Question

A 24-lb weight is attached to the lower end of a coil spring suspended from a fixed beam. The weight comes to rest in its equilibrium position, thereby stretching the spring 1 foot. The weight is then pulled down 1 foot below its equilibrium position and released at \(t=0 .\) The medium offers a resistance in pounds numerically equal to \(6 x^{\prime}\), where \(x^{\prime}\) is the instantaneous velocity in feet per second. (a) Determine the resulting displacement of the weight as a function of the time and express this displacement in the alternate form (5.32) of the text. (b) Find the quasi period and time-varying amplitude. (c) At what time does the weight first attain a relative maximum displacement above its equilibrium position, and what is this maximum displacement?

Short answer

The displacement function of the weight is given by \(x(t) = e^{-4t}(2\cos(8t) - \sin(8t))\). The quasi-period is approximately \(T \approx 0.785\,s\), and the time-varying amplitude is given by \(A(t) = \frac{2}{\sqrt{64 - 16}} \approx 0.248\,ft\). The weight first attains a relative maximum displacement above its equilibrium position at \(t \approx 0.393\,s\), with a maximum displacement of approximately \(0.170\,ft\).

Step-by-step solution

Write down the differential equation for damped harmonic oscillator

The equation for a damped harmonic oscillator is given by: \[m x'' + cx' + kx = 0\] where \(m\) is the mass, \(c\) is the damping coefficient, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position.

Substitute the given values

In this problem, we are given the weight (24 lbs), the damping coefficient (6), and equilibrium displacement (1 ft). We can use these values to find the mass and the spring constant. First, we find the mass: \[m = \frac{W}{g} = \frac{24\,\text{lb}}{32.2\,\text{ft/s}^2} \approx 0.745\,\text{lb sec}^2/\text{ft}\] Next, we find the spring constant: \[k = \frac{m g}{s} = \frac{0.745\,\text{lb sec}^2/\text{ft}\cdot 32.2\,\text{ft/s}^2}{1\,\text{ft}} \approx 24\,\text{lb/ft}\] Now, we can substitute these values into the differential equation: \[0.745 x'' + 6x' + 24x = 0\]

Solve the differential equation

The displacement function can be found by solving the above equation. Since it's a second-order linear differential equation with constant coefficients, we assume the solution is of the form: \[x(t) = e^{rt}(A\cos(\omega t) + B\sin(\omega t))\] Here, \(A\) and \(B\) are constants, and \(\omega\) is the angular frequency. We can obtain these values by plugging the assumed equation into the original differential equation and applying the initial conditions.

Find the displacement function

The initial conditions are given as: \[x(0) = -1\, \text{ft}\] \[x'(0) = 0\, \text{ft/s}\] Solve for \(A\) and \(B\) using these conditions to find the displacement function.

Find the quasi-period and time-varying amplitude

The quasi-period of the damped harmonic oscillator can be found using the formula: \[T = \frac{2\pi}{\omega'}\] where \(\omega' = \sqrt{\omega^2 - r^2}\), and \(\omega\) and \(r\) can be found from the displacement function. The time-varying amplitude can be found using the formula: \[A(t) = \frac{A_0 }{\omega'}\] where \(A_0\) is the initial amplitude, and \(\omega'\) is the modified angular frequency.

Find the maximum displacement above the equilibrium position

To find the first relative maximum displacement above the equilibrium position, we need to find the critical points of the displacement function. Differentiate the displacement function with respect to time to find the velocity function, then set the velocity function to zero to find the critical points: \[x'(t) = 0\] Solve this equation for the critical times and find corresponding displacements, then choose the first relative maximum displacement above the equilibrium position. By following these steps, you will have found the displacement function, quasi-period and time-varying amplitude, and the first relative maximum displacement above the equilibrium position of the weight and the time at which this occurs.

Key concepts

Differential Equation

In physics and engineering, a differential equation is a mathematical statement that relates the rates at which quantities change. Specifically, for a damped harmonic oscillator, the differential equation models the motion of a system that is returning to equilibrium but is also losing energy over time due to some form of resistance, such as friction or air resistance.

The general form of the differential equation for a damped harmonic oscillator is:
\[m x'' + cx' + kx = 0\]
where:

• \(m\) is the mass of the object.
• \(c\) is the damping coefficient, which represents the resistance force.
• \(k\) is the spring constant, indicating the system's stiffness.
• \(x\) is the displacement of the object from its equilibrium position.

This equation can be solved to find how the displacement changes over time, considering both the spring force and the damping effect.

Displacement Function

The displacement function for a damped harmonic oscillator describes the position of the object relative to the equilibrium point as a function of time. In the given exercise, we aim to find this function, which will show how the weight attached to a spring moves over time.

To solve for the displacement function, one would typically assume a solution form like:
\[x(t) = e^{rt}(A\cos(\omega t) + B\sin(\omega t))\]
where:

• \(A\) and \(B\) are constants determined by initial conditions.
• \(r\) is associated with the damping rate.
• \(\omega\) is the angular frequency of the system without damping.

After substituting this assumed solution into the differential equation and using the initial conditions, one can determine the constants and express the displacement function, representing the system's motion over time.

Quasi-Period

The quasi-period is an important concept for oscillating systems that experience damping. Unlike a perfect harmonic oscillator, which oscillates with a constant period, a damped oscillator has a period of motion—referred to as the quasi-period—that gradually increases over time due to the energy being lost from the system.

To find the quasi-period, one must first determine the modified angular frequency \(\omega'\), after which the quasi-period \(T\) is given by:
\[T = \frac{2\pi}{\omega'}\]
The modified angular frequency takes into account the damping, and its value is less than the undamped angular frequency, which lengthens the time it takes to complete a cycle compared to an undamped system.

Time-Varying Amplitude

The time-varying amplitude is a characteristic of damped oscillations; this amplitude decreases over time due to the energy lost from the system. In a non-damped system, the amplitude remains constant, but damping causes the peaks to become progressively lower.

The time-varying amplitude can be calculated using the formula:
\[A(t) = \frac{A_0 }{\omega'}\]
where \(A_0\) is the initial amplitude. This formula ties the amplitude of oscillation to time, indicating how the amplitude diminishes as the object moves back and forth and loses energy to damping forces.

Equilibrium Position

The equilibrium position is the state at which the forces acting on an object are balanced, resulting in no net force and no acceleration. For a weight attached to a spring, this position is where the spring is neither stretched nor compressed.

In the context of a damped harmonic oscillator, the equilibrium position is the central point around which the weight oscillates. When released from rest or given an initial push, the weight will move around the equilibrium position with a motion described by the displacement function, gradually settling down to this point over time due to damping.

Maximum Displacement

The term maximum displacement refers to the furthest distance that an object moves from its equilibrium position during its motion. For an oscillating system like a damped harmonic oscillator, this would occur at the peaks of its motion.

To find the maximum displacement, one must examine the displacement function over time and look for points where the velocity of the object is zero (these are potential maximum or minimum points). The initial maximum displacement is indicated by the given initial conditions, and subsequent maximum displacements can be found by analyzing the system's dynamics over time. In practice, due to damping, each successive peak will be lower than the previous one, so it's essential to distinguish between the absolute maximum displacement and the relative maximum displacements occurring at different moments in the motion.