Modal Analysis

Modal Analysis is the examination of the dynamic properties of a structure.

The modes of a structure can be obtained from two very different approaches:

• Analytical Modal Analysis

• Experimental Modal Analysis

In their most basic form, mathematical models "discretize" a structure by breaking it up into hundreds, thousands, or even millions of masses and springs. This process can be done by the simple lumped mass and lumped spring approach, or it can be done by using the finite element (FE) approach.

Each element in an FE model can be conceptualized as a mass/spring structure. This modeling process simply reduces the complicated structure into many (for example 1000) mass/spring structures. Often this representation is further simplified by selecting a subset of the total problem (called master degrees-of-freedom) and using a matrix condensation technique, such as Guyan reduction, to reduce the size of the problem (for example to 100 degrees-of-freedom).

Analytical Modal Analysis

Analytical Modal Analysis solves an eigenvalue problem to get the frequency and mode shape of each mode for the assumed mass and stiffness distribution. After the modes are defined, another computer run can be performed to examine how that structure will respond to various dynamic inputs. One of the forced response runs that is often used is one in which you input a variable frequency unit force at one point while monitoring the response as a function of frequency at several locations. This data (that is, response per unit input force versus frequency) is a frequency response function (or transfer function).

Experimental Modal Analysis

Experimental Modal Analysis starts out with measured response data (the frequency response functions) and extracts the modes of vibration directly from the frequency response functions without having to make any assumptions about the mass and stiffness distribution. An Experimental Modal Analysis ends up with a set of modes defined by frequency, damping, mode shape, and residue. Certain assumptions and post-processing of the data is necessary to calculate modal mass and stiffness. Even further assumptions and post-processing are necessary to obtain physical stiffness and mass representations.

Equation of motion

The equation of motion for an N degree of freedom structure with viscous damping is:

[M] {}+ [C] {}+ [K] {q}= {f}	Eq. 1

where

[M]	Mass matrix
[C]	Viscous damping matrix
[K]	Stiffness matrix
{f}	Excitation force vector
{q}	Displacement vector
{}	Velocity vector
{}	Acceleration vector

Eigenvalue analysis yields real normal modes when [C] is proportional to [K] and [M], for example if damping is distributed through a structure in the same manner as mass and stiffness. The analysis yields complex modes when [C] is not proportional to [K] and [M], and standard methods of eigenvalue analysis cannot be used conveniently. Lumped damping elements, for example, generally create non-proportional damping and, hence, lead to complex modes.

Real normal modes

To find the real normal modes and natural frequencies in the case when [C] is proportional to [K] and [M], first an N degree-of-freedom undamped free structure is studied. If you seek a solution of the form {q}= {Q}eiωt where ω is the frequency, the matrix form of the linear equations of motion of an N degree of freedom undamped free structure becomes:

- ω2 [M] {Q}+ [K] {Q}= {0}	Eq. 2

This set of equations only has a solution if the determinant of the coefficient matrix is zero:

det {[K] - ω2 [M]} = 0

There is a set of N eigenvalues: ω1, ω2, ω3, ..., ω**N which yield a zero determinant. These are the natural resonant frequencies of the undamped structure.

Associated with each eigenvalue ωr there is an eigenvector {Ψ}r, which is a solution of Eq. 2. There are N eigenvectors which are the real normal modes of this structure.

As the N eigenvectors form a linearly independent set of vectors in the N-space, the following orthogonality conditions apply:

If ωr ≠ ωs,

then:

• {Ψ}rT[M]{Ψ}s = 0 forr ≠ s

• {Ψ}rT[K]{Ψ}s = 0 for r ≠ s

As [C] is proportional to [K] and [M]: [C] = α[K] + β[M], it follows that:

{Ψ}rT[C]{Ψ}s = 0 for r ≠ s

Which means that the N eigenvectors: {Ψ}1, {Ψ}2, {Ψ}3, ..., {Ψ}N are also the real normal modes of the damped structure of N degrees-of-freedom.

This proportional viscous damped system has N eigenvalues: s1, s2, s3, ..., s2N that are complex natural frequencies with an imaginary (oscillatory) part and a real (decay) part.

Complex normal modes

In the case when [C] is not proportional to [K] and [M], the general equation of motion Eq. 1 can be rewritten in the following way:

[A] {} + [B] {y} = {z}	Eq. 3

where

The matrix equation in Eq. 3 is composed of 2N linear equations where N is the number of degrees of freedom of the structure. By setting the right-hand side of Eq. 3 to zero, you can solve for the natural frequencies and mode shapes of the structure. If you seek a solution of the form {y} = {Y}est, Eq. 3 becomes:

s [A] {Y}+ [B] {Y}= {0}	Eq. 4

This set of equations only has a solution if the determinant of the coefficient matrix is zero:

det {[B] + s [A]} = 0

There is a set of 2N eigenvalues: s1, s2, s3, ..., s2N which yield a zero determinant. These are the damped natural frequencies of the structure.

Associated with each eigenvalue sr there is an eigenvector {Ψ}r having 2N components satisfying Eq. 4. There are 2N eigenvectors which are the complex normal modes of this structure.

Both the eigenvalues and the eigenvectors are complex, and they occur in conjugate pairs.

Learn more

Pre-test solution process

Correlation solution process

Look up more details

Converting complex modes to real modes

Accounting for repeated modes

Modal Scale Factor (MSF)

Modal Assurance Criteria (MAC)

Coordinate MAC (COMAC)

Cross-orthogonality (X-Ortho)

Frequency Response Assurance Criterion (FRAC)

Min-MAC algorithm

MODMAC algorithm

Normal Mode Indicator Function (NMIF) algorithm

Driving Point Residue algorithm

Scientific literature references for correlation

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