Question

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

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

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.

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: \(2\ddot{x} + 4x = 0\)

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.

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: \(\omega = \sqrt{\frac{4}{2}} = \sqrt{2}\, \mathrm{rad/s}\)

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.

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: \(x(t) = \cos( \sqrt{2}t)\)

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 \(\sqrt{2} \,\mathrm{rad/s}\).