BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS

8/7/2019 BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS

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DYNAMIC ANALYSIS

The static and dynamic analysis of structural members and machine

components using finite element analysis results in a set of equilibrium

equations of the form

[M]{x}+[C]{x}+[K]{x} = {F} (1)

Where [M], [C] and [K] are the mass, damping and stiffness matrices,

respectively, [F] is the external force vector, and x, x and x are the acceleration,

velocity and displacement vectors respectively. At any instant of time the

equilibrium equation can be written.

{F1} + {FD} + {FE} = {F} (2)

Where {F1} are the inertia forces, {FD} are the damping forces and {FE}

are the elastic forces. All are time dependent and are equal to

{F1} = [M]{x}

{FD} = [C]{x}

{FE} = [K]{x}

Equation (1) represents a system of linear differential equations of

second order. Several solution techniques are available to solve the differential

equations with constant coefficients. Most of the solution procedures for the

general systems of differential equations are very expensive and time

consuming on the computer if the order of the matrices is large say more than

100. The coefficient matrices [M], [C] and [K] of mechanical and structural

engineering problems have special characteristics which can be effectively

utilized in simplifying the analysis. The solution techniques that are considered

here are mainly matrix methods.


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BASIC PROPERTIES OF EIGENVALUES AND EIGENVENCTORS

If [A] is any square matrix of order nxn an if {x} is a non zero vector such

that

[A]{x} = {x}

Where is some number, than {x} is said to be an eigenvector of [A] with

the corresponding eigenvalue . Premultiplying and eigenvector by the

appropriate matrix yields a constant times the eigenvector , where the

constant is the eigenvalue.

It is known that eigenvalues of a square matrix of size nxn satisfy an nth

order polynomial equation. Thus in general there will be n eigenvalues which

are not necessarily distinct and real. They are either real or result in complex

conjugates if all the matrix elements are real. Another property of eigenvalue

problem can be stated as follows. If [A] is a square matrix of size nxn, any

eigenvalue satisfies the nth degree polynomial equation: det(A - I) = 0. This

equation is also known as the characteristic equation of [A]. the proof of this

property is that we seek and non zero {x} such that

[A]{x} = {x}

[A- I}{x} = 0

This represents a system of n homogeneous equations in n unknowns

x1, x2, x3, xn. Hence if we require non zero solutions {x} then [A- I] must be

singular or det[A- I] = 0.


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det[A- I] =

det[A- I] = (a11- )(a22-).(ann-)

The equations det[A- I] = 0 gives a polynomial equation of nth degree

in . There will be n eigenvalues and n corresponding eigenvectors. The

eigenvalues are 1,2,n and the corresponding eigenvectors are {x(1)

},

{x(2)

},. {x(n)

} . that is

[A] {x(r)

} = r {xr} where 1 r n

In general equation (1) provides a procedure for the calculation of

eigenvalues and eigenvectors. The determinant of [A - I] leads to an explicit

polynomial equation. The roots of the polynomial equation give all the

eigenvalues 1,2,n. Substituting the eigen values r into equation (2) yields

n equations, [A- I] {x(r)

}= 0 the solution of which gives {x(r)

}.

a11 - a12 a1n

a22 a22- a23 a2n

an1 an2 ann-


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Example

Find eignevalues and he corresponding eigenvectors of the matrix [A].

[A] =

SOLUTION

[ A-I] =

And

det[ A-I] = (2-0{(2+) +2} 1{0(-)+2} 1{0(1)+{2+}

= -3+=0

Which is the characteristic equation.

The eigenvalues are 0, 1, -1. The corresponding eigenvectors are:

2 1 -10 -2 -2

1 1 0

2- 1 -1

0 -2- -2

1 1 -

BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS

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