2. SINGLE DEGREE OF FREEDOM SYSTEM Equation of Motion, Problem Statement & Solution Methods Pertemuan 19 Matakuliah: Dinamika Struktur & Teknik Gempa Tahun: S0774

3. Systems with two degree of freedom Recap Pendulum systems (double pendulum) Torsional systems String systems Linear systems Definite and semi-definite systems Analysis of various 2DOF systems such as:

4. Systems with two degree of freedom Dynamic coupling Static coupling Principal co-ordinates Co-ordinate coupling

5. Systems with two degree of freedom Problem-1 Obtain the equations of motion of the system shown in the figure. The vibration is restricted in plane of paper m -mass of the system J -mass MI of the system G -centre of gravity G K1K1 K2K2 m,J ab

6. Systems with two degree of freedom G K1K1 K2K2 m,J ab The system has two generalized co-ordinates, x and  Cartesian (x), Polar (  ) Problem-1 (x-a  ) (x+b  )  x G Static equilibrium line a b

7. Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli Systems with two degree of freedom Eqns. of motion The Lagrange’s equation is : generalized co-ordinates Problem-1 (x-a  ) (x+b  )  x G Static equilibrium line a b

8. Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli Systems with two degree of freedom The Lagrange’s equation is : Problem-1

9. Systems with two degree of freedom equations of motion (1 st ) First Eqn. of motion Problem-1

10. Systems with two degree of freedom equations of motion (2 nd ) Problem-1

11. Systems with two degree of freedom equations of motion Second Eqn. of motion equations of motion are: Problem-1 First Second

12. Systems with two degree of freedom equations of motion Matrix form Problem-1

13. Systems with two degree of freedom Matrix form Mass/inertia matrix Stiffness matrix Stiffness matrix shows that co-ordinate x and  are dependent on each other. Any change in x reflects in change in  As seen from the matrix, the equations of motion are coupled with stiffness. This condition is referred as STATIC COUPLING coupling in mass matrix is referred as DYNAMIC COUPLING Problem-1

14. Systems with two degree of freedom Matrix form Mass/inertia matrix Stiffness matrix From the above equations, it can be seen that system do not have dynamic coupling But, it has static coupling Problem-1

15. Systems with two degree of freedom Matrix form Mass/inertia matrix Stiffness matrix To have static uncoupling the condition to be satisfied is: K 1 a=K 2 b Problem-1

16. Systems with two degree of freedom Matrix form The uncoupled Eqns. of motion are Contains only one coordinate, x Contains only one coordinate,  Under such conditions, x and  are referred as PRINCIPAL COORDINATES Problem-1

17. Systems with two degree of freedom Solution to uncoupled Eqns. of motion: From Eqn.1: From Eqn.2: Problem-1

18. Systems with two degree of freedom Obtain the equations of motion of the system shown in the figure. The centre of gravity is away from geometric centre by distance e The vibration is restricted in plane of paper m -mass of the system J -mass MI of the system G -centre of gravity C -centre of geometry C K1K1 K2K2 m,J a b G e Problem-2

19. Systems with two degree of freedom Due to some eccentricity e, the changes are: x=x+e  J=J+me 2 Substitute in Eqns. of motion of earlier problem having e=0: Problem-2 K 1 (x-a  ) K 1 (x+b  )  x G C x+e  Static equilibrium line a b

20. Systems with two degree of freedom equations of motion for system having e=0 Substitute x=x+e  and J=J+me 2= J n in above Eqns. Problem-2

21. Systems with two degree of freedom New equations of motion are Matrix form Dynamic coupling Static coupling Problem-2

22. Systems with two degree of freedom Derive expressions for two natural frequencies for small oscillation of pendulum shown in figure in plane of the paper. Assume rods are rigid and mass less a a a m m K Problem-3

23. Systems with two degree of freedom a a a m m K aa a mg Ka(  2 -  1 ) 11 22 Equilibrium diagram Problem-3

24. Systems with two degree of freedom Equilibrium diagram aa a mg Ka(  2 -  1 ) 11 22 For first mass as  is smaller First Eqn. of motion Problem-3

25. Systems with two degree of freedom Equilibrium diagram aa a mg Ka(  2 -  1 ) 11 22 For second mass as  is smaller Second Eqn. of motion Problem-3

26. Systems with two degree of freedom First Eqn. of motion Second Eqn. of motion Eqns. of motion in matrix form Problem-3 For static coupling Ka=0, which is not possible

27. Systems with two degree of freedom First Eqn. of motion Second Eqn. of motion Problem-3 Solution to governing eqns.: Assume SHM The above equations have to satisfy the governing equations of motions Equations of motion

28. Systems with two degree of freedom In above equations String systems Characteristic Eqns.:

29. Systems with two degree of freedom The above equation is referred as a characteristic determinant Solving, we get : Frequency equation String systems

30. Systems with two degree of freedom Solve the frequency Eqn. for Natural frequencies of the system String systems As the system has two natural frequencies, under certain conditions it may vibrate with first or second frequency, which are referred as principal modes of vibration

31. Summary Dynamic coupling Static coupling Principal co-ordinates Co-ordinate coupling

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