Mountain Bike Rear Shock

What is modelled

The rear wheel follows the ground; the sprung mass (the rider and the frame share on that wheel) rides on a spring and damper. The shock sits at a leverage ratio from the wheel, so the spring that matters at the wheel is much softer than the shock spring. The ground is either one half-sine bump or a train of ripples. Separately, the chain turns the crank cadence into wheel speed through the ratio of the chainring to the chosen cog.

Equations

• Wheel rate kw = kshock / LR², natural frequency ω = √(kw/m), static sag s0 = m g / kw.
• Relative motion u'' + 2ζωu' + ω²u = y'' with y the ground height; compression s = s0 + u.
• Ringing period 2π / (ω√(1 - ζ²)), logarithmic decrement 2πζ / √(1 - ζ²).
• Transmissibility √[(1 + (2ζr)²) / ((1 - r²)² + (2ζr)²)], r = ripple frequency / natural frequency.
• Speed v = cadence x (chainring / cog) x wheel circumference.

Checks run in the page

• The free ringing period and logarithmic decrement from the simulation.
• The settled sag against m g / k.
• Steady shaking amplitude on ripples against the transmissibility formula.
• The distance from counting wheel turns against the gearing formula.

Honest simplifications

The suspension is one linear spring and one linear damper (a real shock has different damping on compression and rebound, an air spring is progressive and the leverage ratio changes through the stroke), only the sprung mass moves, the tyre is rigid, the end stop is a simple bounce and pedalling forces, braking and rider movement are left out. The ground heights in the picture are exaggerated. The 3D exploded view is illustrative and its sliders carry no units.

• Choose Underdamped, pogo: the bike keeps bouncing after the bump.
• Choose Overdamped, harsh: it stops bouncing but passes more of the hit to the rider.
• Choose Washboard at resonance: the ripples arrive at the natural frequency and the rider is shaken more than the ripples are high.