11: The Damped, Driven Pendulum
The simple pendulum is the mathematical idealization of a frictionless pendulum. We now consider the effects of friction as well as an externally imposed periodic force. The frictional force is modeled as
\[F_{f}=-\gamma l \dot{\theta}, \nonumber \]
where the frictional force is opposite in sign to the velocity, and thus opposes motion. The positive parameter \(\gamma\) is called the coefficient of friction. The external periodic force is modeled as
\[\nonumber F_{e}=F \cos \Omega t, \nonumber \]
where \(F\) is the force’s amplitude and \(\Omega\) is the force’s angular frequency. If we also include the gravitational force given by \((10.1)\) , Newton’s equation can then be written as
\[\ddot{\theta}+\lambda \dot{\theta}+\omega^{2} \sin \theta=f \cos \Omega t \nonumber \]
where \(\lambda=\gamma / m, f=F / m l\) , and \(\omega\) is defined in (10.3). An analytical solution of (11.1) is possible only for small oscillations. Indeed, the damped, driven pendulum can be chaotic when oscillations are large.
The linear pendulum
Damped pendulum
Here, we exclude the external force, and consider the damped pendulum using the small amplitude approximation \(\sin \theta \approx \theta\) . The governing equation becomes the linear, second-order, homogeneous differential equation given
\[\ddot{\theta}+\lambda \dot{\theta}+\omega^{2} \theta=0 \nonumber \]
which is usually discussed in detail in a first course on differential equations.
The characteristic equation of \((11.2)\) is obtained by the ansatz \(\theta(t)=\exp (\alpha t)\) , which yields
\[\alpha^{2}+\lambda \alpha+\omega^{2}=0 \nonumber \]
with solution
\[\alpha_{\pm}=-\frac{1}{2} \lambda \pm \frac{1}{2} \sqrt{\lambda^{2}-4 \omega^{2}} \nonumber \]
For convenience, we define \(\beta=\lambda / 2\) so that (11.4) becomes
\[\alpha_{\pm}=-\beta \pm \sqrt{\beta^{2}-\omega^{2}} \nonumber \]
The discriminant of \((11.5)\) is \(\beta^{2}-\omega^{2}\) , and its sign determines the nature of the damped oscillations.
The underdamped pendulum satisfies \(\beta<\omega\) , and we write
\[\nonumber \alpha_{\pm}=-\beta \pm i \omega_{* \prime} \nonumber \]
where \(\omega_{*}=\sqrt{\omega^{2}-\beta^{2}}\) and \(i=\sqrt{-1}\) . In this case, the general solution of (11.2) is a damped oscillation given by
\[\nonumber \theta(t)=e^{-\beta t}\left(A \cos \omega_{*} t+B \sin \omega_{*} t\right) . \nonumber \]
The overdamped pendulum satisfies \(\beta>\omega\) , and the general solution is an exponential decay and is given by
\[\nonumber \theta(t)=c_{1} e^{\alpha_{+} t}+c_{2} e^{\alpha_{-} t} \nonumber \]
where both \(\alpha_{+}\) and \(\alpha_{-}\) are negative.
The critically damped pendulum corresponds to the special case when \(\beta=\omega\) , and with \(\alpha_{+}=\alpha_{-}=\alpha<0\) , the general solution is given by
\[\nonumber \theta(t)=\left(c_{1}+c_{2} t\right) e^{\alpha t} . \nonumber \]
Driven pendulum
Here, we neglect friction but include the external periodic force. The small amplitude approximation results in the governing equation
\[\ddot{\theta}+\omega^{2} \theta=f \cos \Omega t . \nonumber \]
An interesting solution occurs exactly at resonance, when the external forcing frequency \(\Omega\) exactly matches the frequency \(\omega\) of the unforced oscillator. Here, the inhomogeneous term of the differential equation is a solution of the homogeneous equation. With the initial conditions \(\theta(0)=\theta_{0}\) and \(\theta(0)=0\) , the solution at resonance can be determined to be
\[\nonumber \theta(t)=\theta_{0} \cos \omega t+\frac{f}{2 \omega} t \sin \omega t \nonumber \]
which is a sum of a homogeneous solution (with coefficients determined to satisfy the initial conditions) plus the particular solution. The particular solution is an oscillation with an amplitude that increases linearly with time. Eventually, the small amplitude approximation used to derive (11.6) will become invalid.
An interesting computation solves the pendulum equation at resonance-replacing \(\omega^{2} \theta\) in (11.6) by \(\omega^{2} \sin \theta\) -with the pendulum initially at rest at the bottom \(\left(\theta_{0}=0\right)\) . What happens to the amplitude of the oscillation after its initial linear increase?
Damped, driven pendulum
Here, we consider both friction and an external periodic force. The small amplitude approximation of (11.1) is given by
\[\ddot{\theta}+\lambda \dot{\theta}+\omega^{2} \theta=f \cos \Omega t \nonumber \]
The general solution to \((11.7)\) is determined by adding a particular solution to the general solution of the homogeneous equation. Because of friction, the homogeneous solutions decay to zero leaving at long times only the non-decaying particular solution. To find this particular solution, we note that the complex ode given by
\[\ddot{z}+\lambda \dot{z}+\omega^{2} z=f e^{i \Omega t}, \nonumber \]
With \(z=x+i y\) , represents two real odes given by
\[\nonumber \ddot{x}+\lambda \dot{x}+\omega^{2} x=f \cos \Omega t, \quad \ddot{y}+\lambda \dot{y}+\omega^{2} y=f \sin \Omega t, \nonumber \]
where the first equation is the same as (11.7). We can therefore solve the complex ode (11.8) for \(z(t)\) , and then take as our solution \(\theta(t)=\operatorname{Re}(z) .\) With the ansatz \(z_{p}=A e^{i \Omega t}\) , we have from (11.8)
\[\nonumber -\Omega^{2} A+i \lambda \Omega A+\omega^{2} A=f \nonumber \]
or solving for \(A\) ,
\[A=\frac{f}{\left(\omega^{2}-\Omega^{2}\right)+i \lambda \Omega} \nonumber \]
The complex coefficient \(A\) determines both the amplitude and the phase of the oscillation. We first rewrite \(A\) by multiplying the numerator and denominator by the complex conjugate of the denominator:
\[\nonumber A=\frac{f\left(\left(\omega^{2}-\Omega^{2}\right)-i \lambda \Omega\right)}{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}} . \nonumber \]
Now, using the polar form of a complex number, we have
\[\nonumber \left(\omega^{2}-\Omega^{2}\right)-i \lambda \Omega=\sqrt{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}} e^{i \phi}, \nonumber \]
where \(\tan \phi=\lambda \Omega /\left(\Omega^{2}-\omega^{2}\right) .\) Therefore, \(A\) can be rewritten as
\[\nonumber A=\frac{f e^{i \phi}}{\sqrt{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}}} \nonumber \]
With the particular solution given by \(\theta(t)=\operatorname{Re}\left(A e^{i \omega t}\right)\) , we have
\[\begin{align} \theta(t) &=\left(\frac{f}{\sqrt{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}}}\right) \operatorname{Re}\left(e^{i(\Omega t+\phi)}\right) \\ &=\left(\frac{f}{\sqrt{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}}}\right) \cos (\Omega t+\phi) \end{align} \nonumber \]
The amplitude of the pendulum’s oscillation at long times is therefore given by
\[\nonumber \frac{f}{\sqrt{\left(\omega^{2}-\Omega^{2}\right)^{2}+\lambda^{2} \Omega^{2}}} \nonumber \]
and the phase shift of the oscillation relative to the external periodic force is given by \(\phi\) .
For example, if the external forcing frequency is tuned to match the frequency of the unforced oscillator, that is, \(\Omega=\omega\) , then one obtains directly from \((11.9)\) that \(A=f /(i \lambda \omega)\) , so that the asymptotic solution for \(\theta(t)\) is given by
\[\theta(t)=\frac{f}{\lambda \omega} \sin \omega t . \nonumber \]
The oscillator is observed to be \(\pi / 2\) out of phase with the external force, or in other words, the velocity of the oscillator, not the position, is in phase with the force.
The solution given by (11.12) shows that large amplitude oscillations can result by either increasing \(f\) , or decreasing \(\lambda\) or \(\omega\) . As the amplitude of oscillation becomes large, the small amplitude approximation \(\sin \theta \approx \theta\) may become inaccurate and the true pendulum solution may diverge from (11.12).
The nonlinear pendulum
As we already eluded, the fully nonlinear damped, driven pendulum can become chaotic. To study (11.1) numerically, or for that matter any other equation, the number of free parameters should be reduced to a minimum. This usually means that the governing equations should be nondimensionalized, and the dimensional parameters should be grouped into a minimum number of dimensionless parameters. How many dimensionless parameters will there be? The answer to this question is called the Buckingham II Theorem.
The Buckingham I Theorem: If an equation involves \(n\) dimensional parameters that are specified in terms of \(k\) independent units, then the equation can be nondimensionalized to one involving \(n-k\) dimensionless parameters.
Now, the damped, driven pendulum equation (11.1) contains four dimensional parameters, \(\lambda\) , \(f, \omega\) , and \(\Omega\) , and has a single independent unit, namely time. Therefore, this equation can be nondimensionalized to an equation with only three dimensionless parameters. Namely, we nondimensionalize time using one of the dimensional parameters. Here, we choose \(\omega\) , with units of inverse time, and write
\[\nonumber \tau=\omega t \nonumber \]
where \(\tau\) is now the dimensionless time. The damped, driven pendulum equation (11.1) therefore nondimensionalizes to
\[\frac{d^{2} \theta}{d \tau^{2}}+\left(\frac{\lambda}{\omega}\right) \frac{d \theta}{d \tau}+\sin \theta=\left(\frac{f}{\omega^{2}}\right) \cos \left(\left(\frac{\Omega}{\omega}\right) \tau\right), \nonumber \]
and the remaining three dimensionless groupings of parameters are evidently
\[\nonumber \frac{\lambda}{\omega}, \quad \frac{f}{\omega^{2}}, \quad \frac{\Omega}{\omega} \nonumber \]
We may give these three dimensionless groupings new names. Rather than introduce even more named parameters into the problem, I will now call the dimensionless time \(t\) , and reuse some of the other parameter names, with the understanding that the damped, driven pendulum equation that we will now study numerically is dimensionless. We will therefore study the equation
\[\ddot{\theta}+\frac{1}{q} \dot{\theta}+\sin \theta=f \cos \omega t \nonumber \]
with the now dimensionless parameters named \(q, f\) and \(\omega\) .
Equation (11.14) is called a non-autonomous equation. For a differential equation to be called autonomous, the independent variable \(t\) must not appear explicitly. It is possible to write this second-order non-autonomous differential equation as a system of three first-order autonomous equations by introducing the dependent variable \(\psi=\omega t\) . We therefore have
\[\begin{align} \nonumber \dot{\theta} &=u, \\ \dot{u} &=-\frac{1}{q} u-\sin \theta+f \cos \psi, \\ \dot{\psi} &=\omega .\nonumber \end{align} \nonumber \]
The necessary conditions for an autonomous system of differential equations to admit chaotic solutions are (1) the system has at least three independent dynamical variables, and; (2) the system contains at least one nonlinear coupling. Here, we see that the damped, driven pendulum equation satisfies these conditions, where the three independent dynamical variables are \(\theta, u\) and \(\psi\) , and there are two nonlinear couplings, \(\sin \theta\) and \(\cos \psi\) , where we already know that the first nonlinear coupling is required for chaotic solutions.
But what exactly is chaos? What we are considering here is called deterministic chaos, that is chaotic solutions to deterministic equations such as a non-stochastic differential equation. Although there is no definitive definition of chaos, perhaps its most important characteristic is a solution’s sensitivity to initial conditions. A small change in initial conditions can lead to a large deviation in a solution’s behavior. A solution’s sensitivity to initial conditions has been called the Butterfly Effect, where the image of a butterfly appeared in the title of a talk that one of the founders of the field, Edward Lorenz, gave in 1972: "Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?"
We can easily observe that the small amplitude approximation of (11.14) can not admit chaotic solutions. Suppose we consider two solutions \(\theta_{1}(t)\) and \(\theta_{2}(t)\) to the approximate equations, these two solutions differing only in their initial conditions. We therefore have
\[\begin{aligned} &\ddot{\theta}_{1}+\frac{1}{q} \dot{\theta}_{1}+\theta_{1}=f \cos \omega t \\ &\ddot{\theta}_{2}+\frac{1}{q} \dot{\theta}_{2}+\theta_{2}=f \cos \omega t \end{aligned} \nonumber \]
If we define \(\delta=\theta_{2}-\theta_{1}\) , then the equation satisfied by \(\delta=\delta(t)\) is given by
\[\nonumber \ddot{\delta}+\frac{1}{q} \dot{\delta}+\delta=0 \nonumber \]
which is the undriven, damped pendulum equation. Therefore, \(\delta(t) \rightarrow 0\) for large times, and the solution for \(\theta_{2}\) and \(\theta_{1}\) eventually converge, despite different initial conditions. In other words, there is no sensitivity to initial conditions in the solution. Only for large amplitudes where the approximation \(\sin \theta \approx \theta\) becomes invalid, are chaotic solutions possible.
We will next learn some of the concepts and tools required for a numerical exploration of chaos in dynamical systems.