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6.7 Finite Element Method

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In the finite element method, complex structures are replaced by assemblages of simple structural

elements known as finite elements. The elements are connected by joints or nodes. The force and

moments at the ends of the elements are known from structural theory, the joints between the elements

are matched for compatibility of displacement, and the force and moment at the joints are established by

imposing the condition of equilibrium.

The accuracy obtainable from the finite element method depends on being able to duplicate the

vibration mode shapes. Using only one finite element between structure joints or corners gives good

results for the first lowest mode, because the static deflection curve is a good approximation to the lowest

dynamic mode shape. For higher modes, several elements are necessary between structural joints. This

leads to large matrices. The eigenvalues and eigenvectors need to be computed numerically.

This section introduces the basic idea of the finite element method as it applies to the simple vibration

problem.

The basic idea behind the finite element method is to break up the structure into simple component

structures. The structural elements for the bar and the beam are discussed here.

6-20 Vibration and Shock Handbook

6.7.1 Bar Element

The force – displacement relationship for a uniform

rod is

F ¼ ðEA=LÞU

where E is the young’s modulus, A is the cross

sectional area, L is the length of the element and

U is the displacement. Figure 6.8 shows this

one-dimensional element. The two endpoints are

the nodes.

For simplicity, assume the axial displacement at

any point 6 ¼ x=L is linear:

Uðx; tÞ ¼ aðtÞ þ bðtÞx

and U1ðtÞ ¼ Uð0; tÞ; U2ðtÞ ¼ UðL; tÞ: These two

conditions uniquely determine the coefficients

aðtÞ and bðtÞ: They are given by the following:

aðtÞ ¼ U1ðtÞ

and

bðtÞ ¼ ðU2ðtÞ 2 U1ðtÞÞ=L

Using the linear element, the displacement

anywhere along the beam is given by

U ðx; tÞ ¼ ð1 2 x=LÞU1ðtÞ þ x=LU2ðtÞ

¼ w1U1ðtÞ þ w2U2ðtÞ

where

w1 ¼ 1 2 x=L and w2 ¼ x=L

w1 and w2 are known as the mode shape, which can

be seen in Figure 6.9.

These two mode shapes can be superimposed to create a linear function. An example of such a

function is given in Figure 6.10.

The kinetic energy of the bar is given by

T ¼ :5

ðl

u_ 2mdx ¼ :5m

ðl

0 ðð1 26Þu_ 1ðtÞ þ6u_ 2ðtÞÞ2ld6 ¼ :5mlð1=3u_ 21

þ 1=3u_ 1u_ 2 þ 1=3u_ 22

6.7.1.1 Mass Matrix

The generalized mass matrix is derived from the Lagrange equations using the following:

›T

›u_ 1

Given the kinetic energy for the bar, the Lagrange equations become

›T

›u_ 1 ¼ mLð1=3u€ 1 þ 1=6u€ 2Þ

›T

›u_ 2 ¼ mLð1=3u€ 1 þ 1=6u€ 2Þ

FIGURE 6.8 One dimensional element.

FIGURE 6.9 Linear mode shapes.

FIGURE 6.10 Superposition of the linear mode shapes.

Numerical Techniques 6-21

The mass matrix for an axial element with a uniform mass distribution per unit length is given by

2 1

1 2

" #

6.7.1.2 Stiffness Matrix

The force – displacement relationship for a uniform bar is given by

" #

1 21

21 1

" #

6.7.1.3 Variable Properties

A simple approach to problems with variable properties is to use a large number of elements of short

length. The variation of mass or stiffness over each element is small and can be neglected. In this model,

the mass and stiffness for each element is constant and can be placed outside the integral. If a large

number of elements are needed to capture the behavior, this will lead to a large matrix problem.

Example

A tapered rod is modeled as two uniform sections where EA1 ¼ 2EA2 and m1 ¼ 2m2; k2 ¼ 2EA2=L: The

displacement equation for this system is given by

2v2m2

2ð2 þ 1Þ 1

1 2

" #

2EA2

2 þ 1 21

21 1

" #

This can be written as a standard eigenvalue problem where l ¼ v2m2L=12EA2.

The eigenvalue problem is the following

3 2 6l 2ð1 þ lÞ

2ð1 þ lÞ ð1 2 2lÞ

􀀈 􀀈 􀀈 􀀈 􀀈

¼ 0

The solution of the determinant requires computing the roots of a quadratic equation. The two roots are

l ¼ ½0:6140; 1:1088􀀉: The two natural frequencies can be computed from l. They are

v1 ¼ 1:4029

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ððEA2Þ=ðM2LÞÞ

v2 ¼ 3:6477

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ððEA2Þ=ðM2LÞÞ

The modes shapes are calculated by solving the eigenvalue problem. The two eigenvectors are

x1 ¼ ½0:5773; 1􀀉t

and

x2 ¼ ½25258; 1􀀉t

6.7.2 Beam

When the ends of the element are rigidly connected to the adjoining structure, the elements act like a

beam with the moments and lateral forces acting at the joints. Generally, the axial displacement u2 2 u1 is

small compared to the lateral displacement V of the beam.

The local coordinates of the beam element are lateral displacements, V ; and rotation, u, at the two

ends. This results in four coordinates V1; u1; V2; u2: Four constraints uniquely determine a cubic

polynomial. Therefore, the lateral displacement of a beam is assumed to be described by a cubic

6-22 Vibration and Shock Handbook

polynomial

V ðxÞ ¼ p1 þ p26 þ p362 þ p464

where 6 ¼ x=L and pi; the coefficients of the polynomials, are related to the lateral displacement and

the rotation. The lateral displacement at the left-hand side determines V1:

V ð0; tÞ ¼ V1ðtÞ ¼ p1

The rotation at the left-hand end determines the second coefficient, u1:

›V

›x ð0; tÞ ¼ u1 ¼ p2

The remaining two coefficients are combinations of the variables and they are given by the

following two constraints:

V ðL; tÞ ¼ V2ðtÞ and

›V

›x ðL; tÞ ¼ u2

Applying these two conditions, one obtains

P3 ¼ 1=L2ð23v1ðtÞ 2 2u1ðtÞL þ 3v2ðtÞ 2 u2ðtÞLÞ

P4 ¼ 1=L3ð2v1ðtÞ þ u1ðtÞL 2 2v2ðtÞ þ u2ðtÞLÞ

These coefficients can be represented by the following matrix equation:

6666664

7777775

1 0 0 0

0 1 0 0

23 22 3 1

2 1 22 1

6666664

7777775

Lu2

6666664

7777775

The shape functions for the beam elements are determined by equating a single displacement to

one, and all other displacement to zero. The first shape function is derived by letting V1 ¼ 1 and the

remaining three variables zero; i.e., V2 ¼ u1 ¼ u2 ¼ 0: This gives

p1 ¼ 1 p2 ¼ 0 p3 ¼ 23 p4 ¼ 2

The first shape function becomes

w1 ¼ 1 2 362 þ 263

A similar calculation for u1 ¼ 1 and V2 ¼ V1 ¼ u2 ¼ 0 yields

p1 ¼ 0 p2 ¼l p3 ¼ 22l p4 ¼ l

and the shape function becomes

w2 ¼ l6 2 2l62 þ l63

The remaining two shape functions are determined similarly. The four shape functions for the

beam are

w1 ¼ 1 2 362 þ 263

w2 ¼ l6 2 2l62 þ l63

w3 ¼ 362 2 263

w4 ¼ 2l62 þ l63

Numerical Techniques 6-23

6.7.2.1 Mass Matrix

Just as in the case of the bar, the generalized mass mij is given by

Mij ¼

ðl

wiwjmdx

Substitution of the above shape functions and integration yields the following mass matrix for the

uniform beam element in terms of the end displacements:

420

156 22L 54 213L

22L 2L2 13L 23L2

54 13L 156 222L

213L 23L2 222L 4L2

6666664

7777775

6.7.3 Summary of Finite Element Method

1. Linear finite element for a bar w1 ¼ 1 2 x=L and w2 ¼ x=L

2. Mass matrix for the bar

2 1

1 2

" #

3. Stiffness matrix for the bar

" #

1 21

21 1

" #

4. Cubic finite elements for the beam

w1 ¼ 1 2 362 þ 263

w2 ¼ l6 2 2l62 þ l63

w3 ¼ 362 2 263

w4 ¼ 2l62 þ l6

Appendix 6A

Introduction to MATLABw

MATLAB is a software package for numerical computation, visualization, and symbolic manipulation

(also see Appendix 32A). It is an interactive environment with hundreds of built-in functions, which are

in essence subroutines. These functions range from plotting commands, to those for finding the

eigenvalues and eigenvectors of a matrix, to those for solving an ordinary differential equation, and much

more. In addition to the built-in functions, MATLAB contains a programming language, which allows

the user to write their own functions. The name MATLAB is an abbreviation of matrix laboratory. The

original versions of MATLAB concentrated on numerical analysis of linear systems of equation. MATLAB

is available from the Mathworks (www.mathworks.com).

The best way to learn MATLAB is by playing around with the different functions (Pratap, 2001).

The first thing to note when you launch MATLAB is that it is a window-based environment. There are

three windows: the command window, the graphics window, and the edit window. The command

6-24 Vibration and Shock Handbook

window is the main window and it is the one in which you run all functions (built-in or user created).

This is the window which appears when the program is launched, and it has the symbol q as a prompt.

The graphics window is where all of the graphics are displayed, and the edit window is where users create

and save all of their own programs, known as m files.

MATLAB provides routines for all of the basic areas of numerical analysis (numerical linear algebra, data

analysis, Fourier transform, and interpolation), curve fitting, root-finding, numerical solution of ordinary

differential equations, integration, and graphics. There are specialized tool boxes for signal processes and

control systems to name two. With these hundreds of functions, it is imperative to have a good help facility.

In the command window, the command help functionname provides online help. For example, type help

help to get more information about help. There are three other commands for information: lookfor,

helpwin, and helpdesk. Lookfor gives a list of functions with the keyword in their description. Helpwin gives a

help window. Helpdesk is a web browser-based help.

Given the early history of MATLAB as a matrix laboratory, it should not be surprising that one of its

strengths is its ability to manipulate vectors and matrices very well. A row vector is created by typing in

the command window the following:

qx ¼ ½2; 3; 6; 4􀀉

This command will produce

x ¼ 2 3 6 4

A common vector is created by entering the following

qx ¼ ½2; 1; 3; 4􀀉

This produces x ¼

The elements of a vector or matrix are separated by commas or by spaces, and the rows are separated by

semicolons. Printing is suppressed by ending the line with a semicolon. The following command will

produce a 2 £ 2 matrix but it will not print it:

A ¼

2 4

1 5

" #

Providing that the operations make sense, it is easy to do operations on vectors and matrices in

MATLAB. For example, if A and B are two matrices of the same size, the command A þ B adds the

matrices.

There are several MATLAB commands given throughout the text of this chapter. Several books are

available to get you started with MATLAB; for example, Hanselman and Littlefield (2001), Palm (2001),

Pratap (2001), and Recktenwald (2000). The best way to learn more about these commands is to type

help and the command name. This way, you get the most up-to-date information concerning the

function.

References

Atkinson, K. 1978. An Introduction to Numerical Analysis, 2nd ed., Wiley, New York.

Cheney, E. and Kincaid, D. 1999. Numerical Mathematics and Computing, 4th ed., Brooks Cole,

Monterey, CA.

Numerical Techniques 6-25

Hanselman, D. and Littlefield, B. 2001. Mastering MATLAB 6: A Comprehensive Tutorial and Reference,

Prentice Hall, Upper Saddle River, NJ.

Isaacson, E. and Keller, H. 1966. Analysis of Numerical Methods, Wiley, New York.

Palm, W. 2001. Introduction to MATLAB 6 for Engineers, McGraw Hill, Boston.

Pratap, R. 2001. Getting Started with MATLAB 6: A Quick Introduction for Scientists and Engineers, Oxford

University Press, New York.

Recktenwald, G. 2000. Numerical Methods with MATLAB: Implementation and Application, Prentice Hall,

Upper Saddle River, NJ.

Strang, G. 1986. Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA.

Thomson, W. and Dahleh, M. 1998. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper

Saddle River, NJ.

6-26 Vibration and Shock Handbook

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