6. Multi-Degree-of-Freedom Systems and Modal Analysis

Introduction and Context

A mechanical system possesses multiple degrees of freedom (MDOF) when its motion cannot be described by a single independent generalized coordinate. Examples include a multistory building subjected to earthquake excitation, a vehicle suspension with several masses, a turbine rotor supported by flexible bearings, and a discretized finite-element model of a machine structure. Although such systems may contain many coupled coordinates, their vibration response can often be understood as a combination of simpler independent motions called normal modes.

The transition from single-degree-of-freedom analysis to MDOF analysis is conceptually important. In a single-degree-of-freedom system, stiffness, inertia, and damping are represented by scalars. In an MDOF system they become matrices, and coupling appears through off-diagonal terms. The resulting equations are not merely a collection of independent scalar equations; the coordinates interact dynamically. Modal analysis provides the mathematical transformation that converts the coupled equations into a set of uncoupled or partially uncoupled modal equations.

This chapter develops the matrix formulation of undamped and damped MDOF systems, derives the generalized eigenvalue problem, establishes orthogonality properties, and explains modal coordinates, mode superposition, participation factors, and effective modal mass. Particular attention is given to proportional and non-proportional damping, because the validity of classical modal analysis depends strongly on the damping model adopted.

Learning Objectives

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

1. Formulate the mass, damping, and stiffness matrices for linear MDOF systems.

2. Derive and solve the generalized eigenvalue problem for natural frequencies and mode shapes.

3. Explain mass and stiffness orthogonality and normalize mode shapes appropriately.

4. Transform physical coordinates into modal coordinates.

5. Apply mode superposition to free and forced vibration problems.

6. Calculate modal participation factors and effective modal mass.

7. Distinguish between proportional and non-proportional damping models.

8. Interpret coupled modes and identify the limitations of modal truncation.

9. Apply modal analysis to engineering systems and assess modeling assumptions.

From Coupled Coordinates to Matrix Equations

Consider a linear system with n generalized coordinates collected in the vector

q(t) = [q₁(t), q₂(t), …, qₙ(t)]ᵀ.

For a system with constant parameters, the equations of motion are written as

M q̈ + C q̇ + K q = f(t),

where M is the n × n mass matrix, C is the damping matrix, K is the stiffness matrix, and f(t) is the applied-force vector. For a conservative mechanical system, M and K are generally symmetric. The mass matrix is positive definite for a physically meaningful system, while the stiffness matrix is positive definite when rigid-body motions are absent and all relevant constraints have been applied.

The diagonal terms of M and K represent direct inertia and stiffness effects. The off-diagonal terms represent coupling. Coupling may arise because a spring connects two coordinates, because the coordinates are not principal directions of inertia, or because the chosen generalized coordinates have geometric relationships.

For example, consider two masses connected in a line by springs. If x₁ and x₂ are measured from equilibrium, the equations may be

m₁ẍ₁ + (k₁ + k₂)x₁ − k₂x₂ = f₁(t),

m₂ẍ₂ − k₂x₁ + (k₂ + k₃)x₂ = f₂(t).

In matrix form,

[ m₁ 0 ][ẍ₁] + [ c₁+c₂ −c₂ ][ẋ₁] + [ k₁+k₂ −k₂ ][x₁] = [f₁],

[ 0 m₂ ][ẍ₂] [ −c₂ c₂+c₃][ẋ₂] [ −k₂ k₂+k₃][x₂] [f₂].

The negative off-diagonal stiffness terms indicate that displacement of one mass produces a restoring force associated with the displacement of the other mass. Such terms are a mathematical representation of physical interaction.

The matrix equation can also be obtained from Lagrange’s equations. If T is kinetic energy and V is potential energy, then for conservative systems

Mᵢⱼ = ∂²T/∂q̇ᵢ∂q̇ⱼ, Kᵢⱼ = ∂²V/∂qᵢ∂qⱼ.

This energy-based interpretation is particularly useful for finite-element modeling and for systems with non-Cartesian generalized coordinates.

Free Undamped Vibration and the Generalized Eigenvalue Problem

First consider free vibration without damping:

M q̈ + K q = 0.

Assume a harmonic solution of the form

q(t) = φ e^{iωt},

where φ is a constant vector and ω is the circular natural frequency. Substitution gives

(K − ω²M)φ = 0.

For a nontrivial mode shape φ, the coefficient matrix must be singular:

det(K − ω²M) = 0.

This is the generalized eigenvalue equation. Its n roots provide the squared natural frequencies ωᵣ², and the corresponding eigenvectors φᵣ provide the mode shapes. In a stable, unconstrained-free system, rigid-body modes may occur with ωᵣ = 0.

The eigenvalues are usually ordered as

0 ≤ ω₁² ≤ ω₂² ≤ … ≤ ωₙ².

The corresponding mode shapes are not unique in magnitude. If φᵣ is an eigenvector, then aφᵣ is also an eigenvector for any nonzero scalar a. The sign is likewise arbitrary. For meaningful comparison, a consistent normalization must be selected.

For a two-degree-of-freedom system, the characteristic equation is a quadratic in ω². In larger systems, numerical eigensolvers are normally used. Direct computation of the determinant is generally inefficient and numerically less reliable than algorithms designed for symmetric generalized eigenproblems.

Orthogonality of Normal Modes

Suppose φᵣ and φₛ are two distinct mode shapes satisfying

Kφᵣ = ωᵣ²Mφᵣ,

Kφₛ = ωₛ²Mφₛ.

Premultiplying the first equation by φₛᵀ and the second by φᵣᵀ, then using the symmetry of M and K, gives

(ωᵣ² − ωₛ²)φₛᵀMφᵣ = 0.

For distinct frequencies,

φₛᵀMφᵣ = 0.

This is mass orthogonality. Substitution into the eigenvalue relation also gives stiffness orthogonality:

φₛᵀKφᵣ = 0.

Thus, different normal modes are orthogonal with respect to both the mass and stiffness matrices, even though their ordinary Euclidean dot product need not be zero.

If the modal matrix is defined as

Φ = [φ₁ φ₂ … φₙ],

then

ΦᵀMΦ = M* ,

ΦᵀKΦ = K*,

where M and K are diagonal matrices. The diagonal elements are modal mass and modal stiffness:

mᵣ = φᵣᵀMφᵣ,

kᵣ = φᵣᵀKφᵣ = ωᵣ²mᵣ.

A convenient choice is mass normalization, for which

φᵣᵀMφᵣ = 1.

Then

ΦᵀMΦ = I,

ΦᵀKΦ = Ω²,

where Ω² = diag(ω₁², ω₂², …, ωₙ²). Mass-normalized modes simplify many calculations, although other normalizations may be preferable in experimental testing or structural engineering practice.

Repeated eigenvalues require additional care. The eigenvectors associated with a repeated eigenvalue can be selected as mutually orthogonal with respect to M, usually by an orthogonalization procedure within the repeated eigenspace.

Modal Coordinates and Decoupling

The central idea of modal analysis is to express the physical displacement vector as a linear combination of mode shapes:

q(t) = Φη(t),

where η(t) = [η₁, η₂, …, ηₙ]ᵀ is the vector of modal coordinates. Premultiplying the undamped equation by Φᵀ gives

ΦᵀMΦ η̈ + ΦᵀKΦ η = Φᵀf(t).

For mass-normalized modes this becomes

η̈ + Ω²η = Φᵀf(t).

The coupled physical equations have therefore been transformed into n independent modal equations:

η̈ᵣ + ωᵣ²ηᵣ = φᵣᵀf(t).

Each mode behaves mathematically like a single-degree-of-freedom oscillator. The physical response is recovered through

q(t) = Σ φᵣηᵣ(t).

This result is exact when all modes are retained and the system is linear. In practical engineering analysis, only a subset of modes is often used. Such modal truncation is effective when the retained modes dominate the response in the frequency range and loading direction of interest.

For a general normalization, the modal equation is

mᵣη̈ᵣ + kᵣηᵣ = Qᵣ(t),

where

Qᵣ(t) = φᵣᵀf(t)

is the generalized modal force. If the modes are not mass-normalized, division by mᵣ is required before interpreting the equation as a unit-mass oscillator.

Modal Superposition for Forced Response

When harmonic force is applied,

f(t) = Re{F e^{iωt}},

the steady-state response may be written as

q(t) = Re{X e^{iωt}}.

For the undamped system,

(K − ω²M)X = F.

Using modal expansion,

X = Σ φᵣ \frac{φᵣᵀF}{ωᵣ² − ω²}

for mass-normalized modes, provided the excitation frequency does not coincide with an undamped natural frequency. This expression shows that a mode contributes strongly when the forcing frequency is close to its natural frequency and when the modal force φᵣᵀF is large.

The transient response due to initial displacement q(0) and velocity q̇(0) is also obtained mode by mode. For the mass-normalized undamped system,

ηᵣ(0) = φᵣᵀM q(0),

η̇ᵣ(0) = φᵣᵀM q̇(0).

Then

ηᵣ(t) = ηᵣ(0)cos(ωᵣt) + [η̇ᵣ(0)/ωᵣ]sin(ωᵣt).

The complete physical response is the sum of these modal contributions.

Base Excitation and Participation Factors

Many important vibration problems involve motion of the support rather than a directly applied force. Suppose the absolute displacement vector is related to ground displacement y(t) by

q(t) = u(t) + r y(t),

where u is the relative displacement vector and r is an influence vector indicating how each coordinate follows the base. For uniform translation, r is often a vector of ones. The equations in relative coordinates become

M ü + C u̇ + K u = −Mr ÿ(t).

For the undamped system, the modal equation is

η̈ᵣ + ωᵣ²ηᵣ = −Γᵣ ÿ(t),

where the modal participation factor is

Γᵣ = \frac{φᵣᵀMr}{φᵣᵀMφᵣ}.

For mass-normalized modes, Γᵣ = φᵣᵀMr. The participation factor measures how strongly a particular mode is excited by the selected support-motion pattern. A mode may have a low natural frequency but contribute little to a particular excitation if its participation factor is small.

The effective modal mass for base excitation is

Mᵣ^{eff} = \frac{(φᵣᵀMr)^2}{φᵣᵀMφᵣ}.

For a complete set of modes,

Σ Mᵣ^{eff} = rᵀMr.

This identity is a valuable quality check. In seismic analysis and structural dynamics, the cumulative effective modal mass is commonly examined to determine whether enough modes have been retained. The required percentage depends on the governing standard, structural form, and response quantity; it should not be treated as a universal substitute for engineering judgment.

Coupled Modes and Physical Interpretation

A normal mode is not generally the motion of one physical coordinate alone. It is a coordinated deformation pattern involving several coordinates. In a two-mass system, the lower mode often involves masses moving in the same direction, whereas the higher mode often involves them moving in opposite directions. These descriptions are tendencies, not universal rules; the actual pattern depends on mass and stiffness distribution.

The first mode usually has the lowest curvature or deformation complexity and is therefore often dominant under broad, slowly varying excitation. Higher modes may nevertheless control local stresses, floor accelerations, transmitted forces, or response near higher excitation frequencies. A design based only on the first mode can consequently be unsafe for components whose response is governed by higher-frequency deformation.

Modes are properties of the mathematical model, not permanent labels attached to hardware. Changes in mass, boundary conditions, joint stiffness, temperature, or damage can shift frequencies and alter mode shapes. In operational modal analysis, these changes may be used for condition monitoring, but interpretation requires attention to environmental and measurement effects.

Damping: Classical and Non-Classical Models

With damping included, the modal transformation gives

ΦᵀMΦ η̈ + ΦᵀCΦ η + ΦᵀKΦ η = Φᵀf(t).

Although the mass and stiffness terms are diagonal in normal coordinates, the transformed damping matrix

C* = ΦᵀCΦ

is not necessarily diagonal. If it is diagonal, the equations are classically damped or proportionally damped and remain uncoupled:

η̈ᵣ + 2ζᵣωᵣη̇ᵣ + ωᵣ²ηᵣ = Qᵣ(t),

for mass-normalized modes. Here ζᵣ is the modal damping ratio.

A widely used proportional damping model is Rayleigh damping:

C = αM + βK,

where α and β are nonnegative constants. In modal coordinates,

ΦᵀCΦ = αI + βΩ²,

which is diagonal. The modal damping ratio is

ζᵣ = \frac{1}{2}\left(\frac{α}{ωᵣ} + βωᵣ\right).

Rayleigh damping is convenient because α and β can be selected to match damping ratios at two frequencies. However, it may produce unrealistic damping outside the calibration range: the mass-proportional term dominates at low frequencies, while the stiffness-proportional term increases damping at high frequencies.

More general proportional models include Caughey series representations, which can fit damping over a wider frequency range while preserving modal decoupling under suitable assumptions. Nevertheless, mathematical convenience should not be confused with physical accuracy.

In non-proportional damping, C* contains off-diagonal terms, so the real normal modes do not fully decouple the equations. Examples include localized dampers, friction interfaces, viscoelastic treatments, fluid-structure interaction, and systems with substantially different damping mechanisms in different components. The analysis then requires one of several approaches:

• retain the coupled modal equations and solve them simultaneously;
• use complex modes and state-space eigenanalysis;
• employ component-mode synthesis or numerical frequency-response methods;
• approximate the damping as proportional only when the resulting error is acceptable.

For the state-space formulation, define z = [qᵀ q̇ᵀ]ᵀ. Then

ż = A z + Bf(t),

where

A = [ 0 I; −M⁻¹K −M⁻¹C ].

The eigenvalues of A are generally complex and occur in conjugate pairs for real systems. Their real parts describe decay or growth, while their imaginary parts describe damped oscillation frequencies. This formulation is especially useful for non-classically damped systems and for computational control or stability analysis.

A Worked Two-Degree-of-Freedom Illustration

Consider two equal masses m connected by three equal springs k: the first mass is attached to a fixed wall, the masses are connected to each other, and the second mass is attached to another fixed wall. The matrices are

M = m[1 0; 0 1],

K = k[2 −1; −1 2].

The eigenvalue equation is

|2k − ω²m −k |

|−k 2k − ω²m| = 0.

Thus,

(2k − ω²m)² − k² = 0,

which yields

ω₁² = k/m,

ω₂² = 3k/m.

The corresponding mode shapes may be selected as

φ₁ = [1, 1]ᵀ,

φ₂ = [1, −1]ᵀ.

The first mode is an in-phase motion, while the second is an out-of-phase motion. Because the masses are equal, the modes are already orthogonal in the ordinary sense. Their modal masses are 2m. Mass-normalized vectors are

φ₁ = [1, 1]ᵀ/√(2m),

φ₂ = [1, −1]ᵀ/√(2m).

If both masses are subjected to identical base acceleration, the influence vector is r = [1,1]ᵀ. The second mode has

φ₂ᵀMr = 0,

so its participation factor is zero. Uniform base translation excites the first mode but not the antisymmetric second mode. This result illustrates that natural frequency alone does not determine response importance; excitation compatibility is equally significant.

Applied Case Study: Modal Assessment of a Flexible Machine Frame

Background

Consider an illustrative industrial case involving a precision machining frame supporting a spindle, drive motor, and workholding assembly. The frame is modeled with six generalized coordinates representing lateral motion of three concentrated masses in two directions. The objective is to determine whether periodic motor forces near 45 Hz could produce unacceptable vibration.

Situation and Problem

A preliminary static design appears sufficiently stiff, but operators report surface-finish variation during machining. The forcing frequency associated with a motor operating at 2700 revolutions per minute is 45 Hz. The engineering team must determine whether a structural resonance is responsible and whether a design modification is required.

Analysis

Finite-element discretization produces symmetric mass and stiffness matrices. An eigenvalue analysis gives the first four natural frequencies as 31, 48, 76, and 119 Hz. The second mode at 48 Hz is a lateral bending mode involving substantial motion at the spindle support. The first mode is dominated by motion of the lower frame and has limited spindle displacement.

A harmonic-force vector is formed from measured motor imbalance forces. Modal forces φᵣᵀF show that the 48 Hz mode is strongly excited, while the 31 Hz mode is weakly excited. A Rayleigh damping model is initially calibrated to a damping ratio of 1% at the first and third frequencies. Modal superposition predicts a large response near 45–48 Hz.

Approach and Solution

The team evaluates three modifications: increasing frame cross-section, adding a diagonal brace, and installing a tuned viscoelastic support near the spindle. The brace raises the second natural frequency to 61 Hz and changes its mode shape so that spindle motion is reduced. A revised analysis checks the new frequency range, modal participation, local stresses, and the effect of manufacturing tolerances.

Because the added viscoelastic support introduces localized damping, the damping matrix is no longer reliably proportional. The final design is therefore assessed using a state-space frequency-response calculation rather than relying exclusively on uncoupled modal equations.

Outcome

The revised design reduces the predicted spindle vibration substantially and separates the principal excitation frequency from the dominant structural mode. Prototype testing uses impact excitation and accelerometers to compare measured resonant frequencies and mode shapes with the model. Small discrepancies are corrected by updating joint stiffness rather than arbitrarily changing material damping.

Learning Points

The case demonstrates four important principles. First, the mode with the lowest frequency is not necessarily the most important mode. Second, participation depends on the spatial compatibility between excitation and mode shape. Third, structural modifications can alter both frequencies and mode shapes. Fourth, localized damping may invalidate a classical damping assumption and require a more general computational model.

Computational Practice and Engineering Judgment

Modern modal analysis is normally performed with numerical software. A reliable workflow includes defining coordinates and boundary conditions, assembling M, C, and K, checking units and symmetry, solving the generalized eigenproblem, normalizing modes, examining mode shapes visually, and verifying modal mass participation. The analyst should also compare limiting cases and perform mesh or coordinate refinement where appropriate.

Numerical results require physical interpretation. Spurious modes may arise from unconstrained rigid-body motion, poor element formulation, duplicate constraints, or unrealistic joint stiffness. A mode shape that changes dramatically under small modeling changes may indicate an ill-conditioned or poorly characterized system. In experimental correlation, frequency agreement alone is insufficient; mode-shape correlation, modal assurance criteria, and sensitivity to boundary conditions should also be considered.

Modal truncation is another major source of error. Omitting high-frequency modes can underestimate local acceleration or stress, even when the retained modes capture most global displacement. Residual-flexibility or static-correction methods can improve truncated frequency-response calculations. For shock, impact, or highly localized loading, a larger modal basis is generally required than for low-frequency global motion.

Challenges, Limitations, and Common Errors

Several recurring errors weaken MDOF vibration analyses:

• treating a nonsymmetric or incorrectly assembled matrix as physically acceptable without investigation;
• confusing ordinary orthogonality with mass orthogonality;
• comparing mode-shape amplitudes without considering normalization;
• assuming that every mode is significantly excited;
• using Rayleigh damping without checking its behavior outside the calibration frequencies;
• neglecting rigid-body modes or incorrectly constraining them;
• retaining modes based only on frequency and not on participation or response location;
• using proportional damping for systems with strong localized or frequency-dependent damping;
• interpreting finite-element modes without validating joints, supports, and mass distribution.

Linear modal analysis also has physical limits. Nonlinear stiffness, clearance, friction, large deformation, amplitude-dependent damping, and contact can cause mode shapes and frequencies to depend on response amplitude. In these cases, linear modes remain useful for preliminary understanding, but time-domain nonlinear analysis or experimentally identified models may be necessary.

Professional and Ethical Considerations

Modal results can influence safety-critical decisions involving rotating machinery, vehicles, aerospace structures, bridges, and medical equipment. Engineers should document assumptions, boundary conditions, damping choices, normalization, solver settings, and validation evidence. It is professionally unacceptable to adjust damping or stiffness merely to obtain a desired result without physical justification. When model uncertainty is substantial, sensitivity studies and conservative design margins should be reported.

Safety standards may require specific response-combination procedures, damping assumptions, or modal participation criteria. The analyst must use the governing standard for the application rather than transferring criteria from an unrelated industry. Experimental data should be represented honestly, including uncertainty, sensor limitations, and discrepancies between test and simulation.

Emerging Directions

Large-scale finite-element models increasingly use automated eigensolvers, model reduction, and component-mode synthesis. Digital twins combine measured vibration data with computational models to track changes in stiffness, mass, and damping during operation. Machine-learning methods can assist in detecting modal-frequency shifts or classifying damage, but they do not remove the need for physically interpretable models and reliable training data.

Operational modal analysis is useful when excitation is unknown or difficult to measure, such as in bridges, wind turbines, and operating machinery. Experimental modal testing is also becoming more efficient through dense sensor arrays, scanning laser vibrometry, and automated curve fitting. These developments improve identification but introduce challenges involving data quality, sensor placement, environmental variability, and uncertainty quantification.

Best Practices for Modal Analysis

A sound analysis generally follows this sequence:

1. Define physically meaningful coordinates and boundary conditions.

2. Assemble and inspect M, C, and K, checking symmetry, units, and positive definiteness.

3. Solve the generalized eigenvalue problem with an appropriate numerical method.

4. Normalize and inspect mode shapes, including rigid-body modes.

5. Calculate modal forces, participation factors, and effective modal masses for the actual excitation.

6. Select a modal basis based on the response quantity and frequency range, not frequency alone.

7. Justify the damping model and test whether proportional damping is credible.

8. Validate predictions with analytical limiting cases, experiments, or independent computations.

9. Quantify sensitivity to uncertain mass, stiffness, damping, and boundary conditions.

10. Report assumptions and limitations together with the predicted response.

Chapter Summary

An MDOF vibration system is represented by the matrix equation Mq̈ + Cq̇ + Kq = f(t). For undamped free vibration, the assumed harmonic solution leads to the generalized eigenvalue problem (K − ω²M)φ = 0. The eigenvalues provide natural frequencies, while eigenvectors provide mode shapes. Symmetric mass and stiffness matrices produce mass and stiffness orthogonality, allowing the transformation q = Φη to convert coupled undamped equations into independent modal equations.

Modal superposition expresses physical response as a sum of modal responses. Participation factors and effective modal masses determine how strongly a mode is excited by base motion or other spatially distributed loads. Proportional damping, including Rayleigh damping, preserves modal decoupling, whereas non-proportional damping generally requires coupled modal, complex-mode, or state-space methods. Practical analysis must address normalization, modal truncation, model validation, boundary conditions, damping uncertainty, and nonlinear effects. Modal analysis is therefore both a mathematical technique and an exercise in engineering judgment.

Key Terms

Degree of freedom: An independent generalized coordinate required to describe system motion.

Mass matrix: Matrix representing inertial properties and inertial coupling among coordinates.

Stiffness matrix: Matrix representing restoring-force properties and elastic coupling.

Damping matrix: Matrix representing energy dissipation and damping coupling.

Generalized eigenvalue problem: Equation (K − ω²M)φ = 0 used to determine natural frequencies and mode shapes.

Natural frequency: Frequency of free vibration associated with a particular mode.

Mode shape: Relative deformation pattern associated with a natural frequency.

Normal mode: A mode in which all coordinates oscillate at one natural frequency with fixed amplitude ratios.

Mass orthogonality: Orthogonality of distinct modes under the M-weighted inner product.

Stiffness orthogonality: Orthogonality of distinct modes under the K-weighted inner product.

Modal matrix: Matrix whose columns are the system mode shapes.

Modal coordinate: Coordinate describing the amplitude of a particular mode.

Modal mass: Quantity φᵀMφ associated with a selected mode normalization.

Modal force: Generalized force obtained by projecting physical force onto a mode.

Participation factor: Measure of the coupling between an excitation pattern and a mode.

Effective modal mass: Apparent mass associated with a mode under a specified excitation pattern.

Proportional damping: Damping for which the modal damping matrix is diagonal in the undamped modal basis.

Rayleigh damping: Damping model C = αM + βK.

Non-proportional damping: Damping that produces off-diagonal modal coupling.

Modal truncation: Approximation obtained by retaining only a selected subset of modes.

Review Questions

Short Answer Questions

1. What do the off-diagonal terms in a mass or stiffness matrix represent physically?

2. State the generalized eigenvalue equation for an undamped MDOF system.

3. Define mass orthogonality of two mode shapes.

4. What is the purpose of mass normalization?

5. Distinguish between a modal force and a participation factor.

Long Answer Questions

1. Derive the modal transformation for an undamped MDOF system and show how it decouples the equations.

2. Derive the orthogonality relations for mode shapes when M and K are symmetric.

3. Explain modal participation factors and effective modal mass for uniform base excitation.

4. Derive the Rayleigh damping expression for the modal damping ratio and discuss its limitations.

5. Compare classical and non-classical damping, including appropriate solution methods for each.

Analytical / Application Questions

1. A two-degree-of-freedom system has M = diag(m, m) and K = k[[2, −1], [−1, 2]]. Determine its natural frequencies and interpret the associated mode shapes.

2. A machine has a forcing frequency close to its second natural frequency, but the measured response is small. Explain how modal participation could account for this observation and identify what quantities should be evaluated.

3. A finite-element model predicts acceptable global displacement but excessive local stress during an impact. Explain why modal truncation may be responsible and propose an improved analysis strategy.

22. Multiple Choice Questions with Answers

Question 1

For an undamped MDOF system, which equation determines the natural frequencies and mode shapes?

A. det(C − ωM) = 0

B. det(K − ω²M) = 0

C. det(M − ω²K) = 0

D. det(K + ω²M) = 0

Correct Answer: B. det(K − ω²M) = 0

Question 2

The off-diagonal terms of the stiffness matrix primarily represent:

A. Damping ratios

B. Rigid-body mass

C. Elastic coupling between coordinates

D. Modal normalization

Correct Answer: C. Elastic coupling between coordinates

Question 3

For two distinct modes of a system with symmetric M and K, which relation is correct?

A. φᵣᵀMφₛ = 1

B. φᵣᵀMφₛ = 0

C. φᵣᵀKφₛ = ωᵣ²

D. φᵣᵀKφₛ = ωₛ²

Correct Answer: B. φᵣᵀMφₛ = 0

Question 4

If the modes are mass-normalized, the transformed mass matrix is:

A. K

B. Ω²

C. I

D. C

Correct Answer: C. I

Question 5

For uniform base excitation with influence vector r, the participation factor is:

A. φᵀKφ

B. φᵀMr divided by φᵀMφ

C. φᵀMφ divided by φᵀMr

D. φᵀCr

Correct Answer: B. φᵀMr divided by φᵀMφ

Question 6

In the symmetric two-mass example with mode φ₂ = [1, −1]ᵀ, uniform base translation does not excite the second mode because:

A. Its natural frequency is zero

B. Its modal mass is zero

C. Its modal participation factor is zero

D. Its stiffness is zero

Correct Answer: C. Its modal participation factor is zero

Question 7

Which damping model guarantees diagonal modal damping for symmetric M and K?

A. Arbitrary localized damping

B. C = αM + βK

C. C = M + K⁻¹

D. Friction damping at one joint only

Correct Answer: B. C = αM + βK

Question 8

A major limitation of Rayleigh damping is that:

A. It cannot be represented in matrix form

B. It always produces negative damping

C. Its damping behavior may be unrealistic outside the calibration frequencies

D. It eliminates all natural frequencies

Correct Answer: C. Its damping behavior may be unrealistic outside the calibration frequencies

Question 9

When damping is non-proportional, the undamped real modes generally:

A. Always decouple the damped equations completely

B. Produce a diagonal damping matrix automatically

C. Leave off-diagonal modal damping terms

D. Become rigid-body modes

Correct Answer: C. Leave off-diagonal modal damping terms

Question 10

Modal truncation is most likely to underestimate local impact stresses when:

A. Only low-frequency global modes are retained

B. All modes are retained

C. The excitation is zero

D. The mass matrix is symmetric

Correct Answer: A. Only low-frequency global modes are retained