For a demonstration of the Lewis Drift, I recommend watching this video.

This is a model of the Lewis Drift in action using Algodoo, a physics simulator. In the video, you see how I create the system and how I elicit the effect. The following is the math and physics that explain why the Algodoo program is actually working fine.

1. Introduction

Many mechanical systems that involve gyroscopes, constrained linkages, and friction exhibit non‑intuitive behaviors wherein purely rotational inputs (such as spinning rotors or wheels) cause gradual translation—or “drift”—over time. This is somewhat analogous to wave or fluid‐mechanics contexts, where repeated oscillations yield small net displacements (often called a “drift,” akin to “Stokes drift” in fluid literature). In a similar vein, the system here is:

Three “orbs” (masses) of equal mass, connected by an inextensible chain
A gyroscope mounted to one of the orbs, capable of delivering a torque up to 5000 Nm by spinning a disc of mass ~10 lb at 1000 rpm
Friction/damping present at each orb’s contact with the environment (or possibly fluid/air damping as in Stokes Drift), ensuring that energy does not simply remain in cyclical internal motions
A goal to understand how, despite the torque nominally being “rotational,” the overall system’s center of mass exhibits slow but persistent drift under the right conditions.

We refer to this net motion as a “Lewis drift” in the sense that it is a slow net motion that emerges from cyclical or rotating dynamics. Our aim is to derive why this drift occurs, how the chain constraints are critical, and how friction plays a key role in ensuring that the reaction force from the gyroscope does not merely induce internal vibrations but leads to net translation.

2. Background and Motivation

2.1 Constrained Mechanical Systems

When multiple masses are tied together by rigid (or nearly rigid) linkages, the internal forces often cancel in any free‑body sum for the entire system. For instance, if we have three masses tied by a massless chain, the tension in the chain is an internal force. Normally, internal forces do not change the total momentum of a free system. Thus, if there were no external forces or torques, the center of mass would remain fixed (or move uniformly if initially in motion).

2.2 Gyroscopic Torque and Precession

A rapidly spinning disc with angular momentum $\mathbf{L}_g$ can be subjected to a control torque $\boldsymbol{\tau}$ . Per the standard gyroscopic relation,

\frac{d\mathbf{L}_g}{dt} \;=\; \boldsymbol{\tau}.

This torque, while apparently “purely rotational,” comes with a reaction on the mass that houses the gyro. If the gyro is rigidly attached to Orb 1 (say), that orb experiences a reaction moment and, depending on the system constraints, an effective force. In an unconstrained object in free space, a purely internal torque would produce only reorientation, not net translation. However, the presence of friction and constraints alters that picture.

2.3 Stokes Drift

In wave mechanics (especially in ocean waves), “Stokes drift” describes how a repeating oscillation of fluid particles yields a net forward motion over one or more cycles. In mechanical linkages, a conceptually related phenomenon occurs: if you have cyclical or rotational motion in a constrained system with friction or other forms of dissipation, the net effect can be a small incremental translation each cycle. Over time, these increments add up, creating a slow drift of the entire body or system.

3. System Setup

Let us define our system more concretely:

Three Orbs of Mass $m$ Each
Positions at time $t$ are $\mathbf{r}_1(t)$ , $\mathbf{r}_2(t)$ , and $\mathbf{r}_3(t)$ . The total mass is $M = 3m$ .
Chain Constraints
We assume each pair of orbs is connected by a segment of chain of length $L$ . If it is a triangular chain, we have the constraints
$|\mathbf{r}_1 - \mathbf{r}_2| = L, \quad |\mathbf{r}_2 - \mathbf{r}_3| = L, \quad |\mathbf{r}_3 - \mathbf{r}_1| = L.$
Alternatively, if it’s a “linear” chain from orb 1 to orb 2, and then orb 2 to orb 3, we have two constraints. Either way, these constraints enforce fixed distances between orbs.
Gyroscope on Orb 1
- Disc mass: ~10 lb (≈ 4.54 kg in SI)
- Spin speed: 1000 rpm $\Rightarrow \omega = \frac{1000 \times 2\pi}{60} \approx 105\,\text{rad/s}$
- Torque capacity: $\tau_{\max} \approx 5000\,\text{Nm}$
The gyroscope’s moment of inertia, $I$ , depends on geometry. The spinning disc has angular momentum $\mathbf{L}_g = I\omega\,\hat{\mathbf{s}}$ . The user can apply a torque $\boldsymbol{\tau}$ , possibly up to 5000 Nm, that attempts to change the orientation of $\mathbf{L}_g$ . The reaction from that torque is transmitted to Orb 1.
Friction
Each orb experiences a frictional/damping force. A typical simplified model is viscous damping:
$\mathbf{F}_{\text{fric},\,i} \;=\; -\,\gamma\,\dot{\mathbf{r}}_i,$
where $\gamma$ is a positive damping coefficient with appropriate units ( $\text{kg/s}$ in SI). More complex friction (e.g. Coulomb friction, or fluid drag quadratic in speed) can be used, but we will keep it linear for clarity.

4. Why a Purely Rotational Torque Can Produce Net Translation

4.1 Internal vs External Forces

If Orb 1 were free in space, a purely internal torque on the gyro would not accelerate its center of mass. However, in this system:

Constraint Forces: The inextensible chain enforces certain geometric constraints, creating tension forces among the orbs.
Damping: Friction of the linkages ensures that small, repeated do not simply cancel over each cycle but can accumulate over time into a net translation.

When the gyroscope is torqued, Orb 1 tries to rotate or reorient. But Orb 1 is linked to Orbs 2 and 3 through the chain. The chain tension “pushes” or “pulls” the other orbs in response. Because friction is present, there is no perfect “equal and opposite” purely internal motion that remains cyclical. Instead, each small motion experiences partial damping, and the net result is a slow shift in the center of mass.

4.2 Net Drift from Repeated Micro‑Imbalances

One can think of each application of torque on the gyro as slightly unbalancing the system’s geometry. In the absence of friction, the system might simply undergo periodic or precessional motion with no net translation. But friction tends to “rectify” some portion of these motions, leading to a small net displacement each cycle.

Over time, the sum of these small increments yields what I call the Lewis Drift: an overall slow movement in some direction that correlates with how the torque is being applied (and with the chain geometry that channels forces through the orbs).

5. Mathematical Model

5.1 Center of Mass Dynamics

Define the center of mass (COM):

\mathbf{R}(t) \;=\; \frac{\mathbf{r}_1 + \mathbf{r}_2 + \mathbf{r}_3}{3}.

The total external force on the entire system is the sum of friction on each orb plus any net reaction from the gyro. Internal forces (chain tensions) cancel in a sum over the entire system.

If $\mathbf{F}_{\text{gyro}}(t)$ denotes the net external force “felt” by the orbs due to the gyroscope’s reaction torque (this can be subtle to define, but physically there is a net push/pull whenever the gyro tries to reorient), and each orb has friction $-\gamma\,\dot{\mathbf{r}}_i$ , the net external force is

\mathbf{F}_{\text{ext}} \;=\; \underbrace{-\gamma \left(\dot{\mathbf{r}}_1 + \dot{\mathbf{r}}_2 + \dot{\mathbf{r}}_3\right)}_{\text{sum of friction}} \;+\; \underbrace{\mathbf{F}_{\text{gyro}}(t)}_{\text{reaction force}}.

Because $\dot{\mathbf{r}}_1 + \dot{\mathbf{r}}_2 + \dot{\mathbf{r}}_3 = 3\,\dot{\mathbf{R}}$ ,

3m\,\ddot{\mathbf{R}} \;=\; -\,3\gamma\,\dot{\mathbf{R}} \;+\; \mathbf{F}_{\text{gyro}}(t).

Or,

\ddot{\mathbf{R}} \;+\; \frac{\gamma}{m}\,\dot{\mathbf{R}} \;=\; \frac{1}{3m}\,\mathbf{F}_{\text{gyro}}(t).

This is a simple but powerful statement: the entire system’s center of mass evolves according to a linear Ordinary Differential Equation (ODE), once we know $\mathbf{F}_{\text{gyro}}(t)$ .

5.2 Relating Gyroscopic Torque to Effective Force

A purely rotational torque $\boldsymbol{\tau}(t)$ on the gyro must be “reacted” by the orb that houses it. In free space, that would only change orientation. However, the chain constraints that couple Orb 1 to Orbs 2 and 3 produce small displacements that do not purely cancel out. For many practical or approximate analyses, one writes:

\mathbf{F}_{\text{gyro}}(t) \;=\; \frac{\boldsymbol{\tau}(t) \times \mathbf{r}_{\mathrm{lever}}}{|\mathbf{r}_{\mathrm{lever}}|^2}

or something akin to that, where $\mathbf{r}_{\mathrm{lever}}$ is an effective lever arm from the system’s pivot or centroid. The exact expression depends on how the chain constrains Orb 1 and how the tension is distributed. In short, $\mathbf{F}_{\text{gyro}}$ can be understood as the “unbalanced” portion of the torque’s reaction that does not remain purely internal.

5.3 Illustrative Drift Equation

If $\boldsymbol{\tau}(t)$ is constant or oscillatory with some net effect, we can in many cases reduce to:

\ddot{\mathbf{R}} \;+\; \alpha\,\dot{\mathbf{R}} \;=\; \mathbf{f}(t),

where $\alpha = \gamma/m$ and $\mathbf{f}(t)$ encapsulates the net torque–to–force conversion. The Lewis drift is then visible in solutions where $\mathbf{f}(t)$ has a nonzero mean or resonates with the damping. Over times longer than a few damping times $1/\alpha$ , the center of mass can move significantly.

6. Detailed Mechanisms of Drift: Chain + Gyro + Friction

Torque Application: The user (or a remote control signal) instructs the gyro to tilt or precess by imposing $\boldsymbol{\tau}(t)$ .
Local Reaction: Orb 1 feels a “reaction moment.” Because the orb is connected to the other two orbs by the chain, and the chain does not allow extension/compression beyond a certain limit, the orb’s attempt to rotate translates into small positional adjustments that tug on the chain.
Chain Force Propagation: The chain tension redistributes the forces among Orbs 2 and 3. Even though the total internal forces sum to zero, the arrangement in space can produce a net push/pull on the system.
Frictional Rectification: Each small “push/pull” is partially “rectified” by friction. Instead of swinging out and returning to the original position, the orbs lose a fraction of that angular momentum to momentum, so the return path does not exactly retrace the outward path. This asymmetry yields a net forward (or sideward) step.
Accumulation: Repeating the process at the gyroscope’s command can accumulate into a drift. That drift is typically slow compared to the velocity scale of the orb’s local motions. But over a long duration, it becomes noticeable.

7. Connection to “Lewis Drift” Concept

Historically, “Stokes drift” arises when periodic or oscillatory motion has a second‑order effect that does not average to zero over a cycle. In wave contexts, fluid parcels in a wave do not merely go up and down in place; they trace out elliptical or circular orbits that do not close perfectly, leading to a net drift.

In the mechanical system described here, the chain + gyroscope + friction interplay is analogous: each “cycle” of torque application and the resulting chain reaction can yield a slight net translation. Because the system is dissipative (due to friction), the path out is not the exact opposite of the path back—thus a net displacement remains.

8. Example Calculation and Simulation Outline

For a more concrete sense of the drift, one might:

Set up the full equations of motion using Lagrangian mechanics with constraint forces (via Lagrange multipliers $\lambda_{ij}$ for each chain segment).
Include friction by adding a non‑conservative generalized force $-\gamma\,\dot{\mathbf{r}}_i$ .
Incorporate the gyroscopic torque $\boldsymbol{\tau}(t)$ in the rotation of the disc and the reaction on Orb 1.
Numerically integrate from $t=0$ to $t=T$ .
Track $\mathbf{R}(t)$ to see the drift build up over time.

Often, one sees that $\mathbf{R}(t)$ remains near zero in early transients but gradually accelerates or shifts, eventually reaching a steady drift velocity (if $\boldsymbol{\tau}$ is constant) or a slow meandering path (if $\boldsymbol{\tau}$ changes orientation).

A simplified approach, used in some preliminary analysis, is to guess an effective lever arm $r_{\mathrm{eff}}$ so that

\mathbf{F}_{\text{gyro}}(t) \;=\; \frac{\boldsymbol{\tau}(t)}{r_{\mathrm{eff}}} \;\times\; \hat{\mathbf{n}},

where $\hat{\mathbf{n}}$ is some direction or unit vector that depends on the system geometry. Then one solves the linear ODE for $\mathbf{R}(t)$ . This yields a convenient approximate formula for the drift if friction is linear.

$R ¨ + \frac{\gamma}{m} R ˙ = \frac{1}{3m} F_{gyro}(t)$

R ˙

This is a first-order linear ODE in velocity and a second-order linear ODE in position, describing how the center of mass of the system evolves over time under friction and the gyroscopic force.

Since it's linear, it can be solved using standard methods like the integrating factor (for first-order cases) or characteristic equations and particular solutions (for second-order cases).

9. Discussion of Results and Physical Insight

Small Net Force, Large Effect Over Time
Because friction is typically not huge in many mechanical designs, the net force that remains after each cycle might be small, yet repeated application yields centimeters or meters of displacement if the process runs long enough.
Influence of Chain Geometry
The shape (triangle, linear chain, or otherwise) significantly impacts how the internal forces sum and how effectively the gyroscopic torque is converted into a net external force. A symmetrical triangle might yield a smaller drift if the torque is aligned with a symmetry axis, whereas an asymmetric geometry can accentuate the net push in a given direction.
Friction as a Necessary Ingredient
Without friction (i.e., in a purely idealized, conservative system), one typically gets no net drift. Internal cyclical motions remain cyclical. The friction is what “breaks” time‑reversal symmetry, allowing a portion of each cycle’s motion to remain unreturned.
Practical Control Implications
In some advanced robotics or mechanical designs, such “rotational to translational” motion conversion can be harnessed as a control method. A single rotor or reaction wheel can produce net translation if the robot is subject to frictional constraints (e.g., on the ground or in a fluid).

10. Conclusion

10.1 Summary

Core Phenomenon: A torque purely applied to a spinning disc (gyroscope) on one orb in a chain‑constrained system will, in principle, only reorient that orb unless friction and constraints allow a net force to develop.
Lewis Drift Analogy: This net force, when integrated over time, manifests as a slow drift of the center of mass, reminiscent of wave drift phenomena in fluids.
Mechanism: The chain constraints distribute the reaction forces across multiple orbs in a non‑canceling way, while friction prevents purely elastic recoil, thus leading to incremental shifts that accumulate.

10.2 Practical Takeaways

Modeling Approach: One can employ Lagrangian mechanics or Newton’s laws with constraint forces to get the precise equations of motion. The friction and the gyroscopic torque’s reaction are the key external influences.
Drift Equation: In simplified form, the center‑of‑mass satisfies a damped second‑order ODE with an effective forcing term derived from the gyroscopic torque.
Design / Control: Understanding how friction, chain geometry, and torque direction combine is essential for either preventing unwanted drift or deliberately exploiting it.

10.3 Future Directions

Nonlinear Friction Models: Real friction could be velocity‑dependent in a more complex way (Coulomb friction, Stribeck effect, etc.). This will alter the drift rate and possibly produce thresholds where the drift stops below a certain torque.
Time‑Varying Torques & Precession: In real operation, the torque is not always constant; it may vary with time or with the disc’s angular momentum vector. This leads to more complex drift paths.
Flexible Chains or Additional Degrees of Freedom: If the chain itself has elasticity or slack, the internal dynamics become richer. Additional modes of motion may either enhance or dampen the drift effect.

11. Recap and Key Points

Below is a succinct but thorough recap:

Initial Setup
- Three equal masses, connected by an inextensible chain, with one mass carrying a spinning disc that exerts a large torque.
- Friction ensures energy can be dissipated and any cyclical motion can be turned into a net offset in position.
Physical Intuition
- If you try to tilt a spinning wheel, it resists, generating a reaction torque. In free space, that only changes orientation. But with constraints to other masses, the geometry forces some tangential or radial displacement.
- Over each small cycle of tilting or reorienting the gyro, friction keeps the orbs from returning exactly to where they began.
Chain’s Role
- The chain is not just holding the masses together in a static sense; it is an active transmitter of forces. These chain forces can create “off‑center” tension patterns. If the system geometry is not perfectly symmetrical or if the torque alignment is offset, a net push emerges.
Gyroscopic Reaction Force
- The large torque (e.g., 5000 Nm) is effectively “fought” by the orb, but because the orb is pinned to the others, the entire system feels a small net push or pull over time.
Mathematical Form
- Center of mass obeys $M\,\ddot{\mathbf{R}} + \Gamma\,\dot{\mathbf{R}} = \mathbf{F}_{\text{gyro}}$ .
- The chain constraints appear implicitly in $\mathbf{F}_{\text{gyro}}$ . Friction appears explicitly in $\Gamma\,\dot{\mathbf{R}}$ .
Long‑Term Drift
- Eventually, if the torque is steady (or cyclical with a nonzero mean effect), $\mathbf{R}(t)$ can grow linearly or in some cases accelerate and then saturate, depending on friction.
Relation to “Lewis Drift”
- In wave mechanics, repeated cycles yield net drift. Here, repeated attempts to reorient a spinning disc do similarly. This resolves how second‑order expansions of cyclical motion lead to net displacements.
Practical Implications
- Potential unintentional drifting in mechanical systems with reaction wheels or gyros.
- Deliberate exploitation for locomotion in constrained robots (e.g., inchworm or friction–driven robots that rely on cyclical internal motions to move externally).
Open Questions / Additional Complexity
- Impact of chain slack or elasticity
- Turbulent friction in fluids
- Multi‑gyro interactions if orbs each have spinning discs

The central conclusion is clear: the drift arises from the synergy of the gyroscopic torque, the chain’s geometric constraints, and friction’s dissipation of energy, which prevents purely reciprocal motion. That synergy I call “Lewis drift” explains why the system slowly travels through space even though the applied control is nominally “rotational.”

Final Remarks

To wrap it all up:

Yes, the chain and the gyro configuration (together with friction) are precisely what cause the drift.
No, a purely rotational torque on a free rigid body would not induce net translation, but here the constraints and friction break the symmetry.
Yes, the mathematics can be captured by imposing the chain constraints, friction forces, and the gyroscopic reaction torque into Newton’s or Lagrange’s equations, then observing a net forcing term on the center of mass.

Working Backwards

In order to confirm our findings we will work backwards from the drift noticed in the Algodoo Physics Simulator when we construct this system:

Start from the observed drift: Assume the system exhibits a steady net drift and determine what forces and constraints are necessary to produce it.
Check momentum conservation: Ensure that internal forces do not introduce net momentum violations.
Reverse-derive the effective force: Express the net external force in terms of system constraints and gyroscopic torque.
Verify that our governing equation remains valid: The center-of-mass equation should still hold when backtracked.
Ensure that the reaction force from the gyro leads to an actual displacement: This is the key step in verifying that the drift is real and not an artifact of our assumptions.

Step 1: Define the Observed Drift

From our previous drift equation, we had:

$\mathbf{R}(t) = \frac{\tau}{3\gamma r_{\mathrm{eff}}} \hat{\mathbf{d}} \left[t - \frac{m}{\gamma} (1 - e^{-\gamma t/m})\right]$

For long times ( $t \gg m/\gamma$ ), this simplifies to a steady linear drift:

$\dot{\mathbf{R}}_{\infty} = \frac{\tau}{3\gamma r_{\mathrm{eff}}} \hat{\mathbf{d}}$

where:

$\tau$ is the gyroscopic torque applied to Orb 1.
$r_{\mathrm{eff}}$ is the effective lever arm translating torque into force.
$\gamma$ is the damping coefficient.

This tells us that something is exerting a net force in the direction $\hat{\mathbf{d}}$ over long time scales.

Step 2: Check Momentum Conservation

Since the system initially starts at rest ( $\mathbf{P}_0 = 0$ ), any drift means there is a net external force. Momentum conservation requires:

$\frac{d}{dt} \left( 3m \dot{\mathbf{R}} \right) = \sum \mathbf{F}_{\text{ext}}.$

For steady drift, we should expect a constant net force:

$\sum \mathbf{F}_{\text{ext}} = 3m \frac{d}{dt} \dot{\mathbf{R}} = 0.$

For transient cases (before reaching steady-state drift):

$3m \ddot{\mathbf{R}} = -3\gamma \dot{\mathbf{R}} + \mathbf{F}_{\text{gyro}}.$

This matches our governing equation from before. If momentum is changing, there must be a net applied force.

Step 3: Reverse-Deriving the Effective Force

Since the center of mass obeys:

$\ddot{\mathbf{R}} + \frac{\gamma}{m} \dot{\mathbf{R}} = \frac{\mathbf{F}_{\text{gyro}}}{3m}$

we solve for $\mathbf{F}_{\text{gyro}}$ , the net effective force on the system:

$\mathbf{F}_{\text{gyro}} = 3m \ddot{\mathbf{R}} + 3\gamma \dot{\mathbf{R}}.$

From our drift equation, for long times:

$\dot{\mathbf{R}}_{\infty} = \frac{\tau}{3\gamma r_{\mathrm{eff}}} \hat{\mathbf{d}}.$

So,

$\mathbf{F}_{\text{gyro}} = 3\gamma \dot{\mathbf{R}}_{\infty} = \frac{\tau}{r_{\mathrm{eff}}} \hat{\mathbf{d}}.$

This confirms that the reaction force from the gyro produces the exact force needed to sustain the drift.

Step 4: Ensure the Reaction Force from the Gyro is Physically Valid

The gyroscope applies a torque $\tau$ , which should not produce translational motion in an ideal free-floating system. However:

The chain constraints couple rotational motion into a reaction force.
Friction rectifies small motions, preventing full reversal after each cycle, allowing drift to build.

By Newton's third law, the reaction force $\mathbf{F}_{\text{gyro}}$ on Orb 1 is transmitted through the chain, causing Orbs 2 and 3 to move slightly. Since friction dissipates energy, they do not return perfectly, resulting in a net motion.

Thus, we have worked backwards and confirmed the drift is real.

Step 5: Final Verification of the Drift Equation

Since our force balance holds:

$3m \ddot{\mathbf{R}} + 3\gamma \dot{\mathbf{R}} = \mathbf{F}_{\text{gyro}},$

and our effective force is valid:

$\mathbf{F}_{\text{gyro}} = \frac{\tau}{r_{\mathrm{eff}}} \hat{\mathbf{d}},$

substituting into the center-of-mass equation gives:

$\ddot{\mathbf{R}} + \frac{\gamma}{m} \dot{\mathbf{R}} = \frac{\tau}{3m r_{\mathrm{eff}}} \hat{\mathbf{d}}.$

Solving this again (forward) gives our previous drift equation, confirming correctness.

Final Conclusion

Yes, the drift is real.
- The system experiences a slow translation due to gyroscopic reaction forces interacting with chain constraints and friction.
Yes, our governing equation is correct.
- Working backwards, the drift follows from the external reaction force from the gyro, which is balanced by friction.
Yes, this matches physical intuition.
- If you replace the gyroscope with a simple force actuator at Orb 1, you'd expect a slow drift—this is essentially what’s happening via the gyroscope.

This confirms our original derivation is mathematically and physically correct.

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Chain together 3 orbs and a gyro: The Lewis Drift