MDOF Systems & Modal Analysis

1. Introduction to Multi-Degree-of-Freedom (MDOF) Systems

1.1 Structural Dynamics in NVH Engineering

In structural dynamics and Noise, Vibration, and Harshness (NVH) engineering, real-world systems are rarely simple enough to be modeled as single-degree-of-freedom (SDOF) systems. An SDOF system assumes that the entire mass is lumped at a single point and that the system can only move in one direction. Physical structures, such as:

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Automotive bodies (Body-in-White)

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Engine mounts

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Exhaust systems

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Aircraft wings

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Suspension systems

possess distributed mass, stiffness, and damping, meaning they can vibrate in infinite directions and configurations.

To model such systems, we use Multi-Degree-of-Freedom (MDOF) systems. An MDOF system is defined as a system that requires two or more independent coordinates to describe its motion. By discretizing a continuous structure into a finite number of degrees of freedom, we can represent its physical behavior using systems of ordinary differential equations instead of partial differential equations. This discretization is the basis of the Finite Element Method (FEM) and lumped-parameter modeling in dynamic analysis.

1.2 Spatial Discretization and Degrees of Freedom

Spatial discretization is the process of dividing a continuous structure with infinite degrees of freedom into a model with a finite number of coordinates. There are two primary approaches to discretization:

Lumped Parameter Modeling:

We assume the mass of the system is concentrated at discrete rigid bodies connected by massless spring-damper elements. This is useful for preliminary design, suspension analysis, and simplified vehicle models.

Consistent Parameter Modeling (Finite Element Method):

We divide the domain into continuous elements. The mass and stiffness properties are distributed across the elements using interpolation functions (shape functions). This approach provides a much more accurate representation of complex geometries.

The number of degrees of freedom ($N$) of a discretized system is the minimum number of independent kinematic coordinates required to completely define the position of all parts of the system at any instant of time. For a system of $P$ particles moving in three-dimensional space subject to $C$ kinematic constraints, the number of degrees of freedom is:

$$N = 3P - C$$

1.3 Coordinate Systems in MDOF Dynamics

To describe the motion of an MDOF system, we can define several types of coordinate systems:

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Physical Coordinates ($\mathbf{x}$):

These coordinates represent physical displacements (translations or rotations) at specific points (nodes) on the structure. They are directly measurable using sensors such as accelerometers, displacement transducers, or laser vibrometers. Physical coordinate equations of motion are typically highly coupled, meaning the motion of one coordinate depends on the motions of all others.

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Generalized Coordinates ($\mathbf{q}$):

These are any set of independent coordinates that completely describe the system configuration. They do not need to represent physical displacements; they can represent angles, Fourier coefficients, or modal amplitudes.

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Principal (Modal) Coordinates ($\\eta$):

These are a special set of generalized coordinates derived from the system's mode shapes. When the equations of motion are transformed into modal coordinates, they become completely uncoupled. This means the system behaves as a collection of independent SDOF systems, enormously simplifying the solution process.

1.4 Lagrange's Equations of Motion

The equations of motion for complex MDOF systems are most elegantly formulated using energy methods, specifically Lagrange's Equations. For a system defined by $N$ generalized coordinates $q_i$ ($i = 1, 2, \dots, N$), Lagrange's equations are written as:

$$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} + \frac{\partial F}{\partial \dot{q}_i} = Q_i$$

where:

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$T$ is the total Kinetic Energy of the system.

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$V$ is the total Potential Energy (strain energy) of the system.

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$F$ is the Rayleigh Dissipation Function, representing viscous damping energy.

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$Q_i$ is the Generalized Force corresponding to the coordinate $q_i$.

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$q_i$ and $\dot{q}_i$ are the generalized displacement and velocity, respectively.

Let us define these energy quantities for a linear MDOF system:

Kinetic Energy ($T$):

The kinetic energy is a quadratic function of the generalized velocities:

$$T = \frac{1}{2} \sum_{j=1}^N \sum_{k=1}^N M_{jk} \dot{q}_j \dot{q}_k = \frac{1}{2} \dot{\mathbf{q}}^T \mathbf{M} \dot{\mathbf{q}}$$

where $\mathbf{M}$ is the symmetric Mass Matrix.

Potential Energy ($V$):

For a linear elastic system, the potential energy is stored as strain energy and is a quadratic function of the generalized displacements:

$$V = \frac{1}{2} \sum_{j=1}^N \sum_{k=1}^N K_{jk} q_j q_k = \frac{1}{2} \mathbf{q}^T \mathbf{K} \mathbf{q}$$

where $\mathbf{K}$ is the symmetric Stiffness Matrix.

Rayleigh Dissipation Function ($F$):

For viscous damping, the dissipation function is a quadratic function of velocities:

$$F = \frac{1}{2} \sum_{j=1}^N \sum_{k=1}^N C_{jk} \dot{q}_j \dot{q}_k = \frac{1}{2} \dot{\mathbf{q}}^T \mathbf{C} \dot{\mathbf{q}}$$

where $\mathbf{C}$ is the symmetric Damping Matrix.

Applying Lagrange's equations to these quadratic energy forms:

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$\frac{\partial T}{\partial \dot{q}_i} = \sum_{j=1}^N M_{ij} \dot{q}_j \implies \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) = \sum_{j=1}^N M_{ij} \ddot{q}_j$

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$\frac{\partial T}{\partial q_i} = 0$ (since mass properties are independent of displacement for linear systems)

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$\frac{\partial V}{\partial q_i} = \sum_{j=1}^N K_{ij} q_j$

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$\frac{\partial F}{\partial \dot{q}_i} = \sum_{j=1}^N C_{ij} \dot{q}_j$

Substituting these back into Lagrange's equations yields the coupled system:

$$\sum_{j=1}^N M_{ij} \ddot{q}_j + \sum_{j=1}^N C_{ij} \dot{q}_j + \sum_{j=1}^N K_{ij} q_j = Q_i, \quad i = 1, 2, \dots, N$$

In matrix-vector form, this system of second-order differential equations is written as:

$$\mathbf{M} \ddot{\mathbf{q}}(t) + \mathbf{C} \dot{\mathbf{q}}(t) + \mathbf{K} \mathbf{q}(t) = \mathbf{Q}(t)$$

2. Governing Equations of Motion and System Matrices

The matrix equation of motion:

$$\mathbf{M} \ddot{\mathbf{x}}(t) + \mathbf{C} \dot{\mathbf{x}}(t) + \mathbf{K} \mathbf{x}(t) = \mathbf{f}(t)$$

is the cornerstone of structural dynamics. The properties of the matrices $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ determine the physical behavior of the structure, including its natural frequencies, mode shapes, and response to external forces.

2.1 The Mass Matrix ($\mathbf{M}$)

The mass matrix represents the distribution of inertia throughout the system. It relates the acceleration vector $\ddot{\mathbf{x}}$ to the inertial forces.

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Symmetry: $\mathbf{M} = \mathbf{M}^T$.

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Positive Definiteness:

The mass matrix is symmetric positive-definite (SPD). For any non-zero velocity vector $\dot{\mathbf{x}} \neq \mathbf{0}$, the kinetic energy must be strictly positive:

$$T = \frac{1}{2} \dot{\mathbf{x}}^T \mathbf{M} \dot{\mathbf{x}} > 0$$

This implies that all eigenvalues of $\mathbf{M}$ are real and positive, and $\mathbf{M}$ is always invertible (${\mathbf{M}}^{-1}$ exists).

In numerical modeling, two types of mass matrices are common:

Lumped Mass Matrix:

The mass of the system is concentrated at the nodes. This results in a diagonal mass matrix:

$$\mathbf{M} = \begin{bmatrix} m_1 & 0 & \dots & 0 \\ 0 & m_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & m_N \end{bmatrix}$$

Diagonal mass matrices are highly efficient computationally because their inversion is trivial ($M_{ii}^{-1} = 1/m_i$). However, they neglect rotational inertia and cross-coupling between DOFs, which can lead to inaccuracies in high-frequency analysis.

Consistent Mass Matrix:

Formulated using the same shape functions as the stiffness matrix. This leads to a fully populated, non-diagonal mass matrix that accounts for the spatial distribution of mass within elements. Consistent mass matrices are more accurate for bending elements (beams and plates) but require more storage and computational effort.

For example, consider a 1D bar element of length $L$, cross-sectional area $A$, and density $\rho$. The lumped mass matrix is:

$$\mathbf{M}_{lumped} = \frac{\rho A L}{2} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

The consistent mass matrix, derived from linear shape functions $N_1 = 1 - x/L$ and $N_2 = x/L$, is:

$$\mathbf{M}_{consistent} = \int_0^L \rho A \begin{bmatrix} N_1^2 & N_1 N_2 \\ N_1 N_2 & N_2^2 \end{bmatrix} dx = \frac{\rho A L}{6} \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$$

This cross-coupling represents the distributed inertia and provides a more accurate frequency estimation.

2.2 The Stiffness Matrix ($\mathbf{K}$)

The stiffness matrix represents the elastic properties of the system, relating the displacement vector $\mathbf{x}$ to the internal elastic forces.

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Symmetry: $\mathbf{K} = \mathbf{K}^T$.

This symmetry is a consequence of the Maxwell-Betti Reciprocal Work Theorem. It states that the work done by a force system $\mathbf{f}_A$ acting through displacements $\mathbf{x}_B$ is equal to the work done by $\mathbf{f}_B$ acting through displacements $\mathbf{x}_A$.

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Definiteness:

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Positive-Definite:

For a structure that is fully constrained against rigid body motion, any non-zero displacement vector $\mathbf{x} \neq \mathbf{0}$ results in positive stored elastic strain energy:

$$U = \frac{1}{2} \mathbf{x}^T \mathbf{K} \mathbf{x} > 0$$

In this case, $\mathbf{K}$ is positive-definite and invertible.

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Positive-Semidefinite:

If a structure is unconstrained (e.g., an aircraft in flight or a vehicle suspended in mid-air), it can undergo displacement without storing internal strain energy. These displacements are called Rigid Body Modes (translations or rotations). If $\mathbf{u}_{rb}$ is a rigid body displacement vector, then:

$$\mathbf{K} \mathbf{u}_{rb} = \mathbf{0} \implies U = \frac{1}{2} \mathbf{u}_{rb}^T \mathbf{K} \mathbf{u}_{rb} = 0$$

In this case, $\mathbf{K}$ is positive-semidefinite and singular (non-invertible). The eigenvalue problem will yield zero-frequency eigenvalues corresponding to these rigid body modes.

2.3 Coordinate Transformations and Assembly

To construct the global matrices, individual element matrices are transformed from their local coordinate systems to a global coordinate system. For an element, this local-to-global transformation is defined by:

$$\mathbf{x}_{local} = \mathbf{T} \mathbf{x}_{global}$$

where $\mathbf{T}$ is an orthogonal transformation matrix satisfying $\mathbf{T}^T = \mathbf{T}^{-1}$. Substituting this into the element equations of motion yields:

$$\mathbf{M}_{global} = \mathbf{T}^T \mathbf{M}_{local} \mathbf{T}$$

$$\mathbf{K}_{global} = \mathbf{T}^T \mathbf{K}_{local} \mathbf{T}$$

The global matrices are then assembled by summing the overlapping contributions at shared degrees of freedom, enforcing compatibility and equilibrium at the nodes.

2.4 Damping Models in MDOF Systems

The damping matrix $\mathbf{C}$ represents energy dissipation. Unlike mass and stiffness, damping is highly difficult to model from first principles. In structural mechanics, we represent damping using several mathematical models:

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Viscous Damping:

Dissipation forces are proportional to velocity: $\mathbf{f}_d = \mathbf{C} \dot{\mathbf{x}}$. This is the most common model because it preserves the linear second-order structure of the differential equations.

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Hysteretic (Structural) Damping:

Dissipation forces are proportional to displacement but in-phase with velocity. This is modeled using a complex stiffness matrix:

$$\mathbf{K}^* = \mathbf{K} + i \mathbf{D} = \mathbf{K}(1 + i \eta)$$

where $\eta$ is the structural damping factor. This model is valid for harmonic motion only.

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Rayleigh (Proportional) Damping:

To simplify computations, we often assume that the damping matrix $\mathbf{C}$ is a linear combination of the mass and stiffness matrices:

$$\mathbf{C} = \alpha \mathbf{M} + \beta \mathbf{K}$$

where $\alpha$ and $\beta$ are real-valued scalar constants. The major advantage of Rayleigh damping is that it is diagonalized by the same modal transformation that diagonalizes $\mathbf{M}$ and $\mathbf{K}$, allowing the coupled equations of motion to be completely decoupled. However, Rayleigh damping lacks physical justification and can lead to over-damping at high frequencies or under-damping at mid frequencies.

3. The Free Undamped Vibration and the Eigenvalue Problem

To analyze the dynamic characteristics of an MDOF system, we begin by studying its free undamped vibrations. Free vibration occurs when a system is disturbed from its equilibrium position and allowed to vibrate without any external force acting on it ($mathbf{f}(t) = mathbf{0}$). Undamped vibration implies we neglect the dissipation matrix ($\mathbf{C} = \mathbf{0}$). The equation of motion reduces to:

$$\mathbf{M} \ddot{\mathbf{x}}(t) + \mathbf{K} \mathbf{x}(t) = \mathbf{0}$$

This is a system of $N$ coupled linear second-order differential equations. Because the system is linear and elastic, we assume that the free vibration motion is harmonic. This means all degrees of freedom oscillate at the same frequency $\omega$ and pass through their equilibrium positions at the same instant of time. We write the trial solution as:

$$\mathbf{x}(t) = \boldsymbol{\phi} e^{i \omega t}$$

where:

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$\boldsymbol{\phi} = \begin{bmatrix} \phi_1 & \phi_2 & \dots & \phi_N \end{bmatrix}^T$

is a vector of constant amplitudes, called the eigenvector or mode shape.

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$\omega$ is the circular frequency of oscillation (rad/s).

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$i = \sqrt{-1}$.

Taking the second derivative of the trial solution with respect to time:

$$\dot{\mathbf{x}}(t) = i \omega \boldsymbol{\phi} e^{i \omega t}$$

$$\ddot{\mathbf{x}}(t) = -\omega^2 \boldsymbol{\phi} e^{i \omega t}$$

Substituting $\mathbf{x}(t)$ and $\ddot{\mathbf{x}}(t)$ back into the free vibration equation:

$$-\omega^2 \mathbf{M} \boldsymbol{\phi} e^{i \omega t} + \mathbf{K} \boldsymbol{\phi} e^{i \omega t} = \mathbf{0}$$

Since $e^{i \omega t}$ is never zero, we can divide the equation by it, yielding:

$$(\mathbf{K} - \omega^2 \mathbf{M}) \boldsymbol{\phi} = \mathbf{0}$$

This algebraic equation is the Generalized Eigenvalue Problem.

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The scalar values $\lambda = \omega^2$ are the eigenvalues.

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The corresponding vectors $\boldsymbol{\phi}$ are the eigenvectors (mode shapes).

For a non-trivial solution (i.e., $\boldsymbol{\phi} \neq \mathbf{0}$), the coefficient matrix $(\mathbf{K} - \omega^2 \mathbf{M})$ must be singular. Therefore, its determinant must be equal to zero:

$$\det(\mathbf{K} - \omega^2 \mathbf{M}) = 0$$

This equation is known as the Characteristic Equation (or frequency equation). Expanding the determinant results in a polynomial of degree $N$ in terms of $\omega^2$:

$$a_{2N} (\omega^2)^N + a_{2N-2} (\omega^2)^{N-1} + \dots + a_2 (\omega^2) + a_0 = 0$$

Solving this polynomial equation yields $N$ roots: $\omega_1^2, \omega_2^2, \dots, \omega_N^2$.

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The positive square roots $\omega_r$ ($r = 1, 2, \dots, N$) are the natural frequencies of the system.

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By convention, we sort them in ascending order: $\omega_1 \le \omega_2 \le \dots \le \omega_N$.

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$\omega_1$ is called the fundamental natural frequency.

For each natural frequency $\omega_r$, we substitute $\omega^2 = \omega_r^2$ back into the eigenvalue equation:

$$(\mathbf{K} - \omega_r^2 \mathbf{M}) \boldsymbol{\phi}_r = \mathbf{0}$$

Because the matrix $(\mathbf{K} - \omega_r^2 \mathbf{M})$ is singular, we cannot solve for the absolute values of the elements of $\boldsymbol{\phi}_r$. Instead, we can only solve for their relative proportions. This means that the mode shape defines the relative deformation profile of the structure at that frequency. The absolute amplitude of vibration is determined solely by the initial conditions (displacement and velocity) or the magnitude of external excitation.

3.1 Mathematical Properties of Eigenvalues and Eigenvectors

For symmetric, positive-definite mass and stiffness matrices, the eigenvalue problem exhibits key mathematical properties:

Real-Valued Roots:

All eigenvalues $\lambda_r = \omega_r^2$ are real-valued. This guarantees that the natural frequencies $\omega_r$ are either real (representing oscillatory motion) or zero (representing rigid body motion). Complex or imaginary natural frequencies cannot occur in stable, passive, undamped elastic structures.

Positive Roots:

If the stiffness matrix $\mathbf{K}$ is positive-definite (fully constrained system), all eigenvalues $\lambda_r$ are strictly positive ($>0$). If the system is unconstrained, there will be one or more zero-frequency roots (rigid body modes).

3.2 Physical Interpretation of Mode Shapes

Each mode shape $\boldsymbol{\phi}_r$ represents a preferred path of deformation. If an undamped structure is excited with a harmonic force at a frequency exactly equal to a natural frequency $\omega_r$, it will deform according to the shape $\boldsymbol{\phi}_r$. Points on the structure where the mode shape value is zero are called Nodes (or nodal lines/planes). These points remain stationary during vibration in that specific mode. In automotive design, knowledge of these nodal lines is crucial: engine mount locations are ideally positioned near nodal points of major structural modes to minimize force transmission into the passenger cabin.

4. Mode Shape Orthogonality

One of the most important mathematical and physical properties of MDOF systems is the orthogonality of eigenvectors. Orthogonality means that the mode shapes are independent of one another. Specifically, they are orthogonal with respect to the system matrices: the mass matrix $\mathbf{M}$ and the stiffness matrix $\mathbf{K}$.

4.1 Mathematical Proof of Orthogonality

Consider two distinct modes of vibration, mode $r$ and mode $s$, with natural frequencies $\omega_r$ and $\omega_s$ where $\omega_r \neq \omega_s$. Their eigenvalue equations are:

$$(\mathbf{K} - \omega_r^2 \mathbf{M}) \boldsymbol{\phi}_r = \mathbf{0} \implies \mathbf{K} \boldsymbol{\phi}_r = \omega_r^2 \mathbf{M} \boldsymbol{\phi}_r \quad \text{(Eq. A)}$$

$$(\mathbf{K} - \omega_s^2 \mathbf{M}) \boldsymbol{\phi}_s = \mathbf{0} \implies \mathbf{K} \boldsymbol{\phi}_s = \omega_s^2 \mathbf{M} \boldsymbol{\phi}_s \quad \text{(Eq. B)}$$

Pre-multiplying Eq. A by the transpose of eigenvector $s$ ($\boldsymbol{\phi}_s^T$):

$$\boldsymbol{\phi}_s^T \mathbf{K} \boldsymbol{\phi}_r = \omega_r^2 \boldsymbol{\phi}_s^T \mathbf{M} \boldsymbol{\phi}_r \quad \text{(Eq. C)}$$

Next, take the transpose of Eq. B:

$$(\mathbf{K} \boldsymbol{\phi}_s)^T = (\omega_s^2 \mathbf{M} \boldsymbol{\phi}_s)^T \implies \boldsymbol{\phi}_s^T \mathbf{K}^T = \omega_s^2 \boldsymbol{\phi}_s^T \mathbf{M}^T$$

Since $\mathbf{K}$ and $\mathbf{M}$ are symmetric ($\mathbf{K}^T = \mathbf{K}$ and $\mathbf{M}^T = \mathbf{M}$), this simplifies to:

$$\boldsymbol{\phi}_s^T \mathbf{K} = \omega_s^2 \boldsymbol{\phi}_s^T \mathbf{M}$$

Post-multiplying this equation by eigenvector $r$ ($\boldsymbol{\phi}_r$):

$$\boldsymbol{\phi}_s^T \mathbf{K} \boldsymbol{\phi}_r = \omega_s^2 \boldsymbol{\phi}_s^T \mathbf{M} \boldsymbol{\phi}_r \quad \text{(Eq. D)}$$

Subtracting Eq. D from Eq. C:

$$0 = (\omega_r^2 - \omega_s^2) \boldsymbol{\phi}_s^T \mathbf{M} \boldsymbol{\phi}_r$$

Since we assumed that the two frequencies are distinct ($\omega_r \neq \omega_s$), the term $(\omega_r^2 - \omega_s^2)$ cannot be zero. Therefore, the inner product must be zero:

$$\boldsymbol{\phi}_s^T \mathbf{M} \boldsymbol{\phi}_r = 0, \quad \text{for } r \neq s$$

This is the mass orthogonality condition. Substituting this result back into Eq. C or Eq. D, we obtain:

$$\boldsymbol{\phi}_s^T \mathbf{K} \boldsymbol{\phi}_r = 0, \quad \text{for } r \neq s$$

This is the stiffness orthogonality condition.

4.2 Physical Meaning of Orthogonality

The orthogonality of mode shapes has profound physical implications:

Dynamic Independence:

The modes of vibration do not interact with each other. If a system is vibrating purely in mode $r$, the inertial forces generated by this vibration, represented by the vector $\mathbf{f}_{in} = \mathbf{M} \ddot{\mathbf{x}}_r = -\omega_r^2 \mathbf{M} \boldsymbol{\phi}_r$, do no work when acting through the displacements of any other mode $\boldsymbol{\phi}_s$ ($r \neq s$):

$$W = \boldsymbol{\phi}_s^T \mathbf{f}_{in} = -\omega_r^2 \boldsymbol{\phi}_s^T \mathbf{M} \boldsymbol{\phi}_r = 0$$

Energy Partitioning:

The total kinetic and strain energy of an MDOF system undergoing arbitrary motion is simply the sum of the energies in each individual mode. There are no cross-coupling energy terms between different modes. This allows us to partition the response energy-wise, which is the foundation of Statistical Energy Analysis (SEA) and structural troubleshooting.

4.3 Repeated Roots (Degenerate Modes)

If the characteristic equation has repeated roots (e.g., $\omega_r = \omega_s$), the corresponding eigenvectors are not automatically orthogonal. This occurs in symmetric structures, such as circular plates, rings, or square membranes, where multiple mode shapes can exist at the same natural frequency. In this case, any linear combination of these eigenvectors is also an eigenvector. To enforce orthogonality, we use the Gram-Schmidt Orthogonalization Process. If we have a set of non-orthogonal eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots$, we can construct an orthogonal set $\boldsymbol{\phi}_1, \boldsymbol{\phi}_2, \dots$ using:

$$\boldsymbol{\phi}_1 = \mathbf{u}_1$$

$$\boldsymbol{\phi}_2 = \mathbf{u}_2 - \frac{\boldsymbol{\phi}_1^T \mathbf{M} \mathbf{u}_2}{\boldsymbol{\phi}_1^T \mathbf{M} \boldsymbol{\phi}_1} \boldsymbol{\phi}_1$$

$$\boldsymbol{\phi}_k = \mathbf{u}_k - \sum_{j=1}^{k-1} \frac{\boldsymbol{\phi}_j^T \mathbf{M} \mathbf{u}_k}{\boldsymbol{\phi}_j^T \mathbf{M} \boldsymbol{\phi}_j} \boldsymbol{\phi}_j$$

5. Modal Scaling and Normalization

Because eigenvectors represent relative shapes, their magnitudes are mathematically arbitrary. If $\boldsymbol{\phi}_r$ is an eigenvector, then $c \boldsymbol{\phi}_r$ (where $c$ is any non-zero constant) is also an eigenvector. To standardise the mode shapes for computations, we normalize them using one of several schemes.

5.1 Mass-Normalization

This is the most common scaling scheme in structural dynamics, particularly in FEM. The mode shapes are scaled such that the diagonal elements of the modal mass matrix are equal to unity. For eigenvector $r$, we calculate the scalar modal mass $m_r$:

$$m_r = \boldsymbol{\phi}_r^T \mathbf{M} \boldsymbol{\phi}_r$$

The mass-normalized eigenvector $\tilde{\boldsymbol{\phi}}_r$ is then defined as:

$$\tilde{\boldsymbol{\phi}}_r = \frac{1}{\sqrt{m_r}} \boldsymbol{\phi}_r = \frac{1}{\sqrt{\boldsymbol{\phi}_r^T \mathbf{M} \boldsymbol{\phi}_r}} \boldsymbol{\phi}_r$$

When eigenvectors are mass-normalized:

$$\tilde{\boldsymbol{\phi}}_r^T \mathbf{M} \tilde{\boldsymbol{\phi}}_s = \delta_{rs} = \begin{cases} 1 & \text{for } r = s \\ 0 & \text{for } r \neq s \end{cases}$$

$$\tilde{\boldsymbol{\phi}}_r^T \mathbf{K} \tilde{\boldsymbol{\phi}}_s = \omega_r^2 \delta_{rs} = \begin{cases} \omega_r^2 & \text{for } r = s \\ 0 & \text{for } r \neq s \end{cases}$$

where $\delta_{rs}$ is the Kronecker Delta. This scaling simplifies the mathematical expressions for response calculations, particularly for frequency response functions.

5.2 Unity-Maximum Displacement Scaling

In experimental modal analysis (EMA), it is often useful to scale the mode shape such that its largest component has a value of 1.0 (or 100%):

$$\max_{i} |\phi_{ir}| = 1.0$$

This scaling makes the mode shape easy to visualize, as the displacements of all other nodes are expressed relative to the point of maximum motion.

5.3 Unity-DOF Scaling

The eigenvector is scaled such that the displacement at a specific reference degree of freedom $k$ is equal to 1.0:

$$\phi_{kr} = 1.0$$

This is useful when comparing mode shapes from different tests or models, using the same reference node.

Normalization Scheme	Mathematical Definition	Key Advantage	Typical Application
:---	:---	:---	:---
Mass-Normalization	$\tilde{\boldsymbol{\phi}}_r^T \mathbf{M} \tilde{\boldsymbol{\phi}}_r = 1.0$	Decouples and simplifies equations of motion	Finite Element Solvers
Unity-Maximum	$\max_{i} \	\phi_{ir}\	= 1.0$	Excellent for visualization and physical interpretation	Modal Testing / EMA
Unity-DOF	$\phi_{kr} = 1.0$ at ref node $k$	Simplifies calculations relative to a driving point	Localized structural modifications

6. Modal Decoupling and Modal Coordinates

The primary goal of modal analysis is to solve the coupled physical equations of motion:

$$\mathbf{M} \ddot{\mathbf{x}}(t) + \mathbf{C} \dot{\mathbf{x}}(t) + \mathbf{K} \mathbf{x}(t) = \mathbf{f}(t)$$

We can stack all the $N$ mass-normalized eigenvectors side-by-side to form the Modal Matrix $\boldsymbol{\Phi}$:

$$\boldsymbol{\Phi} = \begin{bmatrix} \tilde{\boldsymbol{\phi}}_1 & \tilde{\boldsymbol{\phi}}_2 & \dots & \tilde{\boldsymbol{\phi}}_N \end{bmatrix} = \begin{bmatrix} \phi_{11} & \phi_{12} & \dots & \phi_{1N} \\ \phi_{21} & \phi_{22} & \dots & \phi_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ \phi_{N1} & \phi_{N2} & \dots & \phi_{NN} \end{bmatrix}$$

We introduce a coordinate transformation from physical coordinates $\mathbf{x}(t)$ to modal coordinates $\mathbf{q}(t)$:

$$\mathbf{x}(t) = \boldsymbol{\Phi} \mathbf{q}(t)$$

This means the physical response is a linear combination of the mode shapes, where the modal coordinates $\mathbf{q}(t)$ act as the dynamic scaling factors. This relation is also known as the Modal Expansion Theorem.

Taking time derivatives:

$$\dot{\mathbf{x}}(t) = \boldsymbol{\Phi} \dot{\mathbf{q}}(t)$$

$$\ddot{\mathbf{x}}(t) = \boldsymbol{\Phi} \ddot{\mathbf{q}}(t)$$

Substituting these transformations into the physical equation of motion:

$$\mathbf{M} \boldsymbol{\Phi} \ddot{\mathbf{q}}(t) + \mathbf{C} \boldsymbol{\Phi} \dot{\mathbf{q}}(t) + \mathbf{K} \boldsymbol{\Phi} \mathbf{q}(t) = \mathbf{f}(t)$$

Pre-multiplying the entire equation by the transpose of the modal matrix ($\boldsymbol{\Phi}^T$):

$$(\boldsymbol{\Phi}^T \mathbf{M} \boldsymbol{\Phi}) \ddot{\mathbf{q}}(t) + (\boldsymbol{\Phi}^T \mathbf{C} \boldsymbol{\Phi}) \dot{\mathbf{q}}(t) + (\boldsymbol{\Phi}^T \mathbf{K} \boldsymbol{\Phi}) \mathbf{q}(t) = \boldsymbol{\Phi}^T \mathbf{f}(t)$$

By virtue of the orthogonality relations:

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$\boldsymbol{\Phi}^T \mathbf{M} \boldsymbol{\Phi} = \mathbf{I}$ (Identity Matrix)

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$\boldsymbol{\Phi}^T \mathbf{K} \boldsymbol{\Phi} = \boldsymbol{\Omega}^2 = \begin{bmatrix} \omega_1^2 & 0 & \dots & 0 \\ 0 & \omega_2^2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \omega_N^2 \end{bmatrix}$ (Diagonal Matrix of natural frequencies squared)

If we assume the damping matrix $\mathbf{C}$ is proportional (e.g. Rayleigh damping), then:

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$\boldsymbol{\Phi}^T \mathbf{C} \boldsymbol{\Phi} = \begin{bmatrix} 2 \zeta_1 \omega_1 & 0 & \dots & 0 \\ 0 & 2 \zeta_2 \omega_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 2 \zeta_N \omega_N \end{bmatrix} = 2 \boldsymbol{\Xi} \boldsymbol{\Omega}$

where $\zeta_r$ is the modal damping ratio of the $r$-th mode.

Substituting these diagonalized matrices back, we obtain:

$$\mathbf{I} \ddot{\mathbf{q}}(t) + 2 \boldsymbol{\Xi} \boldsymbol{\Omega} \dot{\mathbf{q}}(t) + \boldsymbol{\Omega}^2 \mathbf{q}(t) = \mathbf{Q}(t)$$

where $\mathbf{Q}(t) = \boldsymbol{\Phi}^T \mathbf{f}(t)$ is the modal force vector. Writing this matrix equation for a single coordinate $q_r(t)$:

$$\ddot{q}_r(t) + 2 \zeta_r \omega_r \dot{q}_r(t) + \omega_r^2 q_r(t) = Q_r(t), \quad r = 1, 2, \dots, N$$

Each equation represents an uncoupled, independent SDOF system! We can solve for $q_r(t)$ using standard analytical or numerical methods for SDOF systems. Once the modal coordinates $q_r(t)$ are calculated, we transform back to physical space:

$$\mathbf{x}(t) = \sum_{r=1}^N \tilde{\boldsymbol{\phi}}_r q_r(t)$$

This process is highly powerful because for large structures (e.g. automobile body models with millions of DOFs), most of the response is governed by the lower-frequency modes. We can truncate the modal matrix, keeping only the first $M$ modes ($M \ll N$), reducing the system from millions of equations to just a few dozen, with minimal loss of accuracy. This is called Modal Truncation.

7. Advanced Damping Formulations

While proportional damping is convenient, real physical systems often exhibit non-proportional damping. This occurs when:

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Damping materials (viscoelastic patches) are applied locally to a structure.

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Different materials (e.g., steel and rubber mounts) are assembled together.

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Fluid-structure interactions occur (shock absorbers, hydraulic dampers).

7.1 State-Space Formulation for Non-Proportional Viscous Damping

When the damping matrix $\mathbf{C}$ is non-proportional, the modal matrix $\boldsymbol{\Phi}$ does not diagonalize it:

$$\boldsymbol{\Phi}^T \mathbf{C} \boldsymbol{\Phi} = \mathbf{C}_{modal} \quad \text{(non-diagonal)}$$

To solve this system, we recast the $N$ second-order coupled differential equations into $2N$ first-order equations. This is the State-Space Formulation (or Hamiltonian form).

We define a state vector $\mathbf{y}(t) \in \mathbb{R}^{2N}$:

$$\mathbf{y}(t) = \begin{bmatrix} \mathbf{x}(t) \\ \dot{\mathbf{x}}(t) \end{bmatrix}$$

The coupled second-order system:

$$\mathbf{M} \ddot{\mathbf{x}}(t) + \mathbf{C} \dot{\mathbf{x}}(t) + \mathbf{K} \mathbf{x}(t) = \mathbf{f}(t)$$

along with the identity:

$$\mathbf{M} \dot{\mathbf{x}}(t) - \mathbf{M} \dot{\mathbf{x}}(t) = \mathbf{0}$$

can be written in the matrix first-order form:

$$\mathbf{A} \dot{\mathbf{y}}(t) + \mathbf{B} \mathbf{y}(t) = \mathbf{F}_{state}(t)$$

where:

$$\mathbf{A} = \begin{bmatrix} \mathbf{C} & \mathbf{M} \\ \mathbf{M} & \mathbf{0} \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} \mathbf{K} & \mathbf{0} \\ \mathbf{0} & -\mathbf{M} \end{bmatrix}, \quad \mathbf{F}_{state}(t) = \begin{bmatrix} \mathbf{f}(t) \\ \mathbf{0} \end{bmatrix}$$

Note that both $\mathbf{A}$ and $\mathbf{B}$ are symmetric matrices of size $2N \times 2N$. Assuming free vibration $\mathbf{y}(t) = \boldsymbol{\psi} e^{s t}$, we obtain the eigenvalue problem:

$$(s \mathbf{A} + \mathbf{B}) \boldsymbol{\psi} = \mathbf{0}$$

Solving this eigenvalue problem yields:

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$2N$ complex conjugate eigenvalues: $s_r, s_r^* = -\zeta_r \omega_r \pm i \omega_r \sqrt{1 - \zeta_r^2}$.

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$2N$ complex conjugate eigenvectors: $\boldsymbol{\psi}_r, \boldsymbol{\psi}_r^*$.

7.2 Physical Meaning of Complex Mode Shapes

In proportional damping, the mode shapes are real-valued, representing standing waves. This means all points on the structure reach their maximum displacements and cross zero at the exact same instant of time (they are either completely in-phase or $180^\circ$ out-of-phase). In non-proportional damping, the eigenvectors are complex-valued, representing traveling waves. Different parts of the structure exhibit phase shifts relative to each other. Consequently, there are no fixed nodal points; instead, nodal regions move along the structure, indicating active transmission of vibration energy through the system.

8. Model Reduction Techniques

In industrial applications, finite element models of vehicle bodies or rocket frames can exceed millions of degrees of freedom. Running transient dynamic simulations on such large models is computationally prohibitive. Therefore, Model Reduction is used to reduce the size of matrices while preserving the low-frequency dynamic characteristics.

8.1 Guyan (Static) Reduction

Guyan reduction partitions the degrees of freedom into master (active, $a$) DOFs and slave (omitted, $o$) DOFs. The static relationship is:

$$\begin{bmatrix} \mathbf{K}_{aa} & \mathbf{K}_{ao} \\ \mathbf{K}_{oa} & \mathbf{K}_{oo} \end{bmatrix} \begin{bmatrix} \mathbf{x}_a \\ \mathbf{x}_o \end{bmatrix} = \begin{bmatrix} \mathbf{f}_a \\ \mathbf{0} \end{bmatrix}$$

Assuming no external forces act on the slave DOFs:

$$\mathbf{K}_{oa} \mathbf{x}_a + \mathbf{K}_{oo} \mathbf{x}_o = \mathbf{0} \implies \mathbf{x}_o = -\mathbf{K}_{oo}^{-1} \mathbf{K}_{oa} \mathbf{x}_a$$

We construct the transformation matrix:

$$\mathbf{x} = \begin{bmatrix} \mathbf{x}_a \\ \mathbf{x}_o \end{bmatrix} = \begin{bmatrix} \mathbf{I} \\ -\mathbf{K}_{oo}^{-1} \mathbf{K}_{oa} \end{bmatrix} \mathbf{x}_a = \mathbf{T}_g \mathbf{x}_a$$

The reduced mass and stiffness matrices are:

$$\mathbf{K}_{red} = \mathbf{T}_g^T \mathbf{K} \mathbf{T}_g = \mathbf{K}_{aa} - \mathbf{K}_{ao} \mathbf{K}_{oo}^{-1} \mathbf{K}_{oa}$$

$$\mathbf{M}_{red} = \mathbf{T}_g^T \mathbf{M} \mathbf{T}_g = \mathbf{M}_{aa} - \mathbf{M}_{ao} \mathbf{K}_{oo}^{-1} \mathbf{K}_{oa} - \mathbf{K}_{ao} \mathbf{K}_{oo}^{-1} \mathbf{M}_{oa} + \mathbf{K}_{ao} \mathbf{K}_{oo}^{-1} \mathbf{M}_{oo} \mathbf{K}_{oo}^{-1} \mathbf{K}_{oa}$$

While stiffness is reduced exactly for static loads, Guyan reduction approximates the mass distribution, which can lead to overestimation of natural frequencies for high-frequency modes.

8.2 Craig-Bampton Method (Component Mode Synthesis)

The Craig-Bampton method is the industry standard for component mode synthesis (CMS). It partitions DOFs into boundary ($b$) DOFs (where the component connects to other substructures) and internal ($i$) DOFs. The coordinate transformation combines two sets of modes:

Constraint Modes ($\\mathbf{\\Phi}_c$):

Deformation shapes of internal DOFs when a unit displacement is applied to one boundary DOF while keeping all other boundary DOFs fixed.

Fixed-Interface Normal Modes ($\\mathbf{\\Phi}_n$):

Vibration mode shapes of the internal DOFs when all boundary DOFs are clamped.

The transformation is written as:

$$\mathbf{x} = \begin{bmatrix} \mathbf{x}_b \\ \mathbf{x}_i \end{bmatrix} = \begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{\Phi}_{ib} & \mathbf{\Phi}_n \end{bmatrix} \begin{bmatrix} \mathbf{x}_b \\ \mathbf{q} \end{bmatrix} = \mathbf{T}_{cb} \mathbf{u}_{cb}$$

where $\mathbf{\Phi}_{ib} = -\mathbf{K}_{ii}^{-1} \mathbf{K}_{ib}$, and $\mathbf{q}$ are the modal coordinates of the fixed-interface modes. Substituting this into the system matrices yields highly reduced, block-diagonal matrices that represent boundary coupling and internal modal dynamics. This allows sub-system models (e.g., an engine block) to be shared between manufacturers and assembled into a vehicle model efficiently.

9. Frequency Response Function (FRF) Matrices

In NVH engineering, we are primarily interested in the system response in the frequency domain. If we excite a structure with a harmonic force:

$$\mathbf{f}(t) = \mathbf{F} e^{i \omega t}$$

The steady-state physical displacement response will also be harmonic at the same frequency:

$$\mathbf{x}(t) = \mathbf{X} e^{i \omega t}$$

where $\mathbf{F}$ and $\mathbf{X}$ are complex vectors containing amplitude and phase information. Substituting these into the equation of motion:

$$(-\omega^2 \mathbf{M} + i \omega \mathbf{C} + \mathbf{K}) \mathbf{X} e^{i \omega t} = \mathbf{F} e^{i \omega t}$$

Dividing by $e^{i \omega t}$ defines the Dynamic Stiffness Matrix $\mathbf{Z}(\omega)$:

$$\mathbf{Z}(\omega) \mathbf{X} = \mathbf{F}$$

$$\mathbf{Z}(\omega) = \mathbf{K} - \omega^2 \mathbf{M} + i \omega \mathbf{C}$$

The inverse of the dynamic stiffness matrix is the Receptance (Displacement FRF) Matrix $\mathbf{H}(\omega)$:

$$\mathbf{X} = \mathbf{H}(\omega) \mathbf{F}$$

$$\mathbf{H}(\omega) = \mathbf{Z}(\omega)^{-1} = (\mathbf{K} - \omega^2 \mathbf{M} + i \omega \mathbf{C})^{-1}$$

9.1 Types of FRFs

Depending on whether we measure displacement, velocity, or acceleration, we define three types of FRFs:

Receptance ($H_{jk} = X_j / F_k$): Displacement response at DOF $j$ due to force at DOF $k$.

Mobility ($Y_{jk} = V_j / F_k = i \omega H_{jk}$): Velocity response at DOF $j$ due to force at DOF $k$.

Inertance/Accelerance ($A_{jk} = A_j / F_k = -\omega^2 H_{jk}$): Acceleration response at DOF $j$ due to force at DOF $k$.

9.2 Modal Expansion of Receptance

Computing the matrix inverse $\mathbf{H}(\omega) = (\mathbf{K} - \omega^2 \mathbf{M} + i \omega \mathbf{C})^{-1}$ directly for a range of frequencies is computationally expensive for large systems. Using the modal matrix, we can express the receptance matrix as a summation over the modes:

$$\mathbf{H}(\omega) = \sum_{r=1}^N \frac{\tilde{\boldsymbol{\phi}}_r \tilde{\boldsymbol{\phi}}_r^T}{\omega_r^2 - \omega^2 + 2 i \zeta_r \omega_r \omega}$$

For an individual element of the matrix, $H_{jk}(\omega)$ (response at node $j$ due to force at node $k$):

$$H_{jk}(\omega) = \sum_{r=1}^N \frac{\phi_{jr} \phi_{kr}}{\omega_r^2 - \omega^2 + 2 i \zeta_r \omega_r \omega}$$

This is the Modal Representation of the FRF. It shows that the transfer function between any two points is a superposition of the contribution of all modes of the structure.

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The numerator $\phi_{jr} \phi_{kr}$ is the product of mode shape displacements at the input and output DOFs.

It is called the modal participation or modal residue.

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The denominator determines the resonance behavior. When the excitation frequency $\omega$

approaches a natural frequency $\omega_r$, the term $(\omega_r^2 - \omega^2)$ becomes zero, and the response is limited only by the damping term $2 i \zeta_r \omega_r \omega$, resulting in a peak in the FRF.

9.3 Driving-Point vs. Cross-Point FRFs

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Driving-Point FRF ($H_{jj}$):

The excitation force and response measurement are at the exact same degree of freedom.

$$H_{jj}(\omega) = \sum_{r=1}^N \frac{\phi_{jr}^2}{\omega_r^2 - \omega^2 + 2 i \zeta_r \omega_r \omega}$$

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Because the numerator is $\phi_{jr}^2$, the residue is always positive.

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This means that in the phase angle plot of a driving-point FRF,

the phase always drops by $180^\circ$ at resonance, and there is a corresponding antiresonance (phase jump of $+180^\circ$) between every pair of resonances. The phase is bounded between $0^\circ$ and $-180^\circ$ for receptance.

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Cross-Point FRF ($H_{jk}$ where $j \neq k$):

The response is measured at a different DOF than the excitation force.

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The residue $\phi_{jr} \phi_{kr}$ can be positive or negative,

depending on whether the two points move in-phase or out-of-phase in that mode.

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Cross-point FRFs may not exhibit antiresonances, and the phase can exceed $-180^\circ$.

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Reciprocity:

Because the matrices $\mathbf{M}$, $\mathbf{K}$, and $\mathbf{C}$ are symmetric, the receptance matrix is also symmetric:

$$H_{jk}(\omega) = H_{kj}(\omega)$$

This means the response at node $j$ due to force at node $k$ is identical to the response at node $k$ due to force at node $j$. This is a vital property for checking the quality of experimental modal testing data.

10. Dynamic Modifications and Structural Tuning

Once a structural dynamic model is established, engineers often need to modify the structure to shift problematic natural frequencies or reduce resonance amplitudes.

10.1 Point Mass and Spring Modifications

Adding a point mass $\Delta m$ at degree of freedom $p$ alters the mass matrix:

$$\mathbf{M}^* = \mathbf{M} + \mathbf{\Delta M}$$

where $\mathbf{\Delta M}$ contains only one non-zero entry: $\Delta M_{pp} = \Delta m$. Using first-order perturbation theory, the shift in natural frequency of mode $r$ can be approximated:

$$\Delta(\omega_r^2) \approx -\omega_r^2 \phi_{pr}^2 \Delta m$$

This formula shows that adding mass at a node where the mode shape displacement is large (antinode) maximally decreases that mode's natural frequency. Adding mass at a node line (where $\phi_{pr} = 0$) has no effect on that frequency.

Similarly, adding a spring of stiffness $\Delta k$ between DOF $p$ and the ground changes the stiffness matrix:

$$\mathbf{K}^* = \mathbf{K} + \mathbf{\Delta K}$$

where $\Delta K_{pp} = \Delta k$. The first-order frequency shift is:

$$\Delta(\omega_r^2) \approx \phi_{pr}^2 \Delta k$$

Adding stiffness at an antinode increases the natural frequency, while adding it at a nodal line has no effect.

10.2 Tuned Mass Dampers (TMDs)

A Tuned Mass Damper is an auxiliary SDOF system (mass $m_d$, spring $k_d$, damper $c_d$) attached to a primary structure to suppress vibrations at a specific resonance frequency. For a single-mode representation of the primary structure (mass $M_p$, stiffness $K_p$), attaching a TMD results in a 2-DOF coupled system. The optimal tuning parameters, derived by Den Hartog, are:

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Mass Ratio: $\mu = m_d / M_p$.

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Optimal Frequency Ratio: $g_{opt} = \omega_d / \omega_p = \frac{1}{1 + \mu}$.

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Optimal TMD Damping Ratio: $\zeta_{d,opt} = \sqrt{\frac{3 \mu}{8 (1 + \mu)^3}}$.

A properly tuned TMD splits the single peak of the primary structure's FRF into two much smaller peaks, significantly reducing the overall vibration levels.

11. Experimental Modal Analysis (EMA) & Model Correlation

11.1 Experimental Setup and Excitation Techniques

Experimental Modal Analysis is the process of extracting modal parameters (natural frequencies, damping ratios, and mode shapes) from physical measurements. The structure is excited using a known force input, and the acceleration response is measured.

Impact Hammer Testing:

A hammer equipped with a force transducer is used to strike the structure, generating a broad-band impulse. This is fast, requires minimal setup, and is suitable for light-to-medium structures.

Shaker Testing:

An electromagnetic shaker is attached to the structure via a thin rod (stinger). The shaker can excite the structure using random noise, burst random, or sine sweep signals. Shakers provide higher energy input and are preferred for heavy structures or detailed damping characterization.

11.2 Signal Processing and Windowing

The measured force and acceleration signals are converted to the frequency domain using the Fast Fourier Transform (FFT). To prevent spectral leakage (leakage of signal energy across frequency bins due to non-periodic window truncation), windowing functions are applied:

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Force Window:

Used in impact testing. It forces the signal to zero after the impulse, removing background noise.

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Exponential Window:

Used on the response channel in impact testing. It adds artificial damping to force the response signal to decay to zero within the measurement window, eliminating leakage but requiring a post-test correction to subtract the added damping.

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Hann Window:

Used in continuous random shaker testing. It smoothly attenuates the signal to zero at both ends of the time window.

11.3 Modal Parameter Estimation (Curve Fitting)

The frequency-domain relationship is modeled using parameter estimation algorithms.

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Circle-Fitting Method:

Near a resonance frequency $\omega_r$, the polar plot (Nyquist plot) of the receptance $H_{jk}(\omega)$ approximates a circle. The center and diameter of the circle are used to extract:

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Natural frequency $\omega_r$ (where the sweep rate along the circle is maximum).

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Damping ratio $\zeta_r$ (from the spacing of frequency points).

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Modal residue (from the diameter of the circle).

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Rational Fraction Polynomial (RFP) Method:

An analytical curve-fitting routine that minimizes the error between the experimental FRF and the ratio of two polynomials representing the transfer function:

$$H(\omega) = \frac{\sum_{j=0}^{m} a_j (i \omega)^j}{\sum_{k=0}^{n} b_k (i \omega)^k}$$

The roots of the denominator polynomial define the poles (natural frequencies and damping), while the numerator determines the residues (mode shapes).

11.4 Modal Assurance Criterion (MAC)

The Modal Assurance Criterion is a statistical indicator used to compare two sets of mode shapes (for example, comparing analytical mode shapes from an FE model $\boldsymbol{\phi}_{FE}$ with experimental mode shapes from an impact test $\boldsymbol{\phi}_{EMA}$). The MAC between mode $r$ and mode $s$ is defined as:

$$\text{MAC}(r,s) = \frac{|\boldsymbol{\phi}_{r,FE}^T \boldsymbol{\phi}_{s,EMA}^*|^2}{(\boldsymbol{\phi}_{r,FE}^T \boldsymbol{\phi}_{r,FE}) (\boldsymbol{\phi}_{s,EMA}^T \boldsymbol{\phi}_{s,EMA}^*) }$$

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A MAC value close to $1.0$ (or 100%) indicates highly correlated, identical mode shapes.

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A MAC value near $0.0$ indicates orthogonal, unrelated mode shapes.

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The results are plotted as a 3D matrix. A successful correlation should show high diagonal values

(approaching $1.0$) and low off-diagonal values (below $0.15$), confirming that the FE model accurately represents the real-world structure's dynamics.