Matakuliah : Dinamika Struktur & Teknik Gempa Tahun : S0774

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

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

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

5. m,J G K1 K2 a b 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

6. m,J Static equilibrium line G b a (x-a) x (x+b) K2 K1 G  a b Systems with two degree of freedom Problem-1 The system has two generalized co-ordinates, x and  Cartesian (x), Polar ()

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

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

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

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

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

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

13. Systems with two degree of freedom Problem-1 Matrix form Stiffness matrix Mass/inertia 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

14. Systems with two degree of freedom Problem-1 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

15. Systems with two degree of freedom Problem-1 Matrix form Mass/inertia matrix Stiffness matrix To have static uncoupling the condition to be satisfied is: K1a=K2b

16. Systems with two degree of freedom Problem-1 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

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

18. m,J C G e K1 K2 a b Systems with two degree of freedom Problem-2 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

19. Static equilibrium line b a K1(x-a) x x+e K1(x+b)  G C Systems with two degree of freedom Problem-2 Due to some eccentricity e, the changes are: x=x+e J=J+me2 Substitute in Eqns. of motion of earlier problem having e=0:

20. Systems with two degree of freedom Problem-2 equations of motion for system having e=0 Substitute x=x+e and J=J+me2=Jnin above Eqns.

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

22. a a K m a m Systems with two degree of freedom Problem-3 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

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

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

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

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

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

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

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

30. Systems with two degree of freedom String systems Solve the frequency Eqn. for Natural frequencies of the system 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 Co-ordinate coupling Static coupling Dynamic coupling Principal co-ordinates

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