Swinging a playground swing
Torque controls for inducing sustained oscillations

The main goal of this paper is to design torque controls for a common model of a playground swing, the model of swinging in the sitting position proposed by Case and Swanson [1], with damping added for more realism. As it turns out, the addition of damping makes the problem much more complex from the analytic standpoint. Modeling human control of playground swings has venerable history going back at least to 1960s, see [1, 2] and references therein. However, even aside from the fact that damping or friction are rarely included, all models in the literature, to the best of our knowledge, take the body’s position as the control input. For swinging in the sitting position it is described by the angle between the body and the swing’s axis (rods, chains or cables attaching the seat to the frame). This is a natural simplification given that human swingers can use their biophysical machinery to generate torques for muscle movements that change and stabilize the requisite body angle. But much complexity is hidden in these built-in mechanisms, and if we want to understand them, or design an electromechanical device that would relieve parents of the chore of swinging their kids, then we can not rely on biophysics to do the work.

Aside from the recreational motivation, torque control of the swing is closely related to control tasks in modern robotics, such as bipedal walking and brachiating for robots with rotating joints [3, 4, 5, 6]. Simplified models of robots, such as the Acrobot and the Pendubot, have been developed to design and analyze controls of similar nature [7, ch. 5],[8, 9, 10], as well as component models such as the reaction wheel pendulum [11, 12, 13, 14] and Furuta pendulum [15]. When there is no damping in the swing’s joints, the equations of motion of the Case-Swanson model with the torque control are identical (up to the signs of constants) to those of the Acrobot, and for the simplified model that we focus on (the balanced swing) to those of the reaction wheel pendulum. Another related task is synchronizing a pair of two connected pendulums [16, 17], where one is equipped with clock escapement.

These systems share common features of underactuation and trigonometric nonlinearity that make them convenient test cases for the methods of nonlinear geometric control theory [7, 18, 19]. Underactuation means that only some degrees of freedom are directly affected by control inputs, which forces one to balance achieving desired behavior with maintaining reasonable internal dynamics of the uncontrolled degrees of freedom. In robotics, underactuation comes up when some joints do not have an associated motor to move them. In human execution, performing such tasks is often associated with dexterity. In part, we use the problem of playground swinging to investigate what sort of controls an underactuated robot would require for some tasks easily performed by humans.

In the Case-Swanson model the body is represented by a rigid dumbbell made of three masses attached to a rigid axis that connects it to a fixed pivot. The axis and the dumbbell can rotate around the upper and the lower joint, respectively. We add damping to both joints, but it is adding it to the unactuated upper joint that complicates the problem considerably. The open loop equations of motion are formally identical to those of the damped double pendulum, and, with the torque control, to those of the damped Acrobot. For analytic simplicity, in most of the paper we consider the case of the balanced swing, where the moments of inertia of the side masses are equal, a variant singled out already in [1]. The equations of motion reduce to those of the damped reaction wheel pendulum. Unbalanced swinging in the sitting position can be treated as a small perturbation of the balanced case (Section “Unbalanced swing”).

While the control system is familiar from robotics, the control task is unusual. It is not stabilization near an equilibrium, but rather inducing sustained oscillations of prescribed amplitude and frequency (by a static state feedback control law). Sometimes such a task is interpreted as a version of trajectory tracking for a selected periodic trajectory [1, 12, 15, 13], but this artificially fixes the phase of oscillations that a swinger does not care about and would not want to spend energy maintaining. Therefore, a more natural control goal is orbital stabilization, where the orbit is a stable limit cycle [20, 21, 22]. Practical considerations lead to further design criteria that the desired control must satisfy:

1. 1.

   In the closed loop system the angle of the swing axis with the vertical approaches a stable limit cycle of oscillations with a single dominant frequency, without beats or chaotic motion.

2. 2.

   The dynamics of the body angle with the swing axis is bounded and oscillatory with a single dominant frequency.

3. 3.

   The transient motion, before the limit cycle is reached, is reasonably short.

Meeting all three desiderata turns out to be surprisingly challenging. The first two criteria are hard to express in analytic form, and we are not aware of general results for nonlinear systems that would guarantee meeting them. Accordingly, our design strategy is semi-analytic. We look for a control ansatz that meets part of our design criteria and contains sufficiently many parameters that are adjusted in simulations to meet the rest. Such hybrid approach is typical for studying systems that are capable of chaotic behavior, like harmonically driven damped pendulum [23, 24]. Our adaptation of it to control design that sustains regular oscillations offers a novel methodology for controlling underactuated mechanical systems with friction that appear in robotics.

Most of the cited work on control of robot arms and pendula concerns the swing up task and stabilization near the unstable equilibrium. Many classical control techniques employed there, such as linearization around a stable equilibrium, pole placement and standard stabilizing controllers such as LQR (including its variants for trajectory tracking), are not suitable for our control task of orbital stabilization. However, the technique of feedback linearization [18, 19], that has been applied to control of swing up in [8, 10] and of the undamped reaction wheel pendulum in [11, 25], is central to our approach. Our control design is closest to that of [20], that uses dynamic state feedback construction to induce limit cycles with bounded zero dynamics in feedback linearizable multi-dimensional systems. Its simplification to a static state feedback form, and application to control of swinging appear to be new. Generally speaking, to make use of feedback linearization one has to get lucky twice. First, the studied system has to be feedback linearizable, or at least close to one in some sense. And second, the new variables have to be related to the original ones in a way that preserves desirable behavior. This second condition is not met in our case by an alternative construction of [22] based on backstepping. It is one of the main observations of this paper that we do get lucky with the balanced swing.

The paper is organized as follows. In Section “Playground swing: equations of motion and naive controls” we show how several natural ideas lead to controls that fail at least one of the design criteria. It also serves as preliminaries, introducing the model of the swing along with some useful background and terminology.

In Section “Feedback linearization and imposing a limit cycle” we generalize the standard construction of a static state feedback law that imposes a limit cycle on a single second order equation [21, 26] to a linear system of any dimension in the Brunovsky canonical form. It is motivated by the dynamic state feedback construction of [20]. The law imposes a limit cycle on a two-dimensional subsystem, while producing bounded zero dynamics on a complementary subsystem. Since the system is linear this dynamics is oscillatory.

In Section “Feedback linearization control: analysis and variations” we analyze the resulting control law for the balanced swing. In principle, our construction includes an arbitrary energy function that determines the shape of the limit cycle, but we always chose it to be quadratic for simplicity, and did not investigate closed loop behavior when it is varied beyond shifting the center of the cycle. So our limit cycle is an ellipse and we have five control parameters to adjust: two describe the ellipse, and three control gains determine transient behavior. Using simulations, we adjust the parameter values to meet our design criteria for realistic playground swings and swingers. Shifting the center of the ellipse is used to incorporate an additional constraint on the range of the body’s angle, which is especially salient for swings with back benches that prevent backward leaning. In simulations, the zero damping law (with minor corrections) turns out to work quite well even when the damping is non-zero but relatively small. But the damped system is no longer feedback linearizable, and the resulting closed loop system is analytically opaque.

To remedy the situation somewhat, in Section “Damping corrections for small amplitudes” we linearize the balanced swing for small amplitudes and run the process of reduction to the Brunovsky canonical form and imposing the limit cycle on it in parallel to the non-linear zero damping case. This allows us to obtain damping corrections analytically, and to track what happens to the closed loop system (for small amplitudes) when the damping increases. The linearized closed loop system can be transformed into an alternative form that makes the physics behind the control more transparent. One can think of it as imposing a limit cycle on a combination of the swing and the body angles and their velocities designed to periodically drive the swing angle, while adding artificial damping (and stiffness) to a complementary degree of freedom. For the linearized system this combination is a linear combination that can be explicitly written for all values of damping and control parameters. But for the non-linear system it exists (in exact form) only in the case of zero damping (as we already know), and also in one other case.

This second case is of independent interest because it does not involve feedback linearization and a small parameter perturbation. We study it in Section “From balanced swing to driven damped pendulum”. Physically, the control can be interpreted as imposing a limit cycle on a linear combination of the angles only, without involving velocities, but the analysis only works for a special choice of the ratio of the last two control gains. The resulting closed loop system for the swing angle turns out to be (in the steady state) the damped pendulum driven by a harmonic excitation. As is well known, this is one of the simplest systems with chaotic behavior, but its appearance is not entirely unexpected since the double pendulum itself is already such a system. If the driving frequency or the driving amplitude is increased, while all other parameters are kept fixed, the pendulum dynamics undergoes a cascade of period doubling bifurcations that ends in regions of chaotic behavior interspersed with returns to regular oscillations at yet higher values [23, 6.3], [24, 3.3.5].

Obviously, meeting our second design criterion requires avoiding chaos, or even period doubling, and having one less control gain makes the task more difficult. It turns out that for realistic values of lengths and masses this is only possible when the damping in the upper joint is relatively large to begin with. We estimate the requisite range of damping values using the results of [27]. Outside of this range in simulations we encountered long transient times, oscillations without dominant frequency (beats), and sometimes numerical instability that indicates sensitivity to initial values characteristic of chaotic dynamics.

In Section “Unbalanced swing” we represent the unbalanced swing as a perturbation of the balanced swing, and show, in simulations, that balanced controls continue to perform well for small imbalances. Finally, in Conclusions we summarize our results, and discuss their limitations and directions of future work they point to. The upshot is that the “well-known” playground swing still holds a lot of unanswered questions.

There is a general agreement that two mechanisms contribute to keeping a playground swing going: moving the body’s center of mass up and down and rotating it around the pivot (angular momentum transfer). In idealized models of swinging these two mechanisms are often studied separately, as swinging in the standing and in the sitting position, respectively. We will start by considering the model of a swing with a human in the sitting position proposed by Case and Swanson [1].

The previous section should suggest to the reader that designing a suitable torque control for the swing is not as simple as it seems, and may require more sophisticated techniques. A more successful approach comes from the observation that the balanced swing system (3) has the same equations of motion as the reaction wheel pendulum with damping. And with the damping turned off this system is feedback linearizable [11, 20, 25]. This means that it can be transformed into a chain of integrators by a change of coordinates and of the control input [18, 4.2][19, 2]. The problem of inducing a limit cycle with bounded zero dynamics in feedback linearizable systems has also received some attention in the recent literature [20, 22].

Feedback linearization transforms the system into one for new state variables $x_{1},y_{1}$, and the original variables $\phi,\theta$ are expressed as explicit functions of them and their derivatives. This is where we get lucky the second time – these functions turn out to be such that imposing linear oscillatory dynamics on $x_{1},y_{1}$ has the effect of driving $\phi$ to a limit cycle with bounded zero dynamics for $\theta$. To introduce the new variables, note that the system (5) with $c=0$ is just

It is feedback linearized by the output $x_{1}$ below, and we define the corresponding state vector $x$ and the new control variable $w$ as follows:

In terms of $x$ the system is transformed into the Brunovsky canonical form for $n=4$:

General approach to inducing stable limit cycles on a two-dimensional subspace of such a system, with a bounded oscillatory dynamics on its complement, is described in [20, 22]. We will simplify the dynamic state feedback construction of [20] to a static state feedback form. The idea is to impose a stable limit cycle on $y=Cx$ by applying the two-dimensional construction (6), where $y=(y_{1},y_{2})^{T}$ is a vector and $C$ is a $2\times n$ matrix, while ensuring stable zero dynamics for the uncontrolled degrees of freedom.

In our case, $\lambda$ has only two gains $\lambda_{1},\lambda_{2}$, so $y_{1}=x_{3}+\lambda_{1}x_{1}+\lambda_{2}x_{2}$, and $x_{3}=\sin\phi$. Hence, explicitly:

Naively, we would prefer to select $\lambda_{i}=0$, so that the limit cycle is imposed directly on $\phi$. This is what we tried with the canceling control in the previous section, and now it becomes clear why it did not work. The polynomial $p(s)=s^{2}+\lambda_{2}s+\lambda_{1}$ is Hurwitz if and only if $\lambda_{i}>0$. To get bounded zero dynamics $\theta,\dot{\phi}$ and $\dot{\theta}$ have to be mixed in into the variable upon which the limit cycle is imposed. The angle of the swing would still undergo bounded oscillations under this refined construction because the entire state vector does, although we do lose some control over their exact shape.

In this section we take a closer look at the swing dynamics under the control law designed in the previous section, and discuss some variations on it. In terms of $x_{1}$, $y_{1}$, the closed loop system is the following:

The second equation imposes a stable limit cycle on $y_{1}$, and in the steady state $y_{1}$ oscillates harmonically with the frequency $\Omega$ and the amplitude $\frac{r\sqrt{2}}{\Omega}$. The first equation then describes a damped harmonic oscillator driven by $y_{1}$. The original angles are recovered by inverting the linearizing transformation (9):

It is the special form of these relations, in addition to the properties of the feedback linearized system, that is responsible for the resulting control being suitable for our purposes. They imply that $\phi$ and $\theta$ undergo bounded anharmonic oscillations with a single dominant frequency $\Omega$ in the steady state. Using the standard results on driven harmonic oscillators [30, 9.10.3], in conjunction with (17), gives the following theorem.

Numerical simulations of the undamped balanced swing (3) with the control law induced by (15) are shown on Figure 5 (a). The physical parameters are selected to match a typical human body and a playground swing, the standard SI units are used throughout the paper. The transient time and the shape of oscillations are quite satisfactory, the amplitudes can be controlled by changing $r$. The anharmonicity is only visible in the shape of the $\theta$ graph. One can also see that the body’s and the swing’s oscillations are exactly anti-phase in the steady state. This is also a direct consequence of the recovery relations (17) for this type of control, because $\ddot{x}_{1}$ and hence $\phi$ oscillates in-phase with $x_{1}$, while $\theta$ is their linear combination with negative coefficients.

These constraints may or may not be relevant for an electromechanical swinging device or a robot, but, in any case, they can be taken into account even by the type of control we proposed, with a small modification. It turns out that all we have to do is shift the center of the limit cycle for $y_{1}$. Changing the energy function $\varphi$ from Theorem 2 to $\varphi_{\sigma}:=\frac{1}{2}\dot{y}_{1}^{2}+\frac{1}{2}\Omega^{2}(y_{1}-\sigma)^{2}$ has the effect of shifting the center of oscillations for $y_{1}$ from $0$ to $\sigma$, and for $x_{1}$ from $0$ to $\frac{\sigma}{\lambda_{1}}$. This does not affect $\phi$ because it is related to $x_{1}$ through the second derivative, but for $\theta$ the center is shifted to $-\frac{Mgl_{1}}{I_{2}}\frac{\sigma}{\lambda_{1}}$. One can adjust $r$ and $\sigma$ so that $\theta$ oscillates between $0$ (or some small negative minimum) and a positive maximum. Numerical simulations of the system with the shift that makes $\theta=0$ the minimal value (no leaning backwards) are shown on Figure 5 (b).

Even though the feedback linearization control was designed for the case of zero damping it works quite well when the damping is turned on as well, with small corrections. The new expression for $\ddot{x}_{1}$ becomes $\ddot{x}_{1}=\sin\phi+\frac{c}{\omega^{2}}\dot{\phi}$ and we have to modify the control $v$ in (15) accordingly:

With these corrections, the closed loop system becomes

Recall that $c=\frac{\alpha}{I_{1}}$ and the value of $I_{1}$ is quite large for human swingers, so $c$ functions as a small parameter, partially explaining why the control still works. However, system (19) is analytically opaque, especially considering that there is no simple expression for $\phi$ in terms of $x_{1}$ and $y_{1}$ anymore.

Numerical simulations of the balanced swing (3) with relatively small damping in the upper joint are shown on Figure 7 (compare to sketches on Figure 4). One can see that the amplitude of the body needed to achieve the same amplitude of the swing increases with the increase of damping, to compensate for the additional energy loss, and the oscillations of the swing and body angles shift away from being exactly anti-phase at zero damping, which does match playground observations better. An animation of the corresponding swing motion, with a shift that allows some backward leaning, is shown on Figure 8. As one would expect, the main difference from human swinging is that the rotation of the body is spread out over the entire swing cycle rather than confined to swift motions near the turning points.

While the damped ($c\neq 0$) reaction wheel pendulum system (5) is not feedback linearizable we can get some insight into its behavior by considering the case of small amplitudes, when $\sin\phi\approx\phi$. The linearized system is analytically tractable, and we can better understand what happens in the closed loop system when a similar control is applied, unlike for the perturbed system (19). As a bonus, it will suggest an alternative control law for the original non-linear system that is applicable when the damping is large.

The linearization of system (5) for small $\phi$ is

and can be written in the matrix form by taking $\xi_{1}:=\psi$, $\xi_{2}:=\dot{\psi}$, $\xi_{3}:=\phi$, and $\xi_{4}:=\dot{\phi}$:

An output that converts it into the Brunovsky canonical form (10) can be chosen as $h\cdot\xi$ [18, 4.1], [19], where $h$ is a vector orthogonal to $B$, $AB$, $A^{2}B$. When we choose $x_{1}$ as below the conversion runs parallel to (9), but with explicit damping corrections:

Indeed, for $c=0$ the expressions for $x_{1}$ and $x_{2}$ are exactly as before, and $x_{3}$ is $\phi$ instead of $\sin\phi$, which are close for small amplitudes. Effectively, the new $x_{1}$ is the approximate second anti-derivative of $\sin\phi$ for $c\neq 0$, in explicit feedback form. The auxiliary control $w:=-\omega^{2}\phi-c\dot{\phi}+v$ that transforms (20) into the Brunovsky form (10), and the auxiliary variable $y_{1}:=\lambda_{1}x_{1}+\lambda_{2}x_{2}+x_{3}$, upon which the limit cycle is imposed according to Theorem 1, also have similar form. The resulting control law is

and the closed loop system in terms of $x_{1}$, $y_{1}$ takes the exact same form as (16):

However, because we now also have $\ddot{x}_{1}=\phi$, and the equations are linear, we can rewrite (23) in terms of $\phi$ directly. Differentiating the first equation twice eliminates $x_{1}$, and solving for $\ddot{y}_{1}$ in the second equation we derive an alternative form of the closed loop system:

The second equation again produces a stable limit cycle consisting of harmonic oscillations with the frequency $\Omega$ and the amplitude $\frac{r\sqrt{2}}{\Omega}$. The first equation is a damped harmonic oscillator driven, in the steady state limit, by a harmonic excitation $-\Omega^{2}y_{1}$.

From the standard results on driven harmonic oscillators [30, 9.10.3] we can derive the following.

This explains why the reaction wheel control law still works (for small amplitudes, at least) with non-zero damping. But if we try to apply linear corrections for $x_{1}$ from (21) to the original non-linear system (5) we do not get $\ddot{x}_{1}$ to come out cleanly as $\sin\phi$, but rather get $\ddot{x}_{1}=\sin\phi+\frac{c}{\omega^{2}}\dot{\phi}(1-\cos\phi)$. The exact second anti-derivative does not exist in a static state feedback form because the system is not feedback linearizable, so there is no exact non-linear analog of $x_{1}$.

However, things are better for $y_{1}$. The control law (22) simplifies already when we take $\lambda_{1}:\lambda_{2}=\omega^{2}:c$, i.e. when the control gains are in the same ratio as the natural gains. Moreover, this choice cancels the derivatives in the definition of $y_{1}$ and reduces it to a linear combination of $\phi$ and $\psi$ only:

The advantage is that we can define $y_{1}$ this way even for the non-linear system, and, although we can no longer write the closed loop system in terms of $x_{1}$ and $y_{1}$, it turns out that the system (24) for $\phi$ and $y_{1}$ does have a non-linear analog, and a well-studied one. This is the approach we pursue in the next section. However, there is a downside. Fixing the gain ratio leaves a single gain parameter to vary and reduces control options. As we will see, such single gain control only works well when the damping in the upper joint of the uncontrolled swing is already large.

As we saw for the linearized balanced swing, for a particular choice of the gain ratio the variable upon which the stable limit cycle is imposed reduces to a linear combination of the swing and the body angles, without time derivatives, and the alternative closed loop system (24) simplifies. We will show that a similar simplification occurs in the original non-linear system, even though, with the damping turned on, it is no longer feedback linearizable. One can see it as implementing the idea that the limit cycle should be imposed on a combination of angles rather than on the swing angle itself to ensure bounded zero dynamics.

It is convenient to rescale the auxiliary variable $y_{1}$ we used before and define

where $\mu$ is a constant. Then from (5) we obtain

We impose a limit cycle on $\chi$ as in (24) by using the control:

and obtain the closed loop system

We need to select $\mu<\frac{I_{1}}{I_{2}}$ for the damping to be positive. In the steady state, $\chi$ is oscillating harmonically with the frequency $\Omega$ and the amplitude $\frac{r\sqrt{2}}{\Omega}$, and drives the equation for $\phi$ with the scaled amplitude $\frac{1}{1-\frac{I_{2}}{I_{1}}\mu}\frac{I_{2}}{I_{1}}r\Omega\sqrt{2}$.

Thus, in the steady-state limit our problem reduces to driving a damped pendulum by a harmonic excitation

In the case of small amplitudes $\sin\phi\approx\phi$ we have a driven harmonic oscillator as in the linearized problem (24). The steady-state oscillations then have the frequency $\Omega$ and the amplitude

We can select any desired frequency $\Omega$ and any desired positive damping coefficient by adjusting $\mu$. Then $r$ can be adjusted to set the amplitude $\Phi$. The energy transfer is most efficient at the resonance, when the expression under the square root is minimized, which gives:

and this is a natural choice for $\Omega$, leaving only $r$ and $\mu$ to select.

As before, we can accommodate constraints on the range of $\theta$. It is easiest to do by choosing $\mu=-1$ which gives $\chi=\theta$, we simply choose $\varphi_{\sigma}=\frac{1}{2}\dot{\theta}^{2}+\frac{1}{2}\Omega^{2}(\theta-\sigma)^{2}$, with some prescribed shifted middle angle $\sigma$, as the energy function. Then the equation of motion (6) for $\theta$ becomes

The amplitude of the steady state oscillations is still given by $\theta_{\max}=\frac{r\sqrt{2}}{\Omega}$, but they are now centered at $\sigma$ rather than at $0$, and hence the range of angles is $[\sigma-\theta_{\max},\sigma+\theta_{\max}]$.

By choosing $\sigma=\theta_{\max}=45^{\circ}$ we get rotations between $0^{\circ}$ and $90^{\circ}$ characteristic for swings with back rests, and by choosing $\sigma=35^{\circ}$, $\theta_{\max}=55^{\circ}$ we get rotations between $-20^{\circ}$ and $90^{\circ}$ that humans can achieve on swings with flat seats without falling over. Perhaps, one can lean even further back by holding on tight to the swing chains or ropes. The corresponding control is given by

and the zero dynamics equation for the swing angle is

Compared to (33) (with $\mu=-1$) we have an additional constant driving term $-\frac{1}{1+\frac{I_{2}}{I_{1}}}\frac{I_{2}}{I_{1}}\,\Omega^{2}\sigma$. Its effect is easy to understand intuitively for small amplitudes when $\sin\phi\approx\phi$. In that case the shift can be subsumed into $\phi$, and we have regular oscillations around a shifted middle position $-\frac{I_{2}}{I_{1}}\,\frac{\Omega^{2}}{\omega^{2}}\sigma$. Because typically $\Omega\approx\omega$ and $I_{2}\ll I_{1}$ the shift in $\phi$ is barely noticeable in practice. This approximation remains valid roughly for $\phi_{\max}\lesssim 0.4$ or $23^{\circ}$.

Damped pendulum driven by a harmonic excitation is one of the simplest, and best studied, mechanical systems that can exhibit chaotic behavior [23, 24]. For small driving amplitudes and frequencies it is well approximated by driven damped harmonic oscillator, which goes into resonance when the driving and the natural frequencies are close to each other. The dynamics remains qualitatively similar even as the approximation is no longer accurate for somewhat larger values. But this does not remain the case forever. As the driving frequency or amplitude are increased, the pendulum starts behaving differently in a dramatic fashion, its motion undergoes a cascade of period doubling bifurcations. Period doubling means that the oscillations no longer have a single dominant frequency. It splits in two, and the graph of the pendulum angle as a function of time looks like a wave of one frequency modulated by another, the pattern known as beats, see Figure 9 (a), (c). As more and more frequencies are superimposed, the motion becomes increasingly erratic. The period doubling cascade ends in fully chaotic behavior, which, surprisingly, gives way back to regular oscillations at still higher values of the frequency or the amplitude, but only for narrow ranges of their values [24, 3.3.5].

Our simulations show that for realistic values of lengths and masses the closed loop driven damped pendulum falls within the range of regular oscillations (and hence satisfies our design criteria) only for relatively large values of the damping coefficient $\alpha$, see Figure 9. For smaller values of $\alpha$, the steady state oscillations display the pattern of beats, a wave with one frequency modulated by another. It appears that the closed loop system undergoes a period doubling bifurcation as $\alpha$ is decreased, past which this type of control is no longer satisfactory.

To interpret our results, we rescale the time variable to $\tau=kt$ with $k^{2}=\frac{\omega^{2}}{1-\frac{I_{2}}{I_{1}}\mu}$ and convert (28) into the dimensionless form

where

In terms of the dimensionless parameters the amplitude formula (29) simplifies to

Note that the choice $\Omega_{D}=1$ corresponds to the resonance in the absence of damping, and (30) reduces to $\Omega_{D}=\sqrt{1-\frac{\delta^{2}}{2}}$. Regular oscillations, for which this formula is valid, take place for small $A_{D}$ and $\delta$, with $\Omega_{D}$ close to $1$.

A detailed semi-analytic investigation of the onset of chaos in a harmonically driven damped pendulum by perturbation methods can be found in [27]. It is based on Fourier approximation with a dominant mode. It turns out that in the next order of approximation the amplitude formula (36) has to be corrected by replacing $1$ in the denominator with an expression that depends on the amplitude $\Phi$, and so it turns into an implicit equation for $\Phi$. This equation (see (38) below) defines curves in the $\Phi$-$\Omega_{D}$ plane called the resonance curves. They are plotted in [27] for some typical parameter values, and have two branches, normal and anomalous.

There is a range of parameter values where the normal branch represents stable regimes and resembles the resonance curve for the driven damped harmonic oscillator given by (36). This is the primary range of regular oscillatory behavior. Increasing $A_{D}$ deforms the resonance curves into multi-valued graphs, and regular oscillations destabilize, setting off the period doubling cascade where the dominant mode approximation is no longer applicable. But it can be used to analytically estimate the threshold values where the onset of the period doubling occurs. We will use the following result from [27].

Up to this point we assumed that the swinger is fully balanced, i.e. her center of mass is on the swing’s axis. When this is not so, rotating the body (modeled by the attached dumbbell), aside from the momentum transfer, has the additional effect of moving the center of mass up and down. This additional energy pumping mechanism is used exclusively when pumping up the swing from the standing position [2, 31]. There is, however, a tradeoff. Pumping the swing up by moving the center of mass is most effective near the midswing, whereas the angular momentum transfer is most effective when it is near the turning points. Since in our dumbbell model both moves can only be accomplished by rotation the two strategies cannot be used simultaneously.

With the $N$ terms included the equations of motion (2) become

Considering the prevalence of the angle sum $\phi+\theta$ and its derivatives in the above equations it is natural to introduce $\xi:=\phi+\theta$ as a new variable. In terms of $\xi$, the equations transform into (we keep $\theta=\xi-\phi$ for brevity)