Problem 25 - Advanced Engineering Mathematics

Question

Consider a pendulum that is released from rest from an initial displacement of \(\theta_{0}\) radians. Solving the linear model (7) subject to the initial conditions \(\theta(0)=\theta_{0}, \theta^{\prime}(0)=0\) gives \(\theta(t)=\theta_{0} \cos \sqrt{g} \| t .\) Theperiod of oscillations predicted by this modelisgivenbythefamiliarformula \(T=2 \pi / \sqrt{g l l}=2 \pi \sqrt{U g}\). The interesting thing about this formula for \(T\) is that it does not depend on the magnitude of the initial displacement \(\theta_{0}\). In other words, the linear model predicts that the time that it would take the pendulum to swing from an initial displacement of, say, \(\theta_{0}=\pi / 2\left(=90^{\circ}\right)\) to \(-\pi / 2\) and back again would be exactly the same time to cycle from, say, \(\theta_{0}=\pi / 360\left(=0.5^{\circ}\right)\) to \(-\pi / 360\). This is intuitively unreasonable; the actual period must depend on \(\theta_{0}\). If we assume that \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) and \(l=32 \mathrm{ft}\), then the period of oscillation of the linear model is \(T=2 \pi \mathrm{s}\). Let us compare this last number with the period predicted by the nonlinear model when \(\theta_{0}=\pi / 4\). Using a numerical solver that is capable of generating hard data, approximate the solution of $$\frac{d^{2} \theta}{d t^{2}}+\sin \theta=0, \quad \theta(0)=\frac{\pi}{4}, \quad \theta^{\prime}(0)=0$$ for \(0 \leq t \leq 2\). As in Problem 24 , if \(t_{1}\) denotes the first time the pendulum reaches the position \(O P\) in Figure 3.11.3, then the period of the nonlinear pendulum is \(4 t_{1} .\) Here is another way of solving the equation \(\theta(t)=0\). Expeniment with small step sizes and advance the time staning at \(t=0\) and ending at \(t=2\). From your hard data, observe the time \(t_{1}\) when \(\theta(t)\) changes, for the first time, from positive to negative. Use the value \(t_{1}\) to determine the true value of the period of the nonlinear pendulum. Compute the percentage relative error in the period estimated by \(T=2 \pi\).

Step-by-Step Solution

Answer

Use numerical data to find the time of zero crossing (\(t_1\)) and compute the nonlinear period as \(4t_1\). Calculate the error as a percentage from \(T = 2\pi\).

Step 1: Setup the differential equation

Begin with the nonlinear differential equation given in the exercise:\[ \frac{d^2 \theta}{dt^2} + \sin \theta = 0 \]This equation represents the motion of a pendulum with an initial angle \(\theta(0) = \frac{\pi}{4}\) and an initial velocity \(\theta'(0) = 0\).

Step 2: Use a numerical solver

Since this is a nonlinear differential equation, we need to apply a numerical solver. Tools like Runge-Kutta or Euler's method can be employed to approximate the solution. Implement or use software to numerically solve \(\theta(t)\) over the interval \(0 \leq t \leq 2\).

Step 3: Identify the time of zero crossing

With your data from the numerical solution, observe when \(\theta(t)\) first crosses zero, i.e., moves from a positive to a negative value. This time is labeled as \(t_1\).

Step 4: Calculate the nonlinear period

The period of the pendulum based on the nonlinear model is four times the zero-crossing time:\[ T_{\text{nonlinear}} = 4t_1 \]

Step 5: Compare the periods

Compare the computed \(T_{\text{nonlinear}}\) with the linear period \(T = 2\pi\). Calculate the percentage relative error using the formula:\[ \text{Percentage Relative Error} = \left(\frac{T_{\text{nonlinear}} - 2\pi}{2\pi}\right) \times 100\% \]

Step 6: Final interpretation

Ensure that the calculated percentage relative error is consistent with expected discrepancies between linear and nonlinear modeling, given different initial displacements \(\theta_0\).

Key Concepts

Pendulum Motion

Pendulum motion is a classic example of a mechanical system that exhibits simple harmonic behavior under small oscillations. A pendulum consists of a weight, or bob, attached to a string or rod of length \( l \), which is fixed at a pivot point. When the pendulum is displaced from its resting position and released, it swings back and forth due to the gravitational force acting upon it.

For small angles, the motion of the pendulum can be approximated by a linear differential equation, leading to simple sinusoidal wave solutions that predict constant oscillation periods. However, as the angle becomes larger, this linear approximation fails, and the behavior becomes distinctly nonlinear. Nonlinear dynamics are characterized by more complex motion, where the period of oscillation is dependent on the initial angle of displacement, \( \theta_0 \). This is where numerical methods are essential for understanding the pendulum's true motion.

Numerical Solvers

Numerical solvers are algorithms or methods used to approximately solve differential equations, especially when analytical solutions are difficult or impossible to obtain. In the context of pendulum motion, numerical solvers are crucial for modeling the nonlinear differential equation \( \frac{d^2 \theta}{dt^2} + \sin \theta = 0 \).

These solvers, such as the Euler method or the Runge-Kutta method, allow us to simulate the pendulum's trajectory by computing successive approximations. By using a numerical solver, you can explore the pendulum's behavior over time and capture the nuances of its nonlinear dynamics. When using numerical solvers, the choice of algorithm and step size can significantly affect the accuracy and efficiency of the simulation. Smaller step sizes generally yield more precise solutions, but they also require more computational power.

Runge-Kutta Method

The Runge-Kutta method, particularly the fourth-order version (RK4), is one of the most popular techniques for solving ordinary differential equations numerically. It is known for its ability to provide highly accurate solutions without excessively small step sizes.

In the case of a pendulum's nonlinear motion, the Runge-Kutta method helps numerically approximate \( \theta(t) \) by taking weighted averages of slopes at various points within a step interval. This approach minimizes errors over each step, making it more precise than simpler methods like Euler's. Implementing the Runge-Kutta method involves calculating multiple estimates of new points and combining them to reach a final value, specifically crafted to balance well between accuracy and computational cost. It is particularly effective for problems with non-stiff dynamics, like the swinging pendulum.

Sinusoidal Functions

Sinusoidal functions arise naturally in the context of pendulum motion, especially under the linear approximation. For small angles, the solution to the pendulum's differential equation resembles a cosine or sine function, leading to predictable, periodic, and simple oscillatory behavior.

These functions have the form \( \theta(t) = \theta_0 \cos(\omega t) \) or \( \theta(t) = \theta_0 \sin(\omega t) \), where \( \omega \) is the angular frequency. Sinusoidal functions are fundamental in analyzing wave motions and harmonic oscillations due to their repetitive pattern and ability to describe smooth periodic effects.

However, the elegance of sinusoidal functions diminishes for larger angles in pendulum motion. The nonlinear effects become significant, and the motion deviates from a simple sine or cosine form. Understanding and modeling these real-world oscillations often requires numerical solvers to address the complexities beyond the basic sinusoidal waves.