Dynamics and Vibrations

Intuition

Vibrations occur when a structure stores more energy than it dissipates. It exchanges kinetic and strain energy each cycle. Small inputs produce large responses driven by forcing frequency and damping. Forcing frequency determines resonance matching. Damping determines per-cycle energy removal. A 10 millinewton harmonic force at resonance displaces a one-kilogram mass more than a 10 newton static load. Static loads do no work per cycle. Harmonic forces deposit energy in phase with velocity each cycle. This gap separates dynamics from statics.

Tacoma Narrows' 1940 collapse is the best-known example, but both versions misstate the cause. The common version blames vortex shedding at torsional resonance. The failure was self-excited flutter in a coupled torsional-vertical mode. It was a negative-damping instability, not forced resonance. The "resonance" label simplifies teaching but hides the feedback between motion and aerodynamics. Ordinary forced resonance destroys machinery. Negative damping and mode coupling are real dangers. Keep the single-degree-of-freedom model as a sketch. Real failures involve self-excitation, nearby modes, or nonlinearities.

Rotating unbalance can cross rotor bending modes. Gear-mesh frequencies excite housing panels. Two-stage isolation with a poorly tuned intermediate mass fails. Tool chatter in machining is regenerative: surface waviness from one pass feeds the next. Chatter is a delay system, not a pure resonance. Engineers detune systems by stiffening or mass-loading. Damping helps but degrades with temperature, age, and amplitude.

First principles

Single degree of freedom: the equation and what each term buys you

The standard lumped system has a mass on a spring and viscous damper, driven by an external force. Three terms shape the response. Inertia resists acceleration. Dissipation resists velocity and does net work. Restoring forces resist displacement and store potential energy. Every vibration model elaborates this triad. The three terms separate cleanly, letting you reason about changes.

Two scalars summarize the free response: the natural frequency and the damping ratio.

\[ \omega_n = \sqrt{k/m}, \qquad \zeta = \frac{c}{2\sqrt{km}} = \frac{c}{c_c}, \qquad c_c = 2\sqrt{km} \]

The damped natural frequency is:

\[ \omega_d = \omega_n\sqrt{1-\zeta^2} \]

An underdamped system rings at the damped natural frequency. Structural systems with damping ratios of 0.01 to 0.05 have a damped natural frequency within a fraction of a percent of the undamped value, so the distinction rarely matters in practice. It does matter when comparing measured peak frequencies to finite-element-analysis eigenvalues. See the FEA modal note.

The free response splits by damping ratio value.

Forced response: magnification, transmissibility, and the two regimes either side of r = sqrt(2)

The steady-state amplitude for a harmonic force of amplitude F0 is:

\[ X = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}, \qquad r = \omega/\omega_n \]

The phase angle is:

\[ \phi = \arctan\!\big(2\zeta r/(1-r^2)\big) \]

Two limits drive engineering decisions. At low frequency ratios the mass follows the force quasi-statically, approaching F0 over k with the spring dominating the response. At high frequency ratios the mass stays approximately stationary in space while the base accelerates underneath it, a principle underpinning seismic instruments and high-frequency isolation. The peak magnification approaches 0.5 over zeta for small damping ratios at a frequency ratio near unity. But the peak frequency falls below the natural frequency, a detail that surprises people who equate resonance with natural frequency without checking.

Transmissibility is the fraction of base motion transmitted to the mass. Reciprocity links this to the force transmitted to the foundation.

\[ T = \sqrt{\frac{1 + (2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}} \]

Isolation only works above a frequency ratio of the square root of two, regardless of damping. Below this ratio the isolator amplifies input. At resonance the isolator transmits strongly. Past the square root of two, the isolator attenuates. Adding damping hurts isolation in this region but helps at resonance, creating the fundamental design tension. Soft mounts produce low natural frequencies for good isolation but also low static stiffness and large sway. Stiff mounts provide good static support but keep the system out of the isolation regime. The practical compromise is a rubber mount with damping ratio of 0.1 to 0.2, where the operating frequency is at least three to four times the mount natural frequency. Data-sheet values represent best-case small-amplitude, room-temperature conditions, so you should derate them before relying on them.

2-DOF, modes, and the eigenvalue problem

Systems with multiple masses are clearest when described by modes, which you find as eigenvectors. You then treat each mode as an independent single-degree-of-freedom system with its own natural frequency and damping ratio. An n-degree-of-freedom conservative system with mass matrix M and stiffness matrix K uses the generalized eigenvalue problem:

\[ (K - \omega_i^2 M)\,\phi_i = 0 \]

This equation gives eigenvalues as omega_i squared and mode shapes as phi_i. Two properties make the machinery tractable. Orthogonality eliminates cross-terms between modes, letting you decouple the equations. You project the forcing onto each mode, solve a single-degree-of-freedom problem for each, and sum the results. Completeness in the continuous limit means that the modes span the response space. You can truncate the mode count to the first N modes and bound the approximation error in energy norm.

The recurring challenge in modal analysis is that damping is not, in general, diagonal in the modal basis. Proportional (Rayleigh) damping assumes a damping matrix equal to alpha M plus beta K, making it diagonal in the modal basis. This assumption is almost never true of an assembled structure but serves as a modeling convenience that decouples modes and lets you specify damping per mode while discarding cross-coupling terms. When two modes are close in frequency (within about 5 to 10 percent), the proportional-damping assumption can give qualitatively wrong results because off-diagonal damping terms then perform real work. See the Pitfalls section for details.

Engineers under-use mode shapes. The amplitude a mode picks up scales with modal forcing, the spatial force distribution dotted with the mode shape. A force applied at a node of mode i does not excite that mode at all, while a force at an antinode excites it maximally. This principle underlies tuned-mass dampers, node-mounted sensors, and the active-control rule to place the actuator at a node of the mode you do not want to excite. See the PID control note.

Rotating unbalance and the rotating-frame subtlety

A rotor with mass unbalance m_e at eccentricity e and angular speed omega exerts a harmonic force of magnitude m_e e omega squared on its bearings. The response amplitude is:

\[ X = \frac{m_e e \omega^2 / k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \]

The numerator grows with the square of omega. At low speed, unbalance force is negligible. At resonance, magnification amplifies it. At high speed (above resonance, in the isolation regime), the amplitude asymptotes to m_e e over m. This result is independent of spring stiffness and damping, showing that the rotor spins about its own mass-centre. These systems balance rotors rather than adding damping. Damping the resonance peak buys only a few decibels at crossing; beyond it, the asymptote does not care about damping.

The single-degree-of-freedom model hides the gyroscopic subtlety. A spinning rotor carries angular momentum, and lateral displacements couple to rotations about the perpendicular axis. The gyroscopic matrix G links these terms, giving the equation M q double-dot plus (C plus omega G) q dot plus K q equals F. The eigenvalues split with spin speed, so forward and backward whirl modes separate. A model that ignores spin predicts a single bending mode where two exist, omitting the gyroscopic contribution from the critical-speed map. This omission is large for overhung rotors and small but non-negligible for centre-hung rotors. See the Bearings and rotordynamics note.

Vibration isolation as a design problem

Isolation is the practical endpoint of the transmissibility curve. Designers usually specify it as a maximum transmissibility at the operating frequency with an allowable static deflection. These two constraints bracket the natural frequency from above and below. Low natural frequencies produce low transmissibility and large deflections because natural frequency squared equals g over delta. The design challenge is finding a mount that fits in this window.

Multi-stage isolation uses mass-spring-mass-spring chains to buy a steeper roll-off of about one over r to the fourth power per stage beyond the corner frequency. The cost is extra intermediate modes that can resonate worse than the single-stage version if mistuned. Tuning the intermediate mass is the most important lever and the most often misapplied in practice.

Worked example: units end-to-end

A steel cantilever carries a 2 kg end mass modeled as a single-degree-of-freedom system. The measured tip stiffness is k equals 8.0 times 10 to the 4th power newtons per meter. A shock tube test gives a log decrement of 0.18 over four cycles. The structure runs at 14 Hz from a reciprocating imbalance that delivers F0 equals 25 newtons at the tip. The goal is to find the natural frequency, the damping ratio, and the steady-state tip amplitude.

1. Natural frequency.

\[ \omega_n = \sqrt{k/m} = \sqrt{8.0\times10^4 / 2.0} = \sqrt{4.0\times10^4} = 200\ \mathrm{rad/s} \] \[ f_n = \omega_n/(2\pi) = 31.83\ \mathrm{Hz} \]

Units check: newtons per meter divided by kilograms gives inverse square seconds, and the square root gives inverse seconds, which is angular frequency in radians per second. A common mistake is to report omega_n in hertz. This radians-per-second versus hertz confusion causes a remarkable fraction of isolation-design errors because transmissibility formulas use the ratio r equals omega over omega_n, and both values must use the same angular units.

2. Damping ratio from log decrement. For small damping ratios:

\[ \zeta \approx \frac{\delta}{2\pi n} = \frac{0.18}{2\pi \cdot 4} = 7.16\times10^{-3} \]

This value is typical for lightly damped welded steel structures. Empirical damping ratios are 0.005 to 0.015 for welded joints, 0.01 to 0.03 for bolted joints, and 0.02 to 0.06 for cast iron. A value of 0.007 is on the low end but plausible for a clean weld. If your measured damping ratio reaches 0.1 on a steel structure, you are probably measuring something else, such as joint slip, air damping on a light mass, or sensor noise.

3. Forcing ratio.

\[ r = \omega/\omega_n = \frac{2\pi \cdot 14}{200} = \frac{87.96}{200} = 0.440 \]

4. Steady-state amplitude.

\[ X = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \] \[ (1-r^2) = 1 - 0.1936 = 0.8064, \quad (2\zeta r) = 2(0.00716)(0.440) = 0.00630 \] \[ X = \frac{25 / (8.0\times10^4)}{\sqrt{0.8064^2 + 0.00630^2}} = \frac{3.125\times10^{-4}}{0.8064} = 3.88\times10^{-4}\ \mathrm{m} \approx 0.39\ \mathrm{mm} \]

The tip moves about 0.39 mm peak. Units: newtons divided by newtons per meter gives meters. The dimensionless denominator provides a magnification of about 1.24. You are below resonance, so the response is spring-dominated and the damping term contributes negligibly. At a frequency ratio of 0.44 and damping ratio of 0.007, the square of 2 zeta r is four orders of magnitude smaller than the square of 1 minus r squared. Dropping damping entirely changes the answer only at the fourth significant figure. In most off-resonance regimes, damping is a rounding error and the response depends on stiffness and inertia alone. Damping only matters near resonance. Engineers rarely measure damping well because most designs never run near a mode, and the incentive structure only drives damping measurement when the system actually runs near a mode. A factor of two change in the damping ratio changes resonant amplitude by factor of two, which can decide whether a sensor resolves or saturates.[fn:: Damping is unmeasurable until it is the only thing that matters.]

5. Sanity checks.

• Static deflection under F0 would be F0 over k equals 0.31 mm. The dynamic amplitude is 0.39 mm, slightly larger than static, consistent with a frequency ratio below unity and a magnification just above unity. The resonant amplitude is about F0 divided by (k times twice zeta), equals 0.31 mm divided by 0.0143, or about 21.6 mm, which would plastically deform the cantilever. This asymmetry is useful: at resonance you fail, off resonance you are fine, and the transition region is where you must think hard.
• A frequency ratio of 0.44 places the forcing in the quasi-static regime where the structure acts essentially as a spring. A designer wanting to reduce amplitude should stiffen the structure, since adding damping produces almost no improvement here.
• At 50 Hz the frequency ratio would be 1.57, past the square root of two into the isolation regime. The amplitude approaches F0 divided by (m omega squared), about 1.27 times 10 to the minus 4 meters, the inertia-dominated asymptote. The only way to reduce amplitude further is to add mass.

Pitfalls

Damping is the thing everyone under-specifies. The dynamicist's dirty secret is that ζ is rarely known to better than ±50% for a built structure, and is often amplitude-, temperature-, and history-dependent. Rubber damping is nonlinear (it rises with amplitude then falls at large strain); bolted joints dissipate through micro-slip and are functions of preload; welds damp less than bolts. A single quoted ζ is a one-point snapshot of a surface. When the design margin depends on ζ, the only honest approach is a sweep — and you will discover that the upper end of the ζ-band sometimes saves you and sometimes doesn't, depending on whether the mode in question is the one being damped.

Treating a real system as SDOF when a nearby mode is close. The SDOF approximation is valid when the nearest other mode is at least ~3× further away in frequency than the mode of interest (a rule of thumb, not a theorem). When two modes are within ~10% of each other, modal coupling — through damping if not through stiffness — means the SDOF magnification formula over-predicts the peak height and /under/-predicts the bandwidth; the actual peak is broader and lower than SDOF suggests, but the off-resonant tail is higher. In rotor-bearing-foundation systems the foundation mode and the rotor bending mode are routinely within this band, and treating them separately is the canonical source of "the rotor ran fine on the test stand but failed in service" stories.

Ignoring gyroscopic terms in rotors. A non-spinning modal analysis of a rotor gives you the static modes, not the operating modes. Forward and backward whirls split from each other as spin speed rises; the critical speeds are where the spin frequency crosses the whirl frequencies, and these are speed-dependent. The error from omitting the gyroscopic matrix is bounded by the angular momentum / lateral stiffness ratio, which is small for low-speed stiff rotors and dominant for high-speed flexible ones — and is exactly the regime where you most need the prediction. See Bearings.

Confusing viscous with structural (hysteretic) damping. The viscous model \(F_d = c\dot{x}\) gives a damping force that grows with frequency at fixed amplitude; the hysteretic (structural, "complex stiffness") model \(F_d = (k \eta)\,x\) gives a damping force that is /amplitude/-dependent and frequency-independent, parameterised by loss factor η. For metal structures at constant amplitude the two can be matched at a reference frequency, but for broadband excitation they give different bandwidths and different off-resonant tails. Using a viscous model when the actual mechanism is material hysteresis systematically over-damps the high-frequency modes and under-damps the low. The practical tell: if your damping ratio comes out frequency-dependent in a way you can't explain by joints, you are probably fitting a viscous model to a hysteretic mechanism.

Confusing "modes" with "natural frequencies". A natural frequency is a scalar attached to a mode; a mode is a shape. Two systems with identical frequency lists can have different responses because the mode shapes project the forcing differently. The frequency-only view (the one a hammer-test gives you immediately) is a necessary but not sufficient characterisation; you need the shapes to know whether the mode of interest is excitable by your forcing. Sensor and actuator placement decisions need the shape, not just the frequency.

Trusting proportional damping. Rayleigh damping \(C = \alpha M + \beta K\) is the lazy default that decouples the modal equations. It is rarely physical: it over-damps either the very-low or very-high modes (depending on the α/β ratio) to fit a target at a single intermediate frequency, and it cannot represent joint-localised dissipation. For structures with concentrated damping treatments (constrained-layer patches, tuned dampers) it is qualitatively wrong; the proper path is a full damping matrix or a state-space model with non-proportional damping and complex modes.

Static condensation that hides a mode. Guyan reduction is standard for substructuring, but it condenses out the mass of the slave DOFs along with their stiffness, and any mode whose kinetic energy lives mainly in those slave DOFs is silently omitted. If a lightweight flexible component is reduced onto a heavy interface, the component's modes vanish from the reduced model and reappear as a surprise in service.

References

• Rao, S. S., /Mechanical Vibrations/, 6th ed., Pearson, 2017 — the standard undergraduate/first-year-graduate text; encyclopaedic, equation-dense, reliable but not opinionated.
• Meirovitch, L., /Fundamentals of Vibrations/, McGraw-Hill, 2001 — cleaner on the modal / analytic-superposition viewpoint than Rao; the place to learn mode-superposition properly.
• Inman, D. J., /Engineering Vibration/, 3rd ed., Pearson, 2006 — more readable than Rao, weaker on continuous systems; good first pass.
• Ewins, D. J., /Modal Testing: Theory, Practice and Application/, 2nd ed., Research Studies Press, 2000 — the reference for experimental modal analysis; essential when you actually have to measure ζ rather than assume it.
• Billah, K. Y. & Scanlan, R. H., "Resonance, Tacoma Narrors Bridge Failure, and Undergraduate Physics Textbooks", Am. J. Phys. 59(2), 1991 — the Tacoma-Narrows correction every dynamicist should read once.
• Tobias, S. A., /Machine-Tool Vibrations/, Blackie, 1965 — the regenerative-chatter classic, still the clearest treatment of the stability-lobes picture.

Related

• Bearings & rotordynamics — gyroscopic effects, critical speeds, fluid-film bearing cross-coupling; the place where "damping" gets genuinely subtle.
• Flexures — compliant mechanisms operate as distributed-parameter systems; their dynamics are the continuous-limit version of this note.
• Actuators — voice-coil and piezo actuators have their own dynamics; bandwidth is set by the first structural mode of the actuator-stack.
• FEA — modal — how eigenvalue extraction is done numerically, and why the Lanczos solver is the workhorse.
• PID control / closed-loop dynamics — a controller closes a loop around a plant whose dynamics are these dynamics; the stability margin is set by the plant's phase at the loop's crossover, which is dominated by the first mode.