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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
Answer: The amplitude of the motion is 1 meter, the angular frequency is √2 rad/s which gives a period of 2π/√2 seconds, and the phase lag is 0.

Step by step solution

01

Write the equation of motion for the system

The equation of motion for a simple vertical spring-mass system is given by: \(m\ddot{x} + kx = 0\) where \(m\) is the mass, \(k\) is the spring stiffness, \(x\) is the displacement, and \(\ddot{x}\) is the acceleration of the mass.
02

Plug in the known values and arrange the equation

Given, \(m = 2\mathrm{~kg}\), \(k = 4\mathrm{~N}/\mathrm{m}\), and \(x_0 = 1\) meter (initial displacement). Plug these values into the equation of motion and rearrange to get the equation in terms of \(x\) and its second derivative: \(2\ddot{x} + 4x = 0\)
03

Solve the differential equation

The solution to this second-order differential equation is given by: \(x(t) = A\cos(\omega t + \phi)\) where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase lag.
04

Find the angular frequency

The angular frequency is given by: \(\omega = \sqrt{\frac{k}{m}}\) Plug in the values of \(k\) and \(m\) to get: \(\omega = \sqrt{\frac{4}{2}} = \sqrt{2}\, \mathrm{rad/s}\)
05

Find the amplitude and phase lag

Since the system is released from rest, its initial velocity (\(v_0\)) is 0. Thus, we have the initial conditions \(x(0) = x_0\) and \(v(0) = 0\). Using the initial conditions and the equation for \(x(t)\), we have: \(x(0) = A\cos(\phi) = 1\) \(v(t) = -A\omega\sin(\omega t + \phi)\) \(v(0) = -A\omega\sin(\phi) = 0\) Since \(\sin(\phi) = 0\), the phase lag, \(\phi = 0\). Thus, the amplitude \(A\) remains equal to the initial displacement, \(x_0 = 1\) meter.
06

Write the final equation for the motion

With the amplitude, angular frequency, and phase lag found, we can write the final equation for the motion: \(x(t) = \cos( \sqrt{2}t)\)
07

Sketch the response history of the system

To sketch the response history of the system, plot the displacement \(x(t)\) as a function of time \(t\). The graph will be a cosine function with an amplitude of 1 meter and an angular frequency of \(\sqrt{2} \,\mathrm{rad/s}\).

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Most popular questions from this chapter

The cranking device shown consists of a mass-spring system of stiffness \(k\) and mass \(m\) that is pin-connected to a massless rod which, in turn, is pin- connected to a wheel at radius \(R\), as indicated. If the mass moment of inertia of the wheel about an axis through the hub is \(I_{O}\), determine the natural frequency of the system. (The spring is unstretched when connecting pin is directly over hub ' \(O\) '.)

The timing device shown consists of a movable cylinder of known mass \(m\) that is attached to a rod of negligible mass supported by a torsional spring at its base. If the stiffness of the spring is \(k_{T}\), where \(k_{T} / m g L>1\), determine the period of small angle motion of the device as a function of the attachment length, \(L\), if the spring is untorqued when \(\theta=0\).

A screen door of mass \(m\), height \(L\) and width \(\ell\) is attached to a door frame as indicated. A torsional spring of stiffness \(k_{T}\) is attached as a closer at the top of the door as indicated, and a damper is to be installed near the bottom of the door to keep the door from slamming. Determine the limiting value of the damping coefficient so that the door closes gently, if the damper is to be attached a distance a from the hinge.

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 \(30 \mathrm{~cm}\) aluminum rod possessing a circular cross section of \(1.25 \mathrm{~cm}\) radius is inserted into a testing machine where it is fixed at one end and attached to a load cell at the other end. At some point during a torsion test the clamp at the load cell slips, releasing that end of the rod. If the \(20 \mathrm{~kg}\) clamp remains attached to the end of the rod, determine the frequency of the oscillations of the rod-clamp system. The radius of gyration of the clamp is \(5 \mathrm{~cm}\). Fig. P2.7 Fig. P2.7
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