Modal analysis is a powerful engineering tool used to determine the most basic dynamic properties of a structure.  These properties include a particular set frequencies and a corresponding set of deformation patterns that can be used to determine how the structure will respond under various dynamic loads.  The deflection patterns are called "mode shapes" and the frequencies are called "modal frequencies" or "natural frequencies".  Every unique structure has its own set of modal frequencies and mode shapes that define that structure dynamically just like a fingerprint.  These modal properties are the frequencies and shapes that are seen during free vibration in the absence of any dynamic loads.  When dynamic loads are applied at frequencies near the modal frequencies then the phenomenon of dynamic amplification can occur which is also called resonance.  The condition of resonance, particularly at lower frequencies, usually generates the maximum dynamic deflections and stresses within a structure.  So the greatest danger of structural failure under dynamic loading usually involves resonance.  Modal analysis is an essential part of the overall dynamic analysis process to ensure that a structure can withstand its required dynamic loading environment and that the effects of resonance can be effectively mitigated. 

The results from a modal analysis typically include the list of relevant modal frequencies and mode shapes along with other important results such as the modal participation factors and modal masses.  These modal results are commonly used as input for a subsequent dynamic analysis in which the actual dynamic response of the structure is predicted under specified dynamic input loads.  So a modal analysis is usually the first step of an overall dynamic analysis to predict the structural response under dynamic loads such as shock and vibration.  The results from the overall dynamic analysis will then include the actual predicted stresses, strains, deflections, velocities, and accelerations that can be used for structural design requirement verification.

The modal properties of a structure depend primarily on two features:  the mass distribution and the stiffness distribution.  That is, the spatial variation of mass and stiffness within the structure determines its modal properties.  In addition, the manner in which the structure is supported or constrained at its boundaries can also be important in a modal analysis.   So a major part of any modal analysis involves properly specifying and computing the mass, geometry, and stiffness properties of the structure along with the proper boundary conditions.   For the most general 3D problems mass distribution really means the full inertia distribution which includes the translational mass and the rotational inertia tensor.  Likewise, the stiffness distribution really means all the 3D translational and rotational stiffness tensor components.

Modal analysis can thus be defined as the process of computing the modal properties of a structure given the mass and stiffness distributions within the structure and any applicable boundary conditions.  In the text that follows I will attempt to provide a brief overview of this process including the most common techniques along with the underlying assumptions and limitations.  I will show several examples of how finite element analysis (FEA) is used for doing modal analysis of simple and complex structures.  I will also show how the results from a modal analysis can then be used to predict structural responses under various dynamic loading environments including shock and vibration.  Finally, I will present a list of essential engineering references for the engineer interested in doing modal analysis.

Doing a modal analysis can be simple or complex depending on the design requirements and the complexity of the structure being analyzed.  The simplest analyses can be done in a matter of minutes using standard engineering formulas.  The most complex analyses can require months and can include large FEA models supplemented by expensive and time-consuming modal test measurements.  No matter how simple or complex, all modal analyses will require as input the geometry, mass, and stiffness of the structure.  The analysis output will include all the requested modal properties that depend on the design requirements.  For the simplest analysis the required output is only the first or lowest natural frequency in any single direction.  For the most complex analyses the required output may include hundreds of natural frequencies, mode shapes, modal masses, and participation factors in 3D.  A modal analysis typically involves the following steps:

1. Review design requirements to determine the frequency range of interest based on the shock and vibration environments that the structure must withstand.

2. Determine the structural domain to be analyzed.

3. Obtain drawings and select the proper dimensions and tolerances

4. Simplify the overall structure into a collection of smaller substructures as necessary while properly accounting for the materials and material boundaries within each substructure.

5. Measure or compute mass and stiffness properties.

6. Properly handle any internal joints and fittings.

7. Properly account for applicable boundary conditions.

8. If necessary build a single or multiple finite element models with proper mesh densities based on the required frequency range of interest.

9. Select the proper formulas or numerical algorithms and then compute the modal results.

All of the above steps require considerable engineering judgment and experience if the analysis is to be done correctly.  From circuit boards to bridges, the consequences of incorrect modal analysis can include catastrophic structural failure and loss of life.  So modal analysis is a vital part of the overall design verification process for most important and high-value structures that are subjected to dynamic environments.

Modal properties are computed from the dynamic equations of motion in the absence of damping and without any external forcing functions.  This results in a system of linear homogeneous second order differential equations with a unique set of eigenvalues and eigenvectors that are directly related to the desired modal properties.  Computing such eigensolutions for complex structures cannot typically be done analytically with engineering formulas and instead usually requires intensive numerical techniques such as 3D finite element analysis (FEA).  Several sets of modal FEA results are presented below.  Many simple structures such as circular plates, pipes, and beams do have closed form analytical formulas for their modal properties.  Ref. 1 provides an extensive collection of modal analysis formulas for many simple geometries. 

Modal analysis and the modal dynamic analysis that is done with the modal analysis results is based on a set of assumptions.  These assumptions include .

Linear problems only with constant load path.

No buckling.

 

Good for frequency domain problems such as random vibration response, shock response, frequency response, and modal transient response.

 

 

Direct transient analysis in time domain

Explicit or implicit time marching

Linear and nonlinear problems

Variable load path including contact and impact problems

Nonlinear material response

Nonlinear buckling

Nonlinear kinematics (large strain, large rotation, stress stiffening)

Thermomechanical coupling

 

 

 

 

 

 

 

 

 

1. Robert D. Blevins, Formulas for Natural Frequencies and Mode Shapes, Krieger, ISBN 1575241846.

2. Dave S. Steinberg, Vibration Analysis for Electronic Equipment, Wiley Interscience, ISBN 047137685X.

3. Allen Pierson, et. al, Harris' Shock and Vibration Handbook, McGraw-Hill,, ISBN 0071508198.