3. Free and Forced Mechanical Vibrations

Introduction and context

Mechanical vibration is oscillatory motion about an equilibrium position. It appears in rotating machinery, vehicle suspensions, machine tools, aircraft structures, turbines, pumps, compressors, and precision instruments. Vibration may be useful, as in vibratory conveyors and ultrasonic machining, but it is often an indicator of excessive stress, looseness, imbalance, resonance, fatigue, noise, or loss of accuracy.

The simplest useful model is the single-degree-of-freedom (SDOF) system. Although real machines generally possess many degrees of freedom, the SDOF model provides the essential ideas needed to understand natural frequency, damping, resonance, frequency response, vibration isolation, and measurement. More elaborate models are often interpreted by reducing a local mode or component to an equivalent mass, stiffness, and damping system.

A vibration problem is governed by three questions: what is the restoring mechanism, what dissipates energy, and what excites the motion? Free vibration results from an initial displacement or velocity after the external disturbance has ceased. Forced vibration persists because an external force or motion continues to act. The distinction is fundamental in design and diagnosis.

Learning objectives

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

1. formulate the equation of motion of an undamped or damped SDOF system;

2. determine natural frequency, damping ratio, logarithmic decrement, and damped frequency;

3. distinguish free, harmonic, transient, and base-excited vibration;

4. derive and interpret frequency-response functions and resonance conditions;

5. explain transmissibility and design principles for vibration isolation;

6. evaluate the influence of imbalance and describe balancing procedures;

7. select suitable vibration sensors and interpret measured data;

8. identify modelling assumptions, practical limitations, and diagnostic risks.

The SDOF model: mass, stiffness, and damping

Consider a mass m constrained to move in one direction. A spring of stiffness k provides a restoring force proportional to displacement x, and a viscous damper of coefficient c provides a force proportional to velocity. With an externally applied force F(t), Newton’s second law gives

m x¨ + c x˙ + kx = F(t).

The three terms represent inertia, damping, and stiffness forces. The equation assumes linear behaviour, small motion, constant properties, and a single dominant coordinate. These assumptions are not universally valid, but they are sufficiently accurate for many preliminary designs and for interpreting measured vibration.

The undamped natural circular frequency is

ωn = √(k/m),

and the corresponding frequency in hertz is fn = ωn/(2π). The critical damping coefficient is

cc = 2mωn = 2√(km),

and the damping ratio is

ζ = c/cc = c/(2mωn).

The value of ζ describes the dynamic character of the system. An undamped system has ζ = 0; an underdamped system has 0 < ζ < 1; critical damping has ζ = 1; and an overdamped system has ζ > 1. Most lightly damped mechanical structures are underdamped.

The ideal spring stores mechanical energy, while the viscous damper dissipates it as heat. Real damping may arise from material hysteresis, friction, fluid resistance, joints, seals, or electromagnetic effects. Consequently, the viscous model is often an equivalent approximation rather than a literal description of the physical mechanism.

Free vibration and its physical interpretation

Undamped free vibration

For free vibration, F(t) = 0. In the undamped case,

m x¨ + kx = 0.

The solution is

x(t) = X cos(ωn t − φ),

where X and φ are determined by the initial displacement and velocity. The motion continues indefinitely in the ideal model because no energy is lost. The mass repeatedly exchanges kinetic and potential energy: kinetic energy is greatest at the equilibrium position, while spring energy is greatest at the extreme positions.

The frequency is independent of amplitude in a linear system. This property is called isochronism. It does not hold exactly for nonlinear springs, large deflections, clearances, or geometrically nonlinear structures.

Damped free vibration

For a damped system,

m x¨ + c x˙ + kx = 0.

The characteristic equation is

ms² + cs + k = 0.

For 0 < ζ < 1, the response is

x(t) = Xe^(−ζωn t) cos(ωd t − φ),

where

ωd = ωn√(1 − ζ²)

is the damped natural frequency. The exponential factor describes decay, whereas the sinusoidal factor describes oscillation. The damped frequency is slightly lower than the undamped natural frequency.

For critical damping, the system returns to equilibrium without oscillating and does so in the shortest ideal time. For overdamping, the response also does not oscillate, but it returns more slowly. Critical damping is therefore important in instruments and suspension systems where overshoot must be avoided, although maximum speed of response may not be the only design objective.

A convenient experimental measure of damping is the logarithmic decrement. If x1 and x2 are successive peak amplitudes separated by one cycle,

δ = ln(x1/x2) = 2πζ/√(1 − ζ²).

For light damping, δ ≈ 2πζ. Using peaks separated by n cycles gives δ = (1/n)ln(x1/xn+1), which is often more reliable in practice because it reduces measurement noise.

Forced vibration under harmonic excitation

Suppose the applied force is harmonic:

F(t) = F0 sin ωt.

The response consists of a transient part and a steady-state part. The transient part depends on initial conditions and decays when damping is present. The steady-state response has the excitation frequency:

x(t) = X sin(ωt − θ).

The displacement amplitude is

X = (F0/k)/√[(1 − r²)² + (2ζr)²],

where r = ω/ωn is the frequency ratio. The phase angle is given by

tan θ = (2ζr)/(1 − r²),

with the quadrant selected correctly from the signs of numerator and denominator.

The dimensionless displacement magnification factor is

M = X/(F0/k) = 1/√[(1 − r²)² + (2ζr)²].

At low frequency, the mass follows the force quasi-statically and M approaches unity. Near r = 1, the response may become large, particularly when damping is small. At high frequency, displacement decreases approximately as 1/r², although acceleration may remain significant.

Resonance is often described as the condition of maximum response. For displacement under force excitation, the peak occurs at

rres = √(1 − 2ζ²),

provided ζ < 1/√2. Thus the maximum displacement occurs slightly below ωn when damping is present. For lightly damped systems the distinction is small, but it matters in accurate testing and parameter identification. The phase changes from approximately 0° at low frequency to 90° near resonance and approaches 180° at high frequency.

The steady-state force and displacement are not generally in phase. This phase information is valuable diagnostically: an amplitude alone may not distinguish stiffness change, damping change, or excitation change, whereas the frequency-dependent phase often provides additional evidence.

Transient excitation and impulse response

Not all disturbances are sinusoidal. A sudden impact, valve closure, gear tooth event, road irregularity, or startup may be represented by a step, impulse, short-duration pulse, or arbitrary time history. The response is obtained through the system’s impulse response or, equivalently, by convolution:

x(t) = ∫0^t h(t − τ)F(τ)dτ,

where h(t) is the displacement response to a unit impulse. For a stable underdamped SDOF system,

h(t) = [1/(mωd)]e^(−ζωn t)sin(ωd t), t ≥ 0.

This relation shows why a short impact can excite a broad range of frequencies. A transient measurement is therefore frequently transformed into the frequency domain using the Fourier transform. The resulting spectrum can reveal natural frequencies, harmonics, sidebands, and broadband disturbances that are difficult to identify from a time trace alone.

A step force produces a response that approaches the static displacement F0/k, but an underdamped system overshoots this value. In control and machine design, the amount of overshoot and settling time are important performance measures. A design that is acceptable under steady operation may still fail during startup, shutdown, emergency stopping, or impact loading.

Frequency response and dynamic compliance

The complex steady-state relation between force and displacement is described by the receptance or dynamic compliance:

H(ω) = X(ω)/F(ω) = 1/[k − mω² + i cω].

Its magnitude is the displacement response per unit force, and its phase expresses the relative timing of response and excitation. Related frequency-response functions include mobility, velocity per unit force, and accelerance, acceleration per unit force. These functions are related by multiplication by iω or −ω² in the frequency domain.

Frequency-response testing can be performed by applying a swept sine, stepped sine, random excitation, or impact. A measured resonance peak provides information about modal frequency and damping, but interpretation requires attention to boundary conditions, sensor mass, excitation location, nonlinearities, and the distinction between an input spectrum and a transfer function.

Base excitation and vibration isolation

Many machines are excited not by a directly applied force but by motion of their support. Let the base displacement be y(t), and let x(t) be the absolute displacement of the mass. The equation is

m x¨ + c(x˙ − y˙) + k(x − y) = 0.

For harmonic base motion, the displacement transmissibility is

T = X/Y = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²].

Here T compares absolute mass motion with base motion. The motion isolation region begins approximately when T < 1, which for small damping occurs when r > √2. Thus an isolator must be designed so that operating frequency is sufficiently above its natural frequency. Increasing damping suppresses the resonance peak, but at high frequency it can increase transmitted motion because the damper directly transmits velocity-dependent force. Isolation design therefore involves a trade-off between resonance control and high-frequency isolation.

Force transmissibility is also important. If a vibrating machine is mounted on isolators, the objective is to reduce the force transmitted to the foundation. If a sensitive instrument is mounted on an oscillating floor, the objective is to reduce motion transmitted to the instrument. The two problems are related but should not be confused; the relevant input and output quantities must be stated explicitly.

Practical isolators include steel springs, elastomeric mounts, wire rope isolators, air springs, and active systems. Selection requires consideration of static deflection, load distribution, creep, temperature, environmental exposure, lateral stability, and the possibility of rocking modes. An isolator that is adequate in a vertical SDOF calculation may perform poorly if horizontal or rotational modes are neglected.

Rotating imbalance and balancing

A rotating mass whose centre of mass does not coincide with the shaft axis produces a centrifugal force. For eccentricity e and rotating mass m, the force amplitude is

F0 = meω².

The force rotates with the shaft and therefore excites the supporting structure at rotational frequency. The response may become severe when the running speed approaches a system natural frequency. Since imbalance force increases with the square of speed, a modest defect at low speed may become dangerous at high speed.

Single-plane balancing is appropriate when the rotor behaves approximately as a rigid disk and correction can be made in one plane. Long or flexible rotors generally require two-plane balancing, because both static imbalance and couple imbalance must be corrected. Field balancing uses measured vibration amplitude and phase to estimate the required correction mass and angular location. Trial-weight procedures remain common, but modern systems use influence coefficients obtained from calibrated measurements.

Balancing does not correct every vibration problem. Misalignment, bent shafts, looseness, bearing defects, aerodynamic forces, electrical unbalance, and resonance can produce similar symptoms. A responsible diagnosis compares vibration frequency, phase, axial and radial components, operating speed, process condition, and machine history before removing or adding balance mass.

Vibration measurement and interpretation

Measurement begins with a clear question: is the objective to detect a developing fault, validate a design, estimate a modal parameter, protect equipment, or assess human exposure? The answer determines sensor type, location, bandwidth, sampling rate, and analysis method.

An accelerometer is widely used because it has broad frequency range and is convenient for portable data collection. Velocity is often useful for general machine-severity assessment over an intermediate frequency range, while displacement is valuable at low frequencies and for shaft motion. Proximity probes measure relative shaft displacement and are common in large turbomachinery. Laser Doppler vibrometers provide non-contact velocity measurement, which is useful for delicate or inaccessible surfaces.

Sensor mounting strongly affects accuracy. Stud mounting generally provides better high-frequency transmission than magnetic or hand-held mounting. Poor cable routing can introduce noise, and an inadequately stiff mounting surface can create a local resonance. The measurement chain should be calibrated, and anti-alias filtering must be applied before analogue-to-digital conversion. Sampling frequency should exceed twice the highest frequency of interest, with practical margin for filter roll-off.

Time records reveal impacts, modulation, clipping, and non-stationarity. Spectra reveal dominant frequencies, harmonics, sidebands, and broadband energy. Order tracking is useful for variable-speed rotating machinery because it relates vibration to rotational order rather than fixed frequency. Envelope analysis can help identify repetitive impacts associated with rolling-element bearings, but it is not a substitute for knowledge of bearing geometry, speed, loading, and measurement bandwidth.

Digital monitoring systems may combine edge acquisition, cloud storage, trend analysis, automated alarms, and machine-learning classifiers. These tools can support predictive maintenance, but an algorithm trained on clean historical data may fail after process changes, sensor replacement, or a new fault mode. Human engineering judgement remains necessary for confirming cause, consequence, and corrective action.

Illustrative case study: resonance in a pump skid

Background

A centrifugal pump mounted on a steel skid exhibited increasing vibration after a process change raised operating speed from 2,900 to 3,450 rpm. Operators reported noise near the motor and occasional loosening of fasteners.

Situation and analysis

The dominant radial vibration component occurred at running speed. A speed sweep showed a pronounced amplitude peak near 3,300 rpm, followed by a rapid phase change. This pattern suggested resonance rather than a simple increase in imbalance. Measurements at the pump, motor, and skid showed that the skid’s vertical mode was close to the new operating speed. Coherence between excitation and response was high near the peak, supporting a structural dynamic explanation.

A preliminary SDOF estimate used the effective skid mass and measured static stiffness to calculate a natural frequency. An impact test with the pump stopped confirmed a mode near the operating region. The response also increased when a guard panel was loosened, indicating that joint stiffness contributed to the effective boundary condition.

Approach and solution

The engineering team first checked alignment, soft foot, foundation bolts, and rotor balance rather than assuming resonance. After these checks, two alternatives were compared: increase skid stiffness and shift the natural frequency upward, or install isolation elements and shift it downward. Because the pump required a rigid alignment relationship, the selected solution was to add structural bracing, improve joint preload, and modify the support geometry. A controlled balance check was then performed to avoid masking a residual imbalance.

Outcome

The modified skid moved its principal vertical mode above the operating range. A repeat speed sweep showed no resonance at the normal speed, and vibration remained stable during startup and shutdown. The improvement was verified under several flow conditions rather than at only one operating point.

Learning points

The case illustrates that a strong running-speed component is not automatically proof of imbalance. Resonance, support stiffness, alignment, and operating speed must be considered together. It also demonstrates the value of phase data, impact testing, controlled speed sweeps, and verification under realistic process conditions.

Limitations, risks, and professional practice

The SDOF model can be misleading when several modes lie within the operating range, when coupling between directions is strong, or when joints and supports are nonlinear. Coulomb friction, backlash, rubbing, fluid-structure interaction, amplitude-dependent stiffness, and time-varying speed may require more advanced models. A fitted damping ratio may depend on the measurement method and frequency range.

Common mistakes include confusing natural frequency with excitation frequency, using static stiffness without considering dynamic boundary conditions, interpreting every 1× running-speed peak as imbalance, ignoring phase, and comparing measurements taken with different sensor locations or operating states. Excessive damping is not always beneficial, particularly in isolation systems. Likewise, changing a machine’s stiffness may shift a resonance into another operating region rather than eliminate the underlying problem.

Safety and professional responsibility require that vibration alarms be linked to credible failure modes and consequences. Measurements on high-speed rotors, electrical equipment, elevated structures, or hot machinery require appropriate access controls and instrumentation. Decisions should be documented with operating speed, load, sensor orientation, calibration status, spectral settings, and corrective-action evidence. Standards such as ISO 20816 may assist in evaluating machine vibration, but standard limits must not replace an engineering assessment of machine design, duty, and failure risk.

Current practice is moving toward permanently installed wireless sensors, higher-resolution acquisition, operational modal analysis, digital twins, and physics-informed condition monitoring. These developments are valuable when data quality and model validity are controlled. The most robust systems combine physical vibration theory with statistical trend detection and expert review.

Best-practice workflow

A technically defensible vibration investigation generally follows this sequence:

1. define the failure or performance question;

2. document machine configuration, speed, load, support, and recent changes;

3. inspect for looseness, damage, misalignment, lubrication problems, and unsafe conditions;

4. select sensors and mounting locations appropriate to the frequency range;

5. acquire time, spectrum, phase, and operating-condition data;

6. identify whether peaks correspond to running speed, harmonics, structural modes, or process excitation;

7. test competing hypotheses using controlled changes such as speed sweeps, impact tests, or load variation;

8. implement the least disruptive effective correction;

9. verify the result under normal and transient operating conditions;

10. preserve the data and update the maintenance or design knowledge base.

Chapter Summary

An SDOF vibration model is governed by mass, stiffness, and damping. Free vibration reveals natural frequency and damping characteristics, while forced vibration describes the response to continuing excitation. Harmonic force excitation produces a frequency-dependent amplitude and phase response, with resonance occurring near the natural frequency when damping is low. Transient disturbances are understood through impulse response and frequency-domain analysis.

Base excitation introduces transmissibility and vibration-isolation design. Isolation requires a sufficiently high frequency ratio, but damping creates a trade-off between resonance suppression and high-frequency force transmission. Rotating imbalance generates a speed-squared centrifugal force, yet imbalance must be distinguished from misalignment, looseness, and resonance. Reliable diagnosis depends on appropriate sensors, calibration, phase information, operating-condition control, and disciplined interpretation. Digital monitoring and analytics extend these capabilities but do not remove the need for physical modelling and professional judgement.

Key Terms

Natural frequency: The frequency at which a system oscillates freely after disturbance.

Damping ratio: The ratio of actual viscous damping to critical damping.

Critical damping: The minimum damping that prevents oscillation during return to equilibrium.

Underdamped system: A system with damping ratio between zero and one.

Damped natural frequency: The oscillation frequency of an underdamped system, lower than its undamped natural frequency.

Logarithmic decrement: The natural logarithm of the ratio of successive free-vibration amplitudes.

Harmonic excitation: A sinusoidally varying external force or motion.

Transient response: The portion of response associated with initial conditions or temporary excitation.

Steady-state response: The persistent response remaining after transients have decayed.

Frequency ratio: Excitation frequency divided by undamped natural frequency.

Resonance: A condition of strongly amplified response near a system’s natural frequency.

Magnification factor: Steady-state displacement divided by static displacement under the same force amplitude.

Frequency-response function: A complex relation between input and output amplitude and phase versus frequency.

Transmissibility: The ratio of output vibration to input vibration or force, defined for a specified isolation problem.

Vibration isolation: Reduction of motion or force transmitted between a machine and its support.

Rotating imbalance: Eccentricity between a rotating mass centre and its axis of rotation.

Order tracking: Analysis that expresses vibration relative to rotational speed.

Receptance: Frequency-domain displacement response per unit applied force.

Accelerometer: A sensor that measures acceleration.

Operational modal analysis: Estimation of modal properties from response data under operating excitation.

Review Questions

Short Answer Questions

1. Define natural frequency and damping ratio for an SDOF system.

2. Why is the damped natural frequency lower than the undamped natural frequency?

3. What is logarithmic decrement, and how can it be measured?

4. State the main difference between free and forced vibration.

5. Why does rotating imbalance produce a force proportional to the square of speed?

Long Answer Questions

1. Derive the steady-state displacement amplitude and phase for an SDOF system subjected to harmonic force.

2. Explain underdamped, critically damped, and overdamped free responses and compare their engineering uses.

3. Derive and interpret the displacement transmissibility for a base-excited system.

4. Explain the principles of single-plane and two-plane rotor balancing, including their limitations.

5. Discuss the complete measurement chain for vibration diagnosis, from sensor selection to spectral interpretation.

Analytical / Application Questions

1. A machine has a natural frequency of 20 Hz and damping ratio 0.05. Explain the likely response when operated at 18 Hz, 20 Hz, and 40 Hz, and identify the most appropriate corrective strategies.

2. A pump’s vibration increases sharply after its speed is raised. Describe an investigation that distinguishes imbalance from structural resonance.

3. An instrument is mounted on a vibrating platform. Explain how you would select stiffness and damping for the isolator, considering both resonance and high-frequency isolation.

22. Multiple Choice Questions with Answers

Question 1

For an SDOF system with mass m and stiffness k, the undamped natural circular frequency is:

A. √(m/k)

B. √(k/m)

C. k/m

D. m/k

Correct Answer: B. √(k/m)

Question 2

A system with damping ratio ζ = 1 is:

A. Undamped

B. Underdamped

C. Critically damped

D. Overdamped

Correct Answer: C. Critically damped

Question 3

For a lightly damped force-excited system, maximum displacement response occurs approximately when the excitation frequency is:

A. Far below the natural frequency

B. Equal to zero

C. Slightly below the natural frequency

D. Much greater than the natural frequency

Correct Answer: C. Slightly below the natural frequency

Question 4

Which quantity is represented by the receptance H(ω)?

A. Force per unit acceleration

B. Displacement per unit force

C. Velocity per unit displacement

D. Damping per unit mass

Correct Answer: B. Displacement per unit force

Question 5

For a base-isolated system, the approximate condition for displacement transmissibility below unity is:

A. r < 0.5

B. r = 1

C. r > √2

D. ζ = 1 only

Correct Answer: C. r > √2

Question 6

If rotor imbalance eccentricity and mass remain constant, the imbalance-force amplitude changes with speed approximately as:

A. ω⁰

B. ω

C. ω²

D. 1/ω

Correct Answer: C. ω²

Question 7

Which measurement is especially useful for determining the relative timing between vibration and a rotating force?

A. Phase

B. Static mass

C. Ambient temperature only

D. DC resistance

Correct Answer: A. Phase

Question 8

A short impact is useful in vibration testing because it:

A. Excites only zero frequency

B. Contains a broad range of frequencies

C. Eliminates all damping effects

D. Makes all modes critically damped

Correct Answer: B. Contains a broad range of frequencies

Question 9

Increasing damping in an isolation system generally:

A. Always improves high-frequency isolation

B. Reduces the resonance peak but can increase high-frequency transmission

C. Makes the natural frequency infinite

D. Eliminates the need for correct stiffness

Correct Answer: B. Reduces the resonance peak but can increase high-frequency transmission

Question 10

A strong 1× running-speed vibration component proves that the machine has imbalance.

A. True in every case

B. True only for flexible rotors

C. False; resonance, misalignment, and other faults can produce similar components

D. False because imbalance never occurs at running speed

Correct Answer: C. False; resonance, misalignment, and other faults can produce similar components