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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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