1. 2 DOF – Torsional System and Coordinate Coupling
ROSLI ASMAWI
Faculty of Mechanical & Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia

2. 1.Torsional System
Consider a torsional system as shown in Fig.5.6. The differential equations of rotational motion for the discs can be derived as
which upon rearrangement become
For the free vibration analysis of the system, Eq.(5.19) reduces to

3. 1. Torsional System
Figure 5.6: Torsional system with discs mounted on a shaft

4. Example 1:Natural Frequencies of a Torsional System
Find the natural frequencies and mode shapes for the torsional system shown in Fig.5.7 for J1 = J0 , J2 = 2J0 and kt1 = kt2 = kt .
Solution: The differential equations of motion, Eq.(5.20), reduce to (with kt3 = 0, kt1 = kt2 = kt, J1 = J0 and J2 = 2J0):
Fig.5.7: Torsional system

5. Example 1 Solution Rearranging and substituting the harmonic solution:
gives the frequency equation:
The solution of Eq.(E.3) gives the natural frequencies

6. Example 1 Solution The amplitude ratios are given by
Equations (E.4) and (E.5) can also be obtained by substituting the following in Eqs.(5.10) and (5.11).

7. 2. Coordinate Coupling and Principal Coordinates
Generalized coordinates are sets of n coordinates used to describe the configuration of the system.
Equations of motion Using x(t) and θ(t).
Fig.5.10

8. 2. Coordinate Coupling and Principal Coordinates
From the free-body diagram shown in Fig.5.10a, with the positive values of the motion variables as indicated, the force equilibrium equation in the vertical direction can be written as
and the moment equation about C.G. can be expressed as
Eqs.(5.21) and (5.22) can be rearranged and written in matrix form as

9. 2. Coordinate Coupling and Principal Coordinates
The lathe rotates in the vertical plane and has vertical motion as well, unless k1l1 = k2l2. This is known as elastic or static coupling.
Equations of motion Using y(t) and θ(t).
From Fig.5.10b, the equations of motion for translation and rotation can be written as

10. 2. Coordinate Coupling and Principal Coordinates
These equations can be rearranged and written in matrix form as
If , the system will have dynamic or inertia coupling only.
Note the following characteristics of these systems:

11. 2. Coordinate Coupling and Principal Coordinates
In the most general case, a viscously damped two degree of freedom system has the equations of motions in the form:
The system vibrates in its own natural way regardless of the coordinates used. The choice of the coordinates is a mere convenience.
Principal or natural coordinates are defined as system of coordinates which give equations of motion that are uncoupled both statically and dynamically.

12. 3. Coupling Coordinate (e.g. vibration of automobile/motorcycle)

13. 3.1 General Equation of Coupling
Dynamic Coupling
Static Coupling
Static Coupling: when stiffness matrix is not diagonal
Dynamic Coupling: when mass matrix is not diagonal

14. 3.2 Selection principal coordinate
There are three possibilities of coordinate
Using the center of gravity
Using an eccentric point off the center of gravity
Using the end point of the model e x 1 2 3

15. Case 1, Using center of gravityl1 l2
xc-x1 xc m x1
xc+x2 θ x2 Io
k1(xc-x1)
k2(xc+x2)

16. Combine Translation and Rotation
Static Coupling If
then uncoupling static and dynamic

17. Case 2, Using eccentric coordinatel1 l2
xp-x1 xp m x1 e
xp+x2 xe IP θ x2 Io
k1(xp-x1)
k2(xp+x2)

18. Combine Translation and Rotation Static and Dynamic CouplingIf
then dynamic coupling

19. Case 3, Pinned at one end l x1 l1 IP m xe
x1+x2 θ x2
k1(x1)
k2(x1+x2)

20. Translation
Rotation
Combine Translation and Rotation
Static and Dynamic Coupling

21. Example 2 Principal Coordinates of Spring-Mass System
Determine the principal coordinates for the spring-mass system shown in Fig.5.4.

22. Example 2 Solution
Approach: Define two independent solutions as principal coordinates and express them in terms of the solutions x1(t) and x2(t).
The general motion of the system shown is
We define a new set of coordinates such that

23. Example 2 Solution
Since the coordinates are harmonic functions, their corresponding equations of motion can be written as

24. Example 2 Solution From Eqs.(E.1) and (E.2), we can write
The solution of Eqs.(E.4) gives the principal coordinates:

25. See you again… in the next lecture