1. Foundations of Mechanical Vibrations

Introduction and Context

Mechanical vibration is the oscillatory motion of a body or system about an equilibrium position. It occurs whenever inertia and a restoring mechanism interact. A vehicle suspension, turbine blade, robotic arm, machine-tool spindle, bridge, building, and microelectromechanical sensor can all vibrate. Vibration may be useful, as in ultrasonic machining and vibratory conveyors, or harmful, as in fatigue failure, excessive noise, loss of precision, and structural instability.

Mechanical vibrations are studied by idealizing a physical system into masses, elastic elements, damping elements, and external forces. The quality of the analysis depends less on mathematical complexity than on selecting an appropriate model. A model that omits an important flexible component may produce a precise but misleading prediction; a model with unnecessary detail may obscure the dominant physics.

This chapter establishes the language and analytical foundations used throughout mechanical vibrations. It reviews kinematics, force and energy methods, harmonic motion, complex notation, degrees of freedom, stiffness, inertia, damping, equilibrium, linearization, and the terminology used to classify vibration problems.

Learning Objectives

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

1. Describe translational and rotational vibratory motion using suitable coordinates.

2. Formulate equations of motion using Newtonian and energy methods.

3. Distinguish equilibrium, static displacement, deformation, and vibration coordinates.

4. Explain mass, stiffness, damping, excitation, and degrees of freedom.

5. Represent harmonic motion using trigonometric and complex notation.

6. Linearize nonlinear mechanical relations about an operating equilibrium.

7. Select an appropriate vibration model for an engineering system.

8. Interpret vibration terminology and recognize common modelling errors.

Motion, Coordinates, and Equilibrium

A mechanical system is described by generalized coordinates. A coordinate may be a linear displacement, an angular displacement, or another independent variable that completely specifies the configuration of the idealized system. For a mass moving along a guide, the coordinate may be x(t). For a rotor, it may be an angular displacement θ(t). The dot notation denotes time differentiation: ẋ is velocity and ẍ is acceleration.

The position of a system is often decomposed into a reference position and a displacement from that position. If x_s is the static equilibrium position and y is a small dynamic displacement, then

x(t) = x_s + y(t).

This distinction is important. Static equilibrium is determined by time-independent forces, whereas dynamic equilibrium concerns the balance of forces during motion. In a vertical spring-mass system, gravity may produce a static extension, but it does not necessarily appear in the final equation when displacement is measured from the static equilibrium position.

An equilibrium configuration is one in which the net generalized force is zero. Stable equilibrium requires that a small displacement produce a restoring tendency. In a conservative one-degree-of-freedom system, stability near an equilibrium is associated with a local minimum of potential energy. An unstable equilibrium corresponds to a local maximum or another configuration in which small disturbances grow rather than being restored.

Kinematics of Vibratory Motion

For translational motion, displacement x, velocity v, and acceleration a are related by

v = dx/dt, a = d²x/dt².

For rotational motion, the corresponding quantities are angular displacement θ, angular velocity ω = dθ/dt, and angular acceleration α = d²θ/dt². The distinction between angular velocity as a kinematic quantity and excitation frequency, commonly also written ω, should be handled carefully. In vibration analysis, Ω is often used for forcing frequency to avoid ambiguity.

The most important ideal motion is simple harmonic motion (SHM):

x(t) = X cos(Ωt + φ),

where X is amplitude, Ω is angular frequency in radians per second, and φ is phase angle. Differentiation gives

ẋ(t) = -ΩX sin(Ωt + φ),

ẍ(t) = -Ω²X cos(Ωt + φ) = -Ω²x(t).

Thus acceleration is proportional to displacement and opposite in direction. The period is T = 2π/Ω and frequency in cycles per second is f = Ω/(2π). Amplitude may be specified as peak, peak-to-peak, or root-mean-square (RMS); these measures must not be confused when comparing measurements or design limits.

A general periodic motion may be represented by a Fourier series, while a nonperiodic motion may require an integral transform or numerical description. Nevertheless, harmonic motion is fundamental because linear systems respond independently to harmonic components.

Physical Elements: Inertia, Stiffness, and Damping

Inertia

Inertia represents resistance to acceleration. For a translating particle or rigid body with mass m, the inertial force in a d’Alembert formulation is -m ẍ. For rotation about a fixed axis, the corresponding quantity is the mass moment of inertia J, and the inertial torque is -Jθ̈. Distributed structures possess distributed inertia and are commonly represented by equivalent or modal masses in reduced-order models.

Stiffness

Stiffness measures the restoring force or torque generated by deformation. For a linear translational spring,

F_s = kx,

where k is stiffness. For a torsional spring,

M_s = k_θ θ.

Stiffness is not restricted to a physical coil spring. It may arise from bending, axial deformation, contact, geometry, magnetic fields, or a combination of components. Series and parallel combinations must be assembled according to compatibility and equilibrium. For two springs in parallel, k_eq = k_1 + k_2. For springs in series, 1/k_eq = 1/k_1 + 1/k_2.

The tangent stiffness is the local slope of a force-displacement curve, k_t = dF/dx. In nonlinear systems, tangent stiffness may vary with displacement and may even become negative in an unstable region.

Damping

Damping dissipates mechanical energy, usually through friction, fluid resistance, material hysteresis, or electromagnetic effects. The viscous damping model is

F_d = c ẋ,

where c is the damping coefficient. The force opposes velocity, so its signed contribution in an equation of motion is commonly -c ẋ. Viscous damping is analytically convenient and often a useful approximation near an operating condition, even when the physical mechanism is more complicated.

Coulomb damping is approximately constant in magnitude and reverses direction with velocity. Structural or hysteretic damping is frequently represented through a complex stiffness or loss factor. Aerodynamic and fluid forces may be nonlinear, depending on velocity squared or other flow variables. Selecting the damping model requires knowledge of the mechanism and the frequency range of interest.

Newtonian Formulation

The equation of motion follows from Newton’s second law. Consider a mass m connected to a spring k and viscous damper c, with external force F(t). Taking displacement x positive to the right,

m ẍ + c ẋ + kx = F(t).

This is the standard single-degree-of-freedom (SDOF) forced vibration model. If F(t) = 0, the system undergoes free vibration. If the initial conditions are nonzero, free motion may still occur without continuing external excitation.

For a rotational SDOF system,

J θ̈ + c_θ θ̇ + k_θ θ = M(t).

Newtonian modelling is direct and especially useful when forces and moments are readily identified. Its main risk is sign inconsistency. A positive coordinate, positive velocity, and positive force direction should be selected before writing each contribution.

For systems with several coordinates, the equations are assembled in matrix form:

[M]{q̈} + [C]{q̇} + [K]{q} = {F(t)}.

Here [M], [C], and [K] are mass, damping, and stiffness matrices, respectively. Their entries reflect coupling between coordinates. In a physically appropriate conservative model, the mass and stiffness matrices are generally symmetric; lack of symmetry may indicate gyroscopic effects, nonconservative forces, or modelling and assembly errors.

Energy Methods

Energy methods provide a second route to equations of motion and are particularly valuable for linked mechanisms, flexible members, and systems in generalized coordinates. The kinetic energy of a translating mass is

T = 1/2 m ẋ²,

and that of a rotating rigid body is T = 1/2 J θ̇². The potential energy stored in a linear spring is

V = 1/2 kx².

For a conservative undamped system, total mechanical energy E = T + V remains constant. Differentiating E for a spring-mass system gives

m ẍ + kx = 0.

For generalized coordinates q_i, Lagrange’s equation is

d/dt(∂L/∂q̇_i) - ∂L/∂q_i = Q_i,

where L = T - V and Q_i represents generalized nonconservative force. Viscous damping may be introduced through a dissipation function

D = 1/2 c q̇²,

so that the equation becomes

d/dt(∂L/∂q̇_i) - ∂L/∂q_i + ∂D/∂q̇_i = Q_i.

Energy methods reduce the need to draw every internal force and naturally accommodate generalized coordinates. However, the energy expressions must include all relevant forms of kinetic and potential energy, including rotational inertia, gravitational potential, and coupling terms.

Free Vibration and Characteristic Quantities

For the undamped SDOF system,

m ẍ + kx = 0,

assume x = Ae^{st}. The characteristic equation is ms² + k = 0, yielding s = ±iω_n, where

ω_n = √(k/m)

is the undamped natural angular frequency. The corresponding natural frequency is f_n = ω_n/(2π). The solution is harmonic, with amplitude and phase determined by initial conditions.

With viscous damping,

m ẍ + c ẋ + kx = 0.

The damping ratio is

ζ = c/(2√(km)),

and the critical damping coefficient is c_c = 2√(km). The system is underdamped when 0 < ζ < 1, critically damped when ζ = 1, and overdamped when ζ > 1. For the underdamped case, the damped natural frequency is

ω_d = ω_n√(1 - ζ²).

These classifications describe the transient response, not the complete behaviour of every nonlinear or multi-degree-of-freedom system.

Harmonic Excitation and Complex Notation

A harmonic force may be written as F(t) = F_0 cos Ωt. In a linear SDOF system, the steady-state response has the same frequency:

x(t) = X cos(Ωt - δ).

The amplitude and phase depend on the frequency ratio r = Ω/ω_n and damping ratio ζ. Complex notation simplifies differentiation and algebra. Represent the force as the real part of a complex quantity:

F(t) = Re{F_0 e^{iΩt}}.

Assume x(t) = Re{X̄e^{iΩt}}, where X̄ is a complex amplitude. Substitution into the equation gives

(k - mΩ² + i cΩ)X̄ = F_0.

The complex dynamic stiffness is k - mΩ² + i cΩ. Its inverse is dynamic compliance. The magnitude of X̄ gives response amplitude and its argument gives phase relative to the force. Complex notation is a mathematical representation; the physical displacement is the real part.

At frequencies near the natural frequency, response may become large, particularly when damping is small. This phenomenon is resonance. In practice, nonlinearities, damping mechanisms, forcing bandwidth, and structural limits determine the actual peak response.

Degrees of Freedom and Model Idealization

The number of degrees of freedom (DOF) is the minimum number of independent coordinates needed to define the configuration of a model. A rigid block constrained to move horizontally has one DOF. Two independently moving masses generally require two DOF. A continuous beam has infinitely many DOF in its exact description, although modal truncation can produce a finite-dimensional approximation.

Model selection is a central engineering decision. A one-DOF model is appropriate when one mode dominates and other components can be treated as rigid or quasi-static. A multi-DOF model is required when modes interact, when multiple measurement locations are important, or when coupling affects design decisions. A finite-element model may contain thousands of coordinates, but model reduction can retain the modes relevant to the operating frequency range.

The coordinates need not be physical displacements. Generalized coordinates, modal coordinates, or relative coordinates may produce simpler equations. Coordinate choice should reflect constraints, energy storage, measurement availability, and the intended use of the model.

Linearization About an Operating Equilibrium

Many mechanical systems are nonlinear. Suppose the restoring force is F(x), and x_e is an equilibrium position. Let x = x_e + y, where y is a small perturbation. A Taylor expansion gives

F(x_e + y) = F(x_e) + F'(x_e)y + 1/2 F''(x_e)y² + ⋯.

Because equilibrium balances the constant term, and because small-motion analysis neglects higher-order terms, the local equation contains the tangent stiffness F'(x_e). For example, a pendulum has restoring torque proportional to sin θ. Near θ = 0, sin θ ≈ θ, producing a linear model; at large amplitude, the approximation becomes inaccurate.

Linearization is useful for natural frequencies, stability, control design, and small-signal response. It cannot predict amplitude-dependent frequency shifts, jump phenomena, nonlinear damping effects, or bifurcations unless the nonlinear terms are retained or treated using an appropriate nonlinear method.

Applied Scenario: Vibration of a Motor–Pump Assembly

Background

Consider an electric motor driving a centrifugal pump through a flexible coupling. The assembly is mounted on a base supported by resilient isolators. During commissioning, operators observe increased vibration as rotational speed approaches a particular range.

Situation and Problem

The initial engineering question is whether the vibration is caused by imbalance, looseness, misalignment, or a structural resonance. A useful first model treats the combined motor-pump mass as an SDOF system supported by equivalent stiffness k and damping c. The measured forcing frequency is the shaft rotational frequency and, if relevant, its harmonics.

Analysis

The model predicts a natural frequency ω_n = √(k/m). If the operating speed is close to this frequency, resonance is plausible. Measurements should include displacement or acceleration amplitude, phase, speed, and vibration at the motor, pump, and base. A phase change near resonance and a response peak that shifts when stiffness or mass is changed support the resonance hypothesis. Spectral lines at rotational frequency and harmonics may also indicate imbalance or misalignment.

Approach and Solution

Engineers first verify the model parameters and inspect the foundation, coupling, fasteners, and isolators. They then compare measured frequency-response data with the SDOF prediction. Possible remedies include balancing the rotor, correcting alignment, increasing structural stiffness, changing isolator stiffness, adding damping, or shifting the operating speed away from the resonant region. The selected remedy must consider transmitted force, fatigue, noise, and process requirements; simply making the support softer may reduce force transmission while increasing displacement.

Outcome

The case illustrates that vibration diagnosis is not achieved by measuring amplitude alone. Frequency, phase, operating conditions, and a physically defensible model are needed to distinguish excitation from system dynamics.

Learning Points

A vibration problem should be approached as an interaction between source, transfer path, and receiver. Mass, stiffness, and damping changes affect the transfer path, while balancing and alignment address the source. A successful solution therefore combines modelling, measurement, and mechanical inspection.

Professional Practice, Limitations, and Emerging Tools

Common errors include confusing static and dynamic displacement, using peak values as RMS values, ignoring rotational inertia, assigning incorrect spring combinations, and linearizing about a non-equilibrium point. Another frequent error is treating a natural frequency as a fixed property when mass, boundary conditions, temperature, preload, or operating speed change it.

Engineering decisions should account for safety factors, fatigue, human exposure, noise, maintenance, and applicable machinery or structural standards. Vibration data can reveal developing faults, but automated alarms must be validated against operating context. Digital sensors, finite-element analysis, operational modal analysis, and machine-learning classifiers now support condition monitoring. These tools do not eliminate the need for physical reasoning: poor sensor placement, nonrepresentative training data, or an incorrect model can produce confident but unsafe conclusions.

Best practice is to define the coordinate system, identify equilibrium, state modelling assumptions, estimate parameter ranges, and check dimensions and limiting cases. Predictions should be compared with measurements, and uncertainty in mass, stiffness, damping, and excitation should be reported when it affects a design decision.

Chapter Summary

Mechanical vibration is oscillatory motion produced by the interaction of inertia, restoring forces, and energy dissipation. Vibration models use coordinates to describe configuration and commonly represent physical systems through mass, stiffness, damping, and excitation. Newton’s law gives direct force-based equations, while Lagrange’s method and energy principles are efficient for generalized systems. Harmonic motion is characterized by amplitude, frequency, period, and phase; complex notation provides a compact method for calculating steady-state response. Natural frequency, damping ratio, resonance, and degrees of freedom are central modelling concepts. Nonlinear systems can often be approximated locally by linearization about a stable equilibrium, but the limitations of that approximation must be recognized. Effective professional analysis links mathematical prediction with measurement, inspection, uncertainty assessment, and appropriate engineering action.

Key Terms

Degree of freedom: An independent coordinate required to specify a system configuration.

Damping: Mechanism by which vibratory energy is dissipated.

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

Dynamic stiffness: Complex frequency-dependent opposition to harmonic displacement.

Equilibrium: Configuration in which the net generalized force is zero.

Excitation: External force, displacement, or motion that drives a system.

Frequency: Number of cycles completed per unit time.

Harmonic motion: Sinusoidal motion described by a single frequency.

Inertia: Resistance to changes in translational or rotational motion.

Linearization: Approximation of a nonlinear model by its local first-order form.

Natural frequency: Frequency of free vibration determined by system properties.

Phase: Relative timing or angular displacement between harmonic quantities.

Resonance: Large response associated with excitation near a natural frequency.

Stiffness: Restoring force or torque per unit deformation near a specified state.

Transient response: Motion that depends on initial conditions and decays or evolves after excitation changes.

Steady-state response: Long-term response remaining after transient effects have diminished.

Tangent stiffness: Local derivative of force with respect to displacement.

Viscous damping: Damping force proportional to velocity.

Review Questions

Short Answer Questions

1. What distinguishes static equilibrium from dynamic equilibrium?

2. Define natural frequency for an undamped SDOF system.

3. Why is complex notation useful in harmonic vibration analysis?

4. What is meant by a system’s degree of freedom?

5. State the physical meaning of damping ratio.

Long Answer Questions

1. Derive the equation of motion of a damped spring-mass system using Newton’s second law and explain each term.

2. Explain how Lagrange’s equation differs from a direct Newtonian formulation and identify situations in which it is advantageous.

3. Discuss mass, stiffness, and damping as idealized physical elements, including common non-ideal damping mechanisms.

4. Explain free vibration, forced vibration, resonance, and the role of damping in each.

5. Describe the procedure and assumptions involved in linearizing a nonlinear vibration system about equilibrium.

Analytical / Application Questions

1. A machine of mass m is supported by an equivalent spring and damper. Explain how you would determine whether a measured peak near operating speed is resonance or an excitation-related fault.

2. Two springs support a rigid platform, with one spring acting in parallel with a series pair. Develop the equivalent stiffness and discuss how it influences the natural frequency.

3. A pendulum operates with small oscillations during normal service but occasionally experiences large angular motion. Decide when a linear model is acceptable and identify phenomena it cannot predict.

22. Multiple Choice Questions with Answers

Question 1

For an undamped spring-mass system, which expression gives the natural angular frequency?

A. k/m

B. √(k/m)

C. m/k

D. √(m/k)

Correct Answer: B. √(k/m)

Question 2

A displacement measured from static equilibrium in a vertical spring-mass system generally has which advantage?

A. It eliminates inertia

B. It removes the need to model damping

C. The constant gravitational term cancels from the dynamic equation

D. It makes the system nonlinear

Correct Answer: C. The constant gravitational term cancels from the dynamic equation

Question 3

Which equation represents a viscous damping force in one-dimensional motion?

A. F = kx

B. F = m ẍ

C. F = c ẋ

D. F = c x²

Correct Answer: C. F = c ẋ

Question 4

For a linear SDOF system subjected to harmonic force at frequency Ω, the steady-state response frequency is generally:

A. Zero

B. The natural frequency only

C. The forcing frequency Ω

D. The damped frequency only

Correct Answer: C. The forcing frequency Ω

Question 5

The damping ratio ζ = 1 corresponds to:

A. Undamped motion

B. Critical damping

C. Coulomb damping

D. Negative damping

Correct Answer: B. Critical damping

Question 6

Which statement best describes complex notation in vibration analysis?

A. It replaces the physical motion with an unmeasurable quantity

B. It represents harmonic quantities compactly; the physical result is obtained from the real part

C. It applies only to nonlinear systems

D. It eliminates the need to know phase

Correct Answer: B. It represents harmonic quantities compactly; the physical result is obtained from the real part

Question 7

A model requires two independent coordinates to describe its configuration. Its number of degrees of freedom is:

A. One

B. Two

C. Three

D. Infinite

Correct Answer: B. Two

Question 8

For two ideal springs connected in parallel, the equivalent stiffness is:

A. k₁k₂/(k₁+k₂)

B. k₁-k₂

C. k₁+k₂

D. 1/(k₁+k₂)

Correct Answer: C. k₁+k₂

Question 9

Linearization of a nonlinear restoring force about equilibrium primarily uses:

A. Its local first derivative or tangent stiffness

B. Its maximum possible value

C. Only its constant term

D. A random stiffness estimate

Correct Answer: A. Its local first derivative or tangent stiffness

Question 10

In diagnosing a motor-pump vibration peak, which combination provides the strongest basis for distinguishing resonance from imbalance?

A. Amplitude alone

B. Machine colour and operating age

C. Frequency, phase, speed dependence, and a physical model

D. Static weight only

Correct Answer: C. Frequency, phase, speed dependence, and a physical model