ArticleFinite Element Analysis: A Numerical Tool For Vibration Analysis
by Robert J. Sayer, P.E.
This paper serves as an introduction to finite element analysis as a tool for the vibration analyst. A general description of the technique is given along with factors that affect its accuracy such as mesh size and boundary conditions. The FEA technique is then compared with experimental modal analysis (EMA) to show differences and to illustrate its strengths. FEA strengths over EMA such as determining natural frequencies and mode shape prior to equipment fabrication and installation, estimating the effects on natural frequencies & mode shapes following structural modifications, and modal studies on very large structures are given.
“Introduction to FEA Method:
Part of any root-cause vibration study should include a solution that addresses the problems identified by the study. Solutions for vibration problems involving the excitation of a natural frequency (resonance) are frequently difficult to obtain based solely upon experimental data. Although an intuitive interpretation of the experimental data may suggest stiffening a component of a machine, the details and the effectiveness of a modification could be difficult to access. Furthermore, in many cases it is difficult to determine the effect of a modification on other natural frequencies that may not currently be a problem, but could become one after implementation of the modification.
The finite element method is a numerical technique that can be used to obtain an approximation of the modal parameters (natural frequencies and mode shapes) of complex structural-mechanical systems. A finite element model is typically much more detailed than an experimental model (operating deflection shape or experimental modal analysis). Finite element models contain thousands of degrees-of-freedom versus an experimental model that usually include hundreds of degrees-of-freedom or less. The half-symmetry finite element model of an autogenous grinding mill included in Figure 1 contained over 100,000 degrees-of-freedom. Because of this level of refinement, a finite element model provides the vibration analysts with an unparalleled approach at evaluating the effect of even the finest details of any mechanical design on the structural dynamic characteristics thereof.
The finite element method focuses on calculating the behavior and response of a continuum consisting of an infinite number of points. In a continuum problem, a field variable such as displacement or velocity contains an infinite number of possible values, since it is a function of each point in the continuum. The task of solving the continuum problem is simplified using a the finite element representation that divides the continuum into a finite number of subdivisions called elements. The elements are connected at nodal points into a mesh or finite element model.
The stiffness and inertial properties of each individual element is defined by a mathematical displacement or interpolation function (linear, quadratic, etc.), the elastic material properties (Modulus of Elasticity, Poisson ratio, and material density), the size of the element and its connectivity and relationship to all other elements in the model. The assemblage of all elements produces an estimation of the stiffness and inertial properties of any complex structural-mechanical system.
It is important to realize that the finite element method is an approximate numerical technique.”
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