

The period of the motion can be found using the formula:$$T = \frac{2 \pi}{\omega_n}$$Using the calculated natural frequency, the period of the motion is:$$T = \frac{2 \pi}{\sqrt{2}} = \frac{2 \pi}{\sqrt{2}} \mathrm{~s}$$
04

Sketch the response history

The response history can be sketched by plotting the displacement formula, \(x(t) = \cos(\sqrt{2} t)\), over the time period \(T\). Since the amplitude is 1 m and the period is \(\frac{2 \pi}{\sqrt{2}} \mathrm{~s}\), the curve will oscillate between +1 m and -1 m with an overall oscillation frequency of \(\sqrt{2} \mathrm{~rad/s}\). Note that the phase lag is 0, so the curve will start at the maximum displacement. To summarize, the amplitude of the motion is 1 m, the period is \(\frac{2 \pi}{\sqrt{2}} \mathrm{~s}\), and the phase lag is 0. The response history of the system can be sketched as a cosine wave with these characteristics.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the block and the surface it moves on are respectively \(\mu_{s}=0.12\) and \(\mu_{k}=0.10\), determine the drop in amplitude between successive periods during free vibration. What is the frequency of the oscillations?

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(2 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 1 meter to the right and released from rest. Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

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.

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.

A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the mass and the surface it moves on are \(\mu_{s}=\mu_{k}=0.1\), and the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\), determine the time after release at which the mass sticks and the corresponding displacement of the mass.
See all solutions

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Read Explanation

Fluids

Read Explanation

Dynamics

Read Explanation

Magnetism

Read Explanation

Nuclear Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free