Calculate ideal wheel torque and its equal-and-opposite carrier reaction from wheel inertia and absolute angular acceleration. A separate assumed pivot resistance and arm inertia give a simplified instantaneous arm-acceleration value, not a prediction of pendulum motion or balance.
Inertia Wheel Pendulum Interactive Calculator
Explore equal-and-opposite wheel reaction torque, an assumed opposing pivot-torque threshold and the resulting idealized arm angular acceleration.
Equation Used
- Entered wheel angular acceleration is an absolute acceleration about its spin axis.
- Equal and opposite motor reaction torque acts on the carrier arm.
- Entered Jarm is the effective inertia used for the arm acceleration division and is not derived from mass geometry.
- Entered pivot resistance is a constant opposing torque magnitude with a simple no-motion threshold.
- No pendulum angle, gravity, arm/wheel coupling, Coriolis terms, motor electrical dynamics, wheel speed/saturation, feedback controller, friction law, flexibility, impact, structural or safety model.
The wheel acceleration is treated as the absolute wheel acceleration used in tau=Iw alpha. The arm calculation omits gravity, wheel/arm acceleration coupling, angle, velocity, motor limits and control. Pivot resistance is applied only as a simple opposing torque magnitude.
Wheel torque and carrier reaction
An ideal motor applies torque to accelerate a wheel, and the carrier receives an equal and opposite reaction torque. This page subtracts an entered opposing pivot-torque magnitude, then divides the remainder by an entered effective arm inertia.
The connected animation shows the arm and wheel responding in opposite senses. It is qualitative and does not integrate the motion.
Use as a sign and unit check
The outputs can check tau=I alpha, reaction sign, millinewton-metre conversion and a simple torque threshold. They cannot predict swing-up, stabilize an inverted pendulum or select a motor, wheel, bearing, encoder or controller.
Instantaneous torque equations
Wheel torque is Iw alpha_w and ideal carrier reaction is its negative. The assumed resisting torque is subtracted from reaction magnitude without reversing it. Remaining torque divided by entered Jarm gives an idealized instantaneous acceleration.
For a real wheel mounted on a moving arm, relative and absolute wheel acceleration must be distinguished and the coupled nonlinear equations include gravity and system geometry.
Worked example
For Iw=0.001 kg·m² and absolute wheel acceleration +50 rad/s², wheel torque is +0.050 N·m and ideal carrier reaction is −50 mN·m. With assumed opposing pivot torque 0.3 mN·m, idealized net arm torque is −49.7 mN·m; dividing by Jarm=0.005 kg·m² gives −9.94 rad/s².
What determines real pendulum motion
Real motion depends on gravity, arm position and velocity, the complete mass distribution and motor limits. Wheel speed and stored angular momentum also limit sustained control torque. A swing-up or balance controller therefore needs a coupled dynamic model and measured system parameters; the instantaneous arithmetic here cannot tune one.
Questions
Will the reported arm acceleration occur on a real pendulum?
Not generally. Gravity, coupling, current state and motor limits are absent.
Does the resistance slider model bearing friction?
Only as a constant opposing threshold; real friction depends on motion, preload, lubrication and other effects.
Can these values tune a controller?
No. A validated dynamic plant model and measured parameters are required.
References & Further Reading
- Wikipedia contributors. Inertia wheel pendulum. Wikipedia
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