Applied Mechanics and Design
This section pairs rigid-body mechanics with the strength-of-materials and machine-design work built on it. Expect free-body reasoning, mechanism kinematics, stress transformation, and a design calculation such as a shaft, spring, bearing or joint.
Statics and dynamics
- Draw the free-body diagram first. For trusses, the method of joints uses two equilibrium equations per joint; the method of sections cuts at most three unknown members.
- At impending slip f = μN, and the angle of repose satisfies tan θ = μ.
- Work–energy and impulse–momentum often replace full time histories.
- Vibration: ω_n = √(k/m) and ζ = c/(2√(km)); resonance occurs at ω = ω_n.
Solid mechanics
- σ = My/I in bending, τ = Tr/J in torsion; in a thin cylinder σ_hoop = pd/2t and σ_long = pd/4t.
- Principal stresses: σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]; Mohr's circle is the same algebra drawn graphically.
- Euler buckling: P_cr = π²EI/L_e², with L_e = 1.0L pinned–pinned, 0.5L fixed–fixed, 2.0L fixed–free.
- Tresca (maximum shear) is conservative relative to von Mises (distortion energy).
Mechanisms and design
- Grübler: F = 3(N−1) − 2J₁ − J₂, where J₁ counts lower pairs; a cam–follower is a higher pair and contributes differently, so check its DOF allowance.
- Shaft design uses maximum shear stress theory with combined bending and torsion; bearings are selected on equivalent dynamic load and L10 life.
Worked example
A thin cylinder has d = 1 m, t = 10 mm, p = 2 MPa. With p = 2 N/mm² and d = 1000 mm: σ_hoop = pd/2t = (2×1000)/(2×10) = 100 MPa, and σ_long = pd/4t = 50 MPa. Hoop stress is twice the longitudinal, so a longitudinal seam is the critical weld.
Traps
Using diameter where radius belongs in pd/2t; omitting L_e in the buckling formula; treating a statically indeterminate truss as determinate; applying τ = Tr/J to a non-circular section.