7. Continuous Systems and Wave Propagation

Introduction and context

The lumped-parameter models introduced for discrete mass–spring–damper systems are useful when the dimensions of a component are small compared with the wavelength of vibration. In many engineering structures this assumption is inadequate. A string, rod, shaft, beam, plate, or elastic solid possesses mass and stiffness at every point. Its deformation therefore varies continuously with position, and the motion is described by partial differential equations rather than by a finite set of ordinary differential equations.

Continuous-system analysis is central to the prediction of natural frequencies, mode shapes, dynamic stresses, noise radiation, vibration transmission, and resonance. It also provides the mechanical foundation for understanding waves: disturbances propagate through an elastic medium with finite speed, reflect at boundaries, interfere with other disturbances, and may be attenuated by damping or material dissipation.

The mathematical treatment follows a recurring pattern. A differential element is isolated, constitutive and equilibrium relations are applied, boundary and initial conditions are specified, and the resulting equation is solved exactly or approximately. Separation of variables, orthogonality, modal expansion, energy methods, and numerical discretization are the principal tools.

Learning objectives

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

1. Distinguish distributed-parameter systems from lumped-parameter models.

2. Derive and interpret the governing equations for strings, rods, torsional shafts, beams, plates, and elastic media.

3. Formulate physically consistent boundary and initial conditions.

4. Determine natural frequencies and mode shapes for standard systems.

5. Explain orthogonality, modal superposition, and wave propagation.

6. Use Rayleigh and Rayleigh–Ritz methods when exact solutions are unavailable.

7. Assess the influence of boundary conditions, discontinuities, damping, and dispersion.

8. Select suitable analytical, experimental, or computational methods for engineering applications.

From finite degrees of freedom to fields

Let a displacement field be denoted by u(x,t), where x is a spatial coordinate and t is time. Unlike a single-degree-of-freedom displacement, u depends on position. For a uniform system, the governing equation commonly has the form

∂²u/∂t² = c² ∂²u/∂x²,

where c is a characteristic wave speed. This equation does not state that every point moves identically; rather, it states how curvature in the spatial displacement field produces acceleration.

A complete continuous-system model consists of:

• a field variable, such as transverse displacement, axial displacement, or rotation;
• material and geometric parameters distributed along the system;
• a governing partial differential equation;
• boundary conditions at supports, ends, or interfaces;
• initial displacement and velocity fields when a transient response is required.

The distinction between a boundary condition and an initial condition is important. A boundary condition applies at a spatial boundary for all time, whereas an initial condition specifies the state throughout the system at a selected time, usually t = 0.

Longitudinal vibration of a uniform rod

Consider a slender rod of area A, density ρ, and Young’s modulus E. Let u(x,t) be its axial displacement. For a differential element of length dx, the axial force is approximately EA ∂u/∂x. Force equilibrium gives

ρA dx ∂²u/∂t² = [EA ∂u/∂x]_{x+dx} − [EA ∂u/∂x]_x.

For a uniform rod,

∂²u/∂t² = c_L² ∂²u/∂x², c_L = √(E/ρ).

The area cancels because both inertia and axial stiffness are proportional to A. The wave speed is therefore controlled by the elastic modulus and density, not directly by the cross-sectional area.

For a rod of length L fixed at both ends, u(0,t)=u(L,t)=0. A separated solution u=X(x)T(t) leads to

X'' + k²X = 0,

with X(0)=X(L)=0. Consequently,

X_n(x)=sin(nπx/L),

and

ω_n = nπc_L/L, n=1,2,3,... .

The mode shapes are sinusoidal and the frequencies are integer multiples of the fundamental frequency. For a free end, the axial force must vanish, so EA u_x=0, or u_x=0. Thus a free–free rod has cosine-type modes, including a zero-frequency rigid-body mode. A fixed–free rod has one fixed-displacement and one zero-force condition, producing quarter-wave-type modes with frequencies proportional to (2n−1)πc_L/(2L).

Transverse vibration of a stretched string

For a taut string with tension T and mass per unit length μ, let y(x,t) denote small transverse displacement. Assuming small slopes and constant tension, transverse force equilibrium gives

μ ∂²y/∂t² = T ∂²y/∂x²,

or

∂²y/∂t² = c_s² ∂²y/∂x², c_s=√(T/μ).

For a string fixed at both ends, the natural modes are

Y_n(x)=sin(nπx/L),

with natural frequencies

ω_n = nπ/L √(T/μ).

The string equation is nondispersive: in the ideal model, all harmonic components travel with the same speed. A plucked string, however, generally contains several modes. Its observed motion is the sum of these modal contributions, and the sound spectrum depends on the initial shape, plucking position, damping, and boundary conditions.

The ideal string model becomes unreliable when bending stiffness, large transverse slopes, variable tension, or significant air and support damping must be considered. In such cases the model must be extended or replaced by a beam or nonlinear-string formulation.

Torsional vibration of shafts

For a slender circular shaft, let θ(x,t) be the angular displacement. The torsional torque is

GJ ∂θ/∂x,

where G is the shear modulus and J is the polar second moment of area. With mass polar moment per unit length ρJ, the equation for a uniform shaft is

ρJ ∂²θ/∂t² = GJ ∂²θ/∂x²,

and hence

∂²θ/∂t² = c_t² ∂²θ/∂x², c_t=√(G/ρ).

For nonuniform shafts, the more general equation is

ρJ(x) θ_{tt} = [GJ(x) θ_x]_x.

At a fixed end θ=0; at a free end the torque GJθ_x is zero. Discontinuities between shaft segments require continuity of angular displacement and torque, unless an attached disk or concentrated inertia introduces a jump condition. This model is used for drive trains, turbine shafts, marine propulsion systems, and torsional vibration absorbers.

Flexural vibration of beams

Euler–Bernoulli beam

For a slender beam undergoing small transverse deflection w(x,t), the Euler–Bernoulli assumption states that plane cross-sections remain plane and normal to the neutral axis. Shear deformation and rotary inertia are neglected. The bending moment is related to curvature by

M = EI w_{xx},

with sign depending on convention. Force and moment equilibrium yield

ρA w_{tt} + EI w_{xxxx}=0

for a uniform, undamped beam.

The fourth spatial derivative distinguishes beam vibration from string and rod vibration. The corresponding flexural wave speed is frequency-dependent, which means that beam waves are dispersive. For harmonic motion, w=W(x)e^{iωt}, the spatial equation is

EI W'''' − ρAω²W=0.

For a simply supported beam, the conditions are w=0 and M=EIw''=0 at both ends. The modes are

W_n(x)=sin(nπx/L),

and

ω_n=(nπ/L)²√(EI/ρA).

Thus beam frequencies vary as n², unlike the n-dependence of an ideal string. For a cantilever, the conditions at the built-in end are w=0 and w'=0, while at the free end M=0 and the shear force EI w'''=0. The characteristic equation is

cosh(βL) cos(βL) = −1,

where β⁴=ρAω²/EI. The first roots are approximately 1.875, 4.694, and 7.855. These roots must be used with care: they apply to the ideal uniform Euler–Bernoulli cantilever and not automatically to beams with concentrated masses, tapered sections, or elastic supports.

Timoshenko beam theory

At higher frequencies, short beams, and thick beams, transverse shear deformation and rotary inertia become important. Timoshenko theory introduces both transverse displacement and cross-sectional rotation as independent variables. It generally predicts lower natural frequencies than Euler–Bernoulli theory in the frequency range where shear flexibility matters. The choice of theory is therefore a modeling decision, not merely a mathematical preference.

Plates and elastic continua

A thin isotropic plate is modeled, under classical Kirchhoff plate assumptions, by

D∇⁴w + ρh w_{tt}=p(x,y,t),

where h is thickness, p is transverse loading, and

D = Eh³/[12(1−ν²)]

is the flexural rigidity. The biharmonic operator ∇⁴ contains fourth-order derivatives in two spatial coordinates. Boundary conditions may specify displacement, slope, bending moment, twisting moment, or effective shear force. Rectangular plates with simply supported edges often admit double sine-series solutions; plates with mixed or free boundaries usually require numerical or approximate methods.

For a three-dimensional isotropic elastic solid, displacement has three components and is governed by the Navier equations. The medium supports longitudinal and shear waves with approximate speeds

c_L = √[(K+4G/3)/ρ], c_S=√(G/ρ),

where K is bulk modulus and G is shear modulus. At interfaces, waves may be reflected, transmitted, mode-converted, or attenuated. These effects underlie ultrasonic inspection, earthquake engineering, elastography, and impact analysis.

Boundary conditions and eigenvalue problems

Natural frequencies are eigenvalues of a boundary-value problem. The spatial function associated with each eigenvalue is an eigenfunction or mode shape. Boundary conditions determine the admissible eigenfunctions and therefore strongly influence the spectrum.

Typical mechanical conditions include:

• fixed or essential condition: displacement or rotation is prescribed;
• free or natural condition: force, torque, moment, or shear is prescribed;
• elastic support: force is proportional to displacement, such as F=k_su;
• interface condition: displacement and force continuity are enforced, subject to concentrated elements.

For conservative self-adjoint systems, distinct modes satisfy an orthogonality relation. For a rod, for example,

∫₀ᴸ ρA X_m X_n dx = 0, m≠n.

The analogous relation for beams uses the mass distribution. Orthogonality permits an arbitrary response to be expanded as

u(x,t)=Σ q_n(t)X_n(x).

Substitution and projection onto each mode converts the continuous problem into uncoupled modal equations for ideal conservative systems:

q̈_n + ω_n²q_n = Q_n(t)/M_n.

Here M_n is the generalized modal mass and Q_n is the generalized force. Damping, nonproportional supports, and nonlinearities may couple the modal coordinates, although modal analysis often remains a powerful approximation.

Wave propagation, reflection, and dispersion

The one-dimensional wave equation admits traveling-wave solutions

u(x,t)=f(x−ct)+g(x+ct).

The first term travels in the positive x-direction and the second in the negative direction. A wave transports energy and momentum, while the medium particles generally oscillate about equilibrium rather than traveling with the wave over long distances.

At a boundary, the reflected wave depends on mechanical impedance. For a string, impedance is approximately Z=√(Tμ). A change in impedance causes partial reflection and transmission. A fixed boundary produces a phase reversal of transverse displacement, whereas a free boundary produces reflection without that reversal in the idealized model. Standing waves arise when incident and reflected waves interfere, producing nodes and antinodes.

A system is nondispersive when phase velocity is independent of frequency. The ideal string and uniform rod models are approximately nondispersive. Beam waves are dispersive because their phase velocity depends on frequency. In a dispersive medium, a wave packet can change shape as its spectral components travel at different group velocities. This distinction is important in transient vibration, impact, and signal interpretation.

Damping may be represented by viscous terms, complex material properties, or frequency-dependent loss factors. In a damped continuous system, wave amplitude decreases with distance and the natural frequencies become complex. Real structures also exhibit radiation losses, friction at joints, viscoelasticity, and aerodynamic damping, none of which should be hidden within an unjustified single damping ratio.

Approximate methods for continuous systems

Exact solutions exist only for idealized geometries and boundary conditions. Approximate methods are therefore essential in engineering practice.

Rayleigh method

For an undamped system with an assumed mode shape φ(x), the fundamental frequency can be estimated from

ω² ≈ U_max/T_max,

where U_max is maximum strain energy and T_max is maximum kinetic-energy expression. For a beam,

ω² ≈ [∫₀ᴸ EI(φ'')²dx]/[∫₀ᴸ ρAφ²dx].

The assumed function must satisfy the essential boundary conditions. For the fundamental frequency, a suitable admissible function generally gives an upper-bound estimate in the classical conservative setting. A poor function can still produce a misleading result, especially when it violates important boundary conditions or fails to represent curvature near supports.

Rayleigh–Ritz method

The displacement is represented as

w(x,t)=Σ a_j(t)φ_j(x),

where each trial function satisfies the essential conditions. Substitution into the energy expressions gives matrix equations

[M]{ä}+[K]{a}={F}.

The eigenvalues of K−ω²M give approximate natural frequencies and mode shapes. Increasing the number of admissible functions generally improves the approximation, but numerical conditioning and basis selection must be considered.

Finite element method

The finite element method divides the structure into elements and approximates the field locally. Assembly produces

[M]{q̈}+[C]{q̇}+[K]{q}={f}.

Mesh refinement, convergence studies, realistic joints, material anisotropy, prestress, and attached equipment can be incorporated. However, a finite element result is not automatically correct: incorrect constraints, excessive element distortion, omitted masses, inappropriate damping, or insufficient mesh density can produce highly precise but physically wrong answers.

Applied scenario: torsional resonance in a flexible drive train

Background

Consider an illustrative two-section steel drive shaft connecting a motor to a compressor. The motor delivers nearly constant speed, but the compressor torque fluctuates twice per revolution because of its compression cycle. The shaft has different diameters in the two sections, and a coupling has a finite torsional stiffness.

Problem

During a speed sweep, vibration increases sharply near one operating speed. A rigid-body model predicts no resonance in the operating range, while field measurements show alternating angular motion at the coupling.

Analysis

The shaft cannot be treated as a single rigid member. The torsional field θ(x,t) must be modeled with piecewise values of GJ and ρJ. At each diameter change, angular displacement and torque continuity are imposed. The motor and compressor inertias provide end conditions, while the coupling contributes a localized stiffness. The fluctuating torque is projected onto the calculated modes. The mode whose frequency is close to twice the running speed has a large modal participation factor.

A finite-element torsional model is first checked against an energy estimate and a two-inertia reduced model. The checks reveal that the initial model omitted the coupling compliance and used an unrealistically rigid compressor connection.

Solution and outcome

The coupling stiffness is measured and included, the shaft is divided into elements near the diameter transition, and a speed-dependent Campbell diagram is constructed. The resonance is confirmed as a torsional mode rather than a bending mode. The design response is reduced by changing the coupling stiffness, shifting the excitation order away from the operating range, and adding a tuned torsional damper. A subsequent analysis includes uncertainty in stiffness and damping because manufacturing tolerances and joint friction affect the resonant peak.

Learning points

The case illustrates that distributed flexibility may be decisive even when a low-order lumped model appears plausible. Boundary and interface conditions must represent the actual machine. Modal frequency alone is insufficient; excitation order, modal participation, damping, and operating speed must also be considered. Experimental data are most valuable when used to validate the physical model rather than merely to fit unexplained parameters.

Engineering practice, limitations, and professional judgment

Continuous-system models are used in bridge and building design, turbine and compressor rotors, acoustic instruments, railway tracks, MEMS resonators, ultrasonic transducers, pipelines, spacecraft structures, and nondestructive evaluation. In each application, the analyst must decide whether small deformation, linear elasticity, uniform properties, negligible damping, and ideal boundary conditions are defensible.

Common errors include confusing a free boundary with zero displacement, using an Euler–Bernoulli beam for a deep high-frequency beam, neglecting rotary inertia, treating a discontinuous shaft as uniform, imposing too many finite-element constraints, and interpreting a mathematical rigid-body mode as a structural failure. Another frequent error is using a natural frequency without checking mode shape and excitation location. A force applied at a modal node may produce little response even when its frequency matches a natural frequency.

Professional analysis should include dimensional checks, limiting-case checks, mesh or basis convergence, sensitivity studies, and comparison with measurements where feasible. For safety-critical structures, assumptions, model revisions, uncertainty, and validation evidence should be documented. Digital twins and automated modal updating can assist monitoring, but they do not remove the need for physical interpretation. Machine-learning methods may identify patterns in vibration data, yet extrapolation outside the training domain and the absence of physically meaningful uncertainty remain important risks.

Current developments include reduced-order models for real-time structural monitoring, operational modal analysis under unknown excitation, wave-based damage detection, metamaterials with tailored dispersion, additive-manufactured lattice structures, and coupled multiphysics models involving thermal, acoustic, fluid, and structural fields. These developments increase capability while making model verification and data governance more important.

Best-practice checklist

A defensible continuous-system analysis should:

• define the field variables, coordinate system, and sign conventions;
• derive or verify the governing equation from equilibrium and constitutive laws;
• state all boundary, interface, and initial conditions explicitly;
• identify the assumptions behind the selected theory;
• check dimensions and limiting cases;
• compare exact, approximate, or numerical results where possible;
• inspect mode shapes as well as frequencies;
• include realistic mass, stiffness, damping, and joints;
• assess uncertainty and sensitivity to parameters;
• validate predictions using experiment, monitoring data, or an independent model.

Chapter Summary

Continuous systems distribute inertia and stiffness throughout space, so their behavior is governed by partial differential equations and boundary-value problems. Strings, rods, and torsional shafts lead to second-order wave equations, while Euler–Bernoulli beams and thin plates lead to fourth-order spatial equations. Boundary conditions determine admissible modes and natural frequencies, and orthogonality enables modal expansion and decoupling. Ideal strings and rods are approximately nondispersive, whereas flexural waves in beams are dispersive. Reflection, transmission, impedance, and standing waves explain the propagation of mechanical disturbances. Rayleigh, Rayleigh–Ritz, and finite-element methods extend analysis to systems for which exact solutions are impractical. Reliable engineering decisions require correct physical assumptions, realistic interfaces and supports, convergence checks, uncertainty assessment, and validation.

Key Terms

Distributed-parameter system: A system whose mass, stiffness, or other properties are distributed continuously in space.

Field variable: A quantity, such as displacement or rotation, that depends on spatial position and time.

Wave speed: The speed at which a disturbance or phase propagates through a medium.

Natural frequency: A frequency of free vibration associated with an admissible mode shape.

Mode shape: The spatial pattern of motion associated with a natural frequency.

Eigenvalue: A characteristic value, commonly ω², obtained from a boundary-value problem.

Orthogonality: A weighted integral property that makes distinct conservative modes independent in modal projection.

Generalized mass: The modal inertia obtained by weighting physical mass with a mode shape.

Modal participation: The extent to which a load or initial condition excites a particular mode.

Impedance: The dynamic opposition governing wave amplitude and reflection at an interface.

Nondispersive wave: A wave whose phase velocity does not vary with frequency.

Dispersive wave: A wave whose phase or group velocity varies with frequency.

Euler–Bernoulli beam: A slender-beam theory neglecting shear deformation and rotary inertia.

Timoshenko beam: A beam theory including transverse shear deformation and rotary inertia.

Rayleigh method: An energy-based estimate of a natural frequency using an assumed mode shape.

Rayleigh–Ritz method: An approximate variational method using several admissible trial functions.

Biharmonic operator: The fourth-order spatial operator ∇⁴ appearing in classical plate equations.

Rigid-body mode: A zero-frequency motion involving no strain energy.

Modal superposition: Representation of a response as a sum of weighted natural modes.

Dispersion relation: A relation connecting wave number and frequency in a propagating medium.

Review Questions

Short Answer Questions

1. Why is a partial differential equation required for a continuous system?

2. State the wave speed for a uniform taut string and define its parameters.

3. What physical conditions are represented by zero bending moment and zero shear force at a free beam end?

4. Explain the difference between a boundary condition and an initial condition.

5. Why does a free–free rod possess a rigid-body mode?

Long Answer Questions

1. Derive the longitudinal vibration equation for a uniform elastic rod and obtain the fixed–fixed natural frequencies.

2. Explain the assumptions of Euler–Bernoulli beam theory and discuss when Timoshenko theory is preferable.

3. Describe modal orthogonality and show how it permits continuous-system equations to be reduced to modal equations.

4. Explain reflection, transmission, standing waves, and impedance using a string or rod as an example.

5. Compare the Rayleigh, Rayleigh–Ritz, and finite-element methods for continuous-system vibration analysis.

Analytical / Application Questions

1. A simply supported beam has its flexural rigidity doubled while its mass per unit length remains unchanged. Determine qualitatively how each natural frequency changes and justify the result.

2. A shaft contains an abrupt diameter reduction. Formulate the interface conditions and explain how the discontinuity affects torsional waves and natural modes.

3. A vibration measurement shows a resonance peak at a predicted natural frequency, but the response is very small when the actuator is placed near a modal node. Explain the observation using modal participation and propose a better actuator location.

22. Multiple Choice Questions with Answers

Question 1

For a uniform taut string, which parameter combination determines the ideal transverse wave speed?

A. √(EA/ρA)

B. √(T/μ)

C. √(EI/ρA)

D. √(GJ/ρJ)

Correct Answer: B. √(T/μ)

Question 2

Which boundary condition applies at the free end of an ideal axial rod?

A. u=0

B. u_x=0

C. u_t=0

D. u_{xx}=0

Correct Answer: B. u_x=0

Question 3

For a simply supported Euler–Bernoulli beam, the natural frequencies vary with mode number approximately as

A. n

B. √n

C. n²

D. 1/n²

Correct Answer: C. n²

Question 4

Which assumption is characteristic of Euler–Bernoulli beam theory?

A. Shear deformation dominates

B. Cross-sections remain plane and normal to the neutral axis

C. Rotary inertia is always dominant

D. The beam has no bending stiffness

Correct Answer: B. Cross-sections remain plane and normal to the neutral axis

Question 5

A free–free uniform rod has a zero-frequency mode because

A. its density is zero

B. its stiffness is infinite

C. it can translate without elastic strain

D. its wave speed is zero

Correct Answer: C. it can translate without elastic strain

Question 6

In a conservative continuous system, modal orthogonality is especially useful because it

A. eliminates all damping

B. converts modal projection into independent equations

C. makes every mode have the same frequency

D. removes the need for boundary conditions

Correct Answer: B. converts modal projection into independent equations

Question 7

Which method estimates a fundamental frequency from the ratio of maximum strain energy to maximum kinetic-energy expression?

A. Rayleigh method

B. Laplace transform only

C. Fourier heat method

D. Rigid-body reduction

Correct Answer: A. Rayleigh method

Question 8

Beam flexural waves are generally dispersive because

A. their phase velocity depends on frequency

B. their density is always variable

C. they cannot reflect at boundaries

D. their displacement is independent of position

Correct Answer: A. their phase velocity depends on frequency

Question 9

At a shaft diameter discontinuity without a concentrated inertia, which pair of quantities is normally continuous?

A. Angular displacement and torque

B. Angular acceleration and density

C. Bending moment and transverse force

D. Temperature and heat flux

Correct Answer: A. Angular displacement and torque

Question 10

If an actuator is located at a mode-shape node, the corresponding mode will usually have

A. maximum modal participation

B. little direct excitation

C. zero natural frequency

D. infinite damping

Correct Answer: B. little direct excitation