Mechanical Vibrations and Machine Condition

Introduction

Mechanical vibration is the oscillatory motion of a machine component, structure, or assembly about an equilibrium position. In theory of machines, vibration analysis connects dynamics, machine design, noise, fatigue, comfort, reliability, and condition monitoring. A rotating shaft may vibrate because of unbalance, misalignment, looseness, rubbing, bearing damage, or resonance. The same measured vibration can therefore represent either normal operation or an early warning of failure, depending on its frequency, amplitude, phase, and operating condition.

A useful vibration model is an idealized representation containing mass, stiffness, damping, excitation, and constraints. Although real machines are continuous systems with many degrees of freedom, single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) models provide the essential analytical foundation.

Learning objectives

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

• formulate free, damped, forced, and torsional vibration models;
• derive and interpret natural frequency, damping ratio, resonance, and critical speed;
• distinguish transient and steady-state response;
• explain vibration isolation and force or motion transmissibility;
• formulate and interpret MDOF modal equations;
• relate vibration frequency components to common machine faults;
• design a basic measurement and condition-monitoring procedure;
• recognize the limitations and safety implications of vibration-based diagnosis.

1. Fundamental concepts and terminology

A vibration cycle consists of a repeated excursion from one extreme position to the other and back. The displacement x(t) is measured from a chosen reference, usually the static or dynamic equilibrium position. Velocity is dx/dt and acceleration is d²x/dt². Amplitude may be expressed as peak, peak-to-peak, or root-mean-square (RMS). For a sinusoid, the relationships are

where X is peak displacement, ω is angular frequency in rad/s, t is time, and φ is phase angle. Frequency f in hertz and angular frequency are related by

For a sinusoidal quantity, x_rms = X/√2 and peak-to-peak displacement is 2X. RMS is particularly useful for assessing energy-related vibration, but crest factor and peak values can be more sensitive to impacts.

Important terms include:

• Free vibration: response after an initial disturbance, with no continuing excitation.
• Forced vibration: response caused by an external time-varying force or motion.
• Damping: energy dissipation, commonly through friction, material hysteresis, fluid resistance, or joints.
• Natural frequency: frequency at which an undamped system oscillates freely.
• Resonance: large response produced when excitation frequency approaches a natural frequency.
• Mode shape: relative motion pattern of an MDOF system during a normal mode.
• Critical speed: rotational speed at which an excitation order coincides with a natural frequency.
• Transmissibility: ratio describing how vibration or force passes through an isolator.

2. Single-degree-of-freedom vibration

2.1 Undamped free vibration

For a mass m connected to a linear spring of stiffness k, Newton’s second law gives

The model assumes small displacement, linear stiffness, a concentrated mass, and no damping. The natural angular frequency and natural frequency are

Here m is in kg, k in N/m, ω_n in rad/s, and f_n in Hz. Dimensional consistency follows because k/m has units N/(m·kg) = 1/s².

The solution is

where A and B are determined by initial displacement and velocity. Equivalently,

The system exchanges kinetic energy and strain energy. In the ideal model, total energy remains constant.

2.2 Viscously damped free vibration

With viscous damping coefficient c, the equation becomes

The critical damping coefficient is

and the damping ratio is

The classification is:

For an underdamped system, the damped natural frequency is

and the response has the form

The exponential factor describes amplitude decay. The equation is valid for linear viscous damping and constant m, c, and k.

A practical estimate of damping uses logarithmic decrement. If successive peak amplitudes are x₁ and x₂,

For an underdamped system, the exact relation is

For small damping, δ ≈ 2πζ. Multiple cycles may be used as δ = (1/n) ln(x₀/x_n), reducing measurement error.

Worked example 1: natural frequency and damping

A machine component has m = 20 kg, k = 80,000 N/m, and c = 400 N·s/m. Determine its undamped natural frequency, damping ratio, and damped natural frequency.

First,

Therefore,

The critical damping coefficient is

Thus,

The system is underdamped, and

or f_d ≈ 9.94 Hz. The small difference between f_n and f_d is characteristic of light damping.

3. Harmonic forced vibration and resonance

For a harmonic force F₀ sin ωt,

The steady-state displacement amplitude is

where r = ω/ω_n is the frequency ratio. The phase lag of displacement behind the force is

The quadrant must be interpreted correctly; in computational work, an atan2-type function is preferable. The static deflection under F₀ is F₀/k. At low frequency, the response approaches this static value. Near r = 1, resonance occurs, although damping prevents the ideal infinite response predicted by an undamped model. At high frequency, displacement amplitude decreases approximately as 1/r².

The exact frequency of maximum displacement for a damped system is r_peak = √(1 − 2ζ²) when ζ < 1/√2. For light damping this is close to r = 1. The maximum amplitude is approximately F₀/(2ζk) for small damping.

Rotating unbalance

A rotating mass m_u at eccentricity e produces a harmonic force with amplitude m_u e ω². If the supported machine mass is M, the unbalance excitation can be represented as

The response increases strongly with speed and may be amplified near a critical speed. Unbalance commonly produces a dominant component at running speed, called the 1× order, with a stable phase when operating conditions are steady.

Base excitation and transmissibility

For a system whose support moves as y(t) = Y sin ωt, the absolute mass motion x(t) is governed by

The displacement transmissibility is

At low frequency, T_d approaches 1: the mass follows the base. Isolation requires sufficiently high frequency ratio, generally r > √2 for the ideal viscous model. Increasing damping reduces resonance but increases high-frequency transmissibility. This is a central design compromise.

For force isolation, the transmitted-force ratio for a force-excited SDOF system is

under the standard assumptions of harmonic excitation and linear viscous damping. The same mathematical form does not mean that every practical installation has identical force and motion boundary conditions; the physical interpretation must be checked.

Figure: Idealized mass–spring–damper system subjected to base motion.

4. Torsional vibration

Torsional vibration is angular oscillation about a shaft axis. For a rotor with polar mass moment of inertia J, torsional stiffness k_t, and torsional damping c_t,

Here θ is angular displacement in radians and T is applied torque in N·m. For a uniform shaft of length L, shear modulus G, and polar second moment of area J_p,

The symbol J_p is a geometric polar second moment and must not be confused with the rotor mass moment J. For a solid circular shaft of diameter d, J_p = πd⁴/32; for a hollow shaft with outer and inner diameters d_o and d_i, J_p = π(d_o⁴ − d_i⁴)/32.

The undamped torsional natural frequency is

Torsional resonance can cause gear-tooth forces, coupling fatigue, crankshaft failure, or excessive angular oscillation. In drivetrain analysis, gears transform stiffness and inertia across a gear ratio, so all quantities must be referred consistently to one shaft before constructing an equivalent model.

5. Multi-degree-of-freedom systems

An MDOF linear system is represented in matrix form by

where [M], [C], and [K] are mass, damping, and stiffness matrices; {x} is the displacement vector; and {F(t)} is the force vector. For free undamped vibration,

Assuming harmonic motion {x} = {φ} sin(ωt) gives

A nontrivial mode shape requires

This eigenvalue equation gives n natural frequencies for an n-degree system. The corresponding eigenvectors {φ} are mode shapes. For classical undamped systems, distinct modes are orthogonal with respect to both [M] and [K].

With modal coordinates {q}, the physical displacement may be approximated by

where [Φ] contains mode shapes. Under proportional damping, the equations decouple into independent modal equations. This permits a complicated structure to be interpreted as several approximately independent SDOF oscillators.

Two-degree example

For two equal masses m connected in a line by equal springs k, with the outer springs attached to ground, the matrices are

The natural frequencies are

The first mode is in phase, with shape proportional to [1, 1]; the second is out of phase, with shape proportional to [1, −1]. The higher mode has greater deformation of the connecting spring and therefore a higher frequency.

6. Critical speeds and rotating machinery

A rotating machine may experience several excitation orders: 1× running speed from unbalance, 2× running speed often associated with angular misalignment, gear-mesh frequency from tooth engagement, blade-pass frequency from rotating blades, and bearing defect frequencies. A critical speed occurs when an excitation frequency intersects a system natural frequency. In an idealized rotor,

where n is an integer order. A Campbell diagram plots natural frequencies and excitation orders against rotational speed. Intersections identify potentially dangerous operating speeds.

Critical-speed control may involve changing shaft stiffness or mass distribution, increasing damping, balancing the rotor, changing support stiffness, or rapidly traversing a resonance when the design permits. Operating continuously near resonance is generally unacceptable because fatigue damage accumulates even if the instantaneous amplitude appears moderate.

7. Vibration measurement and machine condition

7.1 Measurement quantities and sensors

Accelerometers are common for machinery because they have a broad frequency range and are sensitive to bearing and gear impacts. Velocity sensors are useful for general machine severity in an intermediate frequency range. Displacement probes, including eddy-current probes, are valuable for shaft-relative motion in fluid-film bearings. A tachometer or once-per-revolution keyphasor provides speed and phase reference.

Measurement quality depends on sensor mass, mounting stiffness, orientation, cable management, bandwidth, calibration, sampling rate, and operating load. A poorly mounted accelerometer can create a false low-pass response or resonance. Sampling frequency must exceed twice the highest frequency of interest, according to the Nyquist condition; in practice, anti-alias filtering and margin above the target bandwidth are required.

7.2 Time and frequency domains

A time waveform shows impacts, modulation, periodicity, and transient events. The frequency spectrum, usually obtained using a discrete Fourier transform or FFT, separates vibration into frequency components. The frequency resolution is approximately

where f_s is sampling frequency, N is the number of samples, and T is record duration. Narrow frequency resolution requires a sufficiently long record, but excessive record length can mix changing operating conditions.

Phase comparison between bearings is useful for identifying misalignment, structural resonance, and shaft bending. Order tracking expresses frequency relative to rotational speed and is essential when speed varies.

7.3 Common diagnostic patterns

No single spectral peak proves a fault. Diagnosis should combine trend data, phase, waveform, operating state, temperature, lubrication, process variables, and physical inspection.

8. Illustrative case: centrifugal pump condition monitoring

Background. A process plant operates a centrifugal pump at 2,980 rpm, approximately 49.7 Hz. The pump is driven by an induction motor and supported by rolling-element bearings.

Problem. Overall vibration velocity at the non-drive-end bearing increases from 2.1 to 5.8 mm/s RMS over six weeks. Production personnel also report a low-frequency humming sound.

Analysis. A spectrum shows a dominant component at approximately 49.7 Hz, a smaller component near 99.4 Hz, and elevated broadband energy. Phase readings across the coupling show a substantial angular difference. The pump has recently undergone pipework modification, and a soft foot is found at one motor mounting location.

Approach and solution. The maintenance team first verifies sensor calibration, mounting, speed, and operating flow. They correct the soft foot, inspect coupling alignment, and check pipe strain. Alignment is performed at operating temperature where possible. After correction, the 1× and 2× components decrease substantially. A subsequent inspection finds no evidence that bearing replacement is necessary; the broadband component had been amplified by the structural condition rather than by a confirmed bearing defect.

Outcome. Overall vibration returns to 2.4 mm/s RMS and remains stable during a four-week trend. The intervention avoids unnecessary bearing replacement and reduces the risk of coupling and seal damage.

Learning points. Trend changes are more informative than isolated readings; 1× and 2× components are not unique diagnoses; structural faults can amplify ordinary excitation; and corrective action should begin with verification, alignment, mounting, lubrication, and operating conditions before component replacement.

9. Condition-monitoring practice, limitations, and governance

A robust program establishes a baseline during known healthy operation, measures at repeatable locations and operating states, trends both overall and spectral indicators, and defines alert and alarm levels using equipment history, manufacturer guidance, applicable standards, and engineering judgment. ISO 20816 provides a framework for evaluating machine vibration by measurements on non-rotating parts, but standards do not replace machine-specific analysis. Shaft-relative measurements, variable-speed machinery, reciprocating machines, and special structures may require different criteria.

Common limitations include speed variation, changing load, temperature effects, sensor placement, aliasing, spectral leakage, poor phase reference, structural resonance, and insufficient fault history. Automated or AI-based anomaly detection can rank unusual patterns, but it should not be treated as an autonomous safety decision. Training data may be biased toward certain machines, and an algorithm may identify correlation without establishing physical cause. Human review, traceable records, cybersecurity controls, and clearly defined escalation procedures remain essential.

Safety is paramount. Measurements near rotating shafts require guarding, appropriate access, electrical safety, and lockout procedures where physical inspection is needed. Alarm limits should reflect consequences of failure, not merely statistical abnormality. Condition monitoring must support timely engineering decisions rather than encourage operation beyond safe limits.

Best-practice workflow

1. Define the failure modes and critical machine functions.
2. Select sensor type, measurement point, direction, bandwidth, and reference speed.
3. Establish a healthy baseline under documented load and speed.
4. Collect repeatable time waveforms, spectra, phase, and process data.
5. Trend overall values and fault-specific indicators.
6. Confirm suspected faults using independent evidence and physical inspection.
7. Correct the root cause, not only the symptom.
8. Verify the post-maintenance response and update the machine history.

Chapter Summary

Mechanical vibration is modeled by balancing inertia, damping, stiffness, and excitation. The SDOF equation provides the foundation for natural frequency, damping, resonance, forced response, transmissibility, and isolation. Torsional vibration uses rotational inertia and shaft torsional stiffness. MDOF systems are expressed through mass, damping, and stiffness matrices; their eigenvalues give natural frequencies and their eigenvectors give mode shapes. Rotating machinery may encounter critical speeds when excitation orders coincide with structural or rotor natural frequencies.

Condition monitoring combines sensors, time waveforms, spectra, phase, order analysis, process data, and trend interpretation. Unbalance, misalignment, looseness, gear damage, bearing damage, and resonance have characteristic but non-unique signatures. Reliable diagnosis requires sound measurement practice, repeatable operating conditions, physical reasoning, safety controls, and confirmation of root cause.

Key Terms

Vibration: Oscillatory motion about an equilibrium position.

Natural frequency: Frequency of free oscillation of an undamped system.

Damping: Mechanism that dissipates vibration energy.

Damping ratio: Actual viscous damping divided by critical damping.

Resonance: Large response when excitation approaches a natural frequency.

Transmissibility: Ratio describing transmitted motion or force.

Critical speed: Rotational speed associated with resonance of a rotating system.

Torsional vibration: Angular oscillation about a shaft axis.

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

MDOF system: System requiring more than one independent coordinate.

Modal analysis: Analysis using natural frequencies and mode shapes.

Unbalance: Unequal mass distribution relative to a rotation axis.

Order: Frequency expressed as a multiple of rotational frequency.

FFT: Computational transformation from a time record to frequency components.

RMS: Root-mean-square measure of vibration magnitude.

Phase: Relative timing angle between harmonic quantities.

Transducer: Device converting physical vibration into a measurable signal.

Condition monitoring: Assessment of machine health using measurements over time.

Crest factor: Ratio of peak value to RMS value.

Aliasing: False frequency representation caused by inadequate sampling.

Review Questions

Short Answer Questions

1. Define natural frequency and damping ratio.
2. Why does damping limit the amplitude at resonance?
3. Distinguish displacement, velocity, and acceleration measurements.
4. What is meant by a critical speed?
5. Why is a tachometer useful in vibration diagnosis?

Long Answer Questions

1. Derive the natural frequency and describe the response of an undamped SDOF system.
2. Explain the frequency-response amplitude and phase of a viscously damped, harmonically forced system.
3. Derive the torsional natural frequency of a rotor–shaft system and state the assumptions.
4. Explain how the eigenvalue equation produces natural frequencies and mode shapes in an MDOF system.
5. Discuss vibration isolation, including the effects of frequency ratio and damping.

Analytical / Application Questions

1. A 10 kg machine is supported by a spring of 40,000 N/m. Calculate its undamped natural frequency in Hz and explain whether a 20 Hz excitation is likely to be near resonance.
2. A pump spectrum contains strong 1× and 2× running-speed components, with phase variation across the coupling. Propose a diagnostic sequence that distinguishes unbalance, misalignment, and structural looseness.
3. Design a basic condition-monitoring plan for a variable-speed gearbox, specifying sensors, sampling considerations, speed reference, indicators, and safeguards against false diagnosis.

Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system, the natural angular frequency is determined by:

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 = 0.5 is:

A. Underdamped

B. Critically damped

C. Overdamped

D. Undamped

Correct Answer: A. Underdamped

Question 3

For a sinusoidal displacement with peak amplitude X, the RMS displacement is:

A. X

B. 2X

C. X/√2

D. √2X

Correct Answer: C. X/√2

Question 4

The principal excitation frequency produced by ordinary rotor unbalance is usually:

A. 0× running speed

B. 1× running speed

C. 2× running speed only

D. Gear-mesh frequency

Correct Answer: B. 1× running speed

Question 5

For base-motion isolation in the ideal SDOF model, isolation begins approximately when the frequency ratio exceeds:

A. 0.5

B. 1

C. √2

D. 2ζ only

Correct Answer: C. √2

Question 6

The equation det([K] − ω²[M]) = 0 is used to determine:

A. Static deflection only

B. Natural frequencies of an undamped MDOF system

C. Sensor sensitivity

D. RMS acceleration

Correct Answer: B. Natural frequencies of an undamped MDOF system

Question 7

A shaft has torsional stiffness 1,000 N·m/rad and rotor inertia 4 kg·m². Its undamped torsional natural angular frequency is:

A. 15.8 rad/s

B. 250 rad/s

C. 500 rad/s

D. 4,000 rad/s

Correct Answer: A. 15.8 rad/s

Question 8

If a vibration signal is sampled at 1,000 Hz, the Nyquist frequency is:

A. 100 Hz

B. 250 Hz

C. 500 Hz

D. 1,000 Hz

Correct Answer: C. 500 Hz

Question 9

Which instrument is most directly useful for obtaining shaft phase relative to rotation?

A. Thermocouple

B. Once-per-revolution tachometer

C. Pressure gauge

D. Strain-insensitive accelerometer without reference

Correct Answer: B. Once-per-revolution tachometer

Question 10

A spectrum with a gear-mesh component and sidebands most directly suggests investigating:

A. Gear condition and modulation

B. Only static balance

C. Electrical insulation resistance

D. Ambient temperature alone

Correct Answer: A. Gear condition and modulation