Question

A certain coil spring having spring constant \(10 \mathrm{lb} / \mathrm{ft}\) is suspended from the ceiling. A 32 -lb weight is attached to the lower end of the spring and comes to rest in its equilibrium position. Beginning at \(t=0\) an external force given by \(F(t)=\sin t+\frac{1}{4} \sin 2 t+\frac{1}{9} \sin 3 t\) is applied to the system. The medium offers a resistance in pounds numerically equal to twice the instantaneous velocity, measured in feet per second. Find the displacement of the weight as a function of the time, using Chapter 4 , Theorem \(4.10\) to obtain the steady- state term.

Short answer

The displacement of the weight as a function of time is: \(x(t) = x_s(t) = \frac{1}{\sqrt{484}} \sin(t) + \frac{1}{\sqrt{14164}} \sin(2t) + \frac{1}{\sqrt{78880}} \sin(3t)\).

Step-by-step solution

Write the given information in terms of differential equation

Since we have a spring with spring constant 10 lb/ft and a 32-lb weight attached to it, we can set up the mass-spring-damper system as follows: \(mx''(t) + bx'(t) + kx(t) = F(t)\), where m is the mass, x(t) is the displacement, b is the damping constant, k is the spring constant, and F(t) is the external force. Given the information, \(m = 32 \text{ lb}\), \(b = 2 \text{ }(mass)'\), \(k = 10 \text{ lb/ft}\), \(F(t) = \sin(t) + \frac{1}{4} \sin(2t) + \frac{1}{9} \sin(3t)\).

Rewrite the equation using the given values

Now let's rewrite the equation using the given values of m, b, and k: \(32 x''(t) + 2x'(t) + 10x(t) = \sin(t) + \frac{1}{4} \sin(2t) + \frac{1}{9} \sin(3t)\).

Find the steady-state term using Theorem 4.10

According to Theorem 4.10, if we have a second-order linear constant-coefficient inhomogeneous differential equation with a sinusoidal forcing function, the steady-state displacement is given by: \(x_s(t) = \frac{1}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}F(\omega t)\), where ω is the angular frequency of the forcing function. For each sine term in our external force, we will calculate the amplitude of the steady-state displacement, and denote this as \(A_i\): For \(\sin(t)\), \(\omega = 1\), \(A_1 = \frac{1}{\sqrt{(10 - 32)^2 + (2)^2}} = \frac{1}{\sqrt{484}}\), For \(\frac{1}{4}\sin(2t)\), \(\omega = 2\), \(A_2 = \frac{\frac{1}{4}}{\sqrt{(10 - 128)^2 + (4)^2}} = \frac{1}{\sqrt{14164}}\), For \(\frac{1}{9}\sin(3t)\), \(\omega = 3\), \(A_3 = \frac{\frac{1}{9}}{\sqrt{(10 - 288)^2 + (6)^2}} = \frac{1}{\sqrt{78880}}\). So, the steady-state term, \(x_s(t)\) is: \(x_s(t) = A_1 \sin(t) + A_2 \sin(2t) + A_3 \sin(3t)\).

Simplify and find the displacement

Now let's simplify the steady-state term: \(x_s(t) = \frac{1}{\sqrt{484}} \sin(t) + \frac{1}{\sqrt{14164}} \sin(2t) + \frac{1}{\sqrt{78880}} \sin(3t)\). Thus, the displacement of the weight as a function of time is: \(x(t) = x_s(t) = \frac{1}{\sqrt{484}} \sin(t) + \frac{1}{\sqrt{14164}} \sin(2t) + \frac{1}{\sqrt{78880}} \sin(3t)\).

Key concepts

Mass-Spring-Damper System

The mass-spring-damper system is a classic example used in physics and engineering to illustrate how objects move under the influence of various forces. In this context, a mass is attached to a spring, and the whole system is subject to damping, which is resistance that opposes the motion.

When a weight is hung from a spring, the spring stretches until it reaches a point of equilibrium where the force of gravity on the weight is balanced by the force of the spring. This system can be characterized by a second-order linear differential equation, which is a powerful tool for describing the motion of the system over time.

The equation incorporates several key parameters:

• Mass (m): The object's weight, which in our example is 32 pounds.
• Damping constant (b): Represents the medium's resistance, which is proportional to the velocity. Here, it's twice the instantaneous velocity.
• Spring constant (k): The stiffness of the spring; in this case, it's 10 lb/ft.
• Displacement (x(t)): The position of the weight relative to the equilibrium position as a function of time.

By accounting for all these factors, we can model and predict the behavior of the system when subjected to an external force.

Differential Equation Application

Applying differential equations to physical systems allows us to understand and predict the behavior of those systems under various conditions. Differential equations represent the relationship between a function and its derivatives — rates at which things change.

In the context of our mass-spring-damper system, the differential equation models the displacement of the mass over time, given the forces acting on it. The general form of the equation is:
\[ mx''(t) + bx'(t) + kx(t) = F(t) \]
where:

• \( mx''(t) \) captures the mass's acceleration, also known as the inertial force.
• \( bx'(t) \) represents damping, the force that resists motion.
• \( kx(t) \) is the restoring force of the spring, attempting to bring the mass back to the equilibrium position.
• \( F(t) \) denotes the external force applied to the mass, which in our scenario is a complex combination of sinusoidal functions representing time-varying forces.

The deep understanding of each parameter's role in this equation is crucial for accurately solving for the displacement, \( x(t) \), over time.

Sinusoidal Forcing Function

A sinusoidal forcing function is a type of periodic force that varies with time in a sinusoidal manner — similar to the sine or cosine functions from trigonometry.

These forces are common in real-world scenarios, like the push-pull of alternating electrical currents or mechanical vibrations. When applied to a mass-spring-damper system, a sinusoidal force can cause the mass to oscillate. The general formula for such a force is described as \( F(t) = Asin(\theta t) \), where \( A \) is the amplitude, and \( \theta \) is the angular frequency of the sine wave.

Our exercise presents a more complex sinusoidal forcing function that combines several sine waves with different frequencies and amplitudes:
\[ F(t) = \text{sin}(t) + \frac{1}{4} \text{sin}(2t) + \frac{1}{9} \text{sin}(3t) \]
Applying this force to the mass-spring-damper system requires us to consider each component separately to find the steady-state response. This approach allows us to account for the cumulative effect of multiple frequencies on the system’s behavior, leading to a comprehensive solution for the weight’s displacement as a function of time.