Multibody Dynamics and Drive Sizing

Common tools: SolidWorks Motion, Ansys Motion, Simscape Multibody (MATLAB), RecurDyn and MSC Adams.

Every moving part is defined by its mass, its centre of gravity (CoG) and its moment of inertia. The parallel axis theorem moves the inertia to another axis: I = I(cog) + m x d squared. The picture is a plan view with the axis perpendicular to the page. CAD tools export these values (mass properties) for the simulation.

The drive sees the motion profile, not just the distance. A trapezoidal profile has sudden changes in acceleration (high jerk); an S-curve limits jerk and is gentler on the machine, at the cost of slightly higher peak acceleration for the same move time. External loads to include: payload, process forces (cutting, pressing), gravity (the axis orientation), inertia of attached mechanisms and disturbances such as impact or collision. Environmental effects: friction, backlash and compliance, flexible belts and couplings, cable and drag chain forces, and variable load conditions.

How it is calculated: the profile gives speed and acceleration along the move, and the cycle is the move, a dwell, the return move and another dwell. At each instant the load force is mass times acceleration, plus gravity, guide friction and any process force. The screw turns that force into torque (force times lead over 2 pi), adds the screw's own inertia, and the gearbox and efficiencies bring it back to the motor shaft, where the rotor, coupling and gearbox inertia are added. The motor passes if its peak torque covers the peak need times the safety factor, its rated torque covers the continuous (RMS) need with 20 percent margin, the speed is within 3000 rpm and the load to motor inertia ratio is 10 or less. The motor series is generic and illustrative: rated torque is power divided by 3000 rpm, peak is three times rated, and the inertia values are typical, so always use your supplier's catalogue. The gearbox is taken as 95 percent efficient with a small input inertia, and the coupling as a small fixed inertia. Gravity is held by the motor at standstill (a brake would normally do this).

A gearbox reduces the load inertia seen by the motor by the square of the ratio, not by the ratio itself. A load that is 54 times the motor inertia becomes 6 times with a 3:1 gearbox, while the load-side torque is multiplied by three. A ratio equal to the square root of the load to motor inertia ratio makes the reflected load inertia equal to the motor inertia, and gives the smallest torque for a purely inertial load. With friction or gravity in the load, the lowest-torque ratio shifts. Inertia ratio rules of thumb vary between about 5:1 and 10:1 depending on the supplier and how demanding the application is. Higher ratios also mean higher motor speed, regenerative energy and gearbox maintenance.

Reflected inertia: J(ref) = J(load) / i squared. Linear mass through a screw: J = m x (lead / 2 pi) squared. Motor torque = J(total) x acceleration + friction torque + F x lead / (2 pi x efficiency x i). Selected torque = peak torque x safety factor.

The motor and the load are two inertias joined by a spring (the coupling, screw or shaft). That gives an antiresonance frequency, where the motor barely moves while the load oscillates, and a resonance frequency above it, where energy is amplified and a high controller gain can make the drive unstable. A lower inertia ratio moves both higher and brings them closer together. A stiffer coupling raises both. Avoid operating or tuning the servo near these frequencies; use stiffer connections, a lower inertia ratio, or filters. As a common rule of thumb, keep the motion content well below the antiresonance.

Backlash is the gap in a gear mesh or coupling. It does nothing in one direction, but on reversal the load lags until the gap closes. It causes positioning error and, in a servo loop, hunting. Smaller gear backlash (arcmin) and a larger reduction ratio help.