Mechanical Vibrations and Noise

Introduction and Context

Mechanical vibration is oscillatory motion about an equilibrium position. In mechatronic equipment, vibration is produced by rotating imbalance, reciprocating mechanisms, gear and bearing interactions, electromagnetic forces, fluid excitation, control actions, and external disturbances. Vibration is not inherently undesirable: it can be used in feeders, ultrasonic actuators, sensors, and material-processing equipment. However, uncontrolled vibration causes fatigue, inaccurate positioning, noise, loosening of fasteners, bearing damage, poor product quality, and loss of user comfort.

Noise is unwanted sound, usually generated when vibrating surfaces or turbulent flows transmit pressure fluctuations through air or another medium. Vibration analysis and noise control are therefore closely related. A vibrating motor housing may radiate airborne sound, while airborne sound may excite structural vibration. Effective mechatronic design requires both mechanical modelling and measurement-based diagnosis.

Learning Objectives

By the end of this chapter, you should be able to:

1. Model single-degree-of-freedom free, damped, and forced vibration systems.
2. Calculate natural frequency, damping ratio, resonance response, and transmissibility.
3. Explain the physical meaning of stiffness, mass, damping, and phase.
4. Interpret the basic behaviour of multi-degree-of-freedom systems and modes.
5. Select suitable vibration sensors and measurement practices.
6. Apply balancing, isolation, damping, and basic noise-control methods.
7. Diagnose a mechatronic vibration problem using frequency-domain evidence.

1. Fundamental Concepts and Terminology

A vibration system consists of an inertial element, an elastic element, and usually a dissipative element. In a translational model these are represented by mass m, stiffness k, and viscous damping coefficient c. Their SI units are kilograms, newtons per metre, and newton-seconds per metre, respectively. The displacement x(t) is measured from static equilibrium unless stated otherwise.

Important terms include:

• Amplitude: maximum magnitude of an oscillating quantity.
• Period, T: time for one cycle, in seconds.
• Frequency, f: cycles per second, measured in hertz; f = 1/T.
• Angular frequency, ω: rate of phase change, in rad/s; ω = 2πf.
• Natural frequency: frequency at which a system oscillates freely after a disturbance.
• Damping: energy dissipation that reduces vibration amplitude.
• Resonance: large response occurring when excitation is near a natural frequency.
• Phase: angular displacement between two sinusoidal quantities.
• Mode shape: relative displacement pattern associated with a natural frequency.

An ideal one-degree-of-freedom (SDOF) model has one independent coordinate. It is an approximation, but it is often highly useful for a motor mounted on a flexible frame, a sensor package on an isolator, or a robot link vibrating predominantly in one direction.

2. The Single-Degree-of-Freedom Model

Applying Newton’s second law to a mass connected to a spring and viscous damper gives

where ẋ and ẍ are velocity and acceleration, and F(t) is an applied force in newtons. The model assumes linear stiffness, linear viscous damping, small displacement, and a lumped mass. It is valid when geometric nonlinearities, friction nonlinearities, and distributed flexibility are not dominant.

The undamped natural angular frequency is

and the corresponding frequency in hertz is f_n = ω_n/(2π). Dimensional checking gives k/m = (N/m)/kg = 1/s², so ω_n has units of rad/s.

The critical damping coefficient is

The dimensionless damping ratio is

A system is underdamped when 0 < ζ < 1, critically damped when ζ = 1, and overdamped when ζ > 1. Most lightly damped mechanical structures are underdamped.

2.1 Free Vibration

For free vibration, F(t) = 0. When 0 < ζ < 1, the solution can be written

where X₀ and φ depend on initial conditions, and the damped natural frequency is

Damping causes the envelope to decay exponentially. The logarithmic decrement, useful for estimating damping from successive peaks, is

where x_r and x_(r+1) are successive peak amplitudes of an underdamped free response. For very light damping, δ ≈ 2πζ. This approximation should not be used for heavily damped systems.

2.2 Forced Vibration

Suppose the applied force is harmonic:

The steady-state displacement amplitude is

where r = ω/ω_n is the frequency ratio. The static displacement produced by force amplitude F₀ is X_st = F₀/k. Thus the displacement magnification factor is

The phase lag of displacement behind the force is

with quadrant correction applied in software or calculations so that phase changes continuously from approximately 0° at low frequency to approximately 180° at high frequency. At r = 1, the ideal SDOF magnification is M = 1/(2ζ). Consequently, low damping produces a high resonant response.

The maximum displacement magnification does not occur exactly at r = 1 when damping is present. For ζ < 1/√2, its peak occurs at

For damping ratios at or above 1/√2, there is no distinct displacement-amplitude peak above the low-frequency level.

Worked Example 1: Forced Response

A sensor assembly has m = 2 kg, k = 800 N/m, and ζ = 0.05. It is excited by a harmonic force of amplitude F₀ = 10 N at f = 2 Hz.

First,

ω_n = √(800/2) = 20 rad/s,

so f_n = 20/(2π) = 3.18 Hz. The excitation angular frequency is ω = 2π(2) = 12.57 rad/s, giving r = 12.57/20 = 0.628. The static displacement is X_st = 10/800 = 0.0125 m.

The magnification factor is

M = 1/√[(1−0.628²)² + (2×0.05×0.628)²] ≈ 1.65.

Therefore, X = 1.65×0.0125 = 0.0206 m, or approximately 20.6 mm. The response is substantially larger than the static deflection, although the excitation is below resonance.

3. Base Excitation and Transmissibility

Many mechatronic devices are excited through their supports rather than by a direct force. Let the base displacement be y(t) and the mass displacement be x(t). For harmonic base motion, the absolute displacement transmissibility is

where Y is base displacement amplitude. This expression assumes linear SDOF behaviour and harmonic steady state.

For force transmission through the spring and damper, the force transmissibility has the same numerical form:

Isolation requires transmissibility below unity. For the ideal viscously damped model, this begins when

Increasing damping reduces the resonant peak but can increase transmitted force at high frequency. This is a central design trade-off: damping is valuable near resonance, whereas low damping is often preferred for high-frequency isolation. Practical isolators may also have preload, hysteresis, nonlinear stiffness, and frequency-dependent damping, so catalogue data and testing are important.

A useful design sequence is to identify the dominant excitation frequency, estimate the supported mass, select an isolator stiffness that places the natural frequency sufficiently below excitation, and then check static deflection, stability, stroke, creep, temperature, and overload conditions.

4. Rotating Imbalance and Balancing

A rotating mass with eccentricity e produces a centrifugal force with amplitude

where m_e is the unbalanced mass and ω is shaft speed in rad/s. The force is periodic at one times the rotational frequency, commonly called 1× running speed.

Balancing reduces the vector sum of rotating centrifugal forces. Static balancing corrects a single-plane resultant. Couple balancing addresses two equal and opposite forces in separated planes and generally requires two-plane balancing. Field balancing uses vibration amplitude and phase measurements to estimate trial-weight corrections. A balancing solution must also respect rotor strength, allowable residual unbalance, speed range, and the possibility that the observed vibration is caused by misalignment, looseness, resonance, or a bent shaft rather than imbalance.

5. Multi-Degree-of-Freedom Systems and Modal Interpretation

A system with several independent coordinates is represented in matrix form as

Here [[M:[M]]], [[M:[C]]], and [[M:[K]]] are mass, damping, and stiffness matrices; [[M:{x}]] is the displacement vector; and [[M:{F}]] is the force vector. For an undamped free system, assume [[M:{x} = {φ}e^(iωt)]]. Substitution gives

A non-zero mode shape requires

The resulting eigenvalues [[M:ω_r^2]] give natural frequencies and the eigenvectors [[M:{φ_r}]] give mode shapes. In a two-degree-of-freedom system there are generally two natural frequencies and two corresponding modes. The lower mode commonly involves broadly in-phase motion, while the higher mode commonly involves opposite-phase motion, although the exact pattern depends on mass and stiffness distribution.

Modes are not physical parts of a machine; they are mathematical patterns that describe how the structure can vibrate. A measured frequency-response function may show peaks at natural frequencies, but a peak can be hidden by poor sensor location, weak excitation, or modal cancellation. Modal testing therefore uses multiple measurement points and examines both amplitude and phase.

For proportional damping, modal coordinates can approximately decouple the equations into independent SDOF equations. This assumption is convenient but not universal. Joint friction, nonlinear contacts, gyroscopic effects, and fluid interaction can produce non-classical damping or speed-dependent modes.

6. Vibration Measurement and Diagnosis

Common sensors include:

Acceleration is often converted to velocity or displacement in signal processing. For a sinusoid, if displacement amplitude is X, velocity amplitude is ωX, and acceleration amplitude is ω²X. Consequently, a constant acceleration level can correspond to very different displacement amplitudes at different frequencies.

A reliable measurement procedure includes:

1. Define the fault or performance question and measurement direction.
2. Record operating speed, load, temperature, and control state.
3. Select sensor range, bandwidth, sensitivity, and mounting method.
4. Check calibration and avoid cable motion or magnetic interference.
5. Measure a baseline at known healthy conditions.
6. Use time waveforms, spectra, phase, and order analysis where appropriate.
7. Repeat measurements to assess uncertainty and identify operational dependence.

The frequency spectrum is commonly obtained using the discrete Fourier transform (DFT), usually implemented by the fast Fourier transform (FFT). Windowing reduces leakage when the sampled record does not contain an integer number of cycles. Sampling frequency must exceed twice the highest frequency of interest, according to the Nyquist condition; anti-alias filtering is required before digitization.

Characteristic spectral clues include 1× rotational frequency for imbalance, 2× rotational frequency often associated with misalignment or mechanical asymmetry, harmonics and impacts for looseness or bearing faults, and gear-mesh frequency for gear excitation. These are diagnostic clues, not universal proofs. A fault should be confirmed using phase, trend, operating conditions, inspection, and—where possible—an independent measurement.

7. Noise: Sources, Measurement, and Control

Sound pressure level is commonly expressed as

where p_rms is root-mean-square sound pressure in pascals and p₀ = 20 μPa is the reference pressure in air. Sound intensity level is based on intensity rather than pressure and has a different reference; the two quantities should not be casually interchanged.

Decibels are logarithmic. Two uncorrelated sources cannot be added arithmetically in decibels. Their combined level is

For equal independent sources, adding one source increases the total by approximately 3 dB. Weighting networks such as A-weighting approximate human frequency sensitivity and produce dB(A), but unweighted spectra are often more informative for engineering diagnosis.

Noise-control strategies follow the source–path–receiver model:

• Control at source: reduce imbalance, improve gear accuracy, lower turbulence, reduce impact, modify switching waveforms, or select quieter bearings and fans.
• Control along the path: use barriers, enclosures, absorptive liners, damping treatments, resilient mounts, and vibration isolators.
• Control at receiver: increase distance, provide hearing protection, improve operator enclosure, or limit exposure time.

Absorption reduces reflected sound within a space; isolation reduces structural transmission; barriers interrupt direct airborne paths; damping reduces structural vibration and sound radiation. These mechanisms are different and should not be treated as interchangeable.

8. Case Study: Illustrative Robot-Cell Vibration Problem

Background

An illustrative robotic assembly cell uses a servo motor and belt drive mounted on a steel frame. At a particular positioning speed, the end-effector shows blurred camera images and increased placement error. Operators also report a tonal noise near the motor.

Problem

The machine’s accelerometer shows a strong peak near 1× motor speed. A speed sweep reveals that the peak grows sharply at 1,800 rpm. The frame is suspected of having a structural resonance near the operating speed.

Analysis

The rotational frequency at 1,800 rpm is 1800/60 = 30 Hz. A preliminary impact test identifies a frame mode near 29–31 Hz. The motor spectrum contains a dominant 1× component, consistent with imbalance or eccentricity, but the large response at 30 Hz indicates amplification by the frame mode. Inspection finds that a pulley has a small balance error and one frame joint has insufficient preload.

The problem is therefore not explained by a single cause. Excitation at 30 Hz is being amplified by a lightly damped structural mode, while joint looseness and pulley imbalance increase the excitation.

Approach and Solution

The maintenance team first restores joint preload and checks alignment. The pulley is dynamically balanced, and the motor is remounted with controlled fastener torque. A finite-element model is updated using measured modal data. Because the operating range cannot be changed completely, a constrained frame brace is added to shift the mode above the normal speed range. A short-term controller speed notch is also introduced during acceleration through the resonance, after confirming that it does not compromise servo stability.

Outcome

The 1× vibration amplitude decreases substantially after balancing and tightening. The remaining resonance is moved away from the principal operating point, reducing end-effector motion and audible noise. Validation includes a repeated speed sweep, camera-placement accuracy tests, and post-maintenance spectral measurements.

Learning Points

1. A spectral peak identifies a frequency, not automatically a fault.
2. Resonance depends on both excitation and structural dynamics.
3. Mechanical correction should precede compensatory control changes where possible.
4. Vibration, noise, accuracy, and safety should be evaluated together.

9. Design Practice, Limitations, and Contemporary Developments

Good vibration design begins during concept development. Keep excitation frequencies away from structural natural frequencies, provide appropriate stiffness and damping, avoid unnecessary overhung mass, balance rotating components, control tolerances, and design accessible measurement points. Use modal analysis before finalizing thin panels, brackets, covers, and sensor mounts.

Models remain approximations. Linear analysis may fail when clearances close, friction changes state, elastomers become frequency-dependent, or large deflection changes geometry. A finite-element model is only as reliable as its material data, boundary conditions, joint representation, and validation. Sensor data can be corrupted by aliasing, poor mounting, inadequate resolution, temperature drift, or incorrect interpretation of harmonics.

Modern mechatronic systems increasingly combine accelerometers, microphones, encoders, current sensors, and controller data. Order tracking relates spectral components to shaft speed during run-up or coast-down. Operational modal analysis estimates modes from ambient or operating excitation. Machine-learning systems can classify condition-monitoring data, but they require representative training data, traceable features, drift monitoring, and human engineering review. AI should support—not replace—physical reasoning, safety assessment, and maintenance accountability.

Noise and vibration work must also consider occupational exposure, community noise, product standards, electromagnetic compatibility, and machine safety requirements applicable to the jurisdiction and industry. Measurements should use calibrated equipment and documented procedures. Avoid masking a hazardous source merely because a treatment makes it less audible; tonal noise reduction must not compromise alarms, warning signals, or protective functions.

Chapter Summary

Vibration is oscillatory motion generated by mechanical, electrical, fluid, or control-system excitation. The SDOF model uses mass, stiffness, and damping to predict natural frequency, free decay, forced response, resonance, phase, and transmissibility. Damping reduces resonant amplification but may reduce high-frequency isolation performance. Rotating imbalance produces a force proportional to unbalanced mass, eccentricity, and the square of speed.

Multi-degree-of-freedom systems are described by mass, damping, and stiffness matrices. Eigenvalues provide natural frequencies, while eigenvectors provide mode shapes. Measurement requires suitable sensors, calibration, correct sampling, meaningful baselines, and interpretation of time, frequency, phase, and order information. Noise control is organized around source, path, and receiver, with isolation, absorption, barriers, damping, and balancing serving different purposes. Effective engineering combines modelling, measurement, mechanical correction, controls, validation, and attention to safety and regulatory requirements.

Key Terms

Natural frequency: Frequency of free oscillation of a system.

Damping: Dissipation of vibratory energy.

Damping ratio: Damping expressed relative to critical damping.

Resonance: Large dynamic response near a natural frequency.

Transmissibility: Ratio of transmitted response or force to input amplitude.

Frequency ratio: Excitation frequency divided by natural frequency.

Phase: Relative angular timing between sinusoidal quantities.

Imbalance: Uneven mass distribution about a rotating axis.

Static balancing: Correction of a single-plane rotating resultant.

Couple balancing: Correction of separated-plane rotational forces.

Mode shape: Relative displacement pattern of a vibration mode.

Eigenvalue: Quantity whose square-root relationship gives a natural frequency in an undamped modal problem.

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

FFT: Efficient computational implementation of the discrete Fourier transform.

Aliasing: False frequency content caused by insufficient sampling.

Order analysis: Spectrum referenced to rotational speed.

Sound pressure level: Logarithmic level based on rms acoustic pressure.

Isolation: Reduction of vibration transmission between a source and receiver.

Absorption: Reduction of reflected acoustic energy within a space.

Source–path–receiver model: Framework for organizing noise-control interventions.

Review Questions

Short Answer Questions

1. Define natural frequency and damping ratio.
2. Why does low damping produce a large resonant response?
3. Distinguish vibration isolation from acoustic absorption.
4. What information is contained in a mode shape?
5. Why is anti-alias filtering used in vibration measurement?

Long Answer Questions

1. Derive the undamped natural frequency of an SDOF mass–spring system and discuss the assumptions involved.
2. Explain forced vibration response, magnification, phase, and resonance for a viscously damped SDOF system.
3. Compare direct-force excitation with base excitation and explain the design trade-off associated with damping.
4. Explain how matrix equations, eigenvalues, and eigenvectors are used in multi-degree-of-freedom vibration analysis.
5. Develop a source–path–receiver noise-control plan for a servo-driven industrial machine.

Analytical / Application Questions

1. A 5 kg assembly is mounted on a spring of stiffness 2,000 N/m with damping ratio 0.04. Calculate its undamped natural frequency in hertz and explain whether a 4 Hz excitation requires resonance investigation.
2. A motor operates at 1,500 rpm and produces a strong 1× vibration peak. Propose a diagnostic sequence that distinguishes imbalance from structural resonance and looseness.
3. A precision camera is mounted on an isolator. Explain how you would select stiffness and damping when the machine must pass through resonance during startup but operate well above resonance.

Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system, the natural angular frequency is given by ω_n = √(k/m). Which change increases ω_n?

A. Increasing mass while holding stiffness constant

B. Decreasing stiffness while holding mass constant

C. Increasing stiffness while holding mass constant

D. Adding a force with zero average value

Correct Answer: C. Increasing stiffness while holding mass constant

Question 2

A system has damping ratio ζ = 0.2. Which classification is correct?

A. Underdamped

B. Critically damped

C. Overdamped

D. Undamped

Correct Answer: A. Underdamped

Question 3

For harmonic force excitation, the frequency ratio is defined as:

A. r = ω_n/ω

B. r = ω/ω_n

C. r = k/m

D. r = c/k

Correct Answer: B. r = ω/ω_n

Question 4

In the ideal viscously damped base-excitation model, vibration isolation begins when the frequency ratio is approximately:

A. r = 0

B. r = 0.5

C. r = 1

D. r > √2

Correct Answer: D. r > √2

Question 5

A rotating unbalance force has amplitude proportional to:

A. m_e e/ω²

B. m_e eω²

C. k/ω

D. cω²/e

Correct Answer: B. m_e eω²

Question 6

In an undamped multi-degree-of-freedom system, the equation det([K] − ω²[M]) = 0 is used to determine:

A. Natural frequencies

B. Sound pressure reference

C. Sensor sensitivity

D. Static friction coefficient

Correct Answer: A. Natural frequencies

Question 7

Which sensor is generally most appropriate for non-contact measurement of a vibrating surface?

A. Contact accelerometer

B. Laser vibrometer

C. Strain-insensitive thermometer

D. Static pressure gauge

Correct Answer: B. Laser vibrometer

Question 8

The reference sound pressure used in air for sound pressure level is:

A. 20 Pa

B. 2 Pa

C. 20 μPa

D. 2 mPa

Correct Answer: C. 20 μPa

Question 9

Two independent equal sound sources operating together increase the total level by approximately:

A. 1 dB

B. 3 dB

C. 10 dB

D. 20 dB

Correct Answer: B. 3 dB

Question 10

A strong vibration peak exactly at running speed is observed during a speed sweep, and the amplitude rises sharply near one speed. The most appropriate interpretation is:

A. The frequency peak proves only bearing failure

B. The machine may have running-speed excitation amplified by resonance

C. The sensor must be disconnected immediately in every case

D. The peak cannot be related to imbalance

Correct Answer: B. The machine may have running-speed excitation amplified by resonance