Modal analysis using FEM

The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of thecalculations are acceptable. The types of equations which arise from modal analysis are those seen in eigensystems. The physical interpretation of the eigenvalues and eigenvectors which come from solving the system are thatthey represent the frequencies and corresponding mode shapes. Sometimes, the only desired modes are the lowest frequencies because they can be the most prominent modes at which the object will vibrate, dominating all the higher frequencymodes. The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of thecalculations are acceptable. The types of equations which arise from modal analysis are those seen in eigensystems. The physical interpretation of the eigenvalues and eigenvectors which come from solving the system are thatthey represent the frequencies and corresponding mode shapes. Sometimes, the only desired modes are the lowest frequencies because they can be the most prominent modes at which the object will vibrate, dominating all the higher frequencymodes. It is also possible to test a physical object to determine its natural frequencies and mode shapes. This is called an Experimental Modal Analysis. The results of the physical test can be used to calibrate a finite element model to determine if the underlying assumptions made were correct (for example, correct material properties and boundary conditions were used). For the most basic problem involving a linear elastic material which obeys Hooke's Law,the matrix equations take the form of a dynamic three-dimensional spring mass system.The generalized equation of motion is given as: where [ M ] {displaystyle } is the mass matrix, [ U ¨ ] {displaystyle } is the 2nd time derivative of the displacement [ U ] {displaystyle } (i.e., the acceleration), [ U ˙ ] {displaystyle } is the velocity, [ C ] {displaystyle } is a damping matrix, [ K ] {displaystyle } is the stiffness matrix, and [ F ] {displaystyle } is the force vector. The general problem, with nonzero damping, is a quadratic eigenvalue problem. However, for vibrational modal analysis, the damping is generally ignored, leaving only the 1st and 3rd terms on the left hand side: This is the general form of the eigensystem encountered in structuralengineering using the FEM. To represent the free-vibration solutions of the structure harmonic motion is assumed, so that [ U ¨ ] {displaystyle } is taken to equal λ [ U ] {displaystyle lambda } ,where λ {displaystyle lambda } is an eigenvalue (with units of reciprocal time squared, e.g., s − 2 {displaystyle mathrm {s} ^{-2}} ),and the equation reduces to: