Applied Mechanics and Structures II: Vibrations and Machine Design

A mass m on a spring of stiffness k, displaced and released, oscillates at the natural frequency ωₙ = √(k/m) rad/s, fₙ = ωₙ/(2π) Hz. Equivalently ωₙ = √(g/δ_st), where δ_st is the static deflection under the weight. Springs in parallel add, k = k₁ + k₂; in series their flexibilities add, 1/k = 1/k₁ + 1/k₂. The energy (Rayleigh) method equates the maximum kinetic and potential energies and gives ωₙ directly for systems that are awkward to draw.

With a viscous damper c, the damping ratio ζ = c/c_c = c/(2√(km)). If ζ < 1 the system is underdamped and oscillates at the damped frequency ω_d = ωₙ√(1 − ζ²) with an exponentially decaying amplitude; ζ = 1 is critically damped, the fastest return without oscillation; ζ > 1 is overdamped. The logarithmic decrement δ = ln(xₙ/xₙ₊₁) = 2πζ/√(1 − ζ²) measures damping from two successive peaks, so ζ = δ/√(4π² + δ²), close to δ/(2π) for light damping. Amplitudes 10 mm and 8 mm give δ = ln 1.25 = 0.223 and ζ = 0.0355.

Under a harmonic force F₀ sin ωt, the steady-state amplitude is X = (F₀/k)·MF with the magnification factor MF = 1/√[(1 − r²)² + (2ζr)²], r = ω/ωₙ. At resonance (r = 1) MF = 1/(2ζ), unbounded without damping, and the response lags the force by 90°. A rotating unbalance m₀e gives a force m₀eω², so its amplitude grows with speed and tends to m₀e/m well above resonance.

The transmissibility — force transmitted to the foundation over force applied — is TR = √[1 + (2ζr)²]/√[(1 − r²)² + (2ζr)²]. It equals 1 at r = √2 whatever the damping, so isolation (TR < 1) requires r > √2: the mount must be soft enough that the natural frequency is well below the forcing frequency. Above √2, damping increases the transmitted force, so isolators are lightly damped, with just enough damping to pass through resonance on start-up. Undamped at r = 2, TR = 1/(r² − 1) = 1/3.

A system with n degrees of freedom has n natural frequencies, found from det(K − ω²M) = 0, and n mode shapes, the fixed amplitude ratios in which it can vibrate at each. Two masses joined by springs have an in-phase lower mode and an out-of-phase higher mode. The dynamic vibration absorber adds a small mass–spring tuned to the forcing frequency, √(k₂/m₂) = ω, which brings the main mass to rest at that frequency at the cost of two new resonances either side. A rotating shaft whirls at its critical speed ω_c = √(k/m) = √(g/δ), the natural frequency of its lateral vibration; a shaft whose static deflection at the disc is 1 mm has ω_c = √(9.81/0.001) = 99 rad/s, about 946 rpm.

Under static loading, a ductile part is designed against yield with a factor of safety N = S_y/σ (using a failure theory for combined stresses) and a brittle part against the ultimate strength; stress concentrations matter little for a ductile part under static load, because local yielding redistributes the stress, but they matter for brittle parts and for fatigue. Under dynamic (fluctuating) loading, a part fails by fatigue at stresses well below yield. The S–N curve of a steel flattens at the endurance limit Sₑ, below which life is effectively infinite; for many steels the specimen endurance limit is roughly half the ultimate strength, and it is reduced for the real part by surface finish, size, reliability, temperature and load-type (Marin) factors, and by the fatigue stress-concentration factor K_f = 1 + q(K_t − 1), where q is the notch sensitivity.

A fluctuating stress has a mean σₘ and an alternating amplitude σₐ. The Goodman line, σₐ/Sₑ + σₘ/S_ut = 1/N, joins the endurance limit to the ultimate strength; the more conservative Soderberg line, σₐ/Sₑ + σₘ/S_y = 1/N, joins it to the yield strength; the Gerber parabola lies outside both. With σₐ = σₘ = 100 MPa, Sₑ = 300 MPa, S_ut = 600 MPa and S_y = 400 MPa, Goodman gives 1/N = 0.333 + 0.167 = 0.5, N = 2, and Soderberg gives 1/N = 0.333 + 0.25, N = 1.71. Miner’s rule Σnᵢ/Nᵢ = 1 sums the damage of blocks at different stress levels.

A shaft under combined bending M and torque T is sized by the equivalent torque Tₑ = √(M² + T²) (maximum shear stress theory, τ = 16Tₑ/(πd³)) and the equivalent bending moment Mₑ = ½[M + √(M² + T²)] (maximum principal stress theory, σ = 32Mₑ/(πd³)); with M = 3 kN·m and T = 4 kN·m, Tₑ = 5 kN·m and Mₑ = 4 kN·m. Shock and fatigue factors multiply M and T in the design codes, and the shaft must also be checked for deflection and critical speed. Power and torque are related by P = 2πNT/60.

Gears transmit a constant velocity ratio when they satisfy the law of gearing — the common normal at the point of contact passes through the fixed pitch point — which the involute profile does for any centre distance. Pitch diameter d = mz (module m, teeth z), centre distance (d₁ + d₂)/2, velocity ratio N₁/N₂ = z₂/z₁; the Lewis equation F_t = σ_b b m Y sizes the tooth as a cantilever in bending, and the Buckingham and wear checks cover dynamic load and surface fatigue. Spur, helical (smoother, with an axial thrust), double-helical, bevel and worm gears serve parallel, intersecting and crossed shafts; a marine reduction gearbox between a medium-speed engine and the propeller is double-helical.

Rolling-contact bearings are rated by the basic dynamic load rating C, the load giving a life of one million revolutions with 90% reliability; the rating life is L₁₀ = (C/P)ᵏ million revolutions, with k = 3 for ball and 10/3 for roller bearings, P being the equivalent load. Sliding-contact (journal) bearings run on a hydrodynamic oil film built by the shaft’s rotation; Petroff’s equation gives the friction of a lightly loaded bearing, and the Sommerfeld number S = (r/c)²(μN/p) collects the design variables. The propeller shaft’s stern-tube and line-shaft bearings are sliding bearings; the thrust block carries the propeller thrust through tilting pads.

A bolted joint is tightened to a preload so that the members stay in contact under the working load; the bolt then carries only a fraction of the external load, set by the relative stiffness of bolt and members, and fatigue life improves. Stress is taken on the tensile stress area of the thread, and a bolt in shear is checked on its shank area. A riveted joint (lap or butt, single or multiple rows) can fail by shearing of the rivets, tearing of the plate between rivets, and crushing (bearing) of the plate or rivet; its efficiency is the least of these strengths divided by the strength of the solid plate — for plate tearing alone, (p − d)/p, so a 50 mm pitch with 20 mm holes gives 60%.

Welding is the joining technique of modern shipbuilding. A butt weld with full penetration is as strong as the plate; a fillet weld of leg size s fails in shear across its throat t = s cos 45° = 0.707s, so a parallel fillet of length L carries P = 0.707 s L τ. A 10 mm fillet 200 mm long at 100 MPa permissible shear carries 141.4 kN. Welds bring residual stresses, distortion, heat-affected zones and the risk of cracking, controlled by procedure qualification, preheating and inspection.