Problem 43 - Advanced Engineering Mathematics

Question

(a) Show that the general solution of $$ \frac{d^{2} x}{d t^{2}}+2 \lambda \frac{d x}{d t}+\omega^{2} x=F_{0} \sin \gamma t $$ is $$ \begin{aligned} x(t)=& A e^{-\lambda t} \sin \left(\sqrt{\omega^{2}-\lambda^{2} t}+\phi\right) \\\ &+\frac{F_{0}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \sin (\gamma t+\theta) \end{aligned} $$ where \(A=\sqrt{c_{1}^{2}+c_{2}^{2}}\) and the phase angles \(\phi\) and \(\theta\) are, respectively, defined by \(\sin \phi=c_{1} / A, \cos \phi=c_{2} / A\) and $$ \begin{aligned} &\sin \theta=\frac{-2 \lambda \gamma}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \\ &\cos \theta=\frac{\omega^{2}-\gamma^{2}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \end{aligned} $$ (b) The solution in part (a) has the form \(x(t)=x_{c}(t)+x_{p}(t)\). Inspection shows that \(x_{e}(t)\) is ransient, and hence for large values of time, the solution is approximated by \(x_{p}(t)=\) \(g(\gamma) \sin (\gamma t+\theta)\), where $$ g(\gamma)=\frac{F_{0}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} $$ Although the amplitude \(g(\gamma)\) of \(x_{p}(t)\) is bounded as \(t \rightarrow \infty\), show that the maximum oscillations will occur at the value \(\gamma_{1}=\sqrt{\omega^{2}-2 \lambda^{2}}\). What is the maximum value of \(g\) ? The number \(\sqrt{\omega^{2}-2 \lambda^{2}} / 2 \pi\) is said to be the resonance frequency of the system. (c) When \(F_{0}=2, m=1\), and \(k=4, g\) becomes $$ g(\gamma)=\frac{2}{\sqrt{\left(4-\gamma^{2}\right)^{2}+\beta^{2} \gamma^{2}}} $$ Construct a table of the values of \(\gamma_{1}\) and \(g\left(\gamma_{1}\right)\) corresponding to the damping coefficients \(\beta=2, \beta=1, \beta=\frac{3}{4}, \beta=\frac{1}{2}\), and \(\beta=\frac{1}{4}\). Use a graphing utility to obtain the graphs of \(g\) corresponding to these damping coefficients. Use the same coordinate axes. This family of graphs is called the resonance curve or frequency response curve of the system. What is \(\gamma_{1}\) approaching as \(\beta \rightarrow 0\) ? What is happening to the resonance curve as \(\beta \rightarrow 0\) ?

Step-by-Step Solution

Answer

The maximum oscillations occur at \( \gamma_{1} = \sqrt{\omega^2 - 2\lambda^{2}} \), where the system resonates, and \( \gamma_{1} \) approaches \( \omega \) as \( \beta \to 0 \).

Step 1: Identify and solve the homogeneous solution

The differential equation is \( \frac{d^{2} x}{d t^{2}} + 2\lambda \frac{d x}{d t} + \omega^{2}x = 0 \) for the homogeneous part. Assume a solution of the form \( x_{h}(t) = e^{rt} \). Solving the characteristic equation \( r^{2} + 2\lambda r + \omega^{2} = 0 \) gives roots \( r = -\lambda \pm i\sqrt{\omega^2 - \lambda^2} \). The homogeneous solution is \( x_{h}(t) = e^{-\lambda t} (c_{1} \cos(\sqrt{\omega^2 - \lambda^2}t) + c_{2} \sin(\sqrt{\omega^2 - \lambda^2}t)) \)."},{

Step 2: Solve for Particular Solution

We assume a particular solution of the form \( x_{p}(t) = C \sin(\gamma t) + D \cos(\gamma t) \). Substitute \( x_{p} \) into the non-homogeneous differential equation and solve for constants \( C \) and \( D \). After simplification and substitution, we find that \( C \) and \( D \) provide the solution component which will include a phase shift \( \theta \) that satisfies the given trigonometric definitions.

Step 3: Combine Solutions for General Solution

Combine the homogeneous and particular solutions to get the general solution: \( x(t) = x_{h}(t) + x_{p}(t) \). Use the given relations to express \( A \), \( \phi \), and \( \theta \) in the context provided: \( A = \sqrt{c_{1}^{2} + c_{2}^{2}} \) and angles defined by their sine and cosine terms. The general solution is: \[ x(t) = A e^{-\lambda t} \sin(\sqrt{\omega^2 - \lambda^2} t + \phi) + \frac{F_{0}}{\sqrt{(\omega^{2} - \gamma^{2})^{2} + 4 \lambda^{2} \gamma^{2}}} \sin(\gamma t + \theta) \]."

Step 4: Determine Maximum Amplitude and Resonance Frequency

The maximum oscillations occur when the function's amplitude is maximized. Given the expression for \( g(\gamma) \), differentiate and equate to zero to find critical points. The maximum occurs at \( \gamma_{1} = \sqrt{\omega^2 - 2\lambda^{2}} \). The maximum value of \( g(\gamma) \) is determined at this \( \gamma \). "},{

Step 5: Conduct Numerical Analysis for Specific Parameters

For part (c), substitute \( F_{0} = 2, m = 1, \text{ and } k = 4 \) into the expression for \( g(\gamma) \) given the values for \( \beta \) and compute \( \gamma_{1} \) for each condition. Create a table with \( \gamma_{1} \) and corresponding \( g(\gamma_{1}) \). Use a graphing tool to visualize how \( g \) changes with \( \beta \). As \( \beta \to 0 \), observe that \( \gamma_{1} \) approaches \( \omega \), reflecting the enhanced response at less damping. "}],

Step 6: Short Answer

The maximum oscillations occur at \( \gamma_{1} = \sqrt{\omega^2 - 2\lambda^{2}} \), where the system resonates, and \( \gamma_{1} \) approaches \( \omega \) as \( \beta \to 0 \).

Key Concepts

Resonance

In the world of differential equations, resonance refers to a condition where a system experiences maximum oscillation amplitude due to the frequency of the external force matching the system's natural frequency. This phenomenon occurs because the system is absorbing energy in a way that its oscillations increase over time. For a driven harmonic oscillator, given by the differential equation \( \dfrac{d^{2} x}{d t^{2}} + 2 \lambda \dfrac{d x}{d t} + \omega^{2} x = F_{0} \sin \gamma t \), resonance happens at a particular frequency. This frequency, known as the resonance frequency \( \gamma_{1} \), is calculated by \( \sqrt{\omega^{2} - 2\lambda^{2}} \).

• The resonance frequency tells us at what frequency the external force needs to oscillate to achieve the largest possible amplitude.
• Understanding resonance helps in designing systems like bridges and buildings, where uncontrolled resonances could cause structural failures.

When \( \beta \to 0 \), meaning reduced damping, the system’s resonant frequency becomes closer to the natural frequency of the undamped system \( \omega \). This alignment amplifies the oscillation significantly, highlighting the importance of managing resonance in engineering and physics.

Damping Coefficient

The damping coefficient, denoted as \( \lambda \), is a parameter that quantifies the extent to which oscillations decrease in a system. It acts as a counter-effect to resonance, helping control or restrict excessive oscillations that could lead to damage or system failure. In a differential equation, damping is seen in the term \( 2\lambda \dfrac{d x}{d t} \). Here’s why it matters:

• Critical damping precisely stops oscillations in the shortest possible time without oscillating.
• Over-damping results in no oscillations; the system returns to equilibrium slowly.
• Under-damping lets the system oscillate, but these oscillations gradually decrease over time.

By adjusting \( \lambda \), engineers can design systems to behave as needed under different conditions. In cases where minimal damping is desired, the system might respond sharply to inputs like shocks or jolts. Meanwhile, a high damping coefficient ensures stability by preventing oscillation overtaking behavior.

Characteristic Equation

When solving differential equations, the characteristic equation is crucial for finding solutions to the homogeneous part of the equation. It’s derived by assuming a solution of the form \( x(t) = e^{rt} \) for the homogeneous equation \( \frac{d^{2} x}{d t^{2}} + 2\lambda \frac{d x}{d t} + \omega^{2} x = 0 \). Substituting this form into the equation yields the characteristic equation: \( r^{2} + 2\lambda r + \omega^{2} = 0 \).

• This equation is quadratic, and its roots \( r = -\lambda \pm i\sqrt{\omega^{2} - \lambda^{2}} \) represent the behavior of the system’s oscillations.
• The roots can indicate if the system is under-damped, over-damped, or critically damped, based on whether the discriminant \( \omega^{2} - \lambda^{2} \) is positive, negative, or zero.

Solving the characteristic equation provides the fundamental nature of the oscillations, which combines with particular solutions to give the complete solution of the differential system. Understanding this process is essential for analyzing dynamic systems across engineering and physics disciplines.