Problem 16 - Advanced Engineering Mathematics

Question

If we assume that a damping force acts in a direction opposite to the motion of a pendulum and with a magnitude directly proportional to the angular velocity \(d \theta / d t\), the displacement angle \(\theta\) for the pendulum satisfies the nonlinear second-order differential equation $$ m l \frac{d^{2} \theta}{d t^{2}}=-m g \sin \theta-\beta \frac{d \theta}{d t}. $$ (a) Write the second-order differential equation as a plane autonomous system, and find all critical points. (b) Find a condition on \(m, l\), and \(\beta\) that will make \((0,0)\) a stable spiral point.

Step-by-Step Solution

Answer

Critical points: \((n\pi, 0)\); Stable spiral if \(\beta^2 < 4mgl\).

Step 1: Rewrite the Equation

First, rewrite the given second-order nonlinear differential equation as a first-order system. To do this, we introduce a new variable, say \( heta_1 = \theta \), and let \( heta_2 = \frac{d\theta}{dt} \). Then the given differential equation can be expressed as two first-order equations. The system becomes: \[\frac{d\theta_1}{dt} = \theta_2\]\[\frac{d\theta_2}{dt} = -\frac{g}{l} \sin(\theta_1) - \frac{\beta}{ml} \theta_2\]This is an autonomous system because it does not explicitly depend on time.

Step 2: Find Critical Points

To find the critical points for the system, set all derivatives equal to zero and solve for \( \theta_1 \) and \( \theta_2 \). From the equations:\( \theta_2 = 0 \)and\( -\frac{g}{l} \sin(\theta_1) - \frac{\beta}{ml} \theta_2 = 0 \)Substituting \( \theta_2 = 0 \) in the second equation, we obtain:\( -\frac{g}{l} \sin(\theta_1) = 0 \)Thus, critical points are where \( \sin(\theta_1) = 0 \), implying \( \theta_1 = n\pi \), where \( n \) is an integer. Therefore, the critical points are \( (n\pi, 0) \) for any integer \( n \).

Step 3: Analyze Stability at (0,0)

To determine the stability of the point \((0,0)\), we need to linearize the system around this critical point and find the eigenvalues of the Jacobian matrix. The Jacobian matrix for the system is:\[J = \begin{pmatrix}0 & 1 \-\frac{g}{l} \cos(\theta_1) & -\frac{\beta}{ml}\end{pmatrix}\]Evaluating at \((0,0)\), where \( \cos(0) = 1 \), we have:\[J = \begin{pmatrix}0 & 1 \-\frac{g}{l} & -\frac{\beta}{ml}\end{pmatrix}\]The characteristic equation is:\[ \lambda^2 + \frac{\beta}{ml}\lambda + \frac{g}{l} = 0\]

Step 4: Determine Stability of (0,0)

To ensure \((0,0)\) is a stable spiral point, the roots of the characteristic equation must be complex with negative real parts. For complex roots, the discriminant of the quadratic equation needs to be negative:\[\left( \frac{\beta}{ml} \right)^2 - 4\frac{g}{l} < 0\]This simplifies to:\[\beta^2 < 4mgl\]This inequality provides the condition on \(m, l, \beta\) for \((0,0)\) to be a stable spiral point.

Key Concepts

Critical Points

Understanding critical points in nonlinear differential equations is crucial. They represent conditions where the system's state does not change over time, meaning the derivatives are zero at these points. Finding them helps us understand the behavior of the system.

For this pendulum problem, we began by rewriting the given nonlinear differential equation into a set of first-order equations using a substitution method. By introducing new variables for the angle \( \theta_1 \) and its velocity \( \theta_2 = \frac{d\theta}{dt} \), we obtained:

• \(\frac{d\theta_1}{dt} = \theta_2 \)
• \(\frac{d\theta_2}{dt} = -\frac{g}{l} \sin(\theta_1) - \frac{\beta}{ml} \theta_2 \)

To find the critical points, both derivatives were set to zero. This process revealed that critical points occur when \(\sin(\theta_1) = 0\), which simplifies to \(\theta_1 = n\pi\), where \(n\) is an integer. This results in critical points such as \((n\pi, 0)\), indicating states of the system where no change occurs.

Stability Analysis

Stability analysis tells us how a system behaves near its critical points. It's about checking if small disturbances will grow or shrink over time. In the context of the pendulum problem, we want to find out if the point \((0,0)\) is stable.

This involves linearizing the system around the critical point and examining the Jacobian matrix. The Jacobian is a matrix of derivatives that reflects how the system changes near this point. For our system, the Jacobian is:\[J = \begin{pmatrix}0 & 1 \ -\frac{g}{l} & -\frac{\beta}{ml}\end{pmatrix}\]Evaluating its eigenvalues determines stability. The characteristic equation, derived from the Jacobian, is:\[\lambda^2 + \frac{\beta}{ml}\lambda + \frac{g}{l} = 0\]For \((0,0)\) to be a stable spiral, the roots of this equation must be complex with negative real parts, ensuring that disturbances die out over time. By ensuring the discriminant is negative, we confirm the roots are indeed complex. The critical condition \(\beta^2 < 4mgl\) ensures stability.

Autonomous Systems

In differential equations, autonomous systems don't explicitly depend on time. This means the evolution of the system is determined entirely by its current state. Such systems are common in physics and engineering, where behavior does not change over time due to subject-specific external conditions.

For this pendulum, writing our problem as a system of first-order equations made it clear that it was autonomous. The dependencies in the system are solely on the state variables, \(\theta_1\) and \(\theta_2\), not on any time variable.
Key advantages of analyzing autonomous systems include:

• Predicting long-term behavior is simpler.
• They often have conserved quantities or invariants, making them easier to study.
• They frequently appear in natural phenomena, such as in pendulums or planets orbiting stars.

In our solution, identifying the structure of the system allowed us to conduct a stability analysis and gain insights into the motion of the pendulum without time-driven changes.