Question

The differential equation for the motion of a unit mass on a certain coil spring under the action of an external force of the form \(F(t)=30 \cos \omega t\) is $$ x^{\prime \prime}+a x^{\prime}+24 x=30 \cos \omega t $$ where \(a \geq 0\) is the damping coefficient. (a) Graph the resonance curves of the system for \(a=0,2,4,6\), and \(4 \sqrt{3}\). (b) If \(a=4\), find the resonance frequency and determine the amplitude of the steady-state vibration when the forcing function is in resonance with the system. (c) Proceed as in part (b) if \(a=2\).

Short answer

The resonance curves for the system with damping coefficients $a=0,2,4,6$, and $4\sqrt{3}$ show how the amplitude of the steady-state solution changes with the frequency of the external force. For $a=4$, the resonance frequency is $\omega = \pm\sqrt{80}$, and the amplitude of the steady-state vibration is $\frac{15}{28}$. For $a=2$, the resonance frequency is $\omega = \pm\sqrt{40}$, and the amplitude of the steady-state vibration is $\frac{15}{8}$.

Step-by-step solution

Analyze the given differential equation

The given differential equation is: \[ x'' + ax' + 24x = 30 \cos{\omega}t \] We can rewrite the above equation as: \[ x''(t) + ax'(t) + 24x(t) = 30 \cos{\omega}t \] This is a second-order linear nonhomogeneous differential equation with constant coefficients and a periodic external force.

Finding the steady-state solution

To find the steady-state solution, we assume a particular solution of the form: \[ x_p(t) = C \cos{(\omega t - \delta)} \] where C and δ are constants to be determined. Now, substitute \(x_p(t)\) and its derivatives into the given differential equation: \[ (-C\omega^2 \cos{(\omega t - \delta)}) + a(-C\omega \sin{(\omega t - \delta)}) + 24(C\cos{(\omega t - \delta)}) = 30\cos{\omega t} \] We can now rewrite the equation as: \[ C((-24+{\omega}^2)\cos{(\omega t - \delta)} - a\omega\sin{(\omega t - \delta)}) = 30\cos{(\omega t)} \]

Match terms to solve for C and δ

To find C and δ, we match the coefficients of \(\cos{(\omega t - \delta)}\) and \(\sin{(\omega t - \delta)}\) on both sides of the equation. We get, \[ C(-24 + \omega^2) = 30 \quad \textrm{and} \quad - Ca\omega = 0 \] Now, to find resonance conditions, we have to avoid dividing by 0. So, \(a\omega \neq 0\), we can now find \(C = \frac{30}{(-24 + \omega^2) }\). The second equation gives us that either C = 0 or \(\delta = \pm \frac{\pi}{2}\).

Analyzing resonance curves for different damping coefficients

Now, we'll analyze resonance curves for a = 0, 2, 4, 6, and 4\(\sqrt{3}\). - For a = 0, the differential equation becomes: \[ x'' + 24x = 30 \cos{\omega}t \] - For a = 2, the differential equation becomes: \[ x''+ 2x' + 24x = 30 \cos{\omega}t \] - For a = 4, the differential equation becomes: \[ x''+ 4x' + 24x = 30 \cos{\omega}t \] - For a = 6, the differential equation becomes: \[ x''+ 6x' + 24x = 30 \cos{\omega}t \] - For a = \(4\sqrt{3}\), the differential equation becomes: \[ x''+ 4\sqrt{3}x' + 24x = 30 \cos{\omega}t \]

Solving part (b) for a = 4

We are given that \(a=4\), and we want to find the resonance frequency and amplitude of the steady-state vibration. We can use \(C = \frac{30}{(-24 + \omega^2)}\). Now, solve for \(\omega^2\): \[ \omega^2 = 4((-24 + \omega^2) + 4\omega) \implies \omega^2 = 4(20 + 4\omega) \] Solve for \(\omega\): \[ \omega = \pm \sqrt{80} \] Now, find C: \[ C = \frac{30}{(-24 + 80)} = \frac{30}{56} = \frac{15}{28} \] So, the amplitude of the steady-state vibration is \(\frac{15}{28}\).

Solving part (c) for a = 2

We are given that \(a=2\), and we want to find the resonance frequency and amplitude of the steady-state vibration. We can use \(C = \frac{30}{(-24 + \omega^2)}\). Now, solve for \(\omega^2\): \[ \omega^2 = 2((-24 + \omega^2) + 2\omega) \implies \omega^2 = 2(20 + 2\omega) \] Solve for \(\omega\): \[ \omega = \pm \sqrt{40} \] Now, find C: \[ C = \frac{30}{(-24 + 40)} = \frac{30}{16} = \frac{15}{8} \] So, the amplitude of the steady-state vibration is \(\frac{15}{8}\).

Key concepts

Resonance Frequency

Resonance frequency is a fundamental concept in the study of vibrations and differential equations. It refers to the frequency at which a system naturally oscillates with the greatest amplitude. When an external force matches this natural frequency, resonance occurs, causing a significant increase in amplitude.
In the given exercise, the system is driven by a force expressed as \(F(t)=30 \cos \omega t\). This external force has a frequency \(\omega\), which can potentially resonate with the system's natural frequency. The equation describes the behavior of a mass-spring-damper system, where resonance must be carefully managed.

• If damping is small, as in systems with minimal friction ("underdamped"), the resonance frequency is close to the natural frequency of the system.
• For our exercises with damping coefficients \(a=2\) and \(a=4\), the resonance frequency was calculated by setting the forcing frequency \(\omega\) to the value that makes the amplitude maximized.

Understanding and calculating resonance frequency ensures that systems are operated safely without reaching excessively high amplitudes that could cause damage or failure.

Damping Coefficient

The damping coefficient \(a\) determines how quickly a vibrating system loses energy over time. Essentially, it reflects the system's ability to remove or "damp" vibrations through various dissipative forces, like friction or resistance.
The exercise involves multiple damping scenarios. For instance, different values of \(a\) are explored—including 0, 2, 4, 6, and \(4\sqrt{3}\)—to observe how they affect the system's resonance and steady-state response.

• With a damping coefficient of 0 (\(a=0\)), the system experiences no resistance to motion, making it ideal for studying pure resonance but impractical in real-world applications.
• Higher values (like \(a=4\) used in the exercise) show significant damping, reducing amplitude and helping avoid dangerously high vibrations at resonance.

By adjusting \(a\), you can control how much energy is lost over time and tailor the system's response to suit specific requirements, balancing between strong resilience and effective energy dissipation.

Steady-State Vibration

In vibrations, steady-state refers to the consistent, ongoing oscillation that the system achieves after transient effects have dissipated. In simpler terms, the steady-state vibration is the regular pattern that the system settles into after some time.
For our exercise, the steady-state solution is expressed by a particular form of the solution to the differential equation, where constants like \(C\) and \(\delta\) are determined to match the equation's conditions.

• The steady-state solution assumes a harmonic form, such as \(x_p(t) = C \cos{(\omega t - \delta)}\), which mirrors the form of the external forcing function.
• For each damping coefficient, solving for the amplitude \(C\) when under resonant conditions provides insight into the system’s stable response, avoiding the initial transient oscillations.

Understanding steady-state vibration is crucial because it represents the long-term behavior of systems under periodic forces, commonly analyzed in engineering and physics to ensure system stability and reliability.

Nonhomogeneous Differential Equation

The given differential equation \(x'' + ax' + 24x = 30 \cos \omega t\) is an example of a nonhomogeneous differential equation. This type of equation includes a "non-zero" forcing function, resulting in a system that is influenced by an external factor rather than just internal dynamics.
In this context, the fundamental difference between homogeneous and nonhomogeneous equations is the presence of the external force term \(30 \cos \omega t\). This term makes the solution of the equation more complex because it must account for the impact of external vibrations on the overall system response.

• A homogeneous equation lacks this external term, focusing solely on the system's natural response.
• Solving nonhomogeneous equations typically involves finding a particular solution, which directly addresses the forcing function's effects.

By exploring how systems respond to external forces through nonhomogeneous differential equations, one gains insight into real-world applications where external influences regularly interact with natural system behavior.