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Chapter 9: Problem 23

This problem extends Problem 22 to a damped pendulum . The equations of motion are $$ d x / d t=y, \quad d y / d t=-4 \sin x-\gamma y . $$ where \(\gamma\) is the damping coefficient, with the initial conditions \(x(0)=0, y(0)=v\) (a) For \(\gamma=1 / 4\) plot \(x\) versus \(t\) for \(v=2\) and for \(v=5 .\) Explain these plots in terms of the motions of the pendulum that they represent. Also explain how they relate to the corresponding graphs in Problem 22 (a). (b) Estimate the critical value \(v_{c}\) of the initial velocity where the transition from one type of motion to the other occurs. (c) Repeat part (b) for other values of \(\gamma\) and determine how \(v_{c}\) depends on \(\gamma\).

Short Answer

Expert verified
Question: Explain the physical meaning of the plots of x vs t for a damped pendulum with initial velocities v=2 and v=5, and a damping coefficient of γ=1/4. Compare these results with the non-damped pendulum results in Problem 22(a), and discuss the dependence of the critical initial velocity vc on the damping coefficient γ.

Step by step solution

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01

Understand the given problem

We are given the equations of motion for a damped pendulum: dx/dt = y dy/dt = -4sin(x) - γy with initial conditions x(0) = 0 and y(0) = v. Here, x represents the displacement of the pendulum, and y is the rate of change of displacement. γ is the damping coefficient.
02

Plot x vs t for given parameters

We have to plot x versus t for γ = 1/4 and two initial velocities, v=2 and v=5. We can use numerical methods, such as the Euler method or the Runge-Kutta method, to solve the given differential equations. Using any programming language or software that you are comfortable with (Python, MATLAB, Mathematica), you can plot x vs t for the given parameters.
03

Explain the physical meaning of the plots and compare with the non-damped pendulum

After plotting x vs t for v=2 and v=5, analyze the plots to understand the behavior of the pendulum in each case. For v=2, observe whether pendulum oscillation is gradually decreasing over time or not, which represents a simple damped oscillation. In the case of v=5, analyze whether the pendulum moves through multiple cycles, indicating chaotic motion. Compare these plots to the non-damped pendulum plots in Problem 22(a) and examine how damping affects the motion of the pendulum in each case.
04

Estimate the critical initial velocity, vc

To find the critical initial velocity vc, we can use trial and error approach by trying different initial velocities and plotting x versus t. Observe the transition from damped oscillation to chaotic motion. The critical initial velocity vc is the value where the transition between simple damped oscillation and chaotic motion occurs.
05

Repeat part (b) for different values of γ to find the relationship between vc and γ

Repeat part (b) but vary the damping constant γ and calculate the critical initial velocity vc for each value of γ. To find the relationship between vc and γ, plot the values of vc with respect to different γ values. Analyze the graph to observe the pattern or relationship between these two variables. Now we have explained the physical meaning of the plots, compared them with non-damped pendulum plots in Problem 22(a), found the critical initial velocity vc for a given damping coefficient γ, and determined the relationship between vc and γ.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Damping Coefficient
In the context of a damped pendulum, the damping coefficient, represented by the symbol \( \gamma \), is a measure of the force that opposes the motion of the pendulum due to friction or air resistance. This force effectively 'damps' or reduces the amplitude of the oscillations over time, leading to a gradual decrease in the motion of the pendulum until it comes to a stop.

Mathematically, the damping coefficient appears in the equation of motion as a factor multiplied by the velocity term (\( dy/dt \)). It determines how quickly the oscillations' amplitude decreases. A higher value of \( \gamma \), in general, would mean a faster decay of the pendulum's motion, pointing towards more significant resistance. In the exercises, analyzing the effects of varying \( \gamma \) helps students understand how different levels of damping influence the pendulum's behavior.
Critical Initial Velocity
The critical initial velocity, often denoted \( v_c \), in the context of a damped pendulum oscillator, is the minimum velocity that the pendulum must have at the beginning of its motion to exhibit a particular type of behavior, such as moving from a damped oscillatory motion to an underdamped or chaotic condition.

To estimate this critical value, one might employ a numerical experimentation approach, gradually increasing the initial velocity and observing the resulting motion. When the pendulum transitions from predictable (sinusoidal) oscillations to a more erratic pattern, that initial velocity is designated as the critical initial velocity \( v_c \). This concept is crucial for not only theoretical physics but also in practical engineering and technology applications where understanding motion thresholds is key.
Damped Oscillation Analysis
The study of a damped oscillation involves examining how the amplitude and phase of an oscillating system, such as a pendulum, change over time due to the presence of damping forces. The damped pendulum differs from an ideal undamped pendulum because it incorporates the resistance that gradually removes energy from the system.

In the given exercises, analyzing damped oscillation requires us to visually interpret the graphs of displacement versus time, plotting the motion for different initial conditions and damping coefficients. This analysis reveals how the pendulum starts from its initial velocity, oscillates with decreasing amplitude, and eventually comes to rest, exhibiting a behavior governed by a combination of sinusoidal motion influenced by an exponential decay, which is the hallmark of damped systems.
Runge-Kutta Method
The Runge-Kutta method is a sophisticated numerical approach used widely to solve ordinary differential equations (ODEs) that cannot be solved analytically. It's more accurate than simpler methods like Euler's because it takes multiple evaluations of the derivative at different points within the given step interval.

In the exercise, we would use this method to calculate successive points in the motion of the damped pendulum by estimating the displacement and velocity at small time intervals. This technique is particularly useful when you want to ensure higher accuracy in the results, which is critical when examining the behavior near the critical initial velocity where the pendulum's motion transitions from predictable to more complex.
Euler Method
Another numerical technique for solving differential equations is the Euler method. It is simpler than the Runge-Kutta method and requires fewer calculations per step. The method approximates solutions to ODEs by using tangent line slopes to project the next value in the series.

While the Euler method is less accurate and more susceptible to errors, especially over longer timescales or more complex equations, it is still a valuable tool for initial investigations or when computational resources are limited. In our exercise, employing the Euler method could help students grasp the fundamental behavior of the damped pendulum's motion without delving into more complex calculations.

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Most popular questions from this chapter

Consider the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x},\) and suppose that \(\mathbf{A}\) has one zero eigenvalue. (a) Show that \(\mathbf{x}=\mathbf{0}\) is a critical point, and that, in addition, every point on a certain straight line through the origin is also a critical point. (b) Let \(r_{1}=0\) and \(r_{2} \neq 0,\) and let \(\boldsymbol{\xi}^{(1)}\) and \(\boldsymbol{\xi}^{(2)}\) be corresponding eigenvectors. Show that the trajectories are as indicated in Figure \(9.1 .8 .\) What is the direction of motion on the trajectories?

Prove that for the system $$ d x / d t=F(x, y), \quad d y / d t=G(x, y) $$ there is at most one trajectory passing through a given point \(\left(x_{0}, y_{0}\right)\) Hint: Let \(C_{0}\) be the trajectory generated by the solution \(x=\phi_{0}(t), y=\psi_{0}(t),\) with \(\phi_{0}\left(l_{0}\right)=\) \(x_{0}, \psi_{0}\left(t_{0}\right)=y_{0},\) and let \(C_{1}\) be trajectory generated by the solution \(x=\phi_{1}(t), y=\psi_{1}(t)\) with \(\phi_{1}\left(t_{1}\right)=x_{0}, \psi_{1}\left(t_{1}\right)=y_{0}\). Use the fact that the system is autonomous and also the existence and uniqueness theorem to show that \(C_{0}\) and \(C_{1}\) are the same.

By introducing suitable dimensionless variables, the system of nonlinear equations for the damped pendulum [Frqs. (8) of Section 9.3] can be written as $$ d x / d t=y, \quad d y / d t=-y-\sin x \text { . } $$ (a) Show that the origin is a critical point. (b) Show that while \(V(x, y)=x^{2}+y^{2}\) is positive definite, \(f(x, y)\) takes on both positive and negative values in any domain containing the origin, so that \(V\) is not a Liapunov function. Hint: \(x-\sin x>0\) for \(x>0\) and \(x-\sin x<0\) for \(x<0 .\) Consider these cases with \(y\) positive but \(y\) so small that \(y^{2}\) can be ignored compared to \(y .\) (c) Using the energy function \(V(x, y)=\frac{1}{2} y^{2}+(1-\cos x)\) mentioned in Problem \(6(b),\) show that the origin is a stable critical point. Note, however, that even though there is damping and we can epect that the origin is asymptotically stable, it is not possible to draw this conclusion using this Liapunov function. (d) To show asymptotic stability it is necessary to construct a better Liapunov function than the one used in part (c). Show that \(V(x, y)=\frac{1}{2}(x+y)^{2}+x^{2}+\frac{1}{2} y^{2}\) is such a Liapunov function, and conclude that the origin is an asymptotically stable critical point. Hint: From Taylor's formula with a remainder it follows that \(\sin x=x-\alpha x^{3} / 3 !,\) where \(\alpha\) depends on \(x\) but \(0<\alpha<1\) for \(-\pi / 2

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=x+x^{2}+y^{2}, \quad d y / d t=y-x y $$

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=y+x\left(1-x^{2}-y^{2}\right), \quad d y / d t=-x+y\left(1-x^{2}-y^{2}\right) $$
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