Question

A 12 -lb weight is attached to the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 in. Beginning at \(t=0\) an external force given by \(F(t)=2 \cos \omega t\) is applied to the system. (a) If the damping force in pounds is numerically equal to \(3 x^{\prime}\), where \(x^{\prime}\) is the instantaneous velocity in feet per second, determine the resonance frequency of the resulting motion and find the displacement as a function of the time when the forcing function is in resonance with the system. (b) Assuming there is no damping, determine the value of \(\omega\) that gives rise to undamped resonance and find the displacement as a function of the time in this case.

Short answer

The resonance frequency of the motion is 64.39 rad/s. The displacement as a function of time when the forcing function is in resonance with the system (damped) is \(x(t) = \frac{2}{-0.3727(64.39)^2 + 24} \cos 64.39 t\). In the undamped case, the displacement as a function of time is \(x(t) = \frac{2}{24 - 0.3727 \omega^2} \cos \omega t\).

Step-by-step solution

Calculate the mass and spring constant from the given information

From the given information, we can calculate the mass and spring constant. We are given the weight of the object, which is 12 lb. To convert weight to mass (in slugs), we can use the following formula: \[mass (\text{in slugs}) = \frac{weight (\text{in pounds})}{g (\text{acceleration due to gravity - 32.2 ft/s}^2})\] The mass of the object is: \[m = \frac{12}{32.2} = 0.3727 \ \text{slugs}.\] Now, we are given that when the weight is in equilibrium, the spring stretches 6 inches or 0.5 ft. We can use Hooke's Law: \(F_s = kx\), where \(F_s\) is the spring force, \(k\) is the spring constant, and \(x\) is the displacement due to stretching (0.5 ft). The spring force, in equilibrium, is equal to the weight, so: \[k * 0.5 = 12\] The spring constant is: \[k = \frac{12}{0.5} = 24 \ \text{lbf/ft}\]

Write the forced damped harmonic oscillator equation

Now we can write the forced damped harmonic oscillator equation as follows: \[0.3727 x''(t) + 3x'(t) + 24x(t) = 2 \cos \omega t\]

Find the resonance frequency

The resonance frequency, \(\omega_r\), is given by the formula: \[\omega_r = \frac{k}{m}\] So, the resonance frequency is: \[\omega_r = \frac{24}{0.3727} = 64.39 \ \text{rad/s}\]

Find the displacement as a function of time in the damped case

We assume the particular solution to be in the form of \(x_p(t) = A \cos \omega t\). Taking the derivatives and substituting in the damped harmonic oscillator equation, we get: \[0.3727 (-A\omega^2 \cos \omega t) + 3(-A\omega \sin \omega t) + 24(A \cos \omega t) = 2 \cos \omega t\] Comparing the coefficients, we get: \[A \left(-0.3727\omega^2 + 24\right) = 2\] And the amplitude \(A\) is: \[A = \frac{2}{-0.3727\omega^2 + 24}\] The displacement as a function of time, when the forcing function is in resonance with the system, is: \[x(t) = \frac{2}{-0.3727(64.39)^2 + 24} \cos 64.39 t\]

Find the displacement as a function of time in the undamped case

In this case, we can set the damping coefficient \(c\) to 0, and the forced undamped harmonic oscillator equation becomes: \[0.3727 x''(t) + 24x(t) = 2 \cos \omega t\] Comparing coefficients in this case, we get: \[A(24 - 0.3727 \omega^2) = 2\] And the amplitude \(A\) is: \[A = \frac{2}{24 - 0.3727 \omega^2}\] The displacement as a function of time, in the undamped case, is: \[x(t) = \frac{2}{24 - 0.3727 \omega^2} \cos \omega t\] In summary, the resonance frequency is 64.39 rad/s, the displacement as a function of time in the damped case is \(x(t) = \frac{2}{-0.3727(64.39)^2 + 24} \cos 64.39 t\) and the displacement in the undamped case is \(x(t) = \frac{2}{24 - 0.3727 \omega^2} \cos \omega t\).

Key concepts

Forced Damped Harmonic Oscillator

For a forced damped harmonic oscillator, the motion of an object is influenced not only by the restoring force, often obeying Hooke's Law, but also by a damping force and an external, driving force. The damping force typically acts in opposition to the velocity of the object, reducing its motion over time; while the external force adds energy to the system at a specific frequency.

In the scenario presented, the damping force is proportional to the velocity at which the spring stretches or compresses. Mathematically, this situation can be described by a second-order differential equation, which includes terms for the mass of the object, the damping coefficient (here related to velocity by a factor of 3), the spring constant, and the external force varying with time. Solving for the various conditions of the system, such as when it is in resonance, allows one to understand how the displacement varies over time under the influence of these forces.

Resonance Frequency

Resonance frequency is a fundamental concept in the study of oscillatory systems, such as the forced damped harmonic oscillator. Resonance occurs when an external force contributes energy into the system at a frequency that matches the natural frequency of the system, thus causing a significant increase in the amplitude of oscillation.

This frequency is affected by the properties of the system, namely the mass and the spring constant as shown in the exercise. It is crucial for predicting the behavior of the system under external periodic driving forces. When the system operates at resonance frequency, it is exceptionally susceptible to oscillations, and precise determination of this frequency enables the prediction of the maximal amplitude state. In practical applications, understanding and sometimes avoiding resonance is key to maintaining the structural integrity of a system.

Hooke's Law

Hooke's Law is a principle of physics that relates the force needed to extend or compress a spring to the distance it is stretched or compressed. It's a linear approximation that states that the force exerted by a spring is directly proportional to its elongation, assuming the limit of elasticity of the spring is not exceeded.

In the context of this exercise, Hooke's Law is used to determine the spring constant from the displacement caused by the object at rest. This spring constant, denoted by 'k', is a measure of the stiffness of the spring and is a fundamental parameter in both damped and undamped harmonic motion. It's especially crucial in calculating the resonance frequency and the system's behavior under oscillatory conditions. With Hooke's Law, we can bridge the gap between the physical properties of the spring and the mathematical model that describes its motion.

Undamped Harmonic Motion

Undamped harmonic motion describes the idealized oscillation of an object in a system where there is no resistive force to diminish the energy of the system. In such scenarios, the object would continue to oscillate at a fixed amplitude indefinitely, as there would be no energy loss. This contrasts with the damped harmonic motion, where frictional or resistive forces reduce the amplitude of oscillation over time.

The exercise presented inquires into the condition when the system experiences no damping, thereby asking for the determination of a specific \(\omega\) value that will resonate with the system. In essence, this value corresponds to the natural frequency of the system. The study of undamped motion is pivotal in understanding the ideal behavior of oscillatory systems and provides a foundation for analyzing more complex, real-world scenarios where damping cannot be ignored.