8. Computational Vibration Analysis

Introduction and Context

Mechanical vibration problems rarely end with the derivation of a differential equation. In practical engineering systems, the structure may contain many degrees of freedom, non-uniform geometry, distributed mass, joints, damping, nonlinear attachments, or uncertain material properties. Exact analytical solutions are then available only for highly idealized cases. Computational vibration analysis provides the mathematical and numerical means of converting such systems into models that can be solved, interpreted, verified, and improved.

The central computational problem is usually the determination of natural frequencies, mode shapes, forced response, stability, and transient behavior. Depending on the system, the engineer may use an approximate energy method, a transfer-matrix formulation, a finite-element model, or a state-space representation. These methods are not competing algorithms in every situation. They represent different modeling philosophies and are often used together. For example, a Rayleigh estimate may provide a rapid preliminary check, a finite-element model may predict detailed structural modes, and a reduced state-space model may be used for control design.

A computational result is not automatically a reliable engineering result. Numerical eigensolutions may contain spurious modes, mesh refinement may expose modeling errors, and a highly detailed model may still be wrong if boundary conditions or joints are represented poorly. Consequently, computational vibration analysis includes not only solution procedures but also convergence studies, sensitivity analysis, verification, validation, and informed model reduction.

Learning Objectives

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

1. Formulate approximate vibration models using Rayleigh, Rayleigh–Ritz, and Galerkin methods.

2. Explain the principles and practical use of transfer-matrix and finite-element methods.

3. Formulate mechanical vibration systems in matrix and state-space form.

4. Compute and interpret numerical natural frequencies, mode shapes, and forced responses.

5. Apply model reduction while preserving the dynamics relevant to a specified frequency range.

6. Conduct convergence and sensitivity studies and distinguish verification from validation.

7. Identify numerical pathologies, modeling errors, and professional risks in computational vibration analysis.

From a Physical System to a Computational Model

For a linear, time-invariant mechanical system with several coordinates, the standard second-order model is

M q̈ + C q̇ + K q = f(t),

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, q is the vector of generalized coordinates, and f(t) is the applied force vector. For an undamped free-vibration problem,

M q̈ + K q = 0.

Assuming q = φ exp(iωt) gives the generalized eigenvalue problem

(K − ω²M)φ = 0.

Nontrivial solutions exist when det(K − ω²M) = 0. The eigenvalues ω² give the squared natural frequencies and the eigenvectors φ give the corresponding mode shapes. For physically meaningful finite-element models, M and K are normally symmetric, and M is positive definite after rigid-body constraints are treated appropriately.

The mode shapes are not unique in amplitude. A mode may be normalized to unit Euclidean length, unit modal mass, or a specified physical displacement. Modal mass normalization is especially useful because, for mass-normalized modes,

ΦᵀMΦ = I and ΦᵀKΦ = Ω²,

where Φ contains the eigenvectors and Ω² is the diagonal matrix of eigenvalues. In classical proportional damping, the transformation q = Φη decouples the equations into independent modal equations. This provides both physical insight and an efficient computational route.

Important modeling decisions occur before numerical solution. They include selecting coordinates, idealizing supports, deciding whether a component is rigid or flexible, defining concentrated and distributed masses, assigning damping, and specifying contact or joint behavior. A computational method can solve the equations supplied to it, but it cannot correct an invalid physical idealization by itself.

Approximate Energy Methods

The Rayleigh Method

The Rayleigh method estimates the fundamental natural frequency from an assumed mode shape. For a conservative system, the maximum kinetic and potential energies are equated. In continuous form,

ω² ≈ Vmax/Tmax = (∫ EI(x)[y″(x)]² dx)/(∫ ρA(x)y²(x) dx),

for a slender Euler–Bernoulli beam with bending stiffness EI, mass per unit length ρA, and assumed transverse displacement y(x). Boundary conditions that prescribe displacement must be satisfied by the assumed function.

For a discrete system with an assumed displacement vector a,

ω_R² = (aᵀKa)/(aᵀMa).

The Rayleigh quotient has an important property: when the assumed displacement satisfies the essential boundary conditions, the estimate generally provides an upper bound to the exact fundamental frequency for a sufficiently appropriate conservative model. The estimate improves when the assumed shape resembles the actual first mode. A poor assumed shape can still produce a useful order-of-magnitude check, but it should not be treated as a detailed prediction.

The method is valuable in preliminary design, hand verification of finite-element results, and rapid sensitivity studies. It is less suitable when several modes are required or when the system includes strong coupling, complicated constraints, or non-conservative effects.

Rayleigh–Ritz Formulation

The Rayleigh–Ritz method extends the energy idea by expressing the unknown displacement as a linear combination of admissible trial functions:

y(x,t) = Σ a_j(t) ψ_j(x).

The functions ψ_j must satisfy the essential, or geometric, boundary conditions. Substitution into the kinetic and potential energies gives

T = 1/2 ȧᵀM_R ȧ, V = 1/2 aᵀK_R a,

where

(M_R)ij = ∫ ρA ψ_i ψ_j dx,

(K_R)ij = ∫ EI ψ_i″ ψ_j″ dx.

The approximate eigenproblem is

(K_R − ω²M_R)c = 0.

Increasing the number of admissible functions usually improves the approximation and provides upper bounds for the lower eigenfrequencies in many self-adjoint conservative problems. However, convergence may be slow if the trial functions do not represent local curvature, discontinuities, or joint behavior. Choosing functions that satisfy both essential and natural boundary conditions can improve performance, although only the essential conditions are mandatory in the classical Ritz formulation.

Galerkin’s Method

Galerkin’s method begins with the governing differential equation and a residual. If the approximate solution does not satisfy the equation exactly, the residual R is required to be orthogonal to a set of weighting functions w_i:

∫ w_i R dx = 0.

When the weighting functions are chosen to be the same as the trial functions, the method is called Bubnov–Galerkin. For many structural vibration problems, this procedure produces the same algebraic equations as a finite-element or Ritz formulation. Galerkin’s method is particularly useful for beams, plates, shells, and systems governed by partial differential equations. It also extends naturally to problems in which direct energy expressions are inconvenient.

The distinction is conceptual. Ritz emphasizes stationarity of an energy functional, whereas Galerkin emphasizes weighted residuals. For self-adjoint structural vibration equations they often lead to equivalent matrices, but their mathematical applicability differs for non-self-adjoint, gyroscopic, fluid–structure, and dissipative systems.

Transfer-Matrix Analysis

The transfer-matrix method represents a structure as a sequence of elements and relates the state at one location to the state at another. For a beam, a state vector may contain displacement, rotation, bending moment, and shear force. For an element,

z_{j+1} = T_j(ω) z_j,

and for a complete system,

z_n = T_n T_{n-1} … T_1 z_0.

Boundary conditions at the two ends produce a frequency equation. Natural frequencies are found by searching for values of ω for which the boundary conditions are simultaneously satisfied.

Transfer matrices are efficient for chains of beams, shafts, rotor systems, layered members, and systems with repeated components. They are attractive when the number of interfaces is small and the analytical state-transition relations are available. Their principal weakness is numerical conditioning. At high frequencies or for long systems, products of transfer matrices may contain very large and very small terms, causing loss of precision. Riccati, impedance, dynamic-stiffness, or substructure formulations are often used to improve stability.

The method is also sensitive to the choice of state variables and to discontinuities. A concentrated mass, spring, or damper must be represented by an appropriate jump condition. Incorrect ordering of these interface relations can produce plausible but incorrect frequencies.

Finite-Element Formulation

The finite-element method divides a continuous structure into elements connected at nodes. Within each element, displacement is approximated using shape functions:

u(x,t) = N(x)q_e(t).

For a beam element, the element mass and stiffness matrices are obtained from

M_e = ∫ ρA NᵀN dx,

K_e = ∫ BᵀEI B dx,

where B relates nodal coordinates to strain or curvature. Element matrices are assembled into global matrices using connectivity relations. Boundary conditions are then imposed, and the resulting eigenproblem is solved.

The mass matrix may be consistent, obtained from the distributed mass integral, or lumped, in which mass is concentrated at selected nodes. Consistent mass generally represents low- and intermediate-frequency behavior more accurately, while lumped mass may simplify computation and can be useful in explicit time integration. Neither choice is universally superior; the appropriate choice depends on element type, frequency range, and solver.

Finite elements accommodate irregular geometry, material variation, concentrated components, joints, and multidimensional structures. They are therefore the dominant general-purpose method in industrial vibration analysis. Nevertheless, the analyst must recognize that discretization replaces a continuous system with a finite-dimensional approximation. A coarse mesh may miss local modes, stress concentrations, or flexible connections. A refined mesh does not compensate for incorrect constraints or unrealistic material data.

Numerical Eigensolution

The generalized eigenproblem may be solved by dense methods for small systems or iterative methods for large sparse systems. Subspace iteration, Lanczos, Arnoldi, and Krylov-subspace methods are common choices. The solver should be selected according to the number of modes required, matrix symmetry, sparsity, and whether eigenvalues near a specified frequency are sought.

Eigenpair quality should be checked through the residual

r_i = Kφ_i − ω_i²Mφ_i.

A small relative residual indicates that the algebraic equations have been solved accurately, but it does not prove that the physical model is correct. Closely spaced or repeated eigenvalues require care: individual mode shapes may rotate within the eigenspace, so comparison should sometimes be made using modal subspaces rather than component-by-component vectors. The modal assurance criterion is frequently used to compare mode shapes:

MAC(φ_a,φ_b) = |φ_aᵀφ_b|²/[(φ_aᵀφ_a)(φ_bᵀφ_b)].

A value near one indicates strong modal correlation, subject to consistent degrees of freedom and normalization.

State-Space Representation and Dynamic Response

A second-order system can be converted to first-order state space by defining

x = [q; q̇].

For nonsingular M,

ẋ = Ax + Bu,

y = C_s x + Du,

where

A = [0 I; −M⁻¹K −M⁻¹C], B = [0; M⁻¹F].

The eigenvalues of A contain the modal information of the damped system. For a stable system, eigenvalues have negative real parts. Their imaginary parts describe damped oscillation frequencies, while their real parts describe decay rates. State-space form is especially useful for systems with non-proportional damping, feedback control, actuator and sensor models, gyroscopic terms, and coupled mechanical, electrical, or hydraulic subsystems.

For harmonic excitation u(t) = Re(Ue^{iωt}), the steady response is found from the dynamic stiffness matrix

Z(ω) = K + iωC − ω²M,

through

Q(ω) = Z(ω)⁻¹F U.

Near resonance, direct inversion may be ill-conditioned. Modal superposition, complex modes, or carefully scaled numerical solvers are commonly used. The choice of output is important: displacement, velocity, acceleration, transmitted force, and stress may have different resonant peaks and different design significance.

Model Reduction

Large finite-element models may contain millions of degrees of freedom, while the response of interest may be governed by only a limited number of low-frequency modes. Model reduction seeks a lower-dimensional model that retains the relevant input–output behavior.

The simplest approach is modal truncation. If q ≈ Φ_rη, where Φ_r contains selected modes, the reduced matrices are

M_r = Φ_rᵀMΦ_r, C_r = Φ_rᵀCΦ_r, K_r = Φ_rᵀKΦ_r.

Low-frequency modes are often sufficient for low-frequency displacement response, but truncation can be inaccurate for high-frequency force transmission or local stress response. Static correction, residual flexibility, and attachment modes can improve the representation of truncated modes.

Other reduction techniques include Guyan reduction, which eliminates slave coordinates under a static approximation; component mode synthesis, such as Craig–Bampton reduction, which combines fixed-interface normal modes with constraint modes; and Krylov or moment-matching methods, which approximate the transfer function around selected expansion frequencies. The appropriate method depends on whether the objective is eigenvalue prediction, transient response, frequency response, or control design.

Reduction must always be judged against a specified frequency range and set of inputs and outputs. A reduced model that reproduces the first five natural frequencies may still fail to reproduce the force at a sensor location or the response near an unretained local mode.

Convergence, Sensitivity, Verification, and Validation

A convergence study investigates how a numerical result changes as the approximation is enriched. Enrichment may involve mesh refinement, higher-order elements, more Ritz functions, smaller time steps, tighter solver tolerances, or additional retained modes. A typical study records the first several frequencies and relevant response quantities as the model size increases. Convergence should be assessed for the engineering quantity of interest, not merely for displacement plots.

The convergence pattern can reveal modeling problems. Monotonic frequency convergence is common in some Ritz and finite-element settings, but it should not be assumed universally. Non-monotonic behavior may arise from mixed formulations, numerical integration, constraints, or changing mass representation. A converged answer can still be wrong if the model converges to an incorrect idealization.

Sensitivity analysis quantifies the influence of parameters. For an eigenvalue λ = ω² with normalized eigenvector φ, a small parameter change p gives approximately

∂λ/∂p = φᵀ[(∂K/∂p) − λ(∂M/∂p)]φ,

when the eigenvalue is simple and the normalization is handled consistently. This relation shows why stiffness increases generally raise frequencies while added mass generally lowers them. Finite differences, complex-step differentiation, automatic differentiation, and adjoint methods are also useful. Sensitivity analysis supports redesign, uncertainty quantification, tolerance allocation, and experimental planning.

Verification asks whether the equations have been implemented and solved correctly. It includes comparison with analytical benchmark problems, checking matrix symmetry, examining units, testing limiting cases, evaluating residuals, and reproducing known solutions. Validation asks whether the mathematical model represents the physical system adequately. It requires comparison with experiments, operational data, or trusted higher-fidelity evidence. Calibration of uncertain parameters may improve agreement, but calibration should not be confused with independent validation.

A professional verification and validation process should document geometry, material properties, boundary conditions, mesh, element formulation, solver settings, damping assumptions, measured data, uncertainty, and acceptance criteria. Agreement in one frequency does not establish validation of all modes or operating conditions.

Illustrative Applied Case: Flexible Drive Train

Background

Consider an illustrative industrial drive train consisting of a motor rotor, a flexible coupling, an intermediate shaft, and a gear-connected load. The engineering objective is to determine whether torsional resonance may occur near the operating speed range and to identify a practical redesign.

Situation and Initial Model

A preliminary two-degree-of-freedom model represents the motor and load inertias connected by an equivalent torsional stiffness. Its first torsional frequency is estimated using the Rayleigh quotient and then computed exactly for the two-degree-of-freedom system. This calculation provides a rapid screening result but neglects shaft distributed mass and the flexibility of the coupling hub.

A finite-element shaft model is then constructed using torsional beam elements. The motor and load are represented as concentrated inertias, and the coupling is represented by a stiffness and damping element. A transfer-matrix model is independently developed for the uniform shaft so that the finite-element result can be checked against a second formulation.

Analysis

The first finite-element mesh predicts a torsional mode close to the operating excitation range. Mesh refinement changes the first frequency only slightly but shifts a higher mode significantly. This indicates that the first mode is adequately resolved for the immediate operating decision, whereas the higher mode requires further refinement if it is to be used in a broader design study.

A sensitivity calculation shows that the first torsional frequency is more sensitive to coupling stiffness than to a small change in shaft density. The engineering team therefore avoids reducing mass as the primary remedy and instead evaluates a coupling with greater stiffness and controlled damping. A reduced state-space model containing the first two torsional modes is used to compute forced response under harmonic motor torque.

Outcome

The redesigned coupling moves the principal resonance away from continuous operation and reduces the predicted torque amplification. The final model is checked against a measured coast-down test. The measured first frequency lies within the agreed engineering tolerance, while a discrepancy in the second mode leads to investigation of bearing flexibility and joint friction. The model is not accepted merely because the first mode matches; the discrepancy is documented and the model’s domain of validity is stated.

Learning Points

The case illustrates the complementary roles of computational methods. Rayleigh analysis supports early screening, transfer matrices provide an independent shaft calculation, finite elements represent geometric and component detail, sensitivity analysis identifies influential parameters, and state-space reduction supports forced-response prediction. Most importantly, convergence and validation prevent a convenient numerical model from being mistaken for a verified physical representation.

Challenges, Risks, and Common Mistakes

Several errors occur repeatedly in computational vibration work. Applying a displacement boundary condition incorrectly can create artificial stiffness or remove a physical rigid-body mode. Mixing units may shift frequencies by large factors while leaving plots visually plausible. Treating every damping coefficient as modal damping can produce inconsistent energy dissipation. In finite elements, excessively stiff constraints, poor aspect ratios, reduced integration, or unsuitable element formulations may create spurious modes or locking.

Another common mistake is interpreting every eigenvalue as an important physical resonance. Some modes may be numerical artifacts, local modes outside the operating range, or rigid-body modes that are physically expected. Mode-shape scaling and sign are arbitrary, so visual comparison must use correlation measures and consistent coordinates.

Model reduction can hide important attachment or residual-flexibility effects. Similarly, selecting modes solely by lowest frequency may fail when the excitation acts locally or when the output is a force rather than a displacement. Frequency-response calculations should therefore include the relevant excitation and observation locations.

Computational efficiency must not replace engineering judgment. Sparse storage, parallel eigensolvers, adaptive meshing, and automated parameter studies are valuable, but automation can repeat an incorrect boundary condition or an unphysical material definition thousands of times.

Professional and Emerging Practice

Modern vibration analysis increasingly combines finite-element models with test data, digital thread systems, reduced-order models, and uncertainty quantification. Automated model updating can estimate uncertain stiffness, mass, or damping parameters from measured modal data. Machine-learning methods may assist surrogate modeling or anomaly detection, but their predictions require physical constraints, representative training data, and independent validation. They should supplement rather than replace mechanics-based reasoning.

A reliable computational workflow is therefore:

1. Define the engineering question, frequency range, inputs, outputs, and acceptance criteria.

2. Construct the simplest physically credible model.

3. Check dimensions, constraints, limiting cases, and energy consistency.

4. Select an appropriate computational method and solver.

5. Perform residual, mesh, time-step, or basis convergence checks.

6. Conduct sensitivity analysis for influential parameters.

7. Reduce the model only after identifying the dynamics that matter.

8. Compare with analytical solutions, independent formulations, or experiments.

9. Record assumptions, uncertainties, and the model’s valid domain.

10. Communicate results with mode identities, normalization, units, and uncertainty rather than isolated numerical values.

Chapter Summary

Computational vibration analysis transforms continuous or multi-component mechanical systems into solvable numerical models. Rayleigh and Rayleigh–Ritz methods provide efficient energy-based approximations, while Galerkin methods formulate weighted residual equations. Transfer matrices are effective for sequential systems, and finite elements provide a flexible general formulation for complex structures. State-space models support damped systems, transient analysis, control, and coupled subsystems.

The generalized eigenproblem supplies natural frequencies and mode shapes, while dynamic stiffness and modal or state-space methods provide forced response. Model reduction must preserve the behavior relevant to specified frequencies, inputs, and outputs. Convergence studies assess numerical approximation, sensitivity analysis identifies influential parameters, verification checks implementation, and validation compares the model with physical evidence. Reliable practice depends on correct physical idealization, transparent assumptions, independent checks, and disciplined interpretation of numerical results.

Key Terms

Rayleigh quotient: Ratio of maximum potential energy to maximum kinetic energy used to estimate a natural frequency.

Rayleigh–Ritz method: Energy-based approximation using a finite combination of admissible trial functions.

Galerkin method: Weighted-residual method requiring the residual to be orthogonal to selected weighting functions.

Transfer matrix: Matrix relating state variables at one point in a sequential mechanical system to those at another point.

Finite-element method: Discretization technique that represents a continuous structure through interconnected elements and nodal variables.

Generalized eigenproblem: Matrix equation (K − ω²M)φ = 0 for undamped free vibration.

Mode shape: Spatial pattern of deformation associated with a natural frequency.

Modal mass: Inertial coefficient associated with a particular mode under a selected normalization.

Consistent mass matrix: Mass matrix obtained by integrating distributed inertia using the element shape functions.

Lumped mass matrix: Approximation in which mass is concentrated at selected nodes or coordinates.

State-space model: First-order representation of a dynamic system using state variables and system matrices.

Dynamic stiffness: Frequency-dependent matrix K + iωC − ω²M relating harmonic forces and responses.

Model reduction: Process of replacing a high-order model with a lower-order model that preserves specified dynamic behavior.

Modal truncation: Reduction method retaining only selected vibration modes.

Convergence study: Investigation of solution changes as mesh, basis, time step, or solver accuracy is refined.

Sensitivity analysis: Evaluation of how outputs change with respect to model parameters.

Verification: Demonstration that equations and numerical implementation have been solved correctly.

Validation: Demonstration that a model represents the physical system adequately for its intended use.

Modal assurance criterion: Correlation measure used to compare vibration mode shapes.

Residual: Difference between the computed substitution into the governing algebraic or differential equation and zero.

Review Questions

Short Answer Questions

1. State the generalized eigenvalue equation for an undamped finite-dimensional vibration system.

2. Why must Ritz trial functions satisfy essential boundary conditions?

3. What is the principal numerical weakness of conventional transfer-matrix multiplication?

4. Distinguish consistent and lumped mass matrices.

5. What is the difference between verification and validation?

Long Answer Questions

1. Derive the Rayleigh–Ritz matrix equations for a vibrating Euler–Bernoulli beam.

2. Explain the finite-element assembly process for mass and stiffness matrices and discuss the influence of boundary conditions.

3. Describe state-space formulation for a damped multi-degree-of-freedom system and explain the meaning of its eigenvalues.

4. Compare modal truncation, Guyan reduction, and component mode synthesis.

5. Develop a systematic convergence and validation plan for a finite-element vibration model.

Analytical / Application Questions

1. A structure is modeled by M and K matrices, and an assumed displacement vector is available. Explain how the Rayleigh quotient can be used to estimate the first natural frequency and identify the assumptions behind the estimate.

2. A refined mesh changes the first frequency by 0.3% but changes a local higher mode by 12%. Decide whether the model is adequate for low-frequency base excitation and justify the decision.

3. A measured first mode agrees with a simulation, but the second mode does not. Propose a sensitivity, model-updating, and validation strategy rather than immediately calibrating the model to the first mode.

22. Multiple Choice Questions with Answers

Question 1

For an undamped finite-dimensional system, which equation determines the natural frequencies?

A. (K + ωC)φ = 0

B. (K − ω²M)φ = 0

C. (M − ω²C)φ = 0

D. (C − ωM)φ = 0

Correct Answer: B. (K − ω²M)φ = 0

Question 2

A Rayleigh estimate of the fundamental frequency is most directly obtained from which expression?

A. ω² = (aᵀMa)/(aᵀKa)

B. ω² = aᵀ(K + M)a

C. ω² = (aᵀKa)/(aᵀMa)

D. ω² = det(K)/det(M) for any assumed vector

Correct Answer: C. ω² = (aᵀKa)/(aᵀMa)

Question 3

In the classical Ritz method, trial functions must necessarily satisfy which conditions?

A. All natural boundary conditions only

B. Essential or geometric boundary conditions

C. The exact differential equation at every point

D. Zero displacement at every boundary

Correct Answer: B. Essential or geometric boundary conditions

Question 4

Which method relates state variables at one location of a sequential structure to those at another location?

A. Transfer-matrix method

B. Modal truncation

C. Static condensation only

D. Rayleigh quotient

Correct Answer: A. Transfer-matrix method

Question 5

What is a principal advantage of the finite-element method in vibration analysis?

A. It requires no assumptions about material properties

B. It can represent irregular geometry and distributed structural detail

C. It always gives exact continuous-system frequencies

D. It eliminates the need for boundary conditions

Correct Answer: B. It can represent irregular geometry and distributed structural detail

Question 6

For a mass-normalized modal matrix Φ, which relation is expected for an undamped system?

A. ΦᵀMΦ = 0

B. ΦᵀMΦ = I

C. ΦᵀKΦ = I for every normalization

D. ΦᵀMΦ = K

Correct Answer: B. ΦᵀMΦ = I

Question 7

Which statement best distinguishes verification from validation?

A. Verification compares with experiments; validation checks coding only

B. Verification checks implementation and solution accuracy; validation checks physical adequacy

C. They are identical terms

D. Validation is required only for analytical models

Correct Answer: B. Verification checks implementation and solution accuracy; validation checks physical adequacy

Question 8

A reduced model retains the first few modes but gives poor high-frequency force predictions. What is the most likely reason?

A. Low-frequency modes always contain all high-frequency information

B. The retained basis does not represent the relevant high-frequency or local dynamics

C. The mass matrix is necessarily singular

D. State-space models cannot predict forces

Correct Answer: B. The retained basis does not represent the relevant high-frequency or local dynamics

Question 9

Which quantity is commonly used to compare two computed mode shapes?

A. Modal assurance criterion

B. Static safety factor

C. Poisson ratio

D. Damping loss angle only

Correct Answer: A. Modal assurance criterion

Question 10

If stiffness increases while mass remains unchanged, the usual first-order effect on natural frequencies is to:

A. Decrease them

B. Leave them unchanged

C. Increase them

D. Make all modes rigid-body modes

Correct Answer: C. Increase them