reflectedto the rest of the system. Any increase in Phase II is wasted, but the other half of the Rider's angular momentumtransfersto the Trikke and Rider around the turn center in Phase II. These results inform Trikke Riders desiring to improve their riding goals with a solid physical understanding previous unknown in the literature.

*spinning*. Spinning with appropriate arm and leg *tension* with some pushing and pulling drives a Trikke. The Trikke does not spin around it own axis, it rotates around a turn center located to one side or the other over a meter away. Since the Rider is holding onto the Trikke, she is also rotating around the same turn center at the same *angular velocity*. In this role, she is a *Glider*.

*decouple* the relationship (like becoming a "rag doll", waving to someone or jumping on the decks), but unless stated, assume enough tension is present.

Two rotating and one spinning object must account for the *angular momentum* of the system because angular momentum is physically conserved. The two rotating objects have a common angular velocity but the thrid object can spin independently. The Rider is the third object and one of the two rotating objects, the other being the Trikke. Two roles played by the Rider include one to *drive* the Trikke and one to *glide* with the Trikke. When refering to the Rider in this glider role, she will be called the *Glider*. A fourth quantity of angular momentum is a constant; the total amount in the system. Each harbors an amount of angular momentum using the following symbols:

- L
_{r}- Angular momentum due to the driving spin of the Rider. - L
_{t}- Angular momentum due to the reacting rotation of the Trikke around the turn center. - L
_{g}- Angular momentum due to the reacting rotation of the Rider around the turn center; the Rider has become a*Glider*. - L
_{i}- Total angular momentum in the system, often zero or an initial amount.

To be conserved, they must balance like this:

Equation 1: Total Angular Momentum is conserved

$\displaystyle L_{i} = L_{g} + L_{r} + L_{t}$

_{x} may be positive or negative depending on their direction of rotation. Positive for counterclockwise rotation; negative for clockwise. Generally, if one term is non-zero, one or both of the others must also be non-zero and rotating in the opposite direction to balance the relationship. They decompose further using the *Parallel Axis Theorem* and basic definitions.

Equation 2: Angular Momentum actors in terms of Moment of Inertia and Angular Velocity

$\displaystyle L_{r} = I_{r} ω_{r}$

$\displaystyle L_{t} = ω_{t} \left(I_{t} + M_{t} R^{2}\right)$

$\displaystyle L_{g} = ω_{t} \left(I_{r} + M_{r} R^{2}\right)$

_{r} and ω_{t}, which can be either positive or negative. Note the single expression in ω_{r}; the Rider's spin velocity. There are two terms in ω_{t}; the Trikke and Glider's rotation velocity. I_{r} represents the Rider's and Glider's moment of inertia and M_{r} their mass since they are different roles of the same person. I_{t} and M_{t} are the Trikke's moment of inertia and mass, which are much less than the Rider's. R is the distance from the Trikke-Glider center of mass to the the turn center.

Rewriting Equation 1 in terms of the velocities, the result is:

Equation 3: Conservation in terms of moments of inertia and angular velocities

$\displaystyle L_{i} = I_{r} ω_{r} + ω_{t} \left(I_{r} + I_{t} + R^{2} \left(M_{r} + M_{t}\right)\right)$

Equation 4: Trikke-Glider Moment of Inertia

$\displaystyle I_{tg} = I_{r} + I_{t} + R^{2} \left(M_{r} + M_{t}\right)$

_{tg} to make interpretation of the following equations more obvious. For the sake of argument, R, the distance to the turn center, may be considered constant as it will not affect the conservation relations. For Rider spin opposite to Trikke-Glider rotation, the equations are:

*external* at appropriate points in the analysis. It becomes part of the dynamics of the global reference frame and is ignored in the local analysis frame. Global variables, when needed are capital symbols with corresponing local variabls in lower case, except for the variables L, which are local. The local frame rotates with the Trikke-Glider at an angular speed of Ω_{t} due to built up angular momentum.

The two angular velocities Ω_{t} and ω_{r} can either be *aligned* in the same direction or in *opposite* directions. When aligned, Equations 5a and 5b result, otherwise Equations 6a and 6b for opposing directions. The a and b correspond to

- a. L
_{i}= 0 The local system starts with no net angular momentum. - b. L
_{i}≠ 0 The local system starts with some angular momentum in one direction or the other.

Note, initially in a half-cycle phase, local ω_{t} = 0 but its global counterpart, Ω_{t}, may not be.

$\displaystyle \operatorname{sign}{\left(ω_{r} \right)} = - \operatorname{sign}{\left(Ω_{t} \right)}$

Equation 5a: Reflection - Opposite Angular Velocities with no initial Angular Momentum

$\displaystyle I_{r} ω_{r} = - I_{tg} ω_{t}$

Equation 5b: Transfer - Opposite Angular Velocities with initial Angular Momentum

$\displaystyle L_{i} = I_{r} ω_{r} - I_{tg} ω_{t}$

For Rider spin in the same direction as Trikke-Glider rotation, the equations are:

$\displaystyle \operatorname{sign}{\left(ω_{r} \right)} = \operatorname{sign}{\left(Ω_{t} \right)}$

Equation 6a: Reflection - Aligned Angular Velocities with no initial Angular Momentum

$\displaystyle I_{r} ω_{r} = - I_{tg} ω_{t}$

Equation 6b: Transfer - Aligned Angular Velocities with no initial Angular Momentum

$\displaystyle L_{i} = I_{r} ω_{r} + I_{tg} ω_{t}$

Animation 1: Hip and Shoulder Rotation with Synchronous Turns and Turn Radii

*Drive Phase*, from when the Rider begins spinning and turning to when the Trikke is turned straight, the local angular velocities are opposite. When the system has no angular momentum in the local frame, Equation 5a above indicates that opposite velocity orientations cause the momentum terms to grow or to dissipate together, equally. What happens to one is *reflected* in the other. Since Rider spin increases, the Trikke-Glider also increases rotation in the opposite direction. The Rider supplies equal angular momentum to himself and the Trikke-Glider. Half of his energy goes into accelerating the Trikke-Glider and half his own spin.

*transferred* to the Trikke-Glider. In summary, the Rider's angular momentum transferes to the Trikke as he stops spinning.

Animation 2: Hip and Shoulder Rotation with Opposite Turns

Animation 3: Hip and Shoulder Rotation with delayed Turns

Another technique similar to Synchronous Steering, except that steering begins before the Rider spin stops. Phase II completes when the Rider stops spinning or when the Trikke steers past straight. If he stops spinning before passing straight, all of his Phase I angular momentum is transferred because spin and rotation are still alligned. However, there is less time to develop body spin for Phase II. If the Rider has delayed changing body spin beyond passing straight, there is no Phase I and nothing to transfer in Phase II.

Similar deductions about pre-steering plague the Contra technique. If the Rider stops spinning before passing straight, all of his Phase I angular momentum is transferred because spin and rotation are still opposed. The counter momentum built up will likely be less with less time to develop it, so there is less to make up for in Phase II. When waiting too long to start body spin nothing is available to transfer in Phase II.

Unlike the zero-sum exercise that the Contra Steering Technique is, Pre-Steering seems unadvisable for those desiring efficient locomotion.

_{t}, the Trikke and Glider's co-rotation velocity around the turn center, in Equation 3 above. Orientation of ω_{r}, the Rider's velocity, and value of the total angular momentum, L_{i}, determine specific results.

Equation 7: Local Trikke-Glider angular velocity in terms of Rider spin

$\displaystyle ω_{t} = \frac{- I_{r} ω_{r} + L_{i}}{I_{tg}}$

_{t} in terms of ω_{r} without regard to Phase initial conditions.

_{i} = 0, ω_{r} = 0, ω_{t} = 0. The local frame rotates with the global angular velocity Ω_{t} which can be positive (counterclockwise) or negative (clockwise). While the Trikke rolls out straight, ω_{r} increases in a direction opposite to Ω_{t}.

Equation 8: Body spin increases corotation in the other direction

$\displaystyle L_{i} = 0$

$\displaystyle ω_{t} = - \frac{I_{r} ω_{r}}{I_{tg}}$

_{t} and ω_{r} must have opposite signs (all other symbols are positive). Increasing ω_{r} in one direction, increases ω_{t} in the other, the direction of Ω_{t}. A reflection relationship.

Equation 9: local Trikke-Glider angular velocity is zero when steered straight

$\displaystyle ω_{t} = \lim_{R \to \infty}\left(- \frac{I_{r} ω_{r}}{I_{r} + I_{t} + R^{2} \left(M_{r} + M_{t}\right)}\right) = 0$

_{r}^{max}.

Equation 10: Local Trikke-Glider angular velocity during Synchronous Phase II

$\displaystyle \left|{ω_{r}}\right| \leq \left|{ω^{max}_{r}}\right|$

$\displaystyle ω_{t} = \frac{- I_{r} ω_{r} + ω^{max}_{r}}{I_{tg}}$

_{r} = 0 and ω_{t} = ω_{r}^{max}/I_{tg}, in the same direction.

© Copyright 2023 Michael Lastufka