Vibrations basics

In short

Vibration is the to-and-fro motion of a body about its equilibrium position, caused by the interplay of inertia (mass), elasticity (stiffness) and energy loss (damping). Every machine element with mass and stiffness has a natural frequency; when an exciting force matches it, amplitude builds up dangerously — this is resonance, and avoiding it is the core design purpose of the subject. The topic carries guaranteed marks in GATE/ESE (single-DOF free and damped vibration, resonance, transmissibility, critical speed) and appears in interviews at Tata Motors, Maruti, Mahindra, L&T, BHEL, Cummins, Kirloskar, Siemens and ISRO/DRDO labs. Industrially it governs engine mounts, foundation design, rotor balancing, and condition monitoring in refineries and power plants.

Explained

Definitions and classification

• Vibration: periodic motion about a mean position; needs inertia to store kinetic energy and elasticity to store potential energy, which exchange cyclically.
• Degree of freedom (DOF): number of independent coordinates needed to define the configuration. A spring-mass system = 1 DOF; a rotor on two bearings in a plane = 2 DOF; a continuous beam = infinite DOF.
• Free vibration: after an initial disturbance, no external force acts; occurs at the natural frequency ωn.
• Forced vibration: a continuing external excitation; the steady-state motion occurs at the forcing frequency ω, not at ωn.
• Damped vs undamped: undamped is an idealisation; real systems lose energy by viscous, Coulomb (dry friction), structural (hysteretic) or fluid damping.
• Linear vs non-linear: linear if mass, stiffness and damping terms are first-degree (superposition valid).
• Deterministic vs random: road inputs to a vehicle are random; a rotating unbalance is deterministic (harmonic).
• By physical mode: longitudinal (axial), transverse (lateral/bending) and torsional (twisting) vibration.
• Amplitude, time period T, frequency f = 1/T, circular frequency ω = 2πf, phase angle φ. Units: amplitude m or mm, f in Hz, ω in rad/s.

Simple harmonic motion and single-DOF free vibration

• SHM: x = X sin(ωn t + φ); acceleration is proportional to displacement and directed towards the mean position: ẍ = −ωn² x.
• Equation of motion (undamped): m ẍ + k x = 0 → ωn = √(k/m). Derived by three equivalent methods:
  1. Newton/D'Alembert — sum of forces including inertia force = 0.
  2. Energy method — total energy (KE + PE) constant, so d/dt(KE + PE) = 0.
  3. Rayleigh method — equate maximum KE to maximum PE; useful when spring/shaft mass is included.
• A mass hung on a spring causes static deflection δ = mg/k, so ωn = √(g/δ) — the fastest route in exams when δ is given.
• Equivalent stiffness: springs in parallel keq = k1 + k2 (same deflection); in series 1/keq = 1/k1 + 1/k2 (same force). Cantilever keq = 3EI/L³; simply supported central load keq = 48EI/L³; shaft in torsion kt = GJ/L.
• Effect of spring mass: add one-third of the spring mass (ms/3) to m — Rayleigh's correction.
• Transverse vibration of a shaft/beam carrying a load: ωn = √(g/δ) with δ the static deflection at the load point.

Damped free vibration

• Equation: m ẍ + c ẋ + k x = 0; auxiliary roots s = [−c ± √(c² − 4mk)]/2m.
• Critical damping cc = 2√(km) = 2mωn; damping ratio (damping factor) ζ = c/cc.
• Three regimes: | Case | ζ | Motion | |---|---|---| | Under-damped | ζ < 1 | Oscillatory with exponentially decaying amplitude | | Critically damped | ζ = 1 | Returns to rest in the shortest time without oscillating | | Over-damped | ζ > 1 | Sluggish, non-oscillatory return |
• Under-damped response: x = X e^{−ζωn t} sin(ωd t + φ), with damped natural frequency ωd = ωn√(1 − ζ²) — always lower than ωn.
• Logarithmic decrement δL = ln(x1/x2) = 2πζ/√(1 − ζ²) — the standard experimental route to ζ from a decay trace; over n cycles δL = (1/n) ln(x0/xn).
• Coulomb damping gives a linear (straight-line) decay of amplitude, losing 4F/k per cycle, unlike the exponential decay of viscous damping.

Forced vibration, resonance and isolation

• Harmonic force F0 sin ωt: m ẍ + c ẋ + k x = F0 sin ωt. Steady-state amplitude X = F0/√[(k − mω²)² + (cω)²].
• Writing r = ω/ωn and xst = F0/k: magnification factor MF = X/xst = 1/√[(1 − r²)² + (2ζr)²]; phase lag tan φ = 2ζr/(1 − r²).
• Resonance: r ≈ 1. For ζ = 0 amplitude is theoretically infinite; with damping MF ≈ 1/(2ζ) at r = 1. Peak MF actually occurs at r = √(1 − 2ζ²) and equals 1/(2ζ√(1 − ζ²)) — a common GATE one-liner.
• Below resonance (r < 1) the mass moves nearly in phase with the force; above resonance (r > 1) it lags by nearly 180° and inertia dominates.
• Rotating unbalance excitation F0 = m_e e ω², so amplitude keeps growing with speed and then flattens out to e·m_e/M at high r.
• Support (base) excitation and isolation: transmissibility ε = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²] — ratio of transmitted force (or transmitted motion) to the input.
  • ε = 1 at r = 0 and at r = √2 for every ζ.
  • ε < 1 only when r > √2, so isolators must be *soft* — the machine must run well above the mount's natural frequency.
  • For r > √2, more damping *worsens* isolation; damping is still kept to control amplitude while passing through resonance during run-up.
• Isolation efficiency = (1 − ε) × 100%. Isolators: steel helical springs, rubber/elastomeric mounts, cork, felt, air springs, inertia blocks.
• Vibration absorber (tuned, 2-DOF): an auxiliary mass-spring tuned so k2/m2 = ω² makes the main mass stationary at that one frequency.
• Whirling / critical speed of shafts: the shaft's own transverse natural frequency; ωc = √(g/δ), and critical speed Nc = (60/2π)√(g/δ) rpm. Operate well away from it; dynamic deflection y = e/[(ωc/ω)² − 1].

Torsional vibration and multi-DOF basics

• Single rotor on a shaft: ωn = √(kt/I), kt = GJ/L; J = πd⁴/32 for a solid shaft.
• Two-rotor system: the shaft has a node (zero twist) located by I_A l_A = I_B l_B, with l_A + l_B = L; ωn = √(kt_A/I_A) using the length to the node.
• Three-rotor system has two natural frequencies and two nodes — the basis of crankshaft and turbo-generator torsional analysis.
• Equivalent shaft: a stepped shaft is replaced by one of diameter d by L_eq = Σ L_i (d/d_i)⁴.
• Geared systems: inertia and stiffness are referred across the gear by the square of the gear ratio.
• Multi-DOF (n masses) gives n natural frequencies and n mode shapes; found from the frequency (characteristic) equation, or approximately by Dunkerley's and Holzer's methods, Rayleigh's quotient and matrix (eigenvalue) methods.
• Dunkerley's empirical formula: 1/ωn² ≈ 1/ωs² + Σ 1/ωi², where ωs accounts for the shaft's own mass — gives a value slightly lower than the true fundamental; Rayleigh's gives a value slightly higher.
• Continuous systems (bars, beams, strings) have infinite modes; the first mode usually dominates the response.

Measurement, standards and Indian practice

• Transducers: piezoelectric accelerometers (wide range, most common), velocity pick-ups, LVDT/eddy-current proximity probes for shaft displacement, laser vibrometers.
• A seismic instrument acts as a vibrometer (measures displacement) when ωn ≪ ω, and as an accelerometer when ωn ≫ ω.
• FFT analysis converts a time signal to a frequency spectrum; 1× rpm peak indicates unbalance, 2× misalignment, high-frequency bands bearing or gear-mesh defects — the basis of predictive maintenance.
• Codes and standards (verify the current edition before design use): IS 2974 (design of machine foundations, parts for reciprocating, impact and rotary machines), IS 13301 (vibration isolation guidelines), IS 4729 and ISO 10816/20816 (evaluation of machine vibration by measurements on non-rotating parts), ISO 1940/21940 (balance quality grades such as G6.3, G2.5), ISO 2631 (human exposure to whole-body vibration), IS 14817 (hand-arm vibration). The OSH Code, 2020 (in force 21 November 2025, replacing the Factories Act, 1948) and State factory rules treat excessive vibration as a workplace hazard.

Key points

• Vibration - periodic motion needing mass, elasticity and (in reality) damping.
• Natural frequency - ωn = √(k/m) = √(g/δ); depends only on mass and stiffness, never on amplitude or the force.
• Degree of freedom - independent coordinates required; n DOF gives n natural frequencies and n mode shapes.
• Free vibration happens at ωn; forced steady-state vibration happens at the forcing frequency ω.
• Critical damping cc = 2√(km); damping ratio ζ = c/cc separates oscillatory from non-oscillatory motion.
• Damped natural frequency ωd = ωn√(1 − ζ²) is always less than ωn.
• Logarithmic decrement δL = 2πζ/√(1 − ζ²) - measures damping from a decay record.
• Resonance at ω = ωn; amplitude is limited only by damping, MF ≈ 1/(2ζ).
• Magnification factor MF = 1/√[(1 − r²)² + (2ζr)²], r = ω/ωn.
• Transmissibility equals 1 at r = √2 regardless of damping; isolation needs r > √2 (soft mounts).
• Above r = √2 extra damping increases transmitted force - damping helps only near resonance.
• Unbalance force grows as ω², which is why balancing matters more at high speed.
• Critical (whirling) speed Nc = (60/2π)√(g/δ); run well below or well above it, and cross it quickly.
• Torsional node in a two-rotor shaft from I_A l_A = I_B l_B; a three-rotor shaft has two frequencies.
• Equivalent stiffness - parallel adds, series adds reciprocals; cantilever 3EI/L³, simply supported 48EI/L³.
• Dunkerley under-estimates and Rayleigh over-estimates the fundamental frequency.
• Spring mass correction - add ms/3 to the vibrating mass.
• Spectrum diagnosis - 1× unbalance, 2× misalignment, high frequency bearing/gear faults.

Formulas and examples

1. Natural frequency from static deflection. A 20 kg motor on a spring of k = 8,000 N/m.

• ωn = √(8000/20) = √400 = 20 rad/s; fn = 20/6.2832 = 3.18 Hz; T = 0.314 s.
• Check by deflection: δ = 20 × 9.81/8000 = 0.0245 m → √(9.81/0.0245) = 20 rad/s. ✓

2. Damping ratio from a decay trace. The same system's amplitude falls from 12 mm to 3 mm in 4 cycles, c = 120 N·s/m given separately.

• δL = (1/4) ln(12/3) = (1/4) × 1.3863 = 0.3466.
• ζ from 0.3466 = 2πζ/√(1 − ζ²) → ζ ≈ 0.0551 (5.5%).
• Cross-check: cc = 2 × 20 × 20 = 800 N·s/m → ζ = 120/800 = 0.15; the two differ because the decay trace and the quoted c belong to different tests — in an exam use whichever datum is given.
• ωd = 20√(1 − 0.0551²) = 19.97 rad/s — barely below ωn, as expected at light damping.

3. Forced amplitude at resonance and off resonance. A 50 kg machine, k = 2 × 10⁵ N/m, c = 1,000 N·s/m, force 400 N.

• ωn = √(200000/50) = 63.25 rad/s; cc = 2 × 50 × 63.25 = 6,325 N·s/m → ζ = 1000/6325 = 0.158.
• xst = 400/200000 = 2 mm. At resonance (r = 1): MF = 1/(2 × 0.158) = 3.16, X = 6.33 mm.
• At 1,200 rpm (ω = 125.7 rad/s, r = 1.987): (1 − r²) = −2.949, 2ζr = 0.628 → MF = 1/√(8.696 + 0.394) = 0.332, X = 0.66 mm.

4. Vibration isolation of a pump. A 300 kg pump runs at 1,500 rpm on four undamped springs; 85% isolation is required.

• ω = 2π × 1500/60 = 157.1 rad/s; ε = 1 − 0.85 = 0.15.
• Undamped: ε = 1/(r² − 1) → r² = 1 + 1/0.15 = 7.667, r = 2.769.
• ωn = 157.1/2.769 = 56.7 rad/s → total k = m ωn² = 300 × 3217 = 9.65 × 10⁵ N/m, i.e. ≈ 2.41 × 10⁵ N/m per spring.
• Static deflection δ = 300 × 9.81/9.65 × 10⁵ = 3.05 mm — a practical, selectable mount.

5. Critical speed of a shaft. A 60 kg rotor midway on a simply supported steel shaft, L = 1 m, d = 40 mm, E = 200 GPa.

• I = π × 0.04⁴/64 = 1.2566 × 10⁻⁷ m⁴; k = 48EI/L³ = 48 × 200 × 10⁹ × 1.2566 × 10⁻⁷ = 1.206 × 10⁶ N/m.
• δ = 60 × 9.81/1.206 × 10⁶ = 4.88 × 10⁻⁴ m → ωc = √(9.81/4.88 × 10⁻⁴) = 141.8 rad/s.
• Nc = 141.8 × 60/6.2832 ≈ 1,354 rpm — so a 1,500 rpm motor drive would sit uncomfortably close and the shaft must be stiffened.

Where it is used

• Automotive: engine mounts, suspension tuning and NVH work at Maruti Suzuki, Tata Motors, Mahindra, Ashok Leyland and their Tier-1 suppliers; shake-rig and road-load data acquisition.
• Power and process plants: turbine-generator and compressor monitoring at NTPC, BHEL, IOCL and Reliance refineries to ISO 10816/20816 limits; trip settings on shaft vibration.
• Machine foundations and civil-mechanical interface: block and framed foundations for compressors, presses and pumps designed to IS 2974, with isolation to IS 13301.
• Condition monitoring and predictive maintenance: FFT analysers and online sensors; roles such as reliability engineer and vibration analyst (ISO 18436 Category I-III certification is valued).
• Rotating machinery manufacture: balancing of fans, rotors and crankshafts at Kirloskar, Crompton, Cummins and Siemens; critical-speed checks on pump and turbine shafts.
• Aerospace and defence: vibration qualification of satellite and launch-vehicle hardware on shaker tables at ISRO/VSSC and DRDO labs.

Interview questions

• Q: What is resonance and why is it dangerous? A: It is the condition when the forcing frequency equals the natural frequency; the amplitude is then limited only by damping and can grow until the structure fails or bearings are destroyed.
• Q: Does the natural frequency depend on the exciting force? A: No — it depends only on mass and stiffness; the force decides amplitude and the forcing frequency decides where the system operates relative to resonance.
• Q: What is critical damping and where is it wanted? A: c = 2√(km), the least damping that prevents oscillation; it is used in instruments, door closers and gun recoil mechanisms where the quickest return without overshoot is needed.
• Q: Why must vibration isolators be soft rather than stiff? A: Isolation requires the frequency ratio to exceed √2, so the mount's natural frequency must be well below the running speed, which means low stiffness and larger static deflection.
• Q: Is more damping always better? A: No — it helps near resonance and during run-up, but above r = √2 it increases transmitted force, so damping is chosen, not maximised.
• Q: What is the difference between damped and undamped natural frequency? A: ωd = ωn√(1 − ζ²), so damping lowers the frequency; the difference is negligible below about 10% damping.
• Q: How would you find the damping ratio experimentally? A: Give a free decay (bump test), record the trace and compute logarithmic decrement over n cycles, or use the half-power bandwidth of the frequency response.
• Q: What is whirling of a shaft? A: Lateral vibration of a rotating shaft at its transverse natural frequency, where eccentricity makes deflection grow sharply; operation must avoid that speed.
• Q: A machine has to pass through resonance on start-up. What do you do? A: Accelerate quickly through the band so amplitude cannot build, and provide damping or temporary snubbers; never allow a steady dwell there.
• Q: What does a 1× rpm peak in an FFT spectrum mean? A: Mass unbalance, usually; a strong 2× component suggests misalignment and high-frequency bands point to bearing or gear defects.
• Q: How does a tuned vibration absorber work? A: An auxiliary mass and spring tuned to the excitation frequency generates an equal and opposite force on the main mass, reducing its motion to nearly zero at that one frequency.
• Q: Why is a stepped shaft converted to an equivalent shaft in torsional analysis? A: To replace several diameters by one uniform shaft of equal torsional flexibility using L_eq = ΣL_i(d/d_i)⁴, so the standard two- or three-rotor formulae apply.

Common mistakes

• Mixing rad/s with Hz — forgetting the 2π, or using rpm directly in ωn = √(k/m).
• Taking ωd = ωn always, or expecting damping to change the natural frequency appreciably at ζ = 0.05.
• Treating the maximum magnification as occurring exactly at r = 1 when it is at r = √(1 − 2ζ²).
• Assuming heavier damping always improves isolation, and specifying stiff mounts for a low-speed machine.
• Reversing the series and parallel spring rules, or using cantilever stiffness for a simply supported shaft.
• Using the total spring mass instead of ms/3, or ignoring shaft mass where Dunkerley's term matters.
• Confusing free-vibration frequency with forcing frequency when writing the steady-state response.

Related topics

Balancing of rotating and reciprocating masses · Gyroscopic effects · Flywheels and turning moment diagrams · Strength of materials: deflection of beams · Machine design: shafts and springs · Condition monitoring and maintenance engineering