Sprung Arm

A rubber-band-loaded arm on a pivot, pulling cascade slides through a string — the 12993 RoboKings Aurum approach to counterspringing a linear lift. Also called a sprung linkage.

Sprung ArmContinuousCascadeCustom PartsAdvanced

A bundle of rubber bands loads a pivoting beam that hauls the lift's string. The beam's changing moment arm turns the bands' linear pull into a tension that stays nearly flat across a working window — a custom force level from cheap elastics, at the cost of geometry design and tuning.

Best for

Offseason projects and advanced teams who want a custom, tunable force level (matched to a cascade lift) from cheap rubber bands.

Poor fit for

Mid-season adoption, tight packaging, or teams without the CAD and iteration time to design and tune the beam geometry.

ComplexityHigh

Custom geometry, pivots, and structure; the force profile is an engineered output, not a purchased one.

Force accuracyest.Potentially excellent

In theory the profile can be shaped to match gravity almost exactly; achieved accuracy depends entirely on execution and tuning.

Packagingest.Demanding

The beam sweeps through volume as the lift moves; the swept envelope must stay clear.

Tuning effortest.High

Geometry changes require remanufactured parts; expect multiple iterations to converge.

In-season riskest.High

Long, hard-to-bound development time makes this a classic offseason experiment.

Click a metric to see why it’s rated that way. Ratings marked est. are editorial estimates awaiting test data.

Quick recommendation

Use this when

You want a custom force level no COTS spring rating provides, from cheap rubber bands
Your lift is cascade-strung, so the load is near-constant across the stroke
You have offseason time to iterate on geometry and validate with measurements
Structural mounting for the beam pivot and band anchor is available

Avoid this when

It is competition season and the lift must work next week
Packaging around the lift base is already tight — the beam sweeps volume
Pivot slop or frame flex would corrupt the designed force profile
Nobody on the team can own the beam geometry end-to-end

What the mechanism actually is

Forget catalogs for a moment. This counterspring is four parts:

A rigid beam (a metal arm) that pivots about one fixed point near the lift base.
A bundle of rubber bands anchored at one end to a fixed axle, and at the other end to a point on the beam — a distance r out from the beam's pivot. The bands are the spring.
A string that runs from the beam, over a bearing/pulley, down to the linear slides.
The cascade-strung slides themselves.

When the lift is driven down, the beam is rotated and the bands are stretched — energy goes in. When the motor eases off, the stretched bands pull the beam back, the beam hauls on the string, and the string lifts the slides. The bands are, in effect, holding most of the lift's weight so the motor only has to supply the difference.

Watch: 12993 RoboKings Aurum — Behind the Bot (arm counterspring at 11:31)Inline player planned; opens externally for now.

The rubber-band-loaded beam and the string to the slides are visible here; the clearer close-up is the team's own clip.

Why "cascade" matters here

The slides are strung in cascade: every stage is tied to the one before it, so they all extend together and the string at the actuation point sees a load that stays roughly constant over the whole stroke. (Contrast continuous stringing, where the effective load steps as stages hand off.)

That constant target is exactly what makes this arm practical. The design job is simply: keep the string tension roughly flat over the sweep, at the value that cancels the slides' weight. You are matching a flat line, not a staircase.

The physics, worked out

Put the beam pivot O at the origin. The bands anchor at a fixed point A a distance d to one side of the pivot. The bands attach to the beam at point P, a distance r from the pivot, and the beam sits at angle θ. So P = (r·cosθ, r·sinθ) and A = (−d, 0).

Hand derivation: spring force as the square root of (d + r cosθ)² + (r sinθ)², simplifying to k√(d² + 2rd cosθ + r²), and the perpendicular component F_r = F sinθ.

The original napkin derivation from the source discussion — the same result, reached by hand.

Step 1 — how long is the spring at angle θ? The band length is just the distance from A to P:

Band length vs. beam angle

L(\theta) \;=\; \sqrt{(d + r\cos\theta)^2 + (r\sin\theta)^2} \;=\; \sqrt{d^2 + 2\,d\,r\cos\theta + r^2}

In plain terms: As the beam swings, the distance between the fixed anchor and the moving attachment point changes — so the bands stretch and their force grows. This is where the position dependence enters.

Assumes: Bands modeled as a linear spring · Anchor and pivot in one plane

Step 2 — how hard do the bands pull? Rubber bands only pull, and (over their working range) roughly follow Hooke's law from some unstretched length L₀:

Band force

F_s(\theta) \;=\; k\,\bigl(L(\theta) - L_0\bigr)\qquad\text{for } L > L_0

In plain terms: k is the combined stiffness of all the band loops. Adding bands multiplies k; it does not change the shape of the curve.

Step 3 — turn that into a torque on the beam. Only the part of the band force that acts perpendicular to the beam twists it about the pivot. Working out the moment of F_s about O gives a strikingly clean result:

Spring torque about the pivot

\tau_s(\theta) \;=\; F_s(\theta)\,\frac{d\,r\,\sin\theta}{L(\theta)}

In plain terms: The d·r·sinθ / L factor is the spring's moment arm about the pivot. It vanishes at θ = 0° and θ = 180° (the band line passes through the pivot — no leverage) and is largest near θ = 90°.

Step 4 — the payoff. Heavily preloaded rubber bands behave close to a zero-free-length spring (force nearly proportional to total length, L₀ → 0). Substitute F_s = k·L and the length cancels:

String tension (zero-free-length limit)

T(\theta) \;=\; \frac{\tau_s(\theta)}{R} \;=\; \frac{k\,d\,r}{R}\,\sin\theta

In plain terms: R is the string's moment arm (the pulley/bearing radius). In this limit the tension delivered to the slides is a pure sine in the beam angle — set by geometry (d, r, R) and the number of bands (k).

Variable definitions
Symbol	Quantity	Unit	Notes
$\theta$	Beam angle	deg	Sweeps as the slides move
$r$	Band mount radius	mm	Pivot → band attachment on the beam — the main leverage knob
$d$	Pivot → anchor offset	mm	—
$R$	String moment arm	mm	Effective pulley/bearing radius the string leaves on
$k$	Combined band stiffness	N/m	≈ (per-band stiffness) × (number of loops)
$L_0$	Unstretched band length	mm	—

How does the force behave — the questions this answers

Is the output force sinusoidal? Yes — the math genuinely lands on T ∝ sinθ in the zero-free-length limit. It is a geometric sine (a moment-arm effect), not a vibration. Real bands add an L₀ correction that tilts the curve slightly.

Why is the lift only balanced over part of its travel? The match is only good over the window of θ you operate in. Run the beam through a sweep centered on θ = 90°, where sinθ ≈ 1 and changes slowly, and the tension is nearly flat — that is the "sweet spot." Push toward 0° or 180° and the tension collapses.

How does the force change as the beam angle changes? It rises with the beam toward 90° and falls past it, following sinθ. Because sinθ is flattest at its peak, a symmetric sweep about 90° gives the smallest ripple — which is exactly what you want against a constant cascade load.

Can moving the bands out or adding length reduce how many bands I need? Yes, and the equation says how much. Output scales with d·r, so the torque (and tension) grows linearly with r. Move the band mount twice as far from the pivot and you roughly halve the number of bands for the same lift. The trade: a larger r also stretches the bands more per degree of sweep, so the tension varies more across the window — you buy fewer bands with more ripple.

Continuous vs. cascade

The near-constant cascade load is the natural fit. Center the beam's sweep on θ = 90° and the sine is flat enough to hold the lift across its whole stroke with a single band bundle.

Arrangement: beam pivoted at the base, bands from a fixed axle to a point on the beam, string over a bearing to the first stage.
Force profile: near-constant over the working window.
Advantage: all-custom force level — dial it in with band count and mount radius, no catalog quantization.
Limitation: the beam sweeps real volume, and the match degrades if the window drifts away from 90°.

Interactive: sweep the arm, watch the force

Drag the schematic to rotate the beam and watch the bands stretch and the tension change. The graph plots string tension across the whole 0–180° swing; the shaded band is your working window. Tune r, d, band count, and the window to flatten the curve onto the target load.

Sprung-Arm Force Explorer

Interactive statics model & geometry tuner

1. Geometry & Motion Schematic

θ = 90° | T = 46.8 N

Click or drag beam to sweep θ

Sweep presets:

2. String Tension Profile ($T$ vs $\theta$)

Shaded = Working Window [68°–112°]

Flattest near θ = 90° (sine peak). Match target load across window.

Mean Tension in Window

45.3 N

≈ 10.2 lbf

Tension Ripple

18%

Range: 39–47 N

Assist vs. Target Load

101%

Target: 45 N

Band Loops Needed

≈ 6 loops

Current: 6 loops (540 N/m)

Band Stretch: 64 → 38 mmString Payout: ≈ 15 mmPeak Spring Stretch: 64 mm

Mount Radius r?

Anchor Offset d?

String Radius R?

Retracted Angle θ?

Extended Angle θ?

Rubber bands are modeled as linear springs with rest length $L_0$. Real latex exhibits progressive stiffness near full stretch and thermal drift. Use this explorer for geometry design, then fine-tune with a scale on hardware.

Design and sizing

A realistic loop for this arm:

Measure the target load. Pull the assembled cascade slides through travel with a luggage scale; that near-constant number is your T target.
Pick a working window around 90°. Keep the beam a few degrees clear of both 0° and 180° — near the ends the moment arm collapses and the force gets twitchy (the source notes it "gets weird" past ~175°).
Set leverage before band count. Choose r, d, and the string radius R so a reasonable number of bands hits the target — leverage is free, bands are consumable and drift.
Add bands to reach the force. Total stiffness scales with loop count; this is your fine adjustment.
Check the ripple. If the tension varies too much across the window, narrow the window (shorter stroke on the beam), reduce r, or move the window to sit more symmetrically on 90°.

Worked example: Sizing an arm to hold a ~45 N cascade lift

Cascade slides that weigh in at ~45 N at the string. Beam swings 50°→130° over the stroke. Band mount r = 60 mm, anchor offset d = 40 mm, string radius R = 20 mm.

Near θ = 90° the zero-free-length estimate is T ≈ k·d·r / R (sinθ ≈ 1). Solving for the stiffness you need at the peak: k ≈ T·R / (d·r) = 45 × 0.020 / (0.040 × 0.060) ≈ 375 N/m. If each band loop contributes ~90 N/m of effective stiffness in this geometry, that is roughly 4 loops at the flat top — but because the real window dips below the peak toward its edges, you add loops until the window mean hits 45 N, not just the peak. Push the mount out to r = 90 mm and the same load needs about a third fewer loops. Numbers are illustrative — the explorer above lets you retune them, and the built arm gets the final say on a scale.

CAD, manufacturing, and assembly

Onshape sketch of the arm geometry: a triangle formed by the pivot, the anchor, and the beam's swept positions, with dimensions for the string offset (20 mm to the bearing), the 200 mm span, stroke length, and the beam angle.

The source geometry sketch. Green = band lengths; light blue = start/end positions of the bands and linkage; the 20 mm is the string offset to the bearing and 200 = 180 mm stroke + 20; the ~49–175° figures are the beam's angular limits before the sketch degenerates.

The pivot is the product. Slop at the beam pivot is multiplied straight into tension noise — use a bearing or a real bushing, not a bolt in a hole.
Anchor the bands properly. Bands fail at their hooks; run them over a smooth axle or a captive spool, never a sharp screw thread.
Reserve the swept volume. Model the beam's full arc as a solid keep-out in the assembly; nothing else may live there.
Make the mount radius adjustable. Slotted holes or a few mounting positions for the band attachment let you retune r on the robot — the single most useful adjustment.

CAD embed — to be addedSprung-arm reference build — No verified public reference CAD on file yet. The source Onshape sketch above documents the geometry; a full assembly belongs here.

Testing and validation

Profile measurement: motor disconnected, pull the lift through travel with a luggage scale in both directions; plot force vs. height against the flat target. This is the report card for your window choice.
Release test: does the lift float, climb, or settle when let go at several heights? Climbing = over-sprung (drop bands); settling = under-sprung (add bands or push r out).
Slop audit: wiggle the pivot and band anchors before and after cycle testing; log any growth in free play.
Band fatigue watch: measure tension fresh, then after a full match's worth of cycles and after heat — latex softens, so re-check between events.
Re-measure after every tune: any change to r, the window, or band count reshapes the whole curve.

Real-world examples

12993 RoboKings Aurum

Sprung Arm

CENTERSTAGE · cascade stringing

Cascade slides countersprung by a rubber-band-loaded pivoting arm pulling a string over a bearing. Source for this page; measured force profile and band counts still to be documented.

Placeholder card — a real, verified team example belongs here. Contributions welcome.

Common failure modes

Window drifts off 90° — tension sags at one endMedium severity

Cause: Beam sweep centered too low or too high, so one edge of the stroke rides down the steep part of the sine.
Symptoms: Lift floats at mid-height but settles (or climbs) near one end of travel.
Prevention: Center the working window on θ = 90° in CAD; keep both ends clear of 0°/180°.
Fix: Re-index the beam's rest angle or shorten its sweep so it straddles 90°.

Band bundle fatigues and force fadesMedium severity

Cause: Latex creep and heat soften the bands over a match or a season.
Symptoms: A lift tuned to float now needs more motor to hold; float point drops over time.
Prevention: Over-provision slightly; log tension between events; treat bands as consumables.
Fix: Replace the bundle and re-measure; re-tune band count.

Pivot slop corrupts the profileHigh severity

Cause: A loose beam pivot; play is multiplied by the leverage into tension noise.
Symptoms: Audible knock at direction changes; float point wanders match to match.
Prevention: Bearing/bushing at the pivot; slop audit in the maintenance routine.
Fix: Replace the pivot; re-tune afterward.

Beam sweeps into another mechanismMedium severity

Cause: Swept arc not reserved in CAD, or a late part added into it.
Symptoms: Intermittent jams at specific lift heights.
Prevention: Model the sweep as a solid keep-out in the assembly.
Fix: Re-route the other mechanism — the beam's arc is fixed by its geometry.

How this compares

Bungee

A bungee routed along the slides is the shortcut version of this same idea — an elastic pulling the lift directly. The arm adds a pivot so geometry, not just pretension, sets the profile: flatter force, but a real mechanism to build.

Guide

Constant Force

Constant-force springs buy a flat profile straight from a catalog with almost no design. Choose the arm when you want a custom force level from cheap rubber bands, already have the packaging for a beam, or are doing this as an offseason build.

Guide

Constant Torque

Constant torque delivers one flat level from a steel spring at the winch. The arm delivers a tunable level from bands at the base — cheaper and adjustable, but with more to design and a force that ripples across the sweep.

Guide

For the full matrix, force-curve overlay, and decision summary, see Compare Methods.

References

12993 RoboKings Aurum — Behind the Bot (CENTERSTAGE)Robot example
Primary source: the rubber-band-loaded arm counterspring on cascade slides. Clip starts at 11:31.
Source design discussion + Onshape geometry sketchEngineering source
The physics thread and napkin derivation this page is built from; images reproduced above.
Zero-free-length spring balancing (standard mechanism-design result)Engineering source
The T ∝ sinθ result is the classic gravity-balancer statics, reached here with rubber bands and a beam.