1. Virtual displacement is an imaginary displacement, there is no actual displacement occur in the system (the system is still in equilibrium)
2. If a system forces is in equilibrium, the work done by externally applied forces through virtual displacements are ZERO
3. The virtual displacements do not violate the system constraints.

Example

Consider mechanical system bellow

conservative dynamic system

1. Consider a system as a whole, not as individual components
2. Excludes the reaction and constraint forces
3. Formulates dynamic problems in two scalar function; kinetic and potential energy
4. Formulates equation of motion in generalized coordinate and generalized forces which are not necessary physical coordinates and forces.

Definitions:

Degree of Freedom

Set of independent coordinates that required to completely describe the system’s configuration.

Generalized Coordinate

A set of minimum number of coordinates necessary and sufficent to uniquely define the system’s configuration.

Constraints

Any mathematical expression relates the cordinate and limits the system freedom.

Example

Describe the DoF number, generalized coordinates, and constraint functions of the system bellow!

conservative dynamic system

The motion of each mass can be described mathematically;

- pulley (rotating) position:

- disk (rolling): 

- block (translation):

Since the motion of each mass in the system is function of , thus the system is only 1 DoF dynamic system. The mathematical expression of disk and block position are holonomic constraints. In order to simplified the equation of motion, (translation coordinate) is chosen as the generalized coordinate.

]]> http://itsstudy.wordpress.com/2008/12/07/analytical-dynamics/feed/ 0 yunusbillah analytical dynamic example http://itsstudy.wordpress.com/2008/12/04/solve-the-eigenvalue-problem-using-matlab-its-so-easy/ http://itsstudy.wordpress.com/2008/12/04/solve-the-eigenvalue-problem-using-matlab-its-so-easy/#comments Thu, 04 Dec 2008 22:50:06 +0000 yunusbillah http://itsstudy.wordpress.com/?p=26 ]]> When you have any eigenvalue problem bellow;

you can solve it easily using matlab program. This software is really powerfull!

Example:

Given a matrix

Calculate all its eigenvalues and the corresponding eigenvectors using matlab!

Step 1: Describe matrix A

Start your matlab and type matrix A as follow:

>> A=[2 1 -1;1 4 3;-1 3 4]    (enter)

A =

2     1    -1
1     4     3
-1     3     4

Step 2: Calculate the eigenvalues and the corresponding eigenvectors

>> [V D]=eig(A)    (enter)

V =

-0.5774    0.8165    0.0000
0.5774    0.4082    0.7071
-0.5774   -0.4082    0.7071

D =

0.0000         0         0
0    3.0000         0
0         0    7.0000

You will get new matrices D and V. Matrix D represent the eigenvalues of matrix A with matrix V as its corresponding eigenvectors (normalized eigenvectors).

You can practice by yourself, because it is so easy. Try to solve several problem using matlab!

When it said the eigenvalue problem?

Any equations that can be written in matrix form bellow it is said as the eigenvalue problem:

where is n x n constant matrix and is called as eigenvector, and is an any scalar that we called eigenvalue. The solution of eigenvector depend on eigenvalue ().

How to Solve It?

It is easy!. This problem has “Trivial Solution” , but of course we are not interest in that! Than, how could we find the “Non-trivial Solution”?

If we multiplied the right hand side of the eigenvalue problem by identity matrix , we can modified the equation without change it value into the standard eigenvalue problem bellow

This standard form has a unique characteristic. If  (determinant = 0) then it will have non-trivial solution.

Example 1:

Solve for the , then we get and

The eigenvector for the corresponding eigenvalue is

for , then

Use Gauss-Jordan elimination to reduce the equation in the matrix (become single line or triangle)

let , from the matrix we know that

hence, the corresponding eigenvector is

Do the same for then we get

That’s so easy isn’t it?! Try with bigger matrix size and different number you will enjoy while your ability improve! You can use mathematics software like matlab, maple, or mathematica to check your calculation.

You can also learn and practice to use those software in the articles bellow:

- Solve The Eigenvalue Problem Using Matlab (click!)

- Solve The Eigenvalue Problem Using Mathematica

- Solve The Eigenvalue Problem Using Maple

After you mastered this, then we can try to use the eigenvalue problem to solve so many practical problem in various field.

]]> http://itsstudy.wordpress.com/2008/12/04/solve-the-eigenvalue-problem-it-is-easy/feed/ 0 yunusbillah http://itsstudy.wordpress.com/2008/12/04/hello-world/ http://itsstudy.wordpress.com/2008/12/04/hello-world/#comments Thu, 04 Dec 2008 13:48:11 +0000 yunusbillah ]]> Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!