Vibration of Continuous Systems

4.1 Introduction to Continuous Systems

In the analysis of noise, vibration, and harshness (NVH) of mechanical structures, we encounter a fundamental modeling choice: whether to represent a system using a lumped-parameter (discrete) model or a distributed-parameter (continuous) model.

4.1.1 Discrete vs. Continuous Representations

In previous chapters, we analyzed systems with a finite number of degrees of freedom (DOFs), such as single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. In those systems, mass was assumed to be concentrated at discrete points (rigid bodies), and elasticity was represented by massless springs. Such lumped-parameter models are highly effective at low frequencies where the wavelength of the vibration is much larger than the physical dimensions of the components.

However, real physical structures—such as vehicle body panels, engine crankshafts, transmission shafts, and turbine blades—possess mass and elasticity that are continuously distributed throughout their volume. When a system is subjected to high-frequency excitation, the vibration wavelengths become comparable to or smaller than the structural dimensions. In these cases, lumped-parameter models fail to capture the local deformations and high-frequency resonances, making a continuous system approach mandatory.

A continuous system possesses an infinite number of degrees of freedom because its displacement field must be defined at every point in space. Instead of a finite set of generalized coordinates $q_i(t)$ ($i = 1, 2, \dots, N$), the state of a continuous system is described by one or more continuous field variables, such as the transverse displacement $w(x, y, z, t)$, which depend on both space and time.

4.1.2 The Mathematical Limit: From MDOF to Continuous

To understand the transition from a discrete model to a continuous model, let us consider a classical mechanical system: a series of $N$ identical particles, each of mass $m$, spaced at equal intervals $d$ along a light elastic string of total length $L = (N+1)d$, under a constant tension $T$. Let $w_j(t)$ represent the transverse displacement of the $j$-th particle.

The equation of motion for the $j$-th particle (for small displacements) is given by:

Now, let us take the limit as the number of particles $N \to \infty$ while keeping the total length $L$ and total mass $M$ constant. This means the spacing $d \to 0$ and the mass of each particle $m \to 0$. We can define the mass per unit length of the string as:

Substituting $m = \rho_L d$ into the equation of motion yields:

As $d \to 0$, the discrete coordinate $x_j = j \cdot d$ becomes a continuous coordinate $x$, and the displacement $w_j(t)$ becomes a continuous function $w(x, t)$. The term in the brackets on the right-hand side represents the second-order central difference approximation of the second spatial derivative of $w(x, t)$. Specifically, in the limit:

Thus, the discrete system of ordinary differential equations (ODEs) converges to a single second-order partial differential equation (PDE):

This is the classical 1D wave equation, which governs the transverse vibration of a continuous string. This derivation demonstrates that a continuous system is the limiting case of a discrete system as the number of degrees of freedom approaches infinity.

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4.2 One-Dimensional Wave Propagation

Vibration in continuous systems can be physically interpreted as the propagation of waves through a elastic medium. When a local disturbance is introduced in an elastic body, it does not instantly affect the entire body. Instead, the disturbance travels through the medium at a finite speed, known as the wave speed.

4.2.1 Longitudinal Waves in Thin Rods

Consider a thin, uniform elastic rod of cross-sectional area $A$, Young's modulus $E$, and mass density $\rho$, oriented along the $x$-axis. We assume that the cross-sections of the rod remain plane and move only in the axial direction, meaning the axial displacement $u(x, t)$ is uniform across any cross-section.

1. Strain-Displacement and Constitutive Relations

Consider an infinitesimal element of the rod of length $dx$ located between $x$ and $x + dx$. Under axial deformation, the section at $x$ undergoes a displacement $u(x, t)$, and the section at $x + dx$ undergoes a displacement:

The elongation of the element is the difference between the displacements of its two ends:

The normal strain $\epsilon(x, t)$ is defined as the change in length per unit length:

Assuming linear elastic behavior, Hooke's law relates the axial stress $\sigma(x, t)$ to the strain:

The internal axial force $P(x, t)$ acting on the cross-section is the product of stress and area:

2. Dynamic Equilibrium

Let us analyze the forces acting on the infinitesimal element of mass $dm = \rho A dx$. The internal axial force acting on the left face (at $x$) is $P(x, t)$ directed to the left. The force acting on the right face (at $x + dx$) is:

directed to the right.

Applying Newton's second law in the axial direction yields:

Substituting the force-displacement relationship $P = E A \frac{\partial u}{\partial x}$ into the equilibrium equation gives:

For a uniform rod where $E$, $A$, and $\rho$ are independent of $x$, this simplifies to:

where:

is the speed of longitudinal wave propagation in the rod. For steel, with $E \approx 2.1 \times 10^{11}$ N/m^2 and $\rho \approx 7850$ kg/m^3, the longitudinal wave speed is approximately $c \approx 5170$ m/s. For titanium Ti-6Al-4V ($E = 1.14 \times 10^{11}$ N/m^2, $\rho = 4430$ kg/m^3), it is $c \approx 5070$ m/s.

3. D'Alembert's Solution

The governing 1D wave equation is a hyperbolic PDE. The general solution was first formulated by Jean le Rond d'Alembert in 1747:

where $f$ and $g$ are arbitrary twice-differentiable functions determined by the initial conditions of the system.

$x$-direction with speed $c$ (forward-traveling wave).

$x$-direction with speed $c$ (backward-traveling wave).

4. Wave Reflection at Boundaries

When a longitudinal wave encounters a boundary, it reflects back into the rod. The character of the reflection depends on the boundary conditions:

axial force must be zero:

This is a stress-free boundary condition. When a forward wave $f(x - ct)$ hits a free end, it reflects as a backward wave $g(x + ct) = f(2L - (x + ct))$ with the same sign of strain but opposite velocity. The stress waves cancel at the boundary, resulting in a doubling of the displacement amplitude at the tip.

constrained to zero:

When a wave hits a clamped end, it reflects with a sign change in displacement. The displacements cancel at the boundary, leading to a doubling of the stress amplitude.

4.2.2 Torsional Waves in Circular Shafts

Torsional vibration is a critical concern in rotating machinery, such as automotive driveshafts and engine crankshafts. Consider a uniform circular shaft of shear modulus $G$, polar moment of inertia $I_p$, and mass density $\rho$. Let $\theta(x, t)$ represent the angle of twist of the cross-section at position $x$ and time $t$.

1. Torque-Twist Relationship

For a circular shaft, the internal torque $T_s(x, t)$ acting on a cross-section is proportional to the spatial rate of twist:

where $I_p$ is the polar second moment of area (for a solid shaft of diameter $D$, $I_p = \pi D^4 / 32$).

2. Dynamic Equilibrium

Consider an infinitesimal element of the shaft of length $dx$. The polar mass moment of inertia of this element is $dJ = \rho I_p dx$. The torque acting on the left face is $T_s(x, t)$, and the torque on the right face is:

Applying Newton's second law for rotational systems yields:

Substituting the torque-twist relation into the equilibrium equation gives:

For a uniform shaft (constant $G$, $I_p$, $\rho$), this simplifies to:

where:

is the torsional (shear) wave propagation speed. Since the shear modulus $G$ is related to Young's modulus $E$ and Poisson's ratio $\nu$ by $G = \frac{E}{2(1+\nu)}$, the torsional wave speed is always lower than the longitudinal wave speed. For typical metals ($\nu \approx 0.3$), $c_t \approx 0.62 c$.

4.2.3 Transverse Waves in Strings

The study of transverse wave propagation in a stretched string provides the mathematical foundation for understanding transverse vibrations. Consider a thin, flexible string of mass density per unit length $\rho_L$, stretched under a high axial tension force $T$. Let $w(x, t)$ be the transverse displacement.

1. Assumptions

(bending stiffness is zero).

\ll 1$.

2. Dynamic Equilibrium

Consider an infinitesimal element of the string of length $dx$. The transverse forces acting on the element are the transverse components of the tension force at the two ends.

Applying Newton's second law in the vertical direction:

where:

is the transverse wave speed on the string. This equation is identical in form to the longitudinal and torsional wave equations. This class of wave propagation is non-dispersive, meaning all harmonic components travel at the same velocity $c$, preserving the shape of a traveling wave packet.

4.3 Euler-Bernoulli Beam Theory (Thin Beams)

For structural components whose lateral dimensions are small compared to their length (typically a length-to-depth ratio $L/h > 10$), the bending vibration is modeled using the classical Euler-Bernoulli beam theory. This theory is built upon kinematic assumptions that simplify the 3D elasticity equations to a single spatial dimension.

4.3.1 Kinematic Hypotheses (Kirchhoff-Love Assumptions)

The Euler-Bernoulli beam theory rests on the following fundamental assumptions regarding the behavior of the beam under deformation:

perpendicular to the longitudinal axis (neutral axis) remain plane after deformation.

neutral axis. This implies that shear deformation is neglected ($\gamma_{xz} = 0$).

contraction during pure bending.

their own plane. The transverse normal strains in the $y$ and $z$ directions are neglected ($\epsilon_{yy} = \epsilon_{zz} = 0$).

4.3.2 Displacement Field

Let the $x$-axis lie along the neutral axis of the undeformed beam, and let the $z$-axis be the transverse axis directed downwards or upwards (we choose $z$ directed upwards for our derivation). Let $w(x, t)$ be the transverse deflection of the neutral axis in the $z$-direction, and let $u_0(x, t)$ be the axial displacement of the neutral axis. For pure bending, $u_0 = 0$.

Under the Euler-Bernoulli assumptions, the displacement field $(u, v, w)$ at any point $(x, y, z)$ in the beam is completely determined by the deflection $w(x, t)$:

cross-section. Since the cross-section remains plane and perpendicular to the neutral axis, the rotation angle of the cross-section is equal to the slope of the deflection curve, $\theta \approx \frac{\partial w}{\partial x}$. The axial displacement at a distance $z$ from the neutral axis is:

deflection of the neutral axis:

4.3.3 Bending Strain and Stress Relations

The axial strain $\epsilon_{xx}(x, y, z, t)$ is derived from the strain-displacement relations of linear elasticity:

Here, $\frac{\partial^2 w}{\partial x^2}$ is the curvature of the beam (often denoted by $\kappa$). The linear variation of strain with respect to $z$ indicates that strain is zero at the neutral axis ($z = 0$), maximum at the outer fibers, and varies linearly through the depth of the beam.

Assuming a uniaxial stress state ($\sigma_{yy} = \sigma_{zz} = \sigma_{xy} = \sigma_{yz} = \sigma_{xz} = 0$), Hooke's law gives the axial stress $\sigma_{xx}$:

4.3.4 Bending Moment and Curvature

The internal bending moment $M(x, t)$ is the resultant moment of the axial stress distribution over the cross-sectional area $A$:

The integral $\int_A z^2 dA$ is defined as the second moment of area (moment of inertia) $I$ of the cross-section about the neutral axis:

Thus, we obtain the classical moment-curvature relation:

*Note on Sign Convention*: In this convention, a positive bending moment causes compression in fibers with $z > 0$ and tension in fibers with $z < 0$. If we define $M$ as positive for a "smiling" beam (compression at the top, tension at the bottom), and $z$ is positive upwards, then $M = -EI \frac{\partial^2 w}{\partial x^2}$. Let us maintain this formulation consistently, and define the internal shear force $V(x, t)$ using the equilibrium relation:

For a uniform beam, this becomes $V(x, t) = -EI \frac{\partial^3 w}{\partial x^3}$.

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4.4 Timoshenko Beam Theory (Thick Beams)

For short, stubby beams ($L/h < 10$) or when analyzing high-frequency vibrations where the wavelength is short, Euler-Bernoulli theory significantly overestimates the natural frequencies. Under these conditions, two physical effects neglected in the Euler-Bernoulli theory become significant:

neutral axis, meaning the shear strain $\gamma_{xz} \neq 0$.

addition to translational acceleration.

The Timoshenko beam theory, proposed by Stephen Timoshenko in the early 20th century, incorporates both of these effects.

4.4.1 Kinematic Assumptions of Timoshenko Theory

In Timoshenko beam theory, the cross-sections of the beam are still assumed to remain plane after deformation, but they are no longer required to remain perpendicular to the deformed neutral axis.

This introduces a new independent variable, $\psi(x, t)$, which represents the rotation of the cross-section due to bending only. The total slope of the neutral axis, $\frac{\partial w}{\partial x}$, is now a combination of the rotation of the cross-section due to bending ($\psi$) and the shear deformation (shear angle $\gamma$):

The displacement field is written as:

4.4.2 Constitutive Relations and Shear Correction Factor

The axial strain and stress are:

The bending moment is related to the rotation gradient:

The shear strain $\gamma_{xz}$ is:

The shear stress $\tau_{xz}$ is:

Because the shear stress is actually not uniform over the cross-section (e.g., it is parabolic for a rectangular section), we introduce a shear correction factor $k$ (or $\kappa$) to reconcile the simplified 1D shear force expression with the true 3D stress distribution. The internal shear force $V(x, t)$ is written as:

Typical values for the shear correction factor $k$ are:

4.4.3 Coupled Governing Equations

Applying Newton's second law for translational and rotational motions of an infinitesimal element yields the coupled governing equations of motion for a Timoshenko beam:

where $q(x, t)$ is the distributed transverse force per unit length, and $\rho I \frac{\partial^2 \psi}{\partial t^2}$ is the rotatory inertia term.

For a uniform beam, these equations can be combined into a single fourth-order PDE in terms of the transverse displacement $w(x, t)$:

For free vibration ($q = 0$), the governing equation reduces to:

Comparison of Theories

The third and fourth terms in the above equation represent the additions of the Timoshenko theory over the Euler-Bernoulli theory:

effect of rotatory inertia.

(grouped inside the third term) represents the effect of shear deformation.

w}{\partial t^4}$ represents the coupled effect of both rotatory inertia and shear deformation.

As the beam becomes very thin ($I/A \to 0$), these higher-order terms vanish, and the equation simplifies back to the Euler-Bernoulli equation:

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4.5 Boundary Conditions for Beams

Because the governing equation for bending vibration is a fourth-order PDE in space, it requires exactly four boundary conditions (two at each end) to uniquely determine the spatial solution $W(x)$. These boundary conditions represent the physical constraints on the displacements and internal force resultants at the boundaries.

4.5.1 Standard Boundary Conditions

For a beam of length $L$ spanning from $x = 0$ to $x = L$, the standard boundary conditions are:

4.5.2 Elastic Supports and Boundaries

In many engineering systems, the boundaries are not perfectly rigid or free. Instead, they are connected to springs, dampers, or other structural elements.

a spring constant $k_T$.

constant $k_R$.

4.5.3 Concentrated Inertial Elements

When a beam is attached to a heavy component, such as an engine, a flywheel, or a sensor mass at its boundary, the boundary conditions become frequency-dependent.

For harmonic motion $w(x, t) = W(x) e^{i\omega t}$, this becomes:

For harmonic motion, this simplifies to:

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4.6 Self-Adjoint Operators and Eigenvalue Problems

The governing spatial equation of a vibrating beam is an eigenvalue problem. In general, a continuous system's spatial behavior is defined by a differential operator $L$ acting on a field variable $W(x)$, subject to boundary conditions.

4.6.1 The Differential Operator for Beams

For a uniform Euler-Bernoulli beam in free vibration, the spatial equation is:

We can define the fourth-order differential operator $L$:

Then the eigenvalue problem is written as:

where the eigenvalue is $\lambda = \beta^4 = \frac{\rho A \omega^2}{EI}$.

4.6.2 Definition of Self-Adjointness

An operator $L$ is self-adjoint (or symmetric) over a domain $[0, L]$ with respect to a defined inner product if, for any two functions $u(x)$ and $v(x)$ that satisfy the boundary conditions of the system, we have:

Let us test the self-adjointness of the beam bending operator $L = \frac{d^4}{dx^4}$. We evaluate:

Using integration by parts:

Integrating by parts a second time:

By symmetry, if we swap $u$ and $v$:

Subtracting the two expressions yields:

The operator $L$ is self-adjoint if and only if the boundary term on the right-hand side vanishes:

Let us check standard boundary conditions:

at that end are:

terms are:

boundary terms are:

For any combination of clamped, pinned, or free boundary conditions, the boundary term vanishes. Thus, the beam operator $L = \frac{d^4}{dx^4}$ is self-adjoint.

4.6.3 Consequences of Self-Adjointness

The self-adjoint property of the differential operator has profound mathematical consequences that form the basis of modal analysis for continuous systems:

frequencies $\omega_n$) are real numbers. This matches the physical expectation that a conservative mechanical system vibrates at real, physical frequencies.

distinct eigenvalues are orthogonal. That is, if $\lambda_m \neq \lambda_n$:

$f(x)$ satisfying the boundary conditions of the beam can be represented as a unique linear combination of the infinite set of orthogonal eigenfunctions $W_n(x)$:

where the coefficients $a_n$ are determined using the orthogonality property:

This allows us to solve the forced response of continuous systems by decomposing the load and response into the system's normal modes (modal superposition), converting a PDE into an infinite set of decoupled SDOF ODEs.

4.15 Wave Dispersion, Phase, and Group Velocity

In the study of continuous systems, a fundamental distinction arises between wave propagation in rods (longitudinal and torsional waves) and wave propagation in beams (bending/transverse waves). This difference lies in the concept of wave dispersion.

4.15.1 Non-Dispersive Waves: Rods and Strings

For longitudinal waves in a thin rod, the wave speed is $c = \sqrt{E/\rho}$. For torsional waves in a shaft, it is $c_t = \sqrt{G/\rho}$. For transverse waves on a string, it is $c_s = \sqrt{T/\rho_L}$. In all three cases, the wave speed is a constant that depends solely on the material and geometric properties of the medium, not on the frequency of the wave.

Let us consider a harmonic wave of the form:

where $k$ is the wave number (defined as $k = 2\pi/\lambda$, where $\lambda$ is the wavelength) and $\omega$ is the circular frequency. Substituting this wave into the longitudinal wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ yields:

This relation $\omega(k) = c k$ is the dispersion relation for longitudinal waves. The phase velocity $c_p$, which is the speed at which a single wave crest travels, is defined as:

The group velocity $c_g$, which is the speed at which a wave packet (or energy) propagates, is defined as the derivative of frequency with respect to wave number:

Since $c_p = c_g = c$, waves of all frequencies travel at the identical velocity. A complex waveform containing multiple frequency components (such as an impulse or a square wave packet) will propagate through the rod without changing its shape. Media where the phase velocity is independent of frequency are called non-dispersive.

4.15.2 Dispersive Waves: Euler-Bernoulli Beams

Now let us analyze bending waves in a uniform Euler-Bernoulli beam. The governing equation for free vibration is:

Let us assume a harmonic traveling wave solution:

Substituting this into the beam equation:

Assuming $W_0 \neq 0$, we obtain the dispersion relation for Euler-Bernoulli bending waves:

This is a non-linear relationship between frequency $\omega$ and wave number $k$. Let us calculate the phase velocity $c_p$ and group velocity $c_g$:

These equations reveal two critical physical insights:

(or inversely proportional to the wavelength $\lambda$). High-frequency bending waves travel faster than low-frequency bending waves.

that individual wave crests appear at the rear of a wave packet, travel forward through the packet at twice the packet's overall speed, and disappear at the front.

Because the propagation speed depends on the frequency, a complex bending wave packet will spread out as it travels along the beam—a physical phenomenon known as dispersion. High-frequency components of the disturbance will outrun the low-frequency components, causing the wave profile to flatten and stretch. This dispersion is a major consideration in vehicle structural NVH, as it affects how impact energy and structure-borne noise travel through structural frame members.

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4.16 Wave Reflection and Transmission at Junctions

When a structural wave traveling through a continuous medium encounters a change in material or geometry (a junction), it cannot proceed unimpeded. Instead, a portion of the wave is reflected back, and a portion is transmitted across the junction.

4.16.1 Acoustic and Mechanical Impedance

The reflection and transmission characteristics are governed by the **mechanical impedance** of the medium. For longitudinal waves in a rod, the force is $P = EA \frac{\partial u}{\partial x}$, and the particle velocity is $v_p = \frac{\partial u}{\partial t}$. For a harmonic wave traveling in the positive $x$-direction, $u(x, t) = U_0 e^{i(kx

t)}$

The mechanical wave impedance $Z$ is defined as the ratio of the force to the particle velocity (with sign convention adjusted for direction):

The product $Z = \rho A c$ is the characteristic mechanical impedance of the rod. It represents the structure's dynamic resistance to wave motion.

4.16.2 Reflection and Transmission Coefficients

Consider two rods joined at a junction at $x = 0$. Rod 1 has impedance $Z_1 = \rho_1 A_1 c_1$ (for $x < 0$), and Rod 2 has impedance $Z_2 = \rho_2 A_2 c_2$ (for $x > 0$). An incident wave $u_i(x, t) = I_0 e^{i(k_1 x - \omega t)}$ travels through Rod 1 toward the junction. When it hits $x = 0$:

At the junction ($x = 0$), two continuity conditions must be satisfied:

at the interface: $$u_i(0, t) + u_r(0, t) = u_t(0, t) \implies I_0 + R_0 = T_0$$ (Eq. 4.29)

the interface:

Since particle velocities are proportional to displacements: $$Z_1 (I_0 - R_0) = Z_2 T_0$$ (Eq. 4.30)

Solving Equations 4.29 and 4.30 for the reflection coefficient $C_R = R_0/I_0$ and transmission coefficient $C_T = T_0/I_0$:

Physical Implications

$C_R = 0$ and $C_T = 1$. The wave passes through the junction without any reflection.

boundary), $C_R \to -1$ and $C_T \to 0$. The wave is completely reflected with a phase change of $\pi$ in displacement.

\to 1$ and $C_T \to 2$. The wave is reflected with no change in displacement phase, and the displacement at the boundary doubles.

geometry or cross-sectional area (impedance mismatches) are strategically designed to reflect structure-borne noise energy back toward the source, preventing it from reaching the passenger cabin.

4.17 Sturm-Liouville Form of Structural Equations

The governing equations for many one-dimensional continuous vibratory systems (such as strings, bars in longitudinal vibration, shafts in torsional vibration, and beams in bending) can be categorized under the mathematical framework of Sturm-Liouville theory.

4.17.1 Standard Sturm-Liouville Form

A second-order differential equation is in Sturm-Liouville form if it is written as:

where:

Application to Longitudinal Vibration of a Bar

For a bar with variable cross-section $A(x)$ and Young's modulus $E(x)$, the spatial equation of motion is:

Comparing this with the Sturm-Liouville equation:

Thus, the longitudinal vibration of a rod is a classical second-order Sturm-Liouville problem.

4.17.2 Mathematical Properties of Sturm-Liouville Systems

Sturm-Liouville theory guarantees several important mathematical properties:

eigenvalues $\lambda_1 < \lambda_2 < \lambda_3 < \dots \to \infty$. There is a lowest eigenvalue $\lambda_1$, but no upper bound.

linearly independent eigenfunction corresponding to each eigenvalue.

eigenvalue $\lambda_n$ has exactly $n-1$ zeros (nodes) in the open interval $(0, L)$.

in the Hilbert space $L^2([0, L], r(x)dx)$. This justifies using the eigenfunctions as a coordinate basis for modal expansion.

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4.18 Approximate Methods for Continuous Systems

For most realistic engineering structures, the cross-sectional properties ($A(x), I(x)$) or boundary conditions are too complex to admit closed-form analytical solutions. In these cases, approximate analytical techniques are used to estimate the natural frequencies and mode shapes.

4.18.1 The Rayleigh Quotient

The Rayleigh quotient provides a method to estimate the fundamental natural frequency of a system using an assumed displacement shape. For a uniform beam, the potential energy $V_{\text{max}}$ and kinetic energy $T_{\text{max}}$ for a harmonic deflection $w(x, t) = W(x) \cos(\omega t)$ are:

Equating the maximum potential and kinetic energies (conservation of energy for a conservative system):

Solving for $\omega^2$, we define the Rayleigh quotient $R[W(x)]$: $$\omega^2 \le R[W(x)] = \frac{\int_0^L EI \left( W''(x) \right)^2 dx}{\int_0^L \rho A \left( W(x) \right)^2 dx}$$ (Eq. 4.31)

The Rayleigh Principle

The Rayleigh Principle states that if $W(x)$ is an approximation of the true mode shape, the value of the Rayleigh quotient $R[W(x)]$ will always be an upper bound to the square of the true fundamental frequency $\omega_1^2$:

The equality holds if and only if $W(x)$ is the exact first eigenfunction $W_1(x)$. To obtain a good estimate of the natural frequency, the trial function $W(x)$ must satisfy the essential boundary conditions (geometric boundary conditions, such as displacement and slope constraints) of the system. It does not need to satisfy the natural boundary conditions (force and moment constraints like shear force or bending moment, though satisfying them improves accuracy).

4.18.2 The Rayleigh-Ritz Method

The Rayleigh-Ritz method is an extension of the Rayleigh quotient that allows for the estimation of multiple natural frequencies and mode shapes.

Instead of a single trial function, the deflection $W(x)$ is approximated as a linear combination of $N$ independent admissible functions $\phi_i(x)$:

where $c_i$ are unknown coefficients to be determined, and $\phi_i(x)$ are pre-selected functions (such as polynomials or trigonometric functions) that satisfy at least the geometric boundary conditions of the structure.

Substituting this approximation into the Rayleigh quotient:

where:

coefficients)

To find the coefficients $c_i$ that minimize the Rayleigh quotient (seeking the stationary values of $R$), we take the partial derivatives of $R$ with respect to each coefficient $c_k$ and set them to zero:

This minimization leads to a standard symmetric matrix eigenvalue problem: $$\mathbf{K} \mathbf{c} = \lambda \mathbf{M} \mathbf{c}$$ (Eq. 4.32)

where:

$K_{ij}$.

Solving this generalized eigenvalue problem yields $N$ eigenvalues $\lambda_1 \le \lambda_2 \le \dots \le \lambda_N$ and corresponding eigenvectors $\mathbf{c}^{(n)}$. The eigenvalues provide upper-bound estimates to the first $N$ true natural frequencies of the continuous system. As the number of trials functions $N \to \infty$, the approximate eigenvalues converge to the exact eigenvalues of the system. This method is the direct precursor to the Finite Element Method (FEM).