## What is finite element analysis (FEA)?

Finite element analysis or FEA is a computational tool that evaluates the approximate solution for a particular engineering problem. The tools provided by FEA help to evaluate the solutions of partial differential equations (PDEs) computationally following a numerical approach. Solutions of the PDEs can also be performed by following traditional algebraic approaches, but when the problem gets towards the complex side, it becomes very tedious and almost impossible to arrive at the solutions. In such cases, the FEA fulfills the purpose by evaluating the equations following multiple time steps and iterations. The more the time steps and iterations, the more accurate the solution. The solution plotted by FEA analysis and mathematical approaches varies asymptotically.

A variety of problems in mechanics can be well handled by FEA analysis, such as static and dynamic problems. Static problems are those when the body under analysis is assumed to be in equilibrium under the action of forces, there is no motion in the body under analysis. When the body is in motion, the various forces that cause the motion along with the characteristics are analyzed. In dynamics, both motion and cause of motion are considered. The algorithm of FEA is robust to evaluate both problems in statics and dynamics approximately. The algorithm of FEA is extensively applied to software like ABAQUS, which provides powerful meshing and simulation tools that consider the effects of transient and dynamic loads and aid in the solution of many linear and non-linear vibration problems and noise-related complex engineering problems.

## Steps involved in FEA

FEA approaches a particular engineering problem by following a sequence of steps, these steps are broken down in this section and discussed in detail.

**Discretization**

As the methodology of FEA is an approximation, the particular domain to be analyzed is broken down into several sub-parts or sub-domains by the use of finite elements, known as the discretization of the domain. These finite elements are interlinked at the nodes and form a network that covers the whole domain. Depending upon the complexity of the problem, different finite elements such as linear elements, spring elements, shell elements, quadrilateral, triangular elements, can be chosen from a wide database.

**Formation of mass and stiffness matrices**

In any engineering problem, we normally have the given information regarding the external influences like forces or loads, torques, moments, reactions, and so on. Our main goal lies in the determination of the effect that these influences bring on the body, such as deformation, acceleration, vibrations, oscillations, and so on. These unknowns can only be found through an equation if the value of a third quantity is known. That third quantity is normally given by stiffness matrices, mass matrices, gyroscopic matrices, or a combination of all depending on the complexity of the governing equation of the body.

**Assembly of matrices**

Individual matrices for individual elements are derived using the material properties and their shape functions. Depending upon the number of finite elements that the domain is discretized with, the number of element matrices varies proportionally. These individual matrices are assembled into a single global matrix for the calculation.

**Application of boundary conditions**

Boundary conditions are the constraints of the body under analysis. Boundary represents the physical condition of the body regarding fixed points, thermal constraints, and so on. These are necessary to evaluate a boundary value problem and are naturally induced. There are two kinds of boundary conditions, Neumann boundary condition, and Dirichlet boundary condition. When the normal derivate with respect to any boundary is zero, it is a Dirichlet boundary condition. Neumann boundary condition considers the physical condition of the boundary, like in structural problems, the fixed points, which are the so-called degree of freedoms (DOFs), and in thermal problems, the conditions of temperature with respect to any area in the body.

**Solution**

The equation thus formed is solved. Depending on the complexity of the equation and problem, it can be manually calculated or solved with the aid of a computer. The higher complexity of problems requires high processing speeds.

**Post-processing**

The results obtained after the analysis are plotted in a graphical representation and tabular form. This aids in better analysis and gives an in-depth insight into the solutions.

## Modal analysis and the eigenvalue problem

Modal analysis and the eigenvalue problem form the fundamental base of vibration analysis. When a body undergoes vibration, it follows a pattern of mode shapes under its various natural frequencies. the modal analysis determines the shapes and the natural frequencies of the system by forming a generalized eigenvalue problem. Mathematically, eigenvalues are the scalar multipliers of a quantity, which magnify their magnitude, while eigenvectors are the vectors that species the direction of the magnified quantity. In the modal analysis, the eigenvalues represent the natural frequencies of the system, while the eigenvector species represent the mode shapes of the system. If a system under analysis is a lumped system, it will have finite DOFs, hence, a finite number of natural frequencies. And for a continuous system, like a high-speed rotating shaft, it will have an infinite DOF which will account for infinite natural frequency, but to aid a solution, only a few finite natural frequencies are considered.

## Performing a modal analysis

Considering a rotor dynamics problem with a high-speed rotating shaft with a solid circular disc mounted on one of its ends, while the other end is kept free. A Dynamic FEA analysis is performed on this system to extract out the frequencies and the modal shapes.

To begin with, a differential equation in the form of PDE of the system is formed using the principle of mechanics. Newtonian mechanics deals with lumped or finite systems, as this is a case of the continuous system, Lagrangian or Hamiltonian mechanics is applied in forming the PDE of the system.

The order of the PDE is reduced to lower order by applying the Galerkin method, which is a residual approach that uses weighing functions. The lower order PDE is thus known as the weak form of a differential equation. This equation is further used to derive the mass and stiffness matrix of the system using either the Euler Bernoulli beam theory or the Timoshenko beam theory. Due to the disc mounted at the end of the shaft, it will induce gyroscopic effects during its motion, which is also taken into consideration by developing a gyroscopic matrix for the disc.

A governing equation of motion of the system is thus formed, which is represented in the form:

$\left[M\right]\left\{\ddot{X}\right\}-\left[G\right]\left\{\dot{X}\right\}+\left[K\right]\left\{X\right\}=\left\{F\right\}$

This is a basic governing vibration equation.

Here,

- $\left[M\right]$ is the mass matrix of the shaft
- $\left[G\right]$ is the gyroscopic matrix of the shaft due to disc
- $\left[K\right]$ is the stiffness matrix of the shaft
- $\left\{\ddot{X}\right\}$ represents the acceleration vector
- $\left\{\dot{X}\right\}$ represents the velocity vector
- $\left\{X\right\}$ represents the displacement vector, and
- $\left\{F\right\}$ is the external force vector

The following is then reduced to an eigenvalue problem in the form

$\left[K\right]\left\{X\right\}=\lambda \left[M\right]\left\{X\right\}$

The equation is evaluated to determine the natural frequencies of the shaft in terms of eigenvalues, given by $\lambda $. $\lambda $ can be calculated by simply inverting the mass matrix $\left[M\right]$ to form an equation in terms of $\lambda $.

## Post-processing and the Campbell diagram

The solutions or results of the modal analysis are plotted in the form of a Campbell diagram. A Campbell diagram represents the natural frequency or critical speeds of a shaft vs the rotational speed of the shaft. This diagram follows in all rotor dynamics problems. The abscissa (X-axis) represents the shaft's rotational speed and the ordinate (Y-axis) represents the critical speeds. Drawing an X=Y linear line on the diagram denotes the critical speeds at which the shaft will experience resonance.

## Context and Applications

This topic is majorly taught in many undergraduate and postgraduate degree courses of:

- Bachelor of Technology (Mechanical engineering)
- Bachelor of Technology (Civil engineering)
- Masters in Science (Design of machinery)
- Masters in Science (Computer-Aided Design)

## Practice Problems

Q 1. In which of the following problem, modal analysis can be categorized?

- Nonlinear analysis
- Linear analysis
- Static analysis
- Both a and b

**Answer: **Option a

**Explanation: **Modal analysis falls under the category of nonlinear dynamic analysis.

Q 2. What is the full form of FEA?

- Finite element analysis
- Force evaluation analysis
- Finite elemental analysis
- None of these

**Answer:** Option a

**Explanation:** The full form of FEA is finite element analysis.

Q 3. What is the expression $\left[M\right]$ known as?

- Modulus matrix
- Nonlinear matrix
- Mass matrix
- Moment matrix

**Answer:** Option c

**Explanation:** The expression $\left[M\right]$ represents the mass matrix.

Q 4. What is the expression $\left[G\right]$ represents in the governing equation?

- Gyroscopic matrix due to gyroscopic effect
- Gyroscopic matrix due to plasticity of the body
- Gyroscopic matrix due to flexural rigidity
- Both a and b

**Answer:** Option a

**Explanation:** The expression $\left[G\right]$ represents the gyroscopic matrix due to the gyroscopic effect.

Q 5. What is the post-processing diagram in rotor dynamics called?

- Campbell diagram
- Zienkiewicz diagram
- Both a and b
- None of these

**Answer:** Option a

**Explanation:** Rotor dynamic problems are approached by the creation of Campbell diagrams for post-processing.

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