Why use reduced inertia?

For the relative twist coordinate, the two inertias act like one equivalent inertia μ = J1J2/(J1+J2). That gives the torsional mode frequency with the shaft stiffness.

What does backlash change?

Inside the angular gap the shaft transmits no torque, so the response is nonlinear. The simple model shows loss of contact qualitatively, not detailed gear-tooth impacts.

Torsional Drivetrain Resonance

Two-inertia torsional drivetrain: shaft stiffness and damping, twist angle, first natural mode, resonance response, and optional backlash deadzone.

This two-inertia model captures the first torsional mode of shafts, couplings, gears, and drivetrains. Real systems add gear ratios, multiple shafts, nonlinear friction, clutch compliance, and controller interaction.

Live graphs

About this model

Many drivetrains have a low torsional mode where a motor inertia and a load inertia twist against each other through a compliant shaft or coupling. This simulator models two inertias connected by torsional stiffness and damping, with a sinusoidal drive torque. The relative coordinate has natural frequency ω_n = sqrt(k(1/J1 + 1/J2)), and the reduced inertia μ = J1J2/(J1+J2) sets the damping scale. The right-hand response curve shows twist amplification near r = Ω/ω_n = 1; the optional backlash switch removes shaft torque inside a deadzone, illustrating rattle/impact-prone nonlinear behavior. Gear ratios, multi-shaft systems, friction, clutches, and active motor-control loops are omitted.

Who it's for: Machine dynamics, powertrain design, rotating machinery, robotics, and controls introductions.

Key terms

Torsional vibration
Two-inertia drivetrain
Shaft stiffness
Backlash
Natural mode

How it works

Two-inertia torsional drivetrain with shaft stiffness and damping: visualize twist angle, natural torsional mode, resonance response, and optional backlash deadzone.

Key equations

ω_n = sqrt(k(1/J1 + 1/J2)), μ = J1J2/(J1+J2)

J1θ̈1 = Tdrive − Tshaft, J2θ̈2 = Tshaft − Tload; backlash: Tshaft=0 inside gap

Frequently asked questions

Why use reduced inertia?: For the relative twist coordinate, the two inertias act like one equivalent inertia μ = J1J2/(J1+J2). That gives the torsional mode frequency with the shaft stiffness.
What does backlash change?: Inside the angular gap the shaft transmits no torque, so the response is nonlinear. The simple model shows loss of contact qualitatively, not detailed gear-tooth impacts.

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Key equations

Frequently asked questions

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