1. 1. Kinetic energy, potential energy, virtual work.
In mechanical systems, we will study on the subject of kinetic energy, potential energy, and virtual work with an example below.
The system shown in the figure, the bar OA, labeled 2, is attached to a fixed body at O, labeled 1. The bar rotates around O. For the bar, mass of the bar is 2m/3, length of the bar is 3L/4.
Example 1.1: 1 O 2 A D f xB 3 B 2k 3c 2k 3c x4 k c T k c 4

2. Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c x4 k c T k c 4
The other element is a circular disc, labeled 3. The center of the disc is at B. Mass of the disc is m/2 and its radius is L/3.
The elements, labeled 2 and 3, are fixed to the spring and damper elements with the values of 2k and 3c. The disc, labeled 3, is fixed to the body with the other spring and damper elements with the values of 2k and 3c. The disc, labeled 3, rolls without slipping on the bar, labeled 4.
Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c x4 k c T k c 4

3. Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c x4 k c T k c 4
The bar, labeled 4, slips on the body in the horizontal direction and it is fixed to the body from the both sides with the damper and spring elements with the constants of k and c.
The bar, labeled 4, has mass m and length L. Force f acts to the bar at D in the horizontal direction. OD equals to 9L/16. Torque T acts to the disc at B .
Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c x4 k c T k c 4

4. Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c
The generalized coordinates of the system which has 3 degrees of freedom can be chosen as , xb and x4. F and T are the force inputs of the system. x4 which may be one of the generalized coordinates of the system will be taken as the input for this problem and x4 is controlled by the external effect. So, theta and x b are the rest of generalized coordinates of the system
It will be assumed that the radian value of theta is much smaller than one and the expressions sin()   and cos()  1 will be used to model mechanical systems.
Example 1.1: (Continue) 1 O 2 A D f xB 3 B 2k 3c 2k 3c
Inputs: f, T, x4 x4 k c T k c 4

5. Example 1.1: (Continue) 1 O 2 A D f xB Inputs: f, T, x4 3 B 2k 3c 2k
Total kinetic energy of the system can be written as follows.
The first and second terms are written due to translational and rotational motions of the bar, labeled 2. The third and fourth terms are written due to translational and rotational motions of the disc, labeled 3. The fifth term is written due to translational motion of the bar, labeled 4.

6. Example 1.1: (Continue) 1 O 2 A D f xB Inputs: f, T, x4 3 B 2k 3c 2k
Total potential energy of the system can be written as follows.
The first term is written from the spring between the elements, labeled 2 and 3. The second term indicates the potential energy, which is stored in the spring with the constant of 2k between the disc, labeled 3, and the body. The last two terms are the potential energies, which is stored in springs between the bar, labeled 4, and the body.

7. Example 1.1: (Continue) 1 O 2 A D f xB Inputs: f, T, x4 3 B 2k 3c 2k
The expression of virtual work for the system can be written as follows.
The first term is the virtual work done by the external force f. The second term is the virtual work done by the external torque T. The third term is the virtual work of the damping element with the value of 3c between the elements, labeled 2 and 3. The last two terms could be written as the virtual work of the damper elements between the bar, labeled 4 and the body. But in this problem x4 due taken as the input, the virtual displacement of x4 will be zero. The virtual displacement can be applied only to the generalized coordinates.

8. Using the distributive property of virtual change the expression of the virtual work can be arranged as follows.
x4 is zero this is why x4 is an input.
Taking the multipliers of  and xB into parentheses, the following expression is obtained.
The expression in the parentheses, which is the multiplier of  is the generalized force expression of the generalized coordinate . The expression in the parentheses, which is the multiplier of xB is also the generalized force expression of the generalized coordinate xB and they are given above, respectively.

9. KINETIC ENERGY, POTENTIAL ENERGY AND VIRTUAL WORK FOUND FOR EXAMPLE PROBLEM:
As a result, the kinetic energy, potential energy and virtual work expressions for the example problem are given here again.