From Motion to Equations of Motion

Dynamics asks what motion follows from forces and what forces are required for a desired motion. MIT’s Engineering Dynamics course organizes that work around kinematics, force-momentum, energy, angular momentum, equations of motion, and vibration ¹.

Build the model in five moves

1. Define an inertial frame and the independent coordinates.
2. Write position, velocity, and acceleration consistently.
3. Draw the FBD at a general position, not only at rest.
4. Apply \(\sum \mathbf F=m\mathbf a\) and \(\sum M_G=I_G\alpha\), or an energy method.
5. State initial conditions and test limiting cases.

For a translating carriage with mass \(m\), viscous resistance \(c\dot x\), spring stiffness \(kx\), and actuator force \(u(t)\),

The equation is a claim about a boundary. Belt elasticity, bearing friction, motor inertia, and structural flex are omitted or lumped. A useful model is not the most complicated; it is the simplest one that predicts the decision-relevant behavior.

Worked acceleration estimate

A 12 kg carriage must reach 0.8 m/s in 0.25 s with approximately constant acceleration. Ignoring resistance, \(a=3.2\) m/s² and \(F=ma=38.4\) N. If measured resistance is 18 N and the belt-drive efficiency is 85%, requested belt-side input is roughly \((38.4+18)/0.85=66.4\) N. The motion profile may still demand higher peak force when acceleration is not constant.

For rotation, replace mass with mass moment of inertia: \(\sum T=I\alpha\). Reflecting a load inertia through a gear ratio is essential; speed ratio changes the apparent inertia seen by the motor by the square of the ratio under the usual ideal convention.

Sanity checks

• With zero applied force and zero stored energy, motion should not grow.
• Doubling mass should halve acceleration under the same net force.
• Dimensions on every term in one equation must match.
• Energy input over a move should cover kinetic-energy change plus losses.

Practice artifact

Choose a door closer, printer carriage, lift axis, or conveyor. Define one coordinate, list stored-energy and dissipative elements, derive a first-order or second-order equation, and name one experiment that would estimate each uncertain parameter.