Fundamentals of Mechanical Vibration

Introduction and Context

Mechanical vibration is the oscillatory motion of a body or system about an equilibrium position. It occurs when elastic restoring forces, inertia, and, often, energy dissipation interact. Vibration is central to the design and operation of engines, turbines, pumps, vehicles, machine tools, bridges, aircraft, and robotic systems.

Vibration may be useful, as in material screening, ultrasonic machining, and vibratory conveyors, or harmful, as in fatigue failure, excessive noise, loss of precision, bearing damage, and structural resonance. The engineer’s task is therefore not simply to eliminate vibration, but to predict it, control it, and determine when it indicates a developing fault.

This chapter develops the standard single-degree-of-freedom (SDOF) model. Although real structures commonly have many degrees of freedom, the SDOF model provides the essential concepts of natural frequency, damping, resonance, transmissibility, isolation, and rotating imbalance.

Learning Objectives

After studying this chapter, a student should be able to:

1. Formulate the equation of motion for an SDOF system.
2. Determine undamped and damped natural frequencies.
3. Describe free, damped, and forced vibration.
4. Analyze resonance and frequency-response behavior.
5. Calculate displacement, force, and motion transmissibility.
6. Explain vibration isolation and rotating imbalance.
7. Select appropriate vibration measurements and interpret basic condition-monitoring data.
8. Recognize the assumptions and limitations of SDOF models.

1. The Single-Degree-of-Freedom Model

An SDOF system requires one independent coordinate to describe its motion. The classical model consists of a mass m, a spring of stiffness k, and a viscous damper with coefficient c. The displacement x(t) is measured from the static equilibrium position. An external force F(t) may act on the mass.

Figure: Idealized mass–spring–damper model with applied force.

Applying Newton’s second law gives the general equation of motion:

Here, x is displacement in metres, ẋ is velocity in metres per second, ẍ is acceleration in metres per second squared, m is mass in kilograms, c is viscous damping coefficient in N·s/m, k is stiffness in N/m, and F(t) is applied force in newtons. The model assumes linear spring behavior, linear viscous damping, small motion, constant parameters, and one dominant motion coordinate.

The three terms represent inertia force, damping force, and spring restoring force. Dimensional consistency is evident: each term has units of newtons.

For free vibration, F(t) = 0. For harmonic forcing, F(t) = F₀ sin ωt, where F₀ is force amplitude and ω is forcing angular frequency in rad/s.

2. Natural Frequency and Damping

Undamped natural frequency

If c = 0 and F(t) = 0, the equation becomes m ẍ + kx = 0. Its natural angular frequency is

The ordinary natural frequency is

where ωₙ is in rad/s and fₙ is in hertz. The period is Tₙ = 2π/ωₙ seconds. These expressions apply to a linear, undamped SDOF system.

Damping ratio and damped natural frequency

The critical damping coefficient is

The dimensionless damping ratio is

For an underdamped system, 0 < ζ < 1, the damped natural frequency is

Damping is classified as follows:

The viscous damping model is an approximation. Coulomb friction, material hysteresis, aerodynamic damping, and fluid resistance may produce different force–velocity relationships.

3. Free Vibration

Undamped free vibration

For initial displacement x(0) = x₀ and velocity ẋ(0) = v₀, the undamped response is

The motion is periodic and has constant amplitude because no energy is dissipated. The total mechanical energy is

For an ideal undamped system, E remains constant.

Damped free vibration

For 0 < ζ < 1, the response can be written as

where A and B are determined from initial conditions. The exponential factor describes amplitude decay, while the trigonometric terms describe oscillation at ω_d.

A useful measure of damping is logarithmic decrement. If x₁ and x₂ are successive peak amplitudes of an underdamped free response,

This relationship assumes viscous damping and peaks separated by one damped period. For light damping, ζ is approximately δ/(2π).

4. Worked Example: Free Vibration

A machine component has mass 20 kg and stiffness 18,000 N/m. The damping ratio is 0.05. Determine its undamped natural frequency, damped natural frequency, and critical damping coefficient.

First,

ωₙ = √(18,000/20) = 30.0 rad/s

Thus,

fₙ = 30.0/(2π) = 4.77 Hz

The critical damping coefficient is

c_c = 2mωₙ = 2(20)(30.0) = 1,200 N·s/m

The actual coefficient is c = ζc_c = 60 N·s/m. The damped frequency is

ω_d = 30.0√(1 − 0.05²) = 29.96 rad/s

Therefore, the damped frequency is approximately 4.77 Hz. Light damping causes only a small reduction in frequency, but it strongly affects the response amplitude near resonance.

5. Harmonic Forced Vibration

Consider a harmonic force F(t) = F₀ sin ωt. After the transient response decays, the steady-state displacement is harmonic at the forcing frequency:

The displacement amplitude is

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

The quadrant must be selected correctly: for r < 1 the response is approximately in phase with the force; near resonance it lags by approximately 90°; and for r > 1 it approaches 180° lag.

The dimensionless displacement magnification factor is

This formula assumes a linear SDOF system, sinusoidal steady-state forcing, and viscous damping. The term F₀/k is the static displacement that the same force would produce.

Resonance

Resonance is the large dynamic response that occurs when excitation frequency approaches a system’s natural frequency. For a lightly damped system, the maximum displacement occurs close to r = 1. More precisely, if ζ < 1/√2, the displacement peak occurs at

At r = 1, the magnification factor is M = 1/(2ζ). Thus, damping is a major design tool for limiting resonance. Resonance should not be confused with the exact equality of forcing and natural frequencies in every damped response; the peak frequency depends on the response quantity and damping.

6. Base Excitation and Transmissibility

Machines are often mounted on vibrating foundations. Let the base motion be y(t) = Y sin ωt and the mass motion be x(t). The relative displacement is z = x − y. The system equation is

The displacement transmissibility, defined as the amplitude ratio X/Y, is

This is the absolute motion transmissibility for base excitation. Isolation requires T_D < 1. For negligible damping, isolation begins approximately when r > √2. Thus, an isolator must generally have a natural frequency well below the excitation frequency.

Damping has a dual effect. It reduces the resonant peak, but at high frequency it can increase transmitted motion because the numerator contains the damping term. Practical isolation therefore balances resonance control against high-frequency isolation.

For a force transmitted through a spring and damper, the force transmissibility is

For the standard harmonic force model, the same dimensionless expression describes force transmissibility. The physical definition and excitation type must always be stated because “transmissibility” can refer to different input and output quantities.

7. Rotating Imbalance

A rotating mass whose center of mass is offset from the rotation axis generates a periodic centrifugal force. If an imbalance mass mᵤ rotates at radius e and angular speed Ω, the force amplitude is

The force is sinusoidal in a fixed direction for a single-plane rotating imbalance. The resulting displacement amplitude is obtained from the forced-vibration formula by setting ω = Ω.

For rotating machinery, speed is often given in revolutions per minute. Conversion is

where N is rotational speed in rpm and Ω is rad/s. Since imbalance force varies with Ω², a modest speed increase can produce a substantial increase in vibration.

Balancing may be single-plane or two-plane. Flexible rotors may require modal balancing because the rotor deflects in several shapes. Other causes of periodic vibration include shaft misalignment, bent shafts, looseness, gear-mesh forces, and electromagnetic excitation; frequency content and phase help distinguish them.

8. Worked Example: Rotating Imbalance

A 0.020 kg imbalance is located 0.004 m from a shaft centerline. The shaft rotates at 1,800 rpm. Determine the force amplitude.

The angular speed is

Ω = 2π(1,800)/60 = 188.5 rad/s

Therefore,

F₀ = mᵤeΩ² = (0.020)(0.004)(188.5)²

F₀ ≈ 2.84 N

This force is small in absolute terms, but if the supporting structure is lightly damped and its natural frequency is near 30 Hz, resonance can make the resulting displacement and transmitted force much larger than a static calculation would suggest.

9. Vibration Measurement and Condition Monitoring

A vibration measurement system normally includes a sensor, signal-conditioning hardware, data acquisition, analysis software, and a decision rule. Common sensors are:

• Accelerometers: broad frequency range and convenient for machine monitoring; output is acceleration.
• Velocity transducers: useful for general machine severity because velocity often correlates with structural vibration energy over a practical frequency range.
• Proximity probes: measure shaft displacement relative to a bearing or housing and are important for large rotating machinery.
• Tachometers and once-per-revolution probes: provide rotational speed and phase reference.

Acceleration, velocity, and displacement are related for sinusoidal motion. If x(t) = X sin ωt, then velocity amplitude is V = ωX and acceleration amplitude is A = ω²X. Consequently,

These conversions apply to a single-frequency sinusoid and require consistent units. They should not be applied directly to broadband RMS data without considering frequency content and filtering.

Time-domain and frequency-domain analysis

Time waveforms reveal impacts, modulation, transients, and periodicity. The root-mean-square value for a sampled signal xᵢ with n samples is

where x̄ is the mean value. For vibration acceleration, RMS is often used as a general severity indicator, but it does not identify the fault by itself.

A spectrum, commonly obtained using the fast Fourier transform (FFT), displays amplitude versus frequency. Typical interpretations include:

These are diagnostic clues, not universal rules. Installation, load, speed, sensor location, structural resonances, and machine design must be considered.

A practical monitoring procedure

A reliable program establishes baseline measurements under known operating conditions, uses repeatable sensor locations and orientations, records speed and load, trends both overall values and spectra, and investigates changes rather than relying on a single alarm value. Alarm limits should be based on manufacturer guidance, applicable standards, historical baselines, and engineering judgment. Measurements must be traceable and sensors periodically calibrated.

10. Applied Case: Pump-Skid Vibration

Background. An illustrative centrifugal-pump skid operates at 1,800 rpm. Operators report increasing vibration and occasional seal leakage.

Problem. Housing acceleration has increased over three months, but a single overall RMS value does not establish the cause.

Analysis. The maintenance team records vibration at the drive-end and non-drive-end bearings in horizontal, vertical, and axial directions. A tachometer supplies phase and speed. The spectrum shows a dominant 1× component, significant axial vibration, and a smaller 2× component. A baseline comparison indicates that the phase relationship has changed after a pipe-support modification.

Approach and solution. The team first checks safety, foundation bolts, coupling condition, and soft foot. Pipe strain is measured and corrected. The coupling is aligned at operating temperature, and the pump is rebalanced only after alignment and looseness are excluded. Measurements after correction show a substantial reduction in 1× and axial components.

Outcome. Seal leakage ceases, vibration returns close to baseline, and the inspection interval is extended. The case demonstrates why vibration amplitude alone is insufficient: frequency, phase, operating condition, and mechanical inspection must be combined.

Learning points. Correct diagnosis follows a process rather than a single spectral label. Interventions should address the physical cause, not merely mask the vibration with additional damping. Safety procedures, equipment isolation, and documented acceptance criteria are essential.

11. Design Practice, Limitations, and Contemporary Tools

Good vibration design begins with a dynamic model developed before detailed hardware is finalized. Engineers should estimate mass, stiffness, and damping; identify excitation frequencies and their harmonics; avoid operating continuously near resonances; provide adequate damping where resonant passage is unavoidable; and verify the design experimentally.

The SDOF model has important limitations. Real systems may have distributed mass, multiple modes, nonlinear stiffness, clearance, friction, time-varying parameters, fluid–structure interaction, and coupled translational and rotational motion. A single natural frequency may therefore be inadequate. Finite-element modal analysis, experimental modal testing, and multi-degree-of-freedom models are required for complex structures.

Digital twins and machine-learning tools are increasingly used for condition monitoring. They can classify patterns and detect deviations from historical behavior, but they do not replace sensor validation, physical reasoning, or maintenance expertise. Poor training data, changing operating conditions, sensor drift, and unexplained alarms can produce unsafe decisions. Human review, cybersecurity, data governance, and auditable decision rules are important in industrial applications.

Relevant standards and manufacturer limits should be consulted for machine vibration severity, balancing, sensor calibration, and occupational exposure. Standards support consistency but do not eliminate the need for engineering judgment or risk assessment.

Chapter Summary

Mechanical vibration results from the interaction of inertia, stiffness, damping, and excitation. The SDOF equation m ẍ + c ẋ + kx = F(t) is the foundation for analyzing many practical systems. The undamped natural frequency is ωₙ = √(k/m), while damping is expressed through the critical coefficient and damping ratio. Free vibration may be undamped or decaying, and logarithmic decrement provides an experimental estimate of damping.

Harmonic forcing produces a steady-state response governed by frequency ratio and damping. Near resonance, small damping can lead to large amplitudes. Transmissibility analysis explains how vibration passes through mounts and why isolation generally requires an excitation frequency above the isolator’s natural frequency. Rotating imbalance produces a force proportional to speed squared. Effective condition monitoring combines sensors, time waveforms, spectra, phase, operating data, baselines, and engineering inspection. SDOF analysis is powerful but must be extended when multiple modes or nonlinear behavior are important.

Key Terms

Single-degree-of-freedom system: A system whose motion is described by one independent coordinate.

Natural frequency: Frequency of free oscillation determined by system inertia and stiffness.

Damping: Mechanism by which vibration energy is dissipated.

Viscous damping: Damping force proportional to velocity.

Critical damping: Boundary value separating oscillatory and non-oscillatory free response.

Damping ratio: Actual damping divided by critical damping.

Free vibration: Motion after an initial disturbance without continuing external force.

Forced vibration: Motion maintained by an external excitation.

Resonance: Large dynamic response near a system’s natural frequency.

Frequency ratio: Forcing frequency divided by natural frequency.

Magnification factor: Dynamic displacement divided by equivalent static displacement.

Transmissibility: Ratio of output vibration or force to input vibration or force.

Vibration isolation: Reduction of transmitted vibration using compliant supports.

Rotating imbalance: Offset between a rotating body’s mass center and rotation axis.

Phase angle: Relative timing angle between excitation and response.

Accelerometer: Sensor that measures acceleration.

FFT spectrum: Frequency-domain representation commonly calculated from sampled data.

Condition monitoring: Assessment of equipment health using measured operating signals.

Logarithmic decrement: Natural logarithm of the ratio of successive peak amplitudes.

Modal analysis: Study of natural frequencies, mode shapes, and modal damping.

Review Questions

Short Answer Questions

1. What physical effects are represented by the three terms in the SDOF equation of motion?
2. Define natural frequency, critical damping, and damping ratio.
3. Why does damping reduce the resonant peak?
4. What is meant by vibration isolation?
5. Why is phase information useful in rotating-machinery diagnosis?

Long Answer Questions

1. Derive the undamped natural frequency of a mass–spring system and discuss the assumptions used.
2. Explain the free response of an underdamped SDOF system and interpret the role of the exponential decay term.
3. Develop and interpret the magnification factor for harmonic force excitation.
4. Compare displacement, velocity, and acceleration sensors for industrial condition monitoring.
5. Explain the limitations of an SDOF model and identify situations requiring modal or multi-degree-of-freedom analysis.

Analytical / Application Questions

1. A 12 kg machine is supported by springs totaling 10,800 N/m with a damping ratio of 0.08. Calculate its natural frequency in hertz, critical damping coefficient, and damped natural frequency.
2. A machine is excited at a frequency equal to 1.5 times its natural frequency. Explain whether an ideal lightly damped isolator is likely to provide isolation, and identify additional information needed for a quantitative decision.
3. A rotating pump shows rising 1× vibration, strong axial response, and a phase change after maintenance. Propose a logically ordered diagnostic investigation and justify each step.

Multiple Choice Questions with Answers

Question 1

For a linear undamped mass–spring system, the natural angular frequency is ωₙ = ?.

A. √(m/k)

B. √(k/m)

C. k/m

D. m/k

Correct Answer: B. √(k/m)

Question 2

Which damping condition produces oscillatory free motion with decreasing amplitude?

A. ζ = 0

B. ζ = 1

C. 0 < ζ < 1

D. ζ > 1

Correct Answer: C. 0 < ζ < 1

Question 3

A system has m = 10 kg and k = 2,500 N/m. Its undamped natural frequency is closest to:

A. 2.52 rad/s

B. 15.8 rad/s

C. 25.0 rad/s

D. 250 rad/s

Correct Answer: B. 15.8 rad/s

Question 4

At resonance in a lightly damped forced-vibration system, which quantity generally becomes large?

A. Static stiffness only

B. Dynamic displacement response

C. Mass density

D. Rotational inertia only

Correct Answer: B. Dynamic displacement response

Question 5

The force amplitude caused by rotating imbalance is proportional to:

A. Ω

B. 1/Ω

C. Ω²

D. 1/Ω²

Correct Answer: C. Ω²

Question 6

For sinusoidal motion, if displacement amplitude is X, the acceleration amplitude is:

A. ωX

B. X/ω

C. ω²X

D. X/ω²

Correct Answer: C. ω²X

Question 7

Which instrument is especially useful for obtaining a rotational phase reference?

A. Tachometer

B. Thermocouple

C. Strain-insensitive ruler

D. Pressure gauge

Correct Answer: A. Tachometer

Question 8

For an ideal lightly damped system, vibration isolation generally begins when the frequency ratio exceeds approximately:

A. 0.1

B. 0.5

C. 1.0

D. √2

Correct Answer: D. √2

Question 9

A spectrum with a dominant component at 1× running speed can be associated with:

A. Unbalance

B. Zero stiffness in every direction

C. Constant temperature only

D. Static weight with no dynamic effect

Correct Answer: A. Unbalance

Question 10

Which statement best describes a limitation of an SDOF model?

A. It cannot represent any spring force

B. It assumes one dominant independent motion coordinate

C. It always predicts nonlinear behavior exactly

D. It requires no assumptions about damping

Correct Answer: B. It assumes one dominant independent motion coordinate