physics//mechanics

Mechanics is the branch of physics that describes how forces produce motion, and in engineering it is the source of the models every machine starts from: the mass on a spring and damper, the pendulum, the rigid body of a drone, the shaft with friction. Its working law is Newton's second, \(m\ddot p=\sum F\): the acceleration follows from the forces, and the forces depend on where the body is and how fast it moves. That is why a mechanical state is always position plus velocity, the two quantities that carry memory, and why every mechanical model becomes a pair of first-order ODEs once the velocity is named as a state.

Mechanics is the branch of physics that describes how forces produce motion, and in engineering it is the source of the models every machine starts from: the mass on a spring and damper, the pendulum, the rigid body of a drone, the shaft with friction. Its working law is Newton's second, mp¨=∑Fm\ddot p=\sum Fmp¨​=∑F: the acceleration follows from the forces, and the forces depend on where the body is and how fast it moves. That is why a mechanical state is always position plus velocity, the two quantities that carry memory, and why every mechanical model becomes a pair of first-order ODEs once the velocity is named as a state.

The members of this branch are the canonical systems the control and estimation notes keep returning to. The pendulum is the nonlinear one, with a stable and an unstable equilibrium in the same equation. Vibration is what structures and machines do about their equilibria, with natural frequencies, mode shapes and resonance. The linear prototype of both, the mass on a spring and damper mp¨+cp˙+kp=Fm\ddot p+c\dot p+kp=Fmp¨​+cp˙​+kp=F, is written as an oscillator in the mathematics notes, and a mass pushed by a force with nothing to hold it is a double integrator, the model of a drone's altitude.

Friction is where the simple models end. A viscous term cp˙c\dot pcp˙​ is an approximation chosen because it is linear; dry friction, which holds a body still until a threshold and then lets it slip, is what makes a control valve stick and a loop cycle (stiction).

A free rigid body in space needs twelve numbers, position, velocity, orientation and angular velocity on three axes, which is the state of a multirotor (planar drone for the version reduced to a plane).