The behaviour of a forced spherical pendulum operating in a weightless environment

Grundy, R E

doi:10.1093/qjmam/hbad008

Summary

In this article, we show that by subjecting the pivot of a simple inextensible pendulum to small amplitude high frequency rectilinear oscillations it is possible to make it operate in a weightless environment. The axis of vibration of the pivot defines a preferred direction in space and a consequential dynamical structure which is completely absent when the pivot is fixed. Using spherical polar coordinates centred at the pivot, we show that the motion of such a pendulum has fast and slow-scale components which we analyse using the method of multiple scales. The slow scale equation for the polar angle is autonomous, and a phase plane analysis reveals the essential orbital structure including the existence of conical solutions analogous to the terrestrial fixed pivot conical pendulum. In the absence of an azimuthal velocity component, its behaviour can provide a direct simulation of a plane terrestrial simple fixed pivot pendulum with a correspondingly simple form for the small amplitude period. We can also use a two-scale analysis to examine the effects of damping. Here, the slow scale polar equation has two asymptotically stable states, and we employ a combination of numerical and asymptotic analyses to elicit the slow scale orbital trajectories.

1 Introduction and equations of motion

The advent of space flight and platforms such as orbiting space stations has brought into focus new opportunities for observing the effect of weightlessness in many areas including particle dynamics. The weightlessness induced by the balance of gravitational and inertial forces in such settings inhibits many features of terrestrial dynamics but may also introduce new effects [1]. In this article, we concentrate our attention on how one of the simplest of dynamical systems is affected namely an inextensible simple pendulum. Operating in a weightless setting it is clear that such a system offers little by way of dynamics. However, as we shall show this situation changes fundamentally if the pivot is allowed to vibrate.

We should mention that such a device has been used many times in a terrestrial setting most notably to study the curiously counter intuitive properties of the ‘inverted pendulum’. The theory behind the plane problem was initiated in the early 1900’s in a series of papers by Stephenson [2 to 4]. However, it was not until 1951 that Kapitza [5], using averaging methods, derived an ordinary differential equation describing the global behaviour of such a pendulum. Subsequently, other asymptotic methods have been used to analyse this and related problems specifically the method of multiple scales where Kapitza’s original equation can be regarded as the leading term in an asymptotic expansion of the polar angle for small pivot amplitudes [6]. The theory was first extended to spherical geometry by Markeyev [7] and more recently by Grundy [8] and, in a related problem, by Polekhin [9].

In what follows, we show that in a weightless environment a rectilinearly vibrating pivot introduces a preferred axis aligned along the acceleration vector of the pivot thereby simulating a force which drives the system. The dynamics of the resulting spherical pendulum, although different from its terrestrial counterpart described in [7] and [8], will be shown to be significant. Perhaps, the most noteworthy revelation from the analysis is that such a pendulum exists at all.

To take these issues further, we consider a simple spherical pendulum of fixed length L with its pivot vibrating along an arbitrary directed axis in the weightless environment. The arrangement is presented in Fig. 1 for a right handed Cartesian inertial frame of reference (x, y, z) with origin O fixed within the environment—this is the set-up which applies on an orbiting space station.

Fig. 1

Schematic view of the pendulum-pivot system. The pivot P oscillates along the z-axis relative to the fixed origin O. The pendulum arm is of length L with a bob of mass m

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So in terms of polar coordinates $(θ, φ)$ centred at the pivot, the Cartesian coordinates of the pendulum bob with respect to O are given by

\begin{matrix} x (t) = L sin θ (t) cos ϕ (t), \\ y (t) = L sin θ (t) sin ϕ (t), \\ z (t) = z_{0} (t) - L cos θ (t) . \end{matrix}

(1.1)

Here, the positive z-axis is aligned along the axis of vibration of the pivot and $z_{0} (t)$ is the distance of the pivot from O along this axis. In the absence of damping, the cartesian equations of motion of the pendulum bob can then be written in the form

\ddot{x} + \frac{S}{m} sin θ cos ϕ = 0, \ddot{y} + \frac{S}{m} sin θ sin ϕ = 0

(1.2)

and

\ddot{z} - \frac{S}{m} cos θ = 0,

where S is the tension in the pendulum arm and m is the mass of the bob. Substituting for $x (t), y (t)$ and z(t) from (1.1), we can eliminate $\frac{S}{m}$ and thereby write the polar and azimuthal equations of motion as

\ddot{θ} + \frac{sin θ}{L} {\ddot{z}}_{0} - {\dot{ϕ}}^{2} sin θ cos θ = 0,

(1.3)

\dot{ϕ} {sin}^{2} θ + 2 \dot{θ} \dot{ϕ} sin θ cos θ \equiv \frac{d}{d t} (\dot{ϕ} {sin}^{2} θ) = 0,

(1.4)

where dots denote differentiation with respect to t. We solve this system subject to the conditions

θ = a_{0}, \dot{θ} = b_{1}, ϕ = 0, \dot{ϕ} = b_{2} \neq 0

(1.5)

applied at the release point t = 0. Although there is no sense of up or down in a weightless environment in what follows our choice of a positive z-direction does help us distinguish, in a descriptive sense, between positive and negative hemispheres.

The structure of the article is as follows. In section 2, we derive the autonomous system governing the slow time behaviour of the pendulum. On the whole, we follow the analysis in [6] but we believe its inclusion here is useful to the reader and also adds to the coherence of the presentation. In section 3, we identify conical solutions of the slow scale equations and use them to construct conical solutions of the original system (1.3)–(1.5). In section 4, we consider the general initial value problem for both the slow time and exact systems and compare respective outcomes. Section 5 examines the effect of including a damping term into the formulation, and in Section 6, we present a brief discussion of the plane problem.

2 A two-scale analysis of the governing equations

Returning to the system (1.3)–(1.5), we model the high frequency pivot vibrations by putting $z_{0} = δ cos ω t$ so that the release point coincides with an instantaneously stationary point in the pivot cycle. We will say more about this issue later in the article. We now make the change of variable

t = \frac{τ}{ω}

(2.1)

so that (1.3) and (1.4) can be written in the form

\frac{d^{2} θ}{d τ^{2}} - ϵ cos τ sin θ - {(\frac{d ϕ}{d τ})}^{2} sin θ cos θ = 0

(2.2)

and

\frac{d}{d τ} ({sin}^{2} θ \frac{d ϕ}{d τ}) = 0,

(2.3)

where $ϵ = \frac{δ}{L}$ ⁠.

From (1.5), we solve (2.2) and (2.3) subject to the initial conditions

\begin{matrix} θ (0) = a_{1}, \frac{d θ}{d τ} (0) = \frac{ϵ b_{1}}{ϵ ω} = ϵ a_{1} \\ ϕ (0) = 0, \frac{d ϕ}{d τ} (0) = \frac{ϵ b_{2}}{ϵ ω} = ϵ a_{2}, \end{matrix}

(2.4)

where $a_{1} = \frac{b_{1}}{(ϵ ω)}$ and $a_{2} = \frac{b_{2}}{(ϵ ω)}$ ⁠. We remind ourselves at this stage that in asymptotic terms for small amplitude high frequency vibrations of the pivot we will be considering the limit $ϵ \to 0$ with $ϵ ω = O (1)$ ⁠. There are no further conditions on ω.

Turning to the azimuthal equation, we see that (2.3) can be integrated to give

{sin}^{2} θ \frac{d ϕ}{d τ} = h_{0} = ϵ a_{2} {sin}^{2} a_{0} \equiv ϵ H_{0}

(2.5)

using the initial condition (2.4) on $ϕ$ ⁠. Here,

H_{0} = a_{2} {sin}^{2} a_{0} = \frac{b_{2} {sin}^{2} a_{0}}{(ϵ ω)}

(2.6)

again using (2.4).

So far, our equations are exact. However, we will show that the motion of the pendulum consists of two components: a high frequency small amplitude component due to the motion of the pivot together with a slow scale response which is the one we generally perceive. The aim of the following analysis is to confirm this structure and in the process extract the slow scale behaviour. The first step is to eliminate $\frac{d ϕ}{d τ}$ between (2.2) and (2.5) to give the single equation

\frac{d^{2} θ}{d τ^{2}} - ϵ cos τ sin θ - ϵ^{2} μ \frac{cos θ}{{sin}^{3} θ} = 0

(2.7)

for the polar angle. Here, we have replaced $H_{0}^{2}$ by μ so that from (2.6)

μ = \frac{b_{2}^{2} {sin}^{4} a_{0}}{{(ϵ ω)}^{2}} = a_{2}^{2} {sin}^{4} a_{0} > 0.

(2.8)

As a necessary precursor to what follows we construct the so-called straightforward expansion satisfying, the initial conditions (2.4) namely

θ (τ; ϵ) = a_{0} + ϵ θ_{1} (τ) + ϵ^{2} θ_{2} (τ) + \dots

(2.9)

in the limit $ϵ \to 0, τ = O (1)$ together with a corresponding expansion for $ϕ (τ; ϵ)$ ⁠. We are primarily interested in the algebraic terms in $θ_{1} (τ)$ and $θ_{2} (τ)$ so with this in mind inserting (2.9) into (2.7) we can show that

θ_{1} (τ) = sin a_{0} + a_{1} τ + P_{1} (τ)

(2.10)

and

\begin{matrix} θ_{2} (τ) = \frac{7 cos a_{0} sin a_{0}}{8} - τ a_{1} cos a_{0} \\ + [μ \frac{cos a_{0}}{2 {sin}^{3} a_{0}} - \frac{sin a_{0} cos a_{0}}{4}] τ^{2} + P_{2} (τ), \end{matrix}

(2.11)

where $P_{1} (τ)$ and $P_{2} (τ)$ indicate terms which are trigonometric in τ.

Inserting (2.10) and (2.11) into (2.9) with a slight rearrangement and omitting $P_{1} (τ)$ and $P_{2} (τ),$ we can write

θ (τ; ϵ) = a_{0} + (ϵ τ) a_{1} + [μ \frac{cos a_{0}}{2 {sin}^{3} a_{0}} - \frac{sin 2 a_{0}}{8}] {(ϵ τ)}^{2} + \dots + ϵ [sin a_{0} - (ϵ τ) a_{1} cos a_{0}] + O (ϵ^{2})

(2.12)

a form which reveals the essential two-scale nature of the problem and suggests we seek expansions of the form

θ (τ; ϵ) \equiv Θ (T_{0}, T_{1}) = Θ_{0} (T_{1}) + ϵ Θ_{1} (T_{0}, T_{1}) + O (ϵ^{2})

(2.13)

and also concomitantly

ϕ (τ; ϵ) \equiv Φ (T_{0}, T_{1}) = Φ_{0} (T_{1}) + ϵ Φ_{1} (T_{0}, T_{1}) + O (ϵ^{2})

(2.14)

in terms of the respective fast and slow time variables $T_{0} \equiv τ$ and $T_{1} \equiv ϵ τ$ ⁠.

Inserting (2.13) into (2.7) together with the appropriate change of derivative we find that at the $O (ϵ)$ level

Θ_{1} (T_{0}, T_{1}) = Ψ_{1} (T_{1}) - cos T_{0} sin Θ_{0} (T_{1}),

(2.15)

where at this stage $Θ_{0} (T_{1})$ and indeed the function $Ψ_{1} (T_{1})$ are undetermined. Going further however, the elimination of secular terms in T₀ at the $O (ϵ^{2})$ level yields the equation

\frac{d^{2} Θ_{0}}{d T_{1}^{2}} + \frac{sin 2 Θ_{0}}{4} - μ \frac{cos Θ_{0}}{{sin}^{3} Θ_{0}} = 0

(2.16)

for $Θ_{0} (T_{1})$ ⁠. Equation (2.16) extends the validity of the Spherical Kapitza Equation in [7] to a weightless environment: for convenience in this article, we refer to it as the WSK equation. The initial conditions for (2.16) are obtained from (2.12) by isolating the terms in $(ϵ τ) \equiv T_{1}$ so that we can write

Θ (0) = a_{0} + ϵ sin a_{0}

(2.17)

and

\frac{d Θ_{0}}{d T_{1}} (0) = a_{1} (1 - ϵ cos a_{0})

(2.18)

with error $O (ϵ^{2})$ ⁠. Using (2.17) and (2.18) as initial conditions for (2.16) has the effect of absorbing $Ψ_{1} (T_{1})$ in (2.15) into the leading order form $Θ_{0} (T_{1})$ ⁠.

We observe that (2.16) is autonomous wherein the single parameter μ can be computed from (2.8) given the physical variables $(ϵ, ω)$ and the initial data pair (a₀, a₂). With phase plane variables $X = Θ_{0}$ and $Y = \frac{d Θ_{0}}{d T_{1}}$ ⁠, we then have

\frac{d Y}{d X} = \frac{cos X [2 μ - {sin}^{4} X]}{2 Y {sin}^{3} X}

(2.19)

which is applicable for any other pair (a₀, a₂) provided (2.8) holds. We should remember though that if we do want to change a₀ then we must change a₂ in order to keep μ fixed.

As far as the phase plane structure is concerned, there are two cases to consider. First for $0 < μ < \frac{1}{2}$ - typically in Fig. 2—where there are two symmetrically placed stable centres on the X axis at

X_{1} = {sin}^{- 1} (\sqrt[4]{2 μ})

(2.20)

and

X_{2} = π - {sin}^{- 1} (\sqrt[4]{2 μ})

(2.21)

Fig. 2

Phase plane of (2.19) for $μ = 0.1041$ ⁠. The associated conical solutions are $a_{c} = 2.4$ and 0.7416 with $b_{2 c} = 1.4242$ ⁠. The dash and dashdot trajectories correspond respectively to the solutions in Figs 5 and 6. Here and throughout the article unless otherwise stated, we take $ϵ = 0.1$ and $ω = 20.0$ as the physical parameters. The separatrix through $Θ_{0} = \frac{π}{2}$ is referred to as Γ in the text. Here and throughout the arrows indicate directions of increasing time.

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together with a saddle at $X_{3} = \frac{π}{2}$ while for $μ \geq \frac{1}{2}$ in Fig. 3, there is a single stable centre at

X_{4} = \frac{π}{2} .

(2.22)

The phase plane for μ=1.0 with the conical solution ac=π2. The dash curve corresponds to the solution in Fig. 6.

Fig. 3

The phase plane for $μ = 1.0$ with the conical solution $a_{c} = \frac{π}{2}$ ⁠. The dash curve corresponds to the solution in Fig. 6.

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In the language of dynamical systems, there is a subcritical pitchfork bifurcation at $μ = \frac{1}{2}$ ⁠. In both cases, the singular points correspond to stable conical trajectories which we investigate in the next section. Before that however we deal with the azimuthal angle where from (2.5) and the expansion (2.14) we can write

\begin{matrix} \frac{d ϕ}{d τ} = \frac{\partial Φ}{\partial T_{0}} + ϵ \frac{\partial Φ}{\partial T_{1}} = \frac{ϵ H_{0}}{{sin}^{2} Θ} \\ = \frac{ϵ H_{0}}{{sin}^{2} Θ_{0}} - \frac{2 H_{0} cos Θ_{0}}{{sin}^{3} Θ_{0}} ϵ^{2} Θ_{1} + O (ϵ^{3}) \end{matrix}

(2.23)

Now, using (2.15), the right-hand side of (2.23) can be written as

\frac{ϵ H_{0}}{{sin}^{2} Θ_{0}} - \frac{2 H_{0} cos Θ_{0}}{{sin}^{3} Θ_{0}} Ψ_{1} (T_{1}) ϵ^{2} + \frac{2 H_{0} cos Θ_{0}}{{sin}^{2} Θ_{0}} cos (T_{0}) ϵ^{2} + O (ϵ^{3})

(2.24)

whence, again absorbing the term in $Ψ_{1} (T_{1})$ into $Θ_{0} (T_{1})$ ⁠, we can write (2.23) as

\frac{\partial Φ}{\partial T_{0}} + ϵ \frac{\partial Φ}{\partial T_{1}} = \frac{ϵ H_{0}}{{sin}^{2} Θ_{0}} + O (ϵ^{2})

(2.25)

recalling that this absorption process means that $Θ_{0} (T_{1})$ now satisfies the initial conditions (2.17) and (2.18). Finally inserting the expansion (2.14) into (2.25), we have to $O (ϵ)$

\frac{d Φ_{0}}{d T_{1}} = \frac{H_{0}}{{sin}^{2} Θ_{0}}

(2.26)

with $Φ_{0} (0) = 0$ ⁠. This, along with (2.16), completes our two-scale analysis.

To end this section, we return to the issue of the choice of release point. We recall that the pivot velocity is given by $(ϵ ω) sin (ω t)$ which by assumption is O(1). For our t = 0 choice of release point, this is zero. However, for a release at an arbitrary point in the pivot cycle the pendulum will receive an O(1) impulse due to the pivot motion. This can be incorporated into the analysis and results in a modification to the initial conditions (2.17) and (2.18). However, the important point to understand in this regard is that for an arbitrary release point in the pivot cycle the WSK equation (2.16) remains as the governing slow scale equation—only the initial conditions change.

3 Conical solutions

The conical solutions of (2.16) are computed as follows. Given ϵ and ω then for each a_c in $0 < a_{c} < π$ ⁠, excluding $a_{c} = \frac{π}{2}$ ⁠, there exists a conical solution of the WSK equation (2.16) for which $Θ_{0} = a_{c}, \frac{d Θ_{0}}{d T_{1}} = 0$ and $b_{2} = b_{2 c}$ where from (2.20) with (2.8)

μ = \frac{{sin}^{4} a_{c}}{2} = \frac{b_{2 c}^{2} {sin}^{4} a_{c}}{{(ϵ ω)}^{2}} < \frac{1}{2}

(3.1)

implying

| b_{2 c} | = \frac{ϵ ω}{\sqrt{2}}

(3.2)

so that the azimuthal angle is given by

Φ_{0} = b_{2 c} t

(3.3)

in terms of the original variable t. We refer to this as Type I in which $0 < μ < \frac{1}{2}$ and direct the reader to Fig. 2.

There is another family of stable conical solutions for which $Θ_{0} = a_{c} = \frac{π}{2}$ and $\frac{d Θ_{0}}{d T_{1}} = 0$ with $| b_{2 c} |$ arbitrary satisfying

| b_{2 c} | \geq \frac{ϵ ω}{\sqrt{2}}

(3.4)

This is the Type II case presented in Fig. 3 where

μ = \frac{b_{2 c}^{2} {sin}^{4} a_{c}}{{(ϵ ω)}^{2}} = \frac{b_{2 c}^{2}}{{(ϵ ω)}^{2}} \geq \frac{1}{2} .

(3.5)

Again the azimuthal angle is given by (3.3).

We now ask how we use these results to compute ‘conical’ solutions of the exact system. This would involve making appropriate choices for a₀, a₁ and b₂ in the solution of the system (2.2) and (2.3). So, recalling that the conical solutions for $Θ_{0} (T_{1})$ require

Θ_{0} (0) = a_{c} and \frac{d Θ_{0}}{d T_{1}} (0) = 0

then from (2.17) this would mean computing an $a_{0} \equiv A_{0}$ such that

Θ_{0} (0) = a_{c} = A_{0} + ϵ sin A_{0}

(3.6)

and then replacing a₀ by A₀ in (2.18) to compute a modified a₁, denoted by A₁, by solving

0 = \frac{d Θ_{0}}{d T_{1}} (0) = A_{1} (1 - ϵ cos A_{0})

giving

A_{1} = 0.

(3.7)

We now replace a₀ by A₀, a₁ by A₁ and b₂ by $b_{2 c}$ in the initial conditions (2.4) for the exact polar form with, from (3.1)

μ = \frac{b_{2 c}^{2} {sin}^{4} a_{c}}{ϵ^{2} ω^{2}}, 0 < μ < \frac{1}{2}

(3.8)

or from (3.5)

μ = \frac{b_{2 c}^{2}}{{(ϵ ω)}^{2}}, μ \geq \frac{1}{2} .

(3.9)

The resulting exact ‘conical ‘forms are presented in Fig. 4 for the two values of μ considered in Figs 2 and 3. The oscillations due to the pivot are still present of course but there is no slow scale modulation apparent on this scale since by design the error is $O (ϵ^{2})$ justifying the word ‘conical‘in their description.

Fig. 4

In this figure, we take $ϵ = 0.05$ and $ω = 40.0$ ⁠. The upper curve represents the exact conical solution for $θ (t)$ and the horizontal solid line is the corresponding WSK conical solution $a_{c} = 2.4$ ⁠. Using the algorithm in section 3, the initial conditions required to produce this exact solution are $a_{0} \equiv A_{0} = 2.3650, a_{1} \equiv A_{1} = 0.0, b_{2} \equiv b_{2 c} = 1.4142$ giving $μ = 0.1041$ ⁠. For the lower curve $a_{c} = \frac{π}{2}, A_{0} = 1.5209, A_{1} = 0.0, b_{2 c} = 2.0$ so that $μ = 1.0$ ⁠.

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4 The general initial value problem

Given the physical parameters $(ϵ, ω)$ and the initial data $(a_{0}, b_{1}, b_{2})$ in (1.5), we can compute a value of μ from (2.8): in all computations from now on we take $ϵ = 0.1$ and $ω = 20.0$ so that $ϵ ω = 2.0$ ⁠. To examine the structure and outcomes for the slow scale trajectories, we first refer to Fig. 2, where $0 < μ < \frac{1}{2}$ ⁠. In what follows it is convenient to introduce the change of variable

Z_{0} = - cos (Θ_{0})

(4.1)

so that a first integral of (2.16) can be written in the form

{(\frac{d Z_{0}}{d T_{1}})}^{2} = \frac{4 C - 4 μ - 1 + (3 - 4 C) Z_{0}^{2} - 2 Z_{0}^{4}}{4} \equiv F (Z_{0}),

(4.2)

where $C (a_{0}, a_{1}, ϵ)$ is found by applying the initial conditions (2.17) and (2.18). The $(Z_{0}, \frac{d Z_{0}}{d T})$ phase plane is topologically identical to that in Fig. 2 so for (4.2) to represent the separatrix through the saddle at $Z_{0} = 0$ ⁠, we require $C = \frac{4 μ + 1}{4}$ and these trajectories, identified as Γ in the $(Θ_{0}, \frac{d Θ_{0}}{d T})$ phase plane, are given implicitly by

{(\frac{d Z_{0}}{d T_{1}})}^{2} = (1 - 2 μ) Z_{0}^{2} - Z_{0}^{4} = Z_{0}^{2} (1 - 2 μ - Z_{0}^{2}) .

(4.3)

Hence if the initial data point lies inside Γ, $Θ_{0} (t)$ will follow a trajectory which lies entirely in either the positive or negative hemisphere. Otherwise, it will describe a trajectory which visits both hemispheres. In Fig. 5, we plot $Θ_{0} (Φ_{0})$ for a trajectory which lies entirely in the positive hemisphere and compare it with $θ (ϕ)$ for the exact solution. As can be seen, there is excellent agreement between the two. This conformity is retained except where the initial data points are close to the separatrices through the saddle. Plotting $Θ_{0}$ as a function of $Φ_{0}$ enables us to obtain a visual estimate of the slow scale apsidal precession of the orbits. However, we can compute a precise value for the precession rate by combining (4.1) with the azimuthal equation (2.26) written in terms of Z₀ to obtain

{(\frac{d Z_{0}}{d Φ_{0}})}^{2} = \frac{(4 C - 4 μ - 1 + (3 - 4 C) Z_{0}^{2} - 2 Z_{0}^{4}) {(1 - Z_{0}^{2})}^{2}}{4 μ} .

(4.4)

Fig. 5

Plot of $Θ_{0} (Φ_{0})$ together with the exact form $θ (ϕ)$ ⁠. Here, $a_{0} = 2.0, a_{1} = 0.0, μ = 0.1041$ and $b_{2} = 0.7804$ ⁠. The vertical dotted lines represent intervals of $2 π$ in $Φ_{0}$ while the vertical dash lines represent intervals of $2 Δ Φ_{0}$ ⁠. The extrema for $Θ_{0}$ are $α_{1} = 2.0909$ and $α_{2} = 2.5880$ ⁠. For this example, the precession rate from (4.7) is given by $P_{c} = - 1.8312$ and the slow scale period from (4.8) is 3.3559 s

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Now $\frac{d Z_{0}}{d Φ_{0}} = 0$ at the apsides where Z₀ = z₁ and $Z_{0} = z_{2} > z_{1}$ ⁠. Thus, defining

Δ Φ_{0} = 2 H_{0} \int_{z_{1}}^{z_{2}} \frac{d z}{(1 - z^{2}) \sqrt{4 C - 4 μ - 1 + (3 - 4 C) z^{2} - 2 z^{4}}}

(4.5)

then the slow scale apsidal precession per cycle is given by

P_{c} = 2 Δ Φ_{0} - 2 π

(4.6)

In Fig. 6, we present $Θ_{0} (Φ_{0})$ for a trajectory which visits both the positive and negative hemispheres together with the exact solution.

Fig. 6

Plot of $Θ_{0} (Φ_{0})$ together with the exact form $θ (ϕ)$ ⁠. Here, $a_{0} = 2.0, a_{1} = 0.5$ with $μ = 0.1041$ and $b_{2} = 0.7804$ ⁠. The extrema for $Θ_{0}$ are $α_{1} = 0.3921$ and $α_{2} = 2.7495$ ⁠. For this example, the precession rate from (4.7) is given by $P_{c} = 1.3336$ and the slow scale period from (4.8) is 5.0742 s

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The phase plane for the parameter range $μ \geq \frac{1}{2}$ in Fig. 3 indicates precessing trajectories which in all cases inhabit both the positive and negative hemispheres with a typical behaviour presented in Fig. 6. Finally, the period of the slow time oscillations can be obtained in terms of the original time variable from (4.1) as

P = \frac{4}{ϵ ω} \int_{z_{1}}^{z_{2}} \frac{d z}{\sqrt{4 C - 4 μ - 1 + (3 - 4 C) z^{2} - 2 z^{4}}}

(4.7)

For the purposes of this article, we have obtained the precession rates and periods by evaluating the integrals (4.5) and (4.7) numerically. However, we should point out that these can be obtained in terms of Elliptic Integrals together with $Θ_{0} (T_{1})$ itself in terms of Elliptic Functions (see Appendix).

5 The effect of damping

We model damping by a linear velocity dependent term so that (1.2) can be written as

\ddot{x} + K \dot{x} + \frac{S}{m} sin θ cos ϕ = 0, \ddot{y} + K \dot{y} + \frac{S}{m} sin θ sin ϕ = 0

and

\ddot{z} + K \dot{z} - \frac{S}{m} cos θ = 0,

(5.1)

where K is the damping coefficient. Substituting (1.1) into (5.1), making the change of variable (2.1) and eliminating $\frac{S}{m}$ we find that

\frac{d^{2} θ}{d τ^{2}} + ϵ k \frac{d θ}{d τ} - [ϵ cos (τ) + ϵ^{2} k sin (τ)] sin θ - {(\frac{d ϕ}{d τ})}^{2} sin θ cos θ = 0

(5.2)

and

\frac{d}{d τ} (e^{ϵ k τ} {sin}^{2} θ \frac{d ϕ}{d τ}) = 0,

(5.3)

where the parameters are as in section 2 together with $k = \frac{K}{ϵ ω}$ ⁠. The system (5.2) and (5.3) is solved subject to the initial conditions (2.4). Integrating (5.3) we find that

{sin}^{2} θ \frac{d ϕ}{d τ} = h_{0} e^{- ϵ k τ} = ϵ a_{2} {sin}^{2} a_{0} e^{- ϵ k τ} \equiv ϵ H_{0} e^{- ϵ k τ},

(5.4)

where h₀ and H₀ are as in (2.5) and (2.6). Thus substituting for $ϕ (τ)$ into (5.2), we finally have

\frac{d^{2} θ}{d τ^{2}} + ϵ k \frac{d θ}{d τ} - [ϵ cos (τ) + ϵ^{2} k sin (τ)] sin θ - \frac{ϵ^{2} H_{0}^{2} cos θ}{{sin}^{3} θ} e^{- 2 ϵ k τ} = 0

(5.5)

for the polar angle $θ (τ)$ ⁠.

We can now proceed with the two scale analysis of section 2 which results in the equation corresponding to (2.16) in section 2 namely

\frac{d^{2} Θ_{0}}{d T_{1}^{2}} + k \frac{d Θ_{0}}{d T_{1}} + \frac{sin 2 Θ_{0}}{4} - μ (T_{1}) \frac{cos Θ_{0}}{{sin}^{3} Θ_{0}} = 0,

(5.6)

where

μ (T_{1}) = H_{0}^{2} e^{- 2 k T_{1}} = μ_{0} e^{- 2 k T_{1}} .

(5.7)

For the purposes of this article, we refer to (5.6) as the DWSK equation and is solved subject to the initial conditions (2.17) and (2.18). Following the analysis of section 2 with (5.4), the azimuthal equation for $Φ_{0} (T_{1})$ is given by

\frac{d Φ_{0}}{d T_{1}} = \frac{\sqrt{μ (T_{1})}}{{sin}^{2} Θ_{0}}

(5.8)

with $Φ_{0} (0) = 0$ ⁠.

The DWSK equation (5.6) has asymptotically stable equilibria at $Θ_{0} = 0$ and $Θ_{0} = π$ where in the latter case a linear stability analysis reveals that

Θ_{0} \sim π - {[C_{1} + R_{1} cos (\sqrt{A} (T_{1} - α_{1}))]}^{\frac{1}{2}} e^{- \frac{k T_{1}}{2}}

(5.9)

as $T_{1} \to \infty$ ⁠. Here, $A = 2 - k^{2} > 0$ for the ‘light damping’ case we consider in this article. The constants C₁, R₁ and α₁ are computed from the initial data but are required to satisfy the condition

C_{1}^{2} = R_{1}^{2} + \frac{4 μ_{0}}{A} .

(5.10)

The behaviour (5.9) represents a decaying oscillation about a moving centre at $π - C_{0} e^{- \frac{k T_{1}}{2}}$ ⁠. We note that there is a similar behaviour to (5.9) with (5.10) in the event that $Θ_{0} \to 0$ as $T_{1} \to \infty$ ⁠. Finally, we note that $Θ_{0} = \frac{π}{2}$ is an exact solution of (5.6) which is unstable for k > 0.

To illustrate the behaviour of the damped pendulum, we first consider an example where the parameters and initial data are as in Fig. 7 with, by way of illustration, k = 0.01. The results are shown in Fig. 8 from which we see that initially the trajectory follows the undamped form. As t increases however the plane of the orbit steepens and the precession rate increases until after six cycles the pendulum abruptly transitions to a decaying mode leading to the equilibrium at $Θ_{0} = π$ eventually reaching it according to the behaviour (5.9). Although we do not display it, for the data as in Fig. 6 there are two cycles prior to the decay to $Θ_{0} = 0$ while in the example in Fig. 5 the trajectory enters the decay mode to $Θ_{0} = π$ immediately.

An example for which =1.0>12. The initial data are as in Fig. 5 except that the change in μ now requires b2=2.4189. The extrema for Θ0 are α1=1.0507 and α2=2.0909. The precession rate is computed from (4.7) as Pc=1.3810 with a slow scale period of 3.3335 s

Fig. 7

An example for which $= 1.0 > \frac{1}{2}$ ⁠. The initial data are as in Fig. 5 except that the change in μ now requires $b_{2} = 2.4189$ ⁠. The extrema for $Θ_{0}$ are $α_{1} = 1.0507$ and $α_{2} = 2.0909$ ⁠. The precession rate is computed from (4.7) as $P_{c} = 1.3810$ with a slow scale period of 3.3335 s

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Fig. 8

The dash–dot curve represents $Θ_{0} (Φ_{0})$ for WSK equation with parameters and initial data as in Fig. 7. The solid curve is the corresponding solution of the damped DWSK equation with k = 0.01. The vertical dotted lines represent intervals of $2 π$ in $Φ_{0}$

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Although a full study of the basins of attraction for the two stable equilibria of the DWSK equation is beyond the scope of this article, we can see from Fig. 9 that the fate of the damped pendulum can be very sensitive to the initial data—in this case near $a_{1} \approx 0.14$ ⁠. In Table 1, we present a local snapshot of this dependence in the range $- 0.11582 \leq a_{1} \leq 1.5308$ ⁠. Numerical computations reveal that in a global sense the DWSK equation provides a very acceptable simulation of the exact solution being consistent with the implicit $O (ϵ^{2})$ error bound.

Fig. 9

The dash–dot curve represents $Θ_{0} (Φ_{0})$ for the DWSK equation with the data and parameters as in Fig. 8 except that $a_{1} = 0.13867$ ⁠. The solid curve has $a_{1} = 0.13868$ and the horizontal line represents $Θ_{0} = \frac{π}{2}$

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Table 1

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Contiguous basins of attraction for the DWSK equation for the example of Fig. 8 with $- 0.11582 \leq a_{1} \leq 0.32002$ ⁠.

$a_{1}$	Fate
$0.24496 \leq a_{1} \leq 0.32002$	$Θ_{0} = π$
$0.13868 \leq a_{1} \leq 0.24495$	$Θ_{0} = 0$
$- 0.11582 \leq a_{1} \leq 0.13867$	$Θ_{0} = π$

Table 1

Open in new tab

Contiguous basins of attraction for the DWSK equation for the example of Fig. 8 with $- 0.11582 \leq a_{1} \leq 0.32002$ ⁠.

$a_{1}$	Fate
$0.24496 \leq a_{1} \leq 0.32002$	$Θ_{0} = π$
$0.13868 \leq a_{1} \leq 0.24495$	$Θ_{0} = 0$
$- 0.11582 \leq a_{1} \leq 0.13867$	$Θ_{0} = π$

6 The plane problem

If there is no initialazimuthal component of velocity, then we can formally put $b_{2} = 0$ so that, from (2.7) and (2.6), $\frac{d ϕ}{d τ} \equiv 0$ and the polar equation (2.4) becomes

\frac{d^{2} θ}{d τ^{2}} - ϵ cos τ sin θ = 0,

(6.1)

where now $θ (τ)$ is identified with the polar angle in the plane of the trajectory so that $- \infty < θ (τ) < \infty$ ⁠. The azimuthal equation becomes redundant and the initial conditions for (6.1) are

θ (0) = a_{0}, \frac{d θ}{d τ} (0) = \frac{ϵ b_{1}}{ϵ ω} = ϵ a_{1} .

(6.2)

We can follow the same asymptotic analysis as before to arrive at the plane version of the WSK equation (2.16) namely

\frac{d^{2} Θ_{0}}{d T_{1}^{2}} + \frac{sin (2 Θ_{0})}{4} = 0

(6.3)

which is solved subject to the initial conditions (2.17) and (2.18) where now $- π < a_{0} \leq π$ ⁠. This is the double angle version of the equation for a plane simple terrestrial pendulum with a fixed pivot. A typical phase plane structure for (A.3) is shown in Fig. 10. Depending on the initial conditions it is apparent that the trajectories may be confined to $\frac{π}{2} < | Θ_{0} | < \frac{3 π}{2}$ or to $0 < | Θ_{0} | < \frac{π}{2}$ or indeed perform indefinite complete revolutions. To be specific since the separatrices are given by

Y = \pm \frac{cos}{\sqrt{2}}

(6.4)

The phase plane for the plane WSK equation (6.3) with X≡Θ0(T1),Y≡dΘ0dT1. There are stable centres on the X-axis at X = 0 and X=π

Fig. 10

The phase plane for the plane WSK equation (6.3) with $X \equiv Θ_{0} (T_{1}), Y \equiv \frac{d Θ_{0}}{d T_{1}}$ ⁠. There are stable centres on the X-axis at X = 0 and $X = π$

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there will be stable oscillations in $\frac{π}{2} < | Θ_{0} | < \frac{3 π}{2}$ if

| X (0) | = | a_{0} + ϵ sin a_{0} | < \frac{π}{2}

(6.5)

and

| Y (0) | = | a_{1} | (1 - ϵ cos a_{0}) < \frac{| cos (a_{0} + ϵ sin a_{0}) |}{\sqrt{2}}

giving

| a_{1} | < \frac{| cos (a_{0} + ϵ sin a_{0}) |}{\sqrt{2} (1 - ϵ cos a_{0})}

(6.6)

for each a₀. Similar conditions apply for oscillations in $0 < | Θ_{0} | < \frac{π}{2}$ ⁠. We note that (6.3) can be solved in terms of Elliptic Functions but the qualitative essentials are evident from the phase plane in Fig. 10. It is interesting to observe that the period of small amplitude slow scale oscillations about either stable equilibrium can be obtained directly from the linearised form of (6.3) as

P = 2 π \sqrt{\frac{2}{{(ϵ ω)}^{2}}} \approx 4.44 s

(6.7)

for the parameter values in section 4. This is the weightless counterpart of the familiar small amplitude result for a terrestrial plane simple fixed pivot pendulum. It provides a direct simulation if the length L_t of such a terrestrial pendulum is given by

L_{t} = \frac{2 g}{ϵ^{2} ω^{2}} \approx 4.91 m

(6.8)

with the above parameter choices and g = 9.81 m/s/s. As far as the effects of damping are concerned it is clear that, depending on the initial data, a plane pendulum will terminate at either one of the asymptotically stable steady states at $Θ_{0} = π$ or $Θ_{0} = 0$ preceded by any number of complete or partially complete revolutions.

7 Summary

The primary aim of this article is to show that a simple inextensible pendulum can be made to function in a weightless environment by allowing its pivot to undergo rectilinear small amplitude high frequency oscillations. Having established this we then proceeded to analyse the three dimensional dynamics of such a pendulum. Using spherical polar coordinates centred at the pivot, we have shown that in addition to the high frequency component the pendulum trajectory has a slow scale response which we extract using the method of multiple scales with the scaled pivot amplitude being the small parameter. For an undamped pendulum, we study the autonomous slow scale polar equation and, depending on the initial conditions, we show that the pendulum bob may occupy solely the positive hemisphere, solely the negative hemisphere or visit both the positive and negative hemispheres during its trajectory. The slow scale precession and period of all these orbits can be computed numerically using quadratures but can also be expressed in terms of Elliptic Integrals. We have also been able to identify conical solutions of the slow scale polar equation as an aid in computing corresponding solutions of the exact model system. These can be obtained by choosing the initial conditions appropriately and exist in both the positive and negative hemispheres in contrast to the terrestrial fixed pivot spherical pendulum. For the plane problem in the absence of an azimuthal component of velocity, the weightless pendulum can provide a direct simulation of a fixed pivot terrestrial simple pendulum.We can follow the same two-scale analysis to examine the effect of damping. Depending on the initial data the pendulum trajectory may have two distinct phases. At first, it remains close to the undamped orbit until it undergoes a relatively abrupt transition to a decay phase into one of the two asymptotically stable equilibrium states. For other initial data, it can enter the decay orbit directly from the hemisphere from which it starts. All these trajectories are precessional.

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APPENDIX A Formulation in terms of Elliptic Functions and Integrals

The references [10] and [11] have been useful in compiling this Appendix. For us, there are two cases to consider: the first (I) when the biquadratic F(Z) in (4.2) has the real factorisation

\frac{1}{2} (a^{2} - Z_{0}^{2}) (Z_{0}^{2} + b^{2}), Z_{0}^{2} \leq a^{2} < 1

(A.1)

and secondly (II) when the factorisation takes the form

\frac{1}{2} (a^{2} - Z_{0}^{2}) (Z_{0}^{2} - b^{2}), b^{2} \leq Z_{0}^{2} \leq a^{2} < 1

(A.2)

I holds when $μ \geq \frac{1}{2}, C > \frac{4 μ + 1}{4}$ and for $0 < μ < \frac{1}{2}, C > \frac{4 μ + 1}{4}$ exterior to the curve Γ in Fig. 2. II is relevant when $0 < μ < \frac{1}{2}$ ⁠, $C < \frac{4 μ + 1}{4}$ interior to Γ.

The easiest way to proceed in both cases is directly from the differential equation (4.2). In case, I we make the change of variable

Z_{0} = \pm \frac{a b S_{0}}{\sqrt{a^{2} + b^{2} - a^{2} S_{0}^{2}}}

(A.3)

to give

{(\frac{d S_{0}}{d T_{1}})}^{2} = \frac{a^{2} + b^{2}}{2} (1 - S_{0}^{2}) (1 - k_{0}^{2} S_{0}^{2}) ​,

(A.4)

where

k_{0}^{2} = \frac{a^{2}}{a^{2} + b^{2}}

(A.5)

and the choice of sign in (A.3) accords with that of $\frac{d Θ_{0}}{d T_{1}} (0)$ ⁠.

Equation (A.4) together with the initial condition $S_{0} (0) = 0$ defines the Jacobi Elliptic Function sn so that we can immediately write

S_{0} (T_{1}) = s n (\sqrt{\frac{a^{2} + b^{2}}{2}} (T_{1} - T_{0}), k_{0}) ​,

(A.6)

where T₀ is determined from the initial conditions. The solution for $Θ_{0} (T_{1})$ then follows from (A.3). The period P can be obtained by making the change of variable

z = \frac{abs}{\sqrt{a^{2} + b^{2} - a^{2} s^{2}}}

(A.7)

in the integrand of (4.7) with (A.1) to give

P = \frac{4}{ϵ ω} \sqrt{\frac{2}{(a^{2} + b^{2})}} K (k_{0}),

(A.8)

where

K (k_{0}) = \int_{0}^{1} \frac{d s}{\sqrt{(1 - s^{2}) (1 - k_{0}^{2} s^{2})}}

(A.9)

is the Complete Elliptic Integral of the first kind.

Obtaining the precession rate (4.6) in terms of Elliptic Integrals is a little more involved but, by making the change of variable (A.7), we have from (4.5) with (A.1) that

\begin{matrix} Δ Φ_{0} = \frac{4 H_{0}}{\sqrt{2}} \int_{0}^{a} \frac{d z}{(1 - z^{2}) \sqrt{(a^{2} - z^{2}) (z^{2} + b^{2})}} \\ = 4 H_{0} \sqrt{\frac{2}{(a^{2} + b^{2})}} \int_{0}^{1} \frac{(1 - k_{0}^{2} s^{2}) d s}{(1 - k_{1}^{2} s^{2}) \sqrt{(1 - s^{2}) (1 - k_{0}^{2} s^{2})}} \\ = 4 H_{0} \sqrt{\frac{2}{(a^{2} + b^{2})}} (\frac{k_{0}^{2}}{k_{1}^{2}} \int_{0}^{1} \frac{d s}{\sqrt{(1 - s^{2}) (1 - k_{0}^{2} s^{2})}} \\ + \frac{k_{1}^{2} - k_{0}^{2}}{k_{1}^{2}} \int_{0}^{1} \frac{d s}{(1 - k_{1}^{2} s^{2}) \sqrt{(1 - s^{2}) (1 - k_{0}^{2} s^{2})}}) \\ = 4 H_{0} \sqrt{\frac{2}{(a^{2} + b^{2})}} (\frac{k_{0}^{2}}{k_{1}^{2}} K (k_{0}) + \frac{k_{1}^{2} - k_{0}^{2}}{k_{1}^{2}} Π (k_{1}, k_{0})), \end{matrix}

(A.10)

where $Π (k_{1}, k_{0})$ is the Complete Elliptic Integral of the third kind with

k_{1}^{2} = \frac{a^{2} (1 + b^{2})}{(a^{2} + b^{2})} .

Turning to Case II, we make the change of variable

Z_{0} = \pm \frac{a b}{\sqrt{a^{2} - (a^{2} - b^{2}) S_{0}^{2}}}

(A.11)

in (4.2) to give

{(\frac{d S_{0}}{d T_{1}})}^{2} = \frac{a^{2}}{2} (1 - S_{0}^{2}) (1 - k_{2}^{2} S_{0}^{2}),

(A.12)

where

k_{2}^{2} = \frac{a^{2} - b^{2}}{a^{2}}

(A.13)

with solution

S_{0} (T_{1}) = s n (\frac{a}{\sqrt{2}} (T_{1} - T_{0}), k_{2}) .

(A.14)

The form for $Θ_{0} (T_{1})$ then follows from (A.7). Finally, proceeding as in I above the period P is given by

P = \frac{2 \sqrt{2}}{a ϵ ω} K (k_{2})

(A.15)

and

Δ Φ_{0} = \frac{2 \sqrt{2} H_{0}}{a (1 - b^{2})} (\frac{k_{2}^{2}}{k_{3}^{2}} K (k_{2}) + \frac{k_{3}^{2} - k_{2}^{2}}{k_{3}^{2}} Π (k_{3}, k_{2})),

(A.16)

where

k_{3}^{2} = \frac{a^{2} - b^{2}}{a^{2} (1 - b^{2})} .

This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://dbpia.nl.go.kr/pages/standard-publication-reuse-rights)

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Article Contents

The behaviour of a forced spherical pendulum operating in a weightless environment

Summary

1 Introduction and equations of motion

2 A two-scale analysis of the governing equations

3 Conical solutions

4 The general initial value problem

5 The effect of damping

6 The plane problem

7 Summary

References

APPENDIX A Formulation in terms of Elliptic Functions and Integrals

Citations

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Article Contents

The behaviour of a forced spherical pendulum operating in a weightless environment

Summary

1 Introduction and equations of motion

2 A two-scale analysis of the governing equations

3 Conical solutions

4 The general initial value problem

5 The effect of damping

6 The plane problem

7 Summary

References

APPENDIX A Formulation in terms of Elliptic Functions and Integrals

Citations

Views

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