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MEG 361 CAD Dr. Mostafa S. Hbib Finite Element Method

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FEM is powerful numerical technique ….. FEM uses variational and Interpolation methods for modeling and solving BVPs such as DPS (bars, beams, plates, trusses, frames, fluid flow, heat transfer …..)

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…FEM is powerful numerical technique FEM is very systematic and modular. Therefore, it is easy to implement on computers. There are several FE codes packages available (Ansys, Nastran, IDEAS, ADAMS,….)

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…FEM approximates structures in two ways: Structure (Field ) Discretization (into elements called FE’) Use mathematical model if known Example …

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Example: The Bar Letus first review the math model of longitudinal vibrating bar

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The long. Vib. Of a bar gives a simple example of how FEM is constructed and how is used to approximate the vib of a DPS with that of LPS (FEM). Two FEModels (grids of the same beam. a) Single-element and b) Three-element model.

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The static (time independent) displacement of the bar element must satisfy (for 0 ≤x ≤ l): Intergrating (1) to yield: (1) (2)

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The FEM proceeds with two levels: Which model to use ( i.e., which mesh and size of mesh where to put elements and nodes) The choice of polynomials to use in (1) (shape functions) Intergrating (1) to yield: (2) At each node the value of u is allowed to be time dependent, hence we use the labels u 1 (t) and u 2 (t) as boundaries to evaluate the spatial constants in the shape function: At x=0 sub. Into (2):

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Subs. C1 and c2 yields the shape function: If u 1 and u 2 are known then (3) would provide an approximate solutiion to (1). (3) Strain energy: Subs. With u(x,t): Now consider represented by: Where:

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Using u(x,t): Subs. With u(x,t): Where: Using the variational (Lagrangian) approach: Where: I is the I th coordinate of the system which is assumed to have n DOF

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Subs. With u(x,t) in the lagrangian (remember that u 1 = 0 in this case : Using the variational (Lagrangian) approach: Where: I is the I th coordinate of the system which is assumed to have n DOF Again, u(x,t):

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Subs. With u(x,t) in the lagrangian (remember that u 1 = 0 in this case yields: Which can be solved (given IC for u 2 ) yields Exact solution:

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(3) The FEM has a natural freq. We have the shape function: Subs. the FEM solution, we get: Example: Compare the exact solution of the clamped bar and that is derived by the FEM, i. e., (4) (4)

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NB. FEM gives only one mode (One Element Example: Compare the exact solution of the clamped bar and that is derived by the FEM, i. e., (4)

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This example

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Example Same Cantilever Bar 3-Element, 4-Node Mesh Element 2 Element 3

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To use the Lagrangian approach we need to compute:

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Subs. In the Lagrangian we get:

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Is the global mass matrix and the coeffecient Is the global stiffness matrix Example: Compare the natural frequencies of the 3-element FEM with the exact DPS model. the clamped-free bar determined by substituting the global stiffness matrix the global mass matrix into the FEM. …

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Solution: The natural frequencies of the 3-element FEM of the clamped-free bar are determined by substituting the global stiffness matrix and the global mass matrix into the FEM. The natural frequencies of the 3-element FEM of the clamped- free bar are: (5) Solve the EVP:

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The exact natural frequencies of the clamped-free bar are: Exact FE Freq. %Error

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