IESL Part II Mechanical cheat sheet

Thermal efficiency of an ideal Otto cycle: η = 1 − 1/r^(γ−1), where r is compression ratio and γ = cp/cv; efficiency rises with r and with lower γ.

Diesel cycle efficiency: η = 1 − (1/r^(γ−1))·[(ρ^γ − 1)/(γ(ρ − 1))], where ρ = V3/V2 is the cutoff ratio; for ρ = 1 the expression reduces to the Otto result.

Carnot limit: η_max = 1 − T_L/T_H with absolute temperatures in kelvin; no cycle operating between two reservoirs can exceed it.

Steady-flow energy equation: q − w = Δh + ΔKE + ΔPE; for an adiabatic nozzle or diffuser w = 0 and q ≈ 0, so h1 + V1²/2 = h2 + V2²/2.

Entropy change of an ideal gas: Δs = cp·ln(T2/T1) − R·ln(p2/p1); setting Δs = 0 gives the isentropic relation T2/T1 = (p2/p1)^((γ−1)/γ).

Conduction resistance of a plane wall: R = L/(kA); for a cylindrical shell R = ln(r2/r1)/(2πkL). Thermal resistances in series add, and Q̇ = ΔT/R_total.

Log-mean temperature difference for a heat exchanger: ΔT_lm = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2), with ΔT1 and ΔT2 the terminal differences; Q = U·A·ΔT_lm·F.

Reynolds number Re = ρVD/μ = VD/ν; internal flow is laminar below Re ≈ 2300 and turbulent above Re ≈ 4000, with transition between.

Dittus–Boelter correlation for turbulent tube flow: Nu = 0.023·Re^0.8·Pr^n, n = 0.4 for heating and 0.3 for cooling of the fluid; properties at bulk mean temperature.

Bernoulli equation along a streamline for steady, incompressible, frictionless flow: p/ρg + V²/2g + z = constant; pump head is added and turbine or friction head subtracted.

Shaft power from torque: P = T·ω = 2πNT, with ω in rad/s and N in rev/s; if N is in rev/min, P = 2πNT/60.

Gear train ratio: ω_out/ω_in = N_in/N_out = d_in/d_out for a simple pair; for a compound train the ratio is the product of driven teeth over the product of driving teeth.

Fatigue: endurance limit applies only to steels below about 10⁶–10⁷ cycles; aluminium alloys have no true endurance limit, so finite-life design uses S–N data.

Von Mises equivalent stress for a plane stress state: σ' = √(σx² − σxσy + σy² + 3τxy²); compare with yield strength for ductile materials.

Maximum shear stress in a solid circular shaft: τ_max = 16T/(πd³); in a hollow shaft τ_max = 16T·d_o/(π(d_o⁴ − d_i⁴)).

Euler buckling load for a pin-ended column: P_cr = π²EI/L²; other end conditions enter through the effective length factor K, with P_cr = π²EI/(KL)².

S-N and Goodman relation for combined mean and alternating stress: σa/Se + σm/Sut = 1/n, used to estimate the safety factor n against fatigue failure.

First law for a closed system: Q − W = ΔU; for a control volume at steady state: Q̇ − Ẇ = Σṁ_out·h_out − Σṁ_in·h_in when kinetic and potential terms are neglected.

Hooke's law in three dimensions: εx = [σx − ν(σy + σz)]/E, with cyclic permutations for εy and εz; shear strain γ = τ/G and G = E/[2(1 + ν)].

Stress concentration factor Kt multiplies nominal stress at a geometric discontinuity; for fatigue use Kf = 1 + q(Kt − 1), where q is notch sensitivity (0 for ductile static loading in many codes).

Shaft design under combined bending and torsion: use equivalent moment M_e = ½[M + √(M² + T²)] for bending stress, or the ASME equivalent torque T_e = √(M² + T²).

Bearing life: L10 = (C/P)^p million revolutions, with p = 3 for ball bearings and p = 10/3 for roller bearings; C is the basic dynamic load rating and P the equivalent dynamic load.

Bolt preload: F_i ≈ 0.75·S_p·A_t for reusable joints, where S_p is proof strength and A_t the tensile stress area; joint stiffness and member stiffness split external load between bolt and members.

Welded joint throat thickness for a fillet weld: t = 0.707·w, where w is the leg size; shear stress on the throat is τ = P/(0.707·w·L) for a weld of length L.

Spring rate of a helical compression spring: k = G·d⁴/(8·D³·N_a), where d is wire diameter, D mean coil diameter, and N_a the number of active coils.

Cast iron is brittle with high compressive strength and low tensile strength; steel is ductile and used where tensile loading and toughness are required.

Carbon equivalent CE = %C + %Mn/6 + (%Cr + %Mo + %V)/5 + (%Ni + %Cu)/15; higher CE raises hardenability and weld-cracking risk in steels.

Tempering after quenching reduces hardness and increases toughness; austempering and martempering are isothermal treatments that limit distortion and cracking.

Annealing softens and relieves residual stress; normalizing refines grain structure; both are distinct from quenching, which forms martensite for hardness.

Casting: risers must feed the last-solidifying region, so the riser modulus must exceed the casting modulus; directional solidification is achieved by chilling or by tapering thickness.

Hot working occurs above the recrystallization temperature and refines grains; cold working below it causes strain hardening, improved surface finish, and closer tolerances.

Cutting speed in turning: V = π·D·N/1000 with V in m/min, D in mm, and N in rev/min; feed is mm/rev and depth of cut is radial engagement.

Taylor tool life equation: V·T^n = C, where n ≈ 0.1–0.15 for HSS and 0.2–0.4 for carbide tools; higher n means the tool tolerates higher speed for a given life.

Measurement: tolerance is the difference between upper and lower limits; allowance is the intentional difference between mating part sizes (minimum clearance or maximum interference).

A fit designated H7/g6 is a clearance fit because the hole is H (basic hole) and the shaft is g (below basic size); H7/p6 is an interference fit.

Fatigue design factor from the modified Goodman line reduces to n = 1/[(σa/Se) + (σm/Sut)] when the load line is fixed; check yielding separately with σmax < Sy.