Download presentation

Presentation is loading. Please wait.

Published byKrista Geoffrey Modified over 3 years ago

1
MEG 361 CAD Finite Element Method Dr. Mostafa S. Hbib

2
**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 …..)

3
**…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,….)

4
**…FEM approximates structures in two ways:**

Structure (Field ) Discretization (into elements called FE’) Use mathematical model if known Example …

5
**Example: The Bar Letus first review the math model of longitudinal vibrating bar**

6
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.

7
**Intergrating (1) to yield:**

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

8
**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) At each node the value of u is allowed to be time dependent, hence we use the labels u1(t) and u2(t) as boundaries to evaluate the spatial constants in the shape function: (2) Intergrating (1) to yield: At x=0 sub. Into (2):

9
**If u1 and u2 are known then (3) would provide **

Subs. C1 and c2 yields the shape function: (3) If u1 and u2 are known then (3) would provide an approximate solutiion to (1). Strain energy: Subs. With u(x,t): Now consider represented by: Where:

10
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

11
Again, u(x,t): Using the variational (Lagrangian) approach: Where: I is the I th coordinate of the system which is assumed to have n DOF Subs. With u(x,t) in the lagrangian (remember that u1 = 0 in this case :

12
**Subs. With u(x,t) in the lagrangian (remember that u1 = 0 in this case yields:**

Which can be solved (given IC for u2 ) yields Exact solution:

13
**The FEM has a natural freq.**

We have the shape function: (3) (4) 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)

14
**Example: Compare the exact solution of the clamped bar and that is derived by the FEM, i. e., (4)**

NB. FEM gives only one mode (One Element

15
This example

16
**Example Same Cantilever Bar 3-Element, 4-Node Mesh**

17
**To use the Lagrangian approach we need to compute:**

18
**To use the Lagrangian approach we need to compute:**

Subs. In the Lagrangian we get:

19
**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. …

20
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. Solve the EVP: (5) (5) The natural frequencies of the 3-element FEM of the clamped-free bar are:

21
**The exact natural frequencies of the clamped-free bar are:**

%Error FE Freq. Exact

Similar presentations

OK

Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 1 Practical Application of the Rayleigh-Ritz Method to Verify Launch Vehicle Bending Modes.

Vibrationdata AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 1 Practical Application of the Rayleigh-Ritz Method to Verify Launch Vehicle Bending Modes.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on transportation in plants and animals Ppt on multi sectoral approach on ncds Ppt on surfing the net safely Ppt on bucky paper uses Ppt on village life and city life Ppt on quality education Free ppt on autism Class 7 science ppt on light Ppt on pin diode applications Download ppt on solar and lunar eclipse