9.4 Resonance

Definition of resonance

Resonance describes a phenomenon where a system oscillates with dramatically larger amplitude at specific frequencies. It occurs when the frequency of an applied force matches the system's natural frequency, causing efficient energy transfer and amplified motion. Getting a handle on resonance is essential for designing safe, functional dynamic systems.

Natural frequency vs forcing frequency

Natural frequency is the rate at which a system oscillates on its own when displaced from equilibrium. It's determined entirely by the system's physical properties. Forcing frequency is the rate of whatever external force is driving the system.

Resonance occurs when the forcing frequency approaches or matches the natural frequency. The system absorbs energy from the external force most efficiently at this point, and response amplitude grows.

Natural frequency for a simple translational system:

$\omega_n = \sqrt{\frac{k}{m}}$

where $k$ is the spring stiffness (N/m) and $m$ is the mass (kg). Notice that a stiffer system vibrates faster, and a heavier system vibrates slower.

Amplitude magnification factor

The magnification factor $M$ tells you how much larger the dynamic response is compared to the static deflection under the same force:

$M = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$

where $r = \omega / \omega_n$ is the frequency ratio and $\zeta$ is the damping ratio.

• When $r = 0$ (static load), $M = 1$
• As $r \to 1$ (near resonance), $M$ spikes sharply
• For an undamped system ($\zeta = 0$) at $r = 1$, $M$ goes to infinity
• Higher damping flattens and broadens the peak

This factor is one of the most practical tools for predicting whether a system will experience dangerous vibration levels.

Types of resonance

Resonance shows up in mechanical, electrical, and acoustic systems. Each type has distinct physics, but the core idea is the same: energy input at a system's natural frequency produces an amplified response.

Mechanical resonance

This occurs in physical systems with mass and elasticity, such as bridges, buildings, and machinery. Energy cycles back and forth between potential (elastic deformation) and kinetic (motion) forms.

The classic cautionary example is the Tacoma Narrows Bridge collapse (1940), where wind-induced forcing excited a torsional mode of the bridge. On the useful side, devices like seismographs and mechanical filters deliberately exploit mechanical resonance.

Electrical resonance

In circuits containing inductance ($L$) and capacitance ($C$), resonance occurs when inductive and capacitive reactances cancel each other out. The resonant frequency is:

$f = \frac{1}{2\pi\sqrt{LC}}$

At this frequency, impedance is minimized (series circuit) or maximized (parallel circuit), leading to current or voltage amplification. This principle is the basis for radio tuning, signal filtering, and wireless power transfer.

Acoustic resonance

Sound waves reflecting within an enclosed space or object can interfere constructively, forming standing waves at specific frequencies. Organ pipes, guitar bodies, and Helmholtz resonators all rely on acoustic resonance. It can also cause unwanted noise amplification in vehicle cabins or building HVAC ducts.

Resonance in mechanical systems

The analysis approach for resonance depends heavily on how many degrees of freedom the system has.

Single degree of freedom systems

These are the simplest case: one coordinate describes all the motion. A mass on a spring or a simple pendulum are standard examples.

• Natural frequency: $\omega_n = \sqrt{\frac{k}{m}}$ (translational) or $\omega_n = \sqrt{\frac{g}{L}}$ (simple pendulum)
• Resonance occurs when the forcing frequency equals $\omega_n$
• At resonance, response amplitude is limited only by damping

Most textbook resonance problems start here because the math is tractable and the concepts transfer directly to more complex systems.

Multiple degree of freedom systems

Systems with two or more independent coordinates have multiple natural frequencies, each associated with a distinct mode shape (the pattern of motion at that frequency). Resonance can occur at any of these natural frequencies.

• Analysis requires solving coupled differential equations of motion
• Modal analysis decouples these equations by transforming to modal coordinates
• Each mode can be analyzed as an equivalent SDOF system

Continuous systems

Beams, plates, and shells have theoretically infinite degrees of freedom and are governed by partial differential equations. They possess an infinite number of natural frequencies and mode shapes, though in practice the lower modes dominate the response.

Analysis techniques include separation of variables (for simple geometries), Rayleigh-Ritz methods, and finite element discretization.

Resonance curves

Resonance curves plot response amplitude against forcing frequency. They're the primary visual tool for identifying where a system is vulnerable to large-amplitude vibration.

Frequency response function

The frequency response function (FRF), often written $H(\omega)$, is the complex ratio of output to input as a function of frequency.

• The magnitude $|H(\omega)|$ shows how much the system amplifies the input at each frequency
• The phase angle shows the time lag between force and response
• FRFs can be measured experimentally or derived analytically for linear systems
• They form the backbone of most vibration testing and diagnostics

Peak amplitude at resonance

The maximum response amplitude occurs at or near the natural frequency. For a lightly damped SDOF system, the peak amplitude is approximately:

$X_{peak} \approx \frac{F_0 / k}{2\zeta} = \frac{\delta_{static}}{2\zeta}$

So a system with $\zeta = 0.01$ amplifies the static deflection by a factor of 50. That's why even small periodic forces can cause serious problems at resonance.

Bandwidth and quality factor

Bandwidth ($\Delta\omega$) is the frequency range over which the response amplitude stays above $1/\sqrt{2}$ (about 70.7%) of the peak value. These boundaries are called the half-power points because the response energy there is half the peak energy.

The quality factor relates to bandwidth:

$Q = \frac{\omega_n}{\Delta\omega} \approx \frac{1}{2\zeta}$

• High $Q$ → sharp, narrow resonance peak → low damping
• Low $Q$ → broad, flat resonance peak → high damping

Damping effects on resonance

Damping determines how dramatic the resonance peak is and how quickly transients decay. The damping ratio $\zeta$ divides system behavior into three regimes.

Underdamped systems ($\zeta < 1$)

Most real mechanical systems fall here. The response oscillates with gradually decreasing amplitude. Near resonance, the frequency response curve shows a clear peak. Lower $\zeta$ means a taller, sharper peak and longer settling time. The damped natural frequency is:

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

For lightly damped systems, $\omega_d \approx \omega_n$.

Critically damped systems ($\zeta = 1$)

This is the boundary case. The system returns to equilibrium as fast as possible without oscillating. There's no resonance peak in the frequency response. Critically damped behavior is often the target in control systems where you want quick settling with zero overshoot.

Overdamped systems ($\zeta > 1$)

The system returns to equilibrium sluggishly, without oscillating or overshooting. No resonance peak exists. This is slower than critical damping but is used in safety-critical applications where overshoot is unacceptable (e.g., door closers, certain valve actuators).

Forced vibration near resonance

When an external periodic force acts on a system at frequencies close to its natural frequency, the response can grow dramatically. The total response is the sum of transient and steady-state components.

Steady-state response

After initial transients die out, the system settles into a persistent oscillation at the forcing frequency. Key characteristics near resonance:

• Amplitude increases sharply for lightly damped systems
• The phase shift between the applied force and the response approaches 90° at exact resonance
• For an undamped system driven exactly at $\omega_n$, the steady-state solution doesn't exist in the usual form; instead, amplitude grows linearly with time (pure resonance)

The steady-state solution comes from the particular solution of the equation of motion.

Transient response

This is the system's initial behavior before it settles down. It depends on initial conditions and decays at a rate governed by $\zeta \omega_n$. For lightly damped systems near resonance, transients can persist for many cycles and temporarily push the total response above the steady-state amplitude.

The transient comes from the complementary (homogeneous) solution of the equation of motion.

Beat phenomenon

When the forcing frequency is close to but not exactly equal to the natural frequency, the response exhibits beats: a slow, periodic rise and fall in amplitude.

• The beat frequency equals $|\omega - \omega_n|$
• The response looks like a high-frequency oscillation inside a slowly varying envelope
• As $\omega \to \omega_n$, the beat period lengthens and the peak amplitude grows
• Beats can cause fatigue damage because the stress levels cycle between high and low repeatedly

Resonance in rotating machinery

Rotating equipment (turbines, compressors, pumps, motors) is especially susceptible to resonance because the rotation itself generates periodic forcing.

Critical speeds

A critical speed is a rotational speed at which the shaft deflection reaches a maximum. Each critical speed corresponds to a natural frequency of the rotor-bearing system.

• Complex rotors have multiple critical speeds
• Operating at or near a critical speed causes excessive vibration, potential bearing damage, and shaft fatigue
• Engineers typically design operating speeds to be well above or below critical speeds
• Calculation methods include Dunkerley's formula (lower bound estimate) and transfer matrix methods

Shaft whirling

Whirling is the orbital motion of a rotating shaft's centerline. It occurs when the rotational speed excites a lateral natural frequency.

• Forward whirl: the shaft orbit follows the direction of rotation
• Backward whirl: the shaft orbit opposes the direction of rotation
• Whirling can lead to bearing damage, seal rubbing, and catastrophic failure if unchecked
• Gyroscopic effects split the natural frequencies into forward and backward whirl speeds

Balancing techniques

Mass imbalance in a rotor creates a centrifugal force that grows with the square of rotational speed. Balancing reduces this force:

1. Static balancing corrects imbalance in a single plane (sufficient for thin disks)
2. Dynamic balancing corrects imbalance in two or more planes (required for long rotors)
3. Field balancing is performed on-site using vibration measurements and trial weights
4. The influence coefficient method systematically determines correction masses for complex rotors

Resonance avoidance and control

Preventing or mitigating resonance is a core engineering responsibility. Three main strategies are used.

Vibration isolation

Isolation reduces vibration transmission between a source and its surroundings by introducing compliant elements (springs, rubber mounts, air springs) between them.

• The transmissibility ratio $T$ quantifies how much vibration passes through the isolator
• Isolation is effective only when $r = \omega/\omega_n > \sqrt{2}$; below this ratio, the isolator actually amplifies vibration
• Passive systems use fixed-property elements; semi-active and active systems can adjust in real time

Tuned mass dampers

A tuned mass damper (TMD) is an auxiliary mass-spring-damper attached to a structure. It's tuned so its natural frequency matches the problematic frequency of the primary structure.

• At resonance, the TMD oscillates out of phase with the main structure, counteracting its motion
• The famous example is the 730-tonne pendulum TMD in Taipei 101, which reduces wind-induced sway
• Multiple TMDs can target multiple modes
• TMDs are effective over a narrow frequency band, so accurate tuning is important

Active vibration control

Active systems use sensors, actuators, and control algorithms to suppress vibration in real time.

• Feedback control: measures the response and applies corrective forces
• Feedforward control: measures the disturbance before it reaches the structure and applies preemptive forces
• Requires external power and adds system complexity
• Used in precision manufacturing, aerospace structures, and advanced vehicle suspensions

Applications of resonance

Structural analysis

• Modal analysis identifies a structure's natural frequencies and mode shapes, revealing which loading frequencies could trigger resonance
• Resonance testing validates finite element models by comparing predicted and measured dynamic properties
• Operational modal analysis extracts modal parameters from a structure's response to ambient excitation (wind, traffic) without needing a controlled input force
• Results directly inform design decisions to keep natural frequencies away from expected forcing frequencies (wind gusts, seismic excitation, machinery vibration)

Seismic design

Earthquakes generate ground motion with energy concentrated in a range of frequencies (typically 0.1 to 10 Hz). If a building's natural frequency falls in that range, resonance amplifies the shaking.

• Base isolation places flexible bearings beneath a structure, shifting its natural frequency below the dominant earthquake frequencies
• TMDs reduce resonant response in tall buildings during seismic events
• Performance-based design evaluates structural response under multiple hazard levels
• Soil-structure interaction can shift the effective natural frequency, so the foundation conditions matter

Musical instruments

Musical instruments are deliberate, beneficial applications of resonance:

• String instruments (guitars, violins) use resonant cavities to amplify and shape the sound produced by vibrating strings
• Wind instruments (flutes, trumpets) rely on air column resonance to select and sustain specific pitches
• Percussion instruments (drums, bells) vibrate in complex mode shapes that determine their tonal character
• Instrument design is fundamentally an exercise in controlling which frequencies resonate and how strongly

Experimental methods

Experimental techniques validate theoretical predictions and characterize systems that are too complex for purely analytical treatment.

Modal analysis

Experimental modal analysis determines a structure's natural frequencies, mode shapes, and damping ratios. The general procedure:

1. Excite the structure with a known input (shaker, impact hammer, or ambient vibration)
2. Measure the response at multiple points using accelerometers or laser vibrometers
3. Compute frequency response functions from the input-output data
4. Extract modal parameters (frequency, damping, mode shape) using curve-fitting algorithms
5. Compare results against finite element predictions and update the model if needed

Frequency sweep tests

A frequency sweep excites the system across a range of frequencies to map out its resonances.

• Sine sweep: a sinusoidal input whose frequency changes slowly over time; gives clean, high-resolution FRFs
• Random excitation: broadband input that excites all frequencies simultaneously; faster but noisier
• The resulting FRFs reveal resonance peaks, their frequencies, and the associated damping

Impact hammer testing

An instrumented hammer delivers a short impulse that excites a broad frequency range in a single hit.

• A force sensor in the hammer tip measures the input force
• Accelerometers on the structure measure the response
• FRFs are computed from the force and response signals
• Best suited for lightweight to medium structures and field testing where a shaker setup isn't practical
• Tip material (rubber, plastic, steel) controls the frequency content of the impact: harder tips excite higher frequencies

Numerical methods for resonance

Computational methods are essential for analyzing resonance in systems too complex for closed-form solutions.

Finite element analysis

FEA discretizes a continuous structure into small elements, each with simple assumed displacement functions. For resonance analysis:

1. Build a mesh of the geometry
2. Assemble global mass $[M]$ and stiffness $[K]$ matrices (and damping $[C]$ if needed)
3. Solve the eigenvalue problem to find natural frequencies and mode shapes
4. Perform harmonic response analysis to predict the steady-state response at various forcing frequencies
5. Run transient analysis if time-history loading is relevant

FEA handles complex geometries, material nonlinearities, and realistic boundary conditions that analytical methods can't.

Eigenvalue problems

Finding natural frequencies and mode shapes reduces to solving an eigenvalue problem.

For an undamped system:

$[K - \omega^2 M]\{\phi\} = \{0\}$

For a damped system (state-space form):

$[\lambda^2 M + \lambda C + K]\{\phi\} = \{0\}$

Non-trivial solutions exist only for specific values of $\omega^2$ (or $\lambda$), which are the eigenvalues. The corresponding eigenvectors $\{\phi\}$ are the mode shapes. For large systems, iterative solvers like the Lanczos method or subspace iteration are used because direct methods become computationally expensive.

Time domain vs frequency domain

These are two complementary frameworks for vibration analysis:

For linear systems near resonance, frequency domain analysis is usually more efficient. For nonlinear systems or transient events (impacts, earthquakes), time domain analysis is necessary.