Dynamics, Vibration and Rotordynamic Design

Introduction and context

Mechanical systems rarely operate under purely static loading. Reciprocating masses, rotating shafts, gear meshes, unbalanced rotors, fluctuating torque, impacts, and control actions produce time-dependent forces. If these forces coincide with a system’s natural frequencies, vibration amplitudes may become large enough to cause fatigue, noise, loss of precision, seal damage, bearing failure, or catastrophic fracture.

Dynamics, vibration, and rotordynamics therefore form an integrated part of mechanical design. Dynamics establishes the equations of motion; vibration analysis determines response and stability; rotordynamics applies these principles to rotating shafts, bearings, disks, seals, and support structures.

At postgraduate level, the engineer must do more than calculate a natural frequency. The engineer must formulate an appropriate model, identify assumptions, distinguish measured from predicted behavior, assess uncertainty, and convert analytical results into design decisions.

Learning objectives

After studying this chapter, the reader should be able to:

1. Formulate translational and rotational equations of motion using Newtonian and energy methods.
2. Explain free, forced, damped, and self-excited vibration.
3. Derive and interpret single- and multi-degree-of-freedom models.
4. Evaluate resonance, transmissibility, damping, and vibration isolation.
5. Explain critical speeds, gyroscopic effects, whirl, balancing, and rotor stability.
6. Select suitable analytical, numerical, and experimental tools for vibration design.
7. Diagnose vibration problems using frequency-domain and time-domain evidence.
8. Integrate fatigue, reliability, maintenance, safety, and sustainability into dynamic design.

1. Modeling mechanical systems

A dynamic model is an idealized representation of a physical system. Its purpose is not to reproduce every detail, but to retain the features that control the response of interest. A model normally contains:

• inertia, represented by mass or mass moment of inertia;
• stiffness, represented by springs, structural flexibility, or contact compliance;
• damping, representing energy dissipation;
• excitation, such as force, torque, displacement, imbalance, or fluid interaction; and
• constraints and boundary conditions.

A single-degree-of-freedom (SDOF) translational model is written as

where m is mass in kilograms, c is viscous damping coefficient in N·s/m, k is stiffness in N/m, x is displacement in metres, and F(t) is the applied force in newtons. The model assumes linear stiffness, viscous damping, small motion, and a single generalized coordinate.

For rotational motion, the corresponding equation is

where J is mass moment of inertia in kg·m², c_θ is rotational damping, k_θ is torsional stiffness in N·m/rad, θ is angular displacement in radians, and T(t) is torque in N·m.

Energy formulation

For conservative systems, Lagrange’s equation provides a systematic method:

Here L = T − V is the Lagrangian, T is kinetic energy, V is potential energy, qᵢ is a generalized coordinate, and Qᵢ is the generalized nonconservative force. This approach is especially useful for coupled mechanisms, flexible bodies, and systems with several coordinates.

For a linear multi-degree-of-freedom system,

where [M], [C], and [K]] are mass, damping, and stiffness matrices, respectively. Their dimensions are consistent with the selected coordinates. For example, in a translational model, [M] has units kg and [K] has units N/m.

2. Free vibration

For an undamped SDOF system with no external force,

Assuming harmonic motion x = X sin(ωt + φ), the natural circular frequency is

in rad/s, and the natural frequency in hertz is

The dimensional check is important: k/m has units (N/m)/kg = 1/s², so its square root has units rad/s, with radians treated as dimensionless.

For a damped system, define the critical damping and damping ratio as

The damped natural frequency for 0 ≤ ζ < 1 is

Three regimes are distinguished:

Damping is not simply beneficial in every context. It reduces resonance, but excessive damping can increase energy loss, temperature, or actuator demand and can reduce isolation at some frequencies.

3. Forced vibration and resonance

Consider harmonic force F(t) = F₀ sin ωt. The steady-state displacement amplitude is

where r = ω/ωₙ is the frequency ratio. The dimensionless dynamic magnification factor is

This solution assumes linear time-invariant properties, harmonic excitation, and a sufficiently long time for transient motion to decay.

At low frequency, the response approaches the static displacement F₀/k. Near r = 1, resonance occurs and damping controls the maximum amplitude. For small damping, the peak is approximately near r = 1, although the exact peak location is slightly below one. At high frequency, displacement decreases approximately as 1/ω², while acceleration may remain significant.

The phase angle between force and displacement is

The phase changes from approximately zero below resonance to approximately π above resonance. This phase behavior is useful in experimental identification and fault diagnosis.

Worked example: forced SDOF response

A machine of mass 50 kg is supported by a stiffness of 20,000 N/m and viscous damping of 200 N·s/m. It experiences a harmonic force of amplitude 100 N at 2 Hz. Determine the displacement amplitude.

First,

The damping ratio is

The forcing frequency is ω = 2π(2) = 12.57 rad/s, so r = 12.57/20 = 0.6285. The static displacement is F₀/k = 100/20,000 = 0.005 m. Therefore,

Thus,

The result shows why a moderate force can produce a comparatively large response when the operating frequency approaches the natural frequency.

4. Base excitation and vibration isolation

Machines may be excited through their foundations rather than by a direct applied force. For base displacement y(t), the absolute mass displacement x(t) satisfies

For harmonic base motion, the displacement transmissibility is

where Y is base-displacement amplitude. Isolation generally requires r > √2 for the ideal force-free case, although damping modifies the transition and is essential near resonance.

Force transmissibility for a force-excited machine, neglecting external base motion, is often expressed as

The same mathematical form may arise, but the physical interpretation differs. Designers must identify whether they are isolating force transmitted to the foundation or motion transmitted to the machine.

Practical isolation requires attention to static deflection, mount preload, lateral modes, rocking modes, creep, temperature, and installation tolerances. A mount selected only from its vertical stiffness may fail because a horizontal or rotational mode lies within the operating-speed range.

5. Multi-degree-of-freedom vibration

For an undamped free system,

Assume {x} = {φ}e^iωt. The eigenvalue problem becomes

A nontrivial mode shape exists when

The roots provide the squared natural frequencies, and the vectors {φ} provide the associated mode shapes. For symmetric positive-definite mass and stiffness matrices, the natural frequencies are real and the mode shapes can be chosen to be mass-orthogonal:

Modal analysis transforms physical coordinates into modal coordinates. In classical damping, the equations may be decoupled into independent modal oscillators. In nonproportionally damped systems, full state-space or complex modal analysis may be required.

Finite element analysis extends this formulation to many degrees of freedom. However, a large finite element model does not automatically produce a reliable prediction. Boundary conditions, joint stiffness, damping, mesh quality, material data, and contact assumptions often dominate the error.

6. Nonlinear vibration

Linear theory is a local approximation. Nonlinearities arise from geometric deformation, clearance, friction, contact, magnetic forces, fluid films, and material behavior. A representative nonlinear model is

where α is a cubic stiffness coefficient. If α > 0, the system is hardening; if α < 0, it is softening. Nonlinear systems may exhibit amplitude-dependent natural frequency, jump phenomena, subharmonic response, internal resonance, and coexisting stable solutions.

Linear frequency-response methods can be misleading when clearance or friction is important. Appropriate tools include harmonic balance, describing functions, continuation methods, time integration, bifurcation analysis, and experimental testing over the expected amplitude range.

7. Rotordynamic fundamentals

A rotor is a rotating flexible body supported by bearings and interacting with seals, couplings, foundations, and process fluid. Rotordynamics differs from ordinary lateral vibration because rotation introduces gyroscopic effects, speed-dependent stiffness and damping, anisotropic supports, and forward and backward whirl.

Unbalance excitation

If a mass m_u is located at eccentricity e from the shaft centerline, the unbalance magnitude is

with units kg·m. At angular speed Ω, the rotating unbalance force amplitude is

Thus, a small eccentricity can generate a large force at high speed because force increases with the square of speed. The force rotates synchronously with the shaft.

Critical speed

A critical speed occurs when an excitation frequency approaches a rotor-system natural frequency. For a simplified Jeffcott rotor with rotor mass m and lateral stiffness k,

This approximation assumes a rigid disk, massless flexible shaft, isotropic supports, negligible gyroscopic effects, and linear behavior. Real rotors may have several critical speeds and speed-dependent mode shapes.

Gyroscopic effects

For a rotating disk, gyroscopic torque is proportional to polar or transverse inertia and spin speed. In matrix form, a linearized rotor model can be represented as

where [G] is the skew-symmetric gyroscopic matrix and Ω is spin speed. The gyroscopic term splits forward and backward whirl frequencies and causes critical speeds to vary with rotational speed. The sign convention depends on coordinate orientation and whirl definition.

Bearings and seals

Fluid-film bearings are represented by linearized direct and cross-coupled coefficients:

Cross-coupled stiffness can supply energy to forward whirl and may destabilize the rotor. Seal forces, aerodynamic bearings, magnetic bearings, and active control systems require similar linearization about an operating point, but their coefficients may vary strongly with pressure, temperature, speed, and clearance.

Stability and whirl

A rotor is stable if small perturbations decay with time. In eigenvalue analysis, a mode is stable when the real part of its complex eigenvalue is negative under the adopted convention. Instability may appear as subsynchronous whirl, oil whirl, oil whip, aerodynamic excitation, or rub-induced self-excitation.

A Campbell diagram plots modal frequency against shaft speed. Intersections with excitation orders identify possible resonances. It should be supplemented by damping predictions, mode-shape information, operating transients, and uncertainty bands; an intersection alone does not prove unacceptable vibration.

8. Balancing and vibration measurement

Balancing reduces synchronous vibration by correcting mass distribution. Single-plane balancing is appropriate when the rotor behaves approximately as a rigid body. Two-plane or dynamic balancing is needed when couple unbalance and shaft flexibility are significant. Flexible rotors may require modal balancing at several speeds.

A practical vibration measurement system includes an accelerometer, velocity or displacement conversion, a tachometer or keyphasor, signal conditioning, anti-alias filtering, and data analysis. Common representations are:

• time waveform, useful for impacts, looseness, and nonstationarity;
• spectrum, useful for identifying rotational orders and harmonics;
• order spectrum, useful during speed ramps;
• orbit plot, useful for shaft motion relative to bearing clearance; and
• waterfall or Campbell plot, useful for speed-dependent behavior.

Sensor placement matters. A casing accelerometer may not accurately represent shaft motion, while a proximity probe may be sensitive to electrical runout, target geometry, and probe gap. Measurement uncertainty should be documented rather than hidden by excessive numerical precision.

9. Fatigue, reliability, and design criteria

Dynamic stress is often more damaging than static stress because repeated cycles accumulate fatigue damage. A nominal alternating stress may be corrected for surface finish, size, reliability, temperature, notch sensitivity, and mean stress. One commonly used linear cumulative-damage approximation is Miner’s rule:

where nᵢ is the applied number of cycles at stress level i, Nᵢ is the cycles to failure at that stress level from an appropriate fatigue curve, and failure is conventionally estimated near D = 1. The rule is empirical and does not represent sequence effects, overload retardation, or evolving damage accurately in every material.

A robust design checks at least:

1. peak displacement and acceleration;
2. bearing and seal limits;
3. alternating and mean stress;
4. fatigue life and crack-growth sensitivity;
5. resonance crossing during start-up and shutdown;
6. thermal effects and lubricant viscosity;
7. rotor stability margins;
8. tolerance, balance, and assembly errors; and
9. consequences of sensor, actuator, or control failure.

10. Illustrative case: vibration in a high-speed pump rotor

Background. An illustrative centrifugal pump develops increasing vibration near 7,200 rpm. The shaft is supported by fluid-film bearings, and vibration is measured using radial proximity probes.

Problem. The overall vibration is high, but the cause is uncertain. Possible causes include rotor unbalance, misalignment, hydraulic excitation, bearing instability, and a structural resonance.

Analysis. The operating speed is 120 Hz. A spectrum shows a dominant 1× component, with phase that remains nearly constant relative to the keyphasor. During a controlled run-up, the amplitude rises sharply near 7,000–7,400 rpm and decreases after the speed is increased. The orbit is predominantly forward and elliptical. No strong subsynchronous component is present.

A preliminary rotor model predicts a first lateral critical speed near 7,300 rpm. The predicted mode shape has large shaft motion near the impeller. A balance check shows residual unbalance above the specified grade, but the calculated unbalance force alone does not explain the sharp amplification.

Approach and solution. The team first verifies probe calibration, phase reference, bearing clearances, and foundation integrity. A modal test of the stationary assembly identifies a nearby support mode. The rotor model is updated using measured bearing coefficients and support stiffness. The final model predicts a critical-speed crossing at 7,250 rpm with acceptable damping but insufficient separation from the continuous operating range.

The design is improved by increasing support stiffness, correcting the rotor balance, and modifying the operating procedure so that the critical-speed band is crossed rapidly during start-up and shutdown. The pump is not operated continuously near the resonance. A new run-up test confirms reduced amplitude and stable phase behavior.

Outcome. The result is not attributed to “unbalance” alone. Unbalance supplied the synchronous excitation, while rotor-support dynamics determined the amplification. This distinction leads to a more effective solution than simply adding balance weights.

Learning points. Fault diagnosis should combine amplitude, frequency, phase, orbit, speed dependence, and physical inspection. Model updating is most valuable when it uses high-quality measurements and identifies which parameters are uncertain. Operating procedures can be part of the dynamic design, but they do not substitute for adequate structural margin.

11. Computational and experimental methods

The principal methods are complementary:

AI and machine learning can assist anomaly detection, remaining-useful-life estimation, and automated feature extraction from vibration data. However, a predictive model trained on one machine, speed range, or fault population may fail under changed operating conditions. Physics-informed learning, uncertainty quantification, traceable data management, and human review are essential in safety-critical applications.

12. Design governance, safety, and sustainability

Dynamic design is safety-relevant because resonance and instability can escalate rapidly. Design records should identify assumptions, operating envelopes, acceptance criteria, test conditions, sensor calibration, and change control. Applicable machinery, pressure-equipment, balancing, vibration, and occupational-noise standards should be selected according to jurisdiction and machine type rather than cited generically.

Ethically, engineers must not suppress unfavorable test data, broaden alarm limits without technical justification, or declare a rotor safe solely because average vibration is acceptable. Alarm thresholds should consider transient behavior, rate of change, phase, process conditions, and consequences of failure.

Sustainable design favors efficient rotors, low-loss bearings, repairable components, condition-based maintenance, and designs that minimize unnecessary over-sizing. Yet reduced mass or damping should not compromise reliability. Life-cycle assessment should include manufacturing, lubrication, monitoring, replacement, and end-of-life considerations.

Best-practice design workflow

1. Define operating speeds, loads, transients, environment, and failure consequences.
2. Select the minimum model that captures the relevant modes and excitation mechanisms.
3. Establish material, joint, bearing, seal, and support properties with uncertainty ranges.
4. Calculate natural frequencies, mode shapes, forced response, and stability margins.
5. Check start-up, shutdown, overspeed, and fault conditions.
6. Validate key predictions through modal, run-up, balance, or endurance testing.
7. Update the model using measured data, documenting parameter changes.
8. Specify balance quality, tolerances, monitoring locations, alarm limits, and maintenance actions.
9. Review fatigue, reliability, safety, cybersecurity of monitoring systems, and environmental impact.
10. Preserve traceable calculations and reassess the design when operating conditions change.

Chapter Summary

Dynamics describes how inertia, stiffness, damping, and excitation determine motion. The SDOF equation provides a foundation for understanding natural frequency, damping ratio, resonance, phase, and transmissibility. Multi-degree-of-freedom systems require matrix equations and eigenvalue analysis, while nonlinear systems may show amplitude-dependent and multistable behavior.

Rotordynamics extends vibration theory to rotating shafts, bearings, seals, gyroscopic effects, unbalance, forward and backward whirl, critical speeds, and stability. Reliable design requires more than a frequency calculation: it requires model selection, uncertainty assessment, experimental validation, fatigue evaluation, and consideration of transient and fault conditions. Vibration measurements become meaningful when amplitude, frequency, phase, orbit, speed, and physical evidence are interpreted together.

Key Terms

Inertia: Resistance of a body to acceleration.

Stiffness: Force or torque required per unit displacement or rotation.

Damping: Mechanism that dissipates vibrational energy.

Natural frequency: Frequency of free oscillation of a system.

Damping ratio: Actual viscous damping divided by critical damping.

Resonance: Large response caused by proximity between excitation and natural frequency.

Transmissibility: Ratio describing motion or force transferred through a system.

Mode shape: Relative deformation pattern associated with a natural frequency.

Eigenvalue: Mathematical quantity representing squared modal frequency in an undamped system.

Gyroscopic effect: Speed-dependent coupling caused by rotating inertia.

Unbalance: Eccentric mass distribution in a rotating assembly.

Critical speed: Rotational speed near a rotor-system natural frequency.

Whirl: Precessional motion of a rotor centerline.

Forward whirl: Whirl in the same direction as shaft rotation.

Cross-coupled stiffness: Force coefficient coupling one lateral direction to another.

Rotor stability: Ability of perturbations to decay rather than grow.

Campbell diagram: Plot of modal frequencies and excitation orders versus rotational speed.

Modal balancing: Correction of flexible-rotor unbalance using modal information.

Miner’s rule: Approximate linear cumulative-fatigue-damage relation.

Operational modal analysis: Modal identification using response data under operating excitation.

Review Questions

Short Answer Questions

1. State the assumptions behind the linear SDOF vibration model.
2. Define natural frequency, critical damping, and damping ratio.
3. Why does unbalance force increase with the square of rotational speed?
4. Distinguish forward whirl from backward whirl.
5. What information does a Campbell diagram provide?

Long Answer Questions

1. Derive the frequency-response amplitude of a viscously damped SDOF system under harmonic force excitation and interpret its limiting behavior.
2. Explain the eigenvalue formulation for an undamped multi-degree-of-freedom system and discuss mode orthogonality.
3. Discuss how bearing and seal coefficients influence rotor stability.
4. Compare analytical modeling, finite element analysis, and experimental modal analysis for an industrial rotor.
5. Explain why vibration diagnosis should use phase and orbit information in addition to an amplitude spectrum.

Analytical / Application Questions

1. A 40 kg machine is mounted on a stiffness of 16,000 N/m with damping coefficient 160 N·s/m. Calculate its natural frequency and damping ratio, and determine whether operation at 2.5 Hz is close to resonance.
2. A rotor has unbalance U = 0.002 kg·m and rotates at 9,000 rpm. Calculate the unbalance-force amplitude, assuming rigid-rotor behavior.
3. A rotor shows a strong 1× response during run-up and a subsynchronous component after the speed is held constant. Propose a measurement and analysis plan to distinguish simple unbalance from bearing-induced instability.

Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system, the natural circular frequency is:

A. k/m

B. √(k/m)

C. m/k

D. 2π√(k/m)

Correct Answer: B. √(k/m)

Question 2

The damping ratio is defined as:

A. Actual damping divided by critical damping

B. Critical damping divided by stiffness

C. Mass divided by actual damping

D. Stiffness divided by mass

Correct Answer: A. Actual damping divided by critical damping

Question 3

For rotating unbalance U, the force amplitude at angular speed Ω is:

A. UΩ

B. U/Ω²

C. UΩ²

D. Ω/U

Correct Answer: C. UΩ²

Question 4

Which condition generally indicates resonance risk?

A. Excitation frequency far below every natural frequency

B. Excitation frequency close to a natural frequency

C. Zero damping with no excitation

D. Constant static displacement only

Correct Answer: B. Excitation frequency close to a natural frequency

Question 5

In the rotor equation, the gyroscopic matrix is multiplied by:

A. Mass

B. Stiffness

C. Spin speed

D. Displacement amplitude

Correct Answer: C. Spin speed

Question 6

A Campbell diagram primarily plots frequency against:

A. Temperature

B. Rotational speed

C. Bearing pressure only

D. Fatigue life

Correct Answer: B. Rotational speed

Question 7

For a linear system, mass-orthogonality of distinct modes is associated with:

A. Symmetric mass and stiffness matrices under suitable conditions

B. A nonlinear cubic spring only

C. Zero mass

D. Infinite damping

Correct Answer: A. Symmetric mass and stiffness matrices under suitable conditions

Question 8

A dominant synchronous 1× vibration component most directly suggests excitation at:

A. Half the shaft speed

B. The shaft rotational frequency

C. Twice the natural frequency only

D. A random broadband frequency

Correct Answer: B. The shaft rotational frequency

Question 9

Miner’s rule estimates fatigue damage using:

A. The product of stress and frequency only

B. The sum of applied cycles divided by corresponding fatigue lives

C. The square of damping ratio

D. The ratio of stiffness to mass

Correct Answer: B. The sum of applied cycles divided by corresponding fatigue lives

Question 10

Cross-coupled bearing stiffness can be important because it may:

A. Eliminate all gyroscopic effects

B. Supply energy to forward whirl and reduce stability

C. Make mass zero

D. Prevent every critical speed

Correct Answer: B. Supply energy to forward whirl and reduce stability