The Coriolis effect and coupled oscillations in a rotating swings amusement ride

Rotating swings are found in many different versions in parks around the world. They are beautiful examples of the equivalence between gravitational and inertial mass: empty swings and swings with heavy adults hang at the same angle to the vertical. However, sometimes one can notice empty swings moving in a different pattern in an outdoor ride, where wind can induce additional motion, sideways or back to front—in addition to any oscillations caused by a tilted carousel head. This paper focuses on oscillations for the simpler case of a non-tilting roof. Even for this case, the oscillating motion is found to be complicated by the Coriolis effect, which leads to a gyroscopic coupling between sideways oscillations and back-to-front oscillations. This coupling is illustrated for a few special cases.


Introduction
Rotating swings (chain flyers) are found in many different versions in parks around the world, both with and without tilted carousel head (roof) as shown in figure 1.They are popular textbook examples in connection with circular motion.With the angle of the chains independent of the mass of the rider, chain flyers illustrate the principle of equivalence between inertial and gravitational mass [1,2].They can form the basis of many exercises, theoretical, observational and experimental [3], and can expose incomplete student understanding of concepts related to force and acceleration [4][5][6].
As the carousel turns, the swings hang out with an angle θ which depends on the angular velocity Ω.The swings, with chain length L are suspended in a circle with diameter D r = 2R.The diameter of the circle of swings changes to when the ride is in motion.
In Wave Swinger rides, like the Kättingflygaren shown in figure 1, the whole roof tilts while the carousel is in motion, introducing a wavy motion.Sometimes an empty swing in front of you seems to move out of phase with your swing.Could it be related to slightly different natural frequencies for empty swings?If the centripetal acceleration could be accounted for as if it were a linear acceleration, the frequency of the induced wave motion would be relatively close to the natural frequency of the swings.However, the resonance frequencies for the swings change due to rotation.This paper focuses on oscillations within the rotating system, but without considering the tilting of the roof.Notations used for this analysis is shown in figure 2.Here θ 0 denotes the fixed angle reached when the ride turns with constant angular velocity, Ω = Ωe Z .We also refer to θ 0 as an 'equilibrium' angle or 'steady state' angle although a swing in circular motion is accelerating and not strictly in equilibrium (in this work, we avoid using the inertial centrifugal force).

Rotation, acceleration and angles
When a chain flyer ride starts to turn, the swings hang out, more and more, as the ride rotates faster and faster, since the circular motion involves a centripetal acceleration.The centripetal acceleration needs a horizontal component of the force F c from the chains and the angle θ 0 is related to the acceleration, q = a g tan h 0 as indicated in figure 3. Note that this expression does not involve the mass-the angle is the same for all swings at the same distance from the axis of rotation.The angle gives a direct visual estimate of the acceleration.
The centripetal acceleration is purely horizontal and given by q with notations introduced in figure 2. In the coordinate system rotating with the carousel, e x points out from the centre of the ride.The angular velocity is given by Ω = 2π/T, where T is the time for a full turn.The inner swings, with their smaller values for R, hang at slightly smaller angles.
We are now interested in the equations of motion for small oscillations around the equilibrium angle.We denote the small sideways angular deviations by δ, as indicated in figure 4.
Figure 2. Notations used to analyse the motion in chain flyer rides.The whole ride rotates around an axis, parallel to the Z axis in the XYZ coordinate system which is fixed.The xyz system, used for the analysis in section 4, rotates along with the carousel, with the x axis pointing out from the center and the z axis coinciding with the fixed Z axis.The focus of the paper is on the motion relative to the points Q at the 'equilibrium' locations of the swings within the rotating system.However, before treating that situation in a rotating system, we consider the simpler case of oscillations during constant acceleration in linear motion.

Oscillations in a system in linear motion with constant acceleration
As a first step, consider a situation with constant horizontal acceleration, a h , without rotation.The force diagram in figure 3 still applies.In the absence of oscillations we find If the angle deviates from θ 0 by a small angle δ, as indicated in figure 4, we get q d = ̈̈and where we used the relation in (1) together with the small-angle approximation for δ.Inserting the expression q = a g tan which differs from the expression for a pendulum in a non-accelerating system through the factor q cos in the denominator, resulting in a slightly shorter period: cos cos 0 0 0 , where T 0 is the period for a pendulum in a non-accelerating system.

Oscillations orthogonal to the horizontal acceleration
The swings may also oscillate in a direction orthogonal to the horizontal acceleration of the system.The motion can be described by an angle f.In a rotating swing ride, this would be in the direction of motion, as indicated in figure 5, whereas for linear motion, it would involve a sideways motion.Newton's second law gives , where L is the length of the chains and R is the distance from the suspension points of the chains to the center of the ride.
with coordinates as indicated in figures 6 and 7. Keeping only terms in the f direction gives , which can be rewritten as an equation for f ̈:  When the system is rotating, the horizontal acceleration is given by the centripetal acceleration when the chains are at the equilibrium angle a h = a c,0 (see also figure 7, where the coordinate system is viewed from a different angle).
Recalling again that q = a g tan h 0 , dividing the equation by L and using f f » sin we rewrite the equation in the same way as equation (2) as For a constant acceleration the period is thus independent of the direction of the oscillations.
A consequence of the equivalence principle is that a linear acceleration cannot be distinguished from a gravitational field in the opposite direction, giving an 'effective gravitational field', g − a h .A term −ma h can be viewed as an inertial force.In a rotating system, this inertial force is often referred to as a 'centrifugal force'.Although conceptually different, the mathematics is equivalent.
As shown below, the situation changes for a rotating system: The horizontal acceleration is then given by the centripetal acceleration, which depends on the distance from the axis of rotation, as indicated in figure 4. In addition, motion relative to a rotating system involves Coriolis effects.

Rotation without tilted roof
We now consider swinging in the rotating system, but without considering the tilted roof.Below, we first discuss the corrections to oscillations restricted to either θ or f directions.

Oscillations towards and away from the center
Without tilted roof, but taking the centripetal acceleration into account, we find If the swing hangs at an angle θ = θ 0 + δ, we find q d = ̈̈which is zero at the equilibrium angle θ 0 For small deviations δ we can write In analogy with the case of constant acceleration, we use the relation This gives Note that the R dependence is hidden in the angle θ 0 .Note also the additional term due to the rotation of the whole carousel.

Swinging backward and forward
The swinging motion can also be backward and forward as indicated in figures 5-7.Note that the angle f is defined as motion within the plane spanned by the chain and the direction of motion, resulting in a relative velocity L .An angle f gives a horizontal displacement f L sin , which gives an additional term in the centripetal acceleration to be subtracted from the right hand side of equation (4).This leads to a different equation for f f Note the additional term Ω 2 due to the rotation of the whole carousel and that it differs from the additional term for sideways oscillations given in (7).

Comments
The expressions in (7) and (8) both include an additional term, compared to the constant acceleration equation (2).Since this term has opposite sign, it leads to a slight reduction of the oscillation frequency, which makes sense.When the angle is larger than θ 0 , the centripetal acceleration is larger, thereby reducing the size of the angular acceleration.Similarly, when the angle is smaller than θ 0 the centripetal acceleration is smaller, again reducing the angular acceleration of θ.The additional terms can also be thought about as resulting from a 'centrifugal force'.

The Coriolis effect
New students often know that the Coriolis effect is important for weather systems [7], but may view it as an abstract phenomenon that is hard to observe directly.Still, even young children can be fascinated by simple investigations, like throwing a ball across a slow-moving carousel, or watching a small pendulum swing as an illustration of the Foucault pendulum [2,8].Boccherini and Straulino [9] used a video of the Foucault pendulum at their university allowing for a detailed analysis.One of the IYPT problems in 2017 dealt with a heavy ball falling through a liquid in a rotating cylinder, as discussed by Fu [10].Students taking accelerometer data in large pendulum rides with riders seated around a rotating disc or circle, or other rides with several rotation axes, may discover that the Coriolis effect influences the forces experienced by the riders [11,12].Most modern smartphones include MEMS sensors, that can also measure rotation, e.g. by using 'butterfly gyros' with two small wings connected to an outer frame using three flexible beams.Using electrodes, the wings are made to vibrate so that the masses on opposite sides of the beams will vibrate towards each other.When the whole structure is rotated around the axis between the wings, the Coriolis effect leads to a torque, resulting in an electric signal [13][14][15][16][17]. Easy access to rotation sensors can help students understand the concept of rotation around different axes and opens many possibilities for education (see e.g.[18][19][20][21][22][23][24][25]).
The oscillations of a swing in a rotating swing ride involve a motion relative to the rotating system.As the swing moves inward, it tends to move slightly forward, due to the larger tangential speed when the swing is further away from the center.Similarly, a motion out from the center is accompanied with a motion slightly backward.For a counterclockwise rotation, this leads to a deviation to the right.
The Coriolis effect thus leads to a coupling with the sideways and forward/backward oscillations If you have ever tried swinging your legs in a carousel moving fast in a horizontal plane, you recognize the resulting elliptical motion.

Mathematical treatment
The acceleration for motion relative to a point Q in a rotating system can be written as (e.g.[26], p 400) In the case of the waveswingers, the main rotation Ω = Ωe Z is around a vertical axis (for (most) waveswingers, the rotation is clockwise, corresponding to a negative value for Ω).The point Q is chosen to be the imagined location of a non-oscillating swing moving around the ride with chains at the equilibrium angle θ 0 , as indicated in figure 2. The acceleration a Q of point Q and is directed towards the axis of rotation with the magnitude 2 .We introduce a coordinate system, cθf, moving together with the rider (and a sensor taken along, e.g. a smartphone), where e c points to the suspension point, e θ to the side and e f points forward.This coordinate system can also be expressed in the xyz system rotating together with the ride around the Z axis, as indicated in figure 2. The corresponding angular velocity can be expressed as Ω = Ωe Z = Ωe z q q q q = -= + =

Relative motions in the rotating system
The sideways oscillation discussed above involves a motion relative to the rotating system, as studied by Coriolis.For a counterclockwise rotation-positive Ω-this leads to a deviation to the right.The Coriolis effect leads to a coupling with the sideways and forward/backward oscillations, sometimes referred to as a gyroscopic coupling.
The velocity relative to the rotating system can be written as rel This relative velocity results in a relative acceleration within the rotating system, which can be written as 2 cos cos .

Cor y x
Using q q = q ( ) e e e cos sin x c 0 , and f » cos 1, this gives q q f q q = -W -- 2 cos cos sin .

Cor c 0
The final term gives an additional contribution to the force from the chain, which would show up in radial accelerometer data, but does not affect the angular motions.

Collecting all terms orthogonal to the chain
The sideways components involving e θ give while the forward-backward component (e f ) gives The last term in these two equations lead to a coupling between the sideways and forwardbackward motions: as the swing moves inward (q d = <   0) it tends to move slightly forward, causing an angular acceleration f > ̈0-for positive Ω.This can be ascribed to the larger tangential speed when the swing is further away from the center.Similarly, a motion out from the center is accompanied with a motion slightly backward.A motion forward also involves an outward motion in the rotating system, whereas a backward motion seems to cause an inward acceleration.For a system rotating counterclockwise, all motions are associated with an apparent force to the right, within the rotating system.
In a more compact form the equations for the angular motions can be written as with the coefficients given as Table 1 shows estimated values of the different parameters in (12) for the Zierer waveswinger shown in figures 1 and 8, together with resulting solutions to the coupled equations in (11), as described in section 5.

Ansatz and diagonalization
We now make the simplest possible Ansatz to solve the coupled equations in (11) (one could also include a possible phase, or choose to work with complex functions, e ±iωt ).
Insertion into equation (11) gives Using equation (15), we get b = Cωa/(B − ω 2 ) which can be inserted into equation (14), Combining terms gives ω 4 − (A + B + C 2 )ω 2 + AB = 0, with the solutions for ω 2 : We note that if ω n is a solution, also −ω n will be a solution.However these two solutions will be identical, since also b will then change sign and w w --= ( ) t t sin sin .There are thus only two distinct solutions, apart from a possible phase shift, α which can be included by changing ωt to ωt + α.

Eigenfunctions and initial conditions
The eigenfunctions resulting from the diagonalization can be written as where we have used Table 1 shows numerical results for the Zierer waveswinger in figure 1, Using these values, we find the two eigenvalues ω 1 ≈ ±2.35 s −1 and ω 2 ≈ ±0.91 s −1 .illustrating that the gyroscopic coupling between sideways and back to front motions changes the oscillation frequencies far away from the results without this coupling, and both far from the frequency of induced oscillations due to the tilted roof.That empty swings sometimes seem to move with opposite phase from your own swing can thus not be accounted for by sightly different natural frequencies.
The numerical results for the eigenvalues give a relation between the coefficients in the Ansatz (13): b 1 /a 1 ≈ 0.90 and b 2 /a 2 ≈ −1.32, both corresponding to counterclockwise elliptical orbit for the Zierer waveswinger with its clockwise main rotation.The eigenfunctions can, of course, also be used to generate more complex motions for different initial conditions, as discussed below.

Holding onto swing in to the side
A first example is the situation where you start by holding a swing to the side and then letting go.This would give conditions δ = δ 0 < 0 and f = 0 with both angular velocities zero.As the swing starts to move outward, the angular velocity d  becomes positive.This in turn leads to a positive acceleration for f, since C is negative for the clockwise motion, as found for the simulated values shown in figure 9.A positive f  implies a backward motion, as seen from the notations in figure 5.

Holding onto swing in front
A second example is the situation where you first hold on to a swing in front of you and letting go, where only f differs from zero at the start.For clockwise motion, like in the waveswinger in figure 1, holding onto a swing in front gives a negative initial value for f.When the swing is free to move, f  becomes positive (as the swing moves backward).This in turn leads to a negative value for d ̈, (and then d  and δ), according to (11), as can be seen also from figure 10.The swing thus starts to move inwards.For an overall clockwise motion, the Coriolis effect causes motion to deviate to the left.

Tilted roof
While a wave swinger ride is in motion, the roof tilts, as shown in figure 1(a).This leads to a wavy motion of the swings.The swings then also accelerate up and down, with a period just below 4 s, close to the period for purely lateral oscillations.However, the Coriolis effect leads to more complex motion, as seen from the results of the simulations shown in figures 9 and 10.In fact, the gyroscopic coupling between the different modes leads to large shifts of the different resonance frequencies, as seen in table 1.

Discussion
Weather systems may be the most well-known examples of the Coriolis effect, but the original mathematical description by Coriolis involved motion in rotating machines.This paper gives an example where the Coriolis effect is needed to understand observations, in this case of the motion of empty swings in a chain flyer ride.Although the equivalence principle makes the motion of the swings essentially independent of mass, empty swings are more sensitive to perturbations by the wind, which can initiate oscillations both sideways and backto-front.This can account for the counterclockwise motion of an empty swing observed during a ride, as indicated in figures 8 and in a video provided as supplementary material.More detailed analysis will be possible for an indoor ride without tilted roof, which could be a rewarding extended student project.

Figure 1 .
Figure 1.Two chain flyers: the Zierer wave swinger Kättingflygaren at Gröna Lund in Stockholm and the Star flyer Himmelskibet at Tivoli Gardens in Copenhagen.

Figure 3 .
Figure 3. Forces and acceleration in a rotating swing ride, where the chains form an angle θ 0 to the vertical and the horizontal acceleration is given by q = a gtan h

Figure 4 .
Figure 4. Change of position for a small change δ in the angle θ to the vertical, where θ 0 is the angle for constant angular velocity without any additional swinging.For constant angular velocity, Ω around a vertical axis, Z, the horizontal acceleration is given by q = + W ( ) a R Lsin

Figure 5 .
Figure5.The Kättingflygaren wave swinger at Gröna Lund, together with angles and axes used for the analysis.Sideways motion involves changes in θ whereas back-tofront motion involves changes in f.Note that f is defined in the plane spanned by the chain and the direction of motion of the swing.

Figure 6 .
Figure 6.Orientation of the vectors in the comoving coordinate systems xyz and θfc.When the system is rotating, the horizontal acceleration is given by the centripetal acceleration when the chains are at the equilibrium angle a h = a c,0 (see also figure7, where the coordinate system is viewed from a different angle).

Figure 7 .
Figure 7.The comoving coordinate systems as viewed from the side (see also figure6). d

Figure 8 .
Figure 8. Screen shots from a video taken during a ride in the Wave Swinger Kättingflygaren at Gröna Lund.Note how the different swings move relative to each other.The video (included as supplementary material) indicates a counterclockwise elliptical orbit for the empty swing (close to the end of the ride, when the roof is no longer tilted).

Figure 9 .
Figure 9. Example of motion starting with an initial angle θ deviating by δ from θ 0 (as can happen if you hold a swing to the side and then let go), using values from table 1.The dotted curves represent the time derivatives of the angles.

Figure 10 .
Figure 10.Example of motion starting with an initial angle f deviating from 0 (as can happen if you hold a swing in front of you and let go).The dotted curves represent the time derivatives of the angles.

Table 1 .
Oscillation frequencies for different situations.The numerical values are based on an estimate of a period of 4.5 s for the swings at rest, corresponding to a chain length of 5 m˙.Earlier work estimated that the angle θ 0 ≈ 50°, corresponding to a centripetal The time for a full turn was 5.6 s, giving Ω ≈ −1.11 s −1 , where the negative sign denotes clockwise direction of motion of the swings around the center.