Pancyprian Appl. Mech. Sci. III cheat sheet

In an ideal planar model: pin has two force components, roller one normal reaction, fixed support two force components and a moment.

I = b·h³/12, W = b·h²/6 for a rectangle

Simply supported beam, central load: M_max = F·L/4

Euler: P_cr = π²EI/L_k², with L_k depending on the end conditions

Planar DOF: G = 3(n−1) − 2j₁ − j₂

For the defined ideal lossless gear pair, speed and torque-magnitude ratios are inverse; losses reduce actual output torque.

p = F/A; 1 bar = 10⁵ Pa; 1 MPa = 10 bar

Q = A·v links flow, area and speed

Retraction force uses the annulus (bore minus rod)

Flow control depends on load and circuit; overrunning-load safety requires a properly engineered load-control system.

Hydraulic and pneumatic performance depends on the specified pressure, fluid, components, losses and operating conditions.

Read ISO 1219 symbols by the number of ports and positions

Use kelvin in all gas-law and kinetic-theory calculations

Q = mcΔθ for temperature change; Q = mL for change of state

Mixtures: heat lost by hot bodies = heat gained by cold bodies plus calorimeter

pV = nRT and pV = ⅓Nm⟨c²⟩; mean KE = (3/2)kT

For a fixed amount of an ideal gas, internal energy depends only on temperature

First law: ΔU = Q − W with W defined as work done by the gas

Sensitivity = Δoutput/Δinput; resolution = smallest detectable change

Closed-loop error compares reference and measurement; integral action can remove appropriate steady-state error only under suitable stability and operating assumptions.

PLC scan: read inputs → run program → update outputs

4–20 mA live zero: (I − 4)/16 of the span

Repeatability ≠ accuracy, resolution ≠ sensitivity

Robot DOF come from its joints; workspace is the reachable volume