Rate, curve, and energy

In a linear model, the slope of force versus travel is k; for torsion, torque versus angle gives angular rate. Initial tension shifts an extension spring's curve. Contact, friction, or changing geometry can make it nonlinear.

The area under the load curve represents mechanical work exchanged over that travel. For a linear response from zero it is triangular; with preload, initial work must be included. It is not necessarily all energy available to the system.

For compression from zero load, U=½·k·s². To open a helix with initial tension F₀, U=F₀·s+½·k·s². With N and mm, the result is N·mm: divide by 1000 to express it in joules.

With k=4.8400879 N/mm and s=10 mm, compression gives U≈0.242004 J. If F₀=15 N, extension work is U≈0.392004 J. This is external work from the stated travel start, not all historical energy in the assembly.

For linear torsion, U=½·M·α·π/180, with M in N·mm and α in degrees; the intermediate result is N·mm and is divided by 1000 for J. Convert degrees to radians because work uses angular displacement in radians.

For M=31.02495 N·mm at 5°, U≈0.00135372 J. This belongs to the linear body model; it excludes installation friction and energy dissipated in the system.