1. A PPLIED M ECHANICS Lecture 06 Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. VIBRATION OF MULTI-DOF SYSTEM Many real structures can be represented by a single degree of freedom model. Most actual structures have several bodies and several restraints and therefore several degrees of freedom. The number of DOF that a structure possesses is equal to the number of independent coordinates necessary to describe the motion of the system. Since no body is completely rigid, and no spring is without mass, every real structure has more than one DOF, and sometimes it is not sufficiently realistic to approximate a structure by a single DOF model.

3. VIBRATION OF MULTI-DOF SYSTEM The deployment of the structure at its lowest or first natural frequency is called its first mode, at the next highest or second natural frequency it is called the second mode, and so on. Two DOF structure will be considered initially. This is because the addition of more DOF increases the labour of the solution procedure but does not introduce any new analytical principles. Equations of motion for 2-DOF model, natural frequencies and corresponding mode shapes will be obtained.

4. VIBRATION OF MULTI-DOF SYSTEM Examples of 2-DOF models of vibrating structures a) b) c) d) e) f) a) horizontal motion - x 1, x 2 ; b) shear frame - x 1, x 2 ; c) combined translation and rotation - x,  ; d) rotation plus translation - y,  ; e) torsional system -  1,  2 ; f) coupled pendula -  1,  2.

5. TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING Model of vibrating structures with 2-DOF m1m1 m2m2 k1k1 k2k2 k3k3 x 1, x 1, x 1... x 2, x 2, x 2... m1m1 m2m2 k1x1k1x1 k 2 (x 1 - x 2 ) x 1, x 1, x 1... x 2, x 2, x 2... k 2 (x 1 - x 2 )k3x2k3x2 The equations of motion for x 1  x 2 Equations can be solved for the natural frequencies and mode shapes by assuming a solution of the form

6. TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING Substituting these solutions into the equations of motion System algebraic equations A 1 and A 2 can be eliminated  

7. TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING The frequency equation Considering the case: k 1 = k 2 = k 3 = k, m 1 = m 2 = m. Frequency equation: The frequencies  01 and  02 and the corresponding mode shapes rad/s rad/s.

8. TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING The first mode and second mode of free vibration

9. TWO-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE The mechanical model The equations of motion

10. TWO-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE The solution is supposed in the form Substituting the solutions into the equations of motion The frequency equation

11. N-DOF SYSTEM The mechanical model F2F2 k1k1 x 1, x 1, x 1... m1m1 b1b1 k2k2 b2b2 m2m2 kiki bibi mimi k n+1 b n+1 x 2, x 2, x 2...   k i+1 b i+1 mnmn x i, x i, x i... x n, x n, x n... F1F1 FiFi FnFn

12. N-DOF SYSTEM The equations of motion in matrix notation - displacement, velocity, acceleration - vector of time depending exciting forces - mass matrix,

13. N-DOF SYSTEM - matrix of damping - stiffness matrix

14. N-DOF SYSTEM UNDAMPED FREE VIBRATION Free undamped vibrations are described by equation. The solution - vector of amplitudes of harmonic motion Ω - circular frequency Equation of motion for the assumption of harmonic motion

15. N-DOF SYSTEM UNDAMPED FREE VIBRATION For non-trivial solution - the determinant must be equal to zero. The determinant is called the frequency determinant. Developing the determinant - the frequency equation of n order for While the matrices are positive and definite the roots of this equations are real values

16. N-DOF SYSTEM UNDAMPED FREE VIBRATION Substituting the natural frequency, the set of homogenous equations is obtained. Therefore it is necessary to divide each equation by one element of the amplitude vector u ri. We get for example The vectors v r gives the shape of the vibrating system but not the absolute value of the displacements of its members. Therefore these vectors are called modal vectors. This process is called normalization. The normalization is possible to carry out by using one of the following procedure

17. N-DOF SYSTEM UNDAMPED FREE VIBRATION The displacements that belong to r mode are given resp. The general solution is given by linear combination of all modes are complex integration constants The integration constants C r,  r or A r, B r for r = 1,2,…,n are determined from initial conditions.

18. N-DOF SYSTEM UNDAMPED FREE VIBRATION The modal vectors is possible arrange in modal matrix The natural circular frequencies are arranged in spectral matrix

19. N-DOF SYSTEM UNDAMPED FREE VIBRATION Orthogonality of vibration modes. For Multiply the first equation by the vector,the second one by The second of these equations will be transposed After arrangement

20. N-DOF SYSTEM UNDAMPED FREE VIBRATION Because it has been supposed that These equations are the orthogonality relationships between natural modes of distinct natural frequencies. and for It is also possible to say: The mode vectors belonging to various natural frequencies are orthogonal with respect to the mass matrix as well the stiffness matrix. The quadratic forms are called generalized stiffness and generalized mass of mode r.

21. N-DOF SYSTEM UNDAMPED FREE VIBRATION The orthogonality relationships - in more complex form where V is called the modal matrix. The matrices M y and K y are diagonal. We notice that the mass matrix is positive definite. Therefore all generalized masses are positive. The modal matrix is possible to use to define the main or normal coordinates. The normal coordinates y we obtain by modal transformation or

22. N-DOF SYSTEM UNDAMPED FORCED VIBRATION Let us consider the undamped system after substituting for x Multiplying this equation from left by modal transformed matrix V T or

23. N-DOF SYSTEM UNDAMPED FORCED VIBRATION The matrices M y and K y are diagonal - n independent equations If the modal vectors have been normalized for r = 1,2,..., n. the equation of motion and