Dynamics of structures

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1 Dynamics of structures 4.Multiple degree of freedom systems Arnaud Deraemaeker 1 Outline of the chapter *Multiple degree of freedom systems in real life Hypothesis Examples *Response of a multiple degree of freedom (mdof) system Free response Forced response Influence of the damping *MDOF application : The tuned mass damper Mass-spring TMD Pendulum TMD 2 1

2 Multiple degree of freedom systems in real life 3 Reduction of a system to a mdof system 2

3 Reduction of a system to a mdof system Reduction of a system to a mdof system 3

4 Response of a mdof system 7 Conservative system : equations of motion Mass matrix Stiffness matrix 4

5 Equations of motion: general solution Admits a non trivial solution if r 2 is negative (K and M are positive definite matrices) General solution: eigen frequencies and modeshapes Generalized eigenvalue problem (- 2 ) If the system has n degrees of freedom, there exist n values of - 2 for which this equation is satisfied. These are the n eigenvalues which correspond to n eigenfrequencies n eigen vectors are associated to these eigenfrequencies. They correspond to the n mode shapes of the structure The general solution is written in the form: 5

6 Orthogonality of the modeshapes Property : Proof : (1) (2) = Premultiply (1) by,(2) by and substract taking into account symmetry of K ( ) and M ( ) 11 Orthogonality of the modeshapes Define = Matrix notation 12 6

7 Example : two degrees of freedom system Second order equation in 2 Eigen frequencies and modeshapes for for 7

8 Eigen frequencies and modeshapes Eigen frequencies and modeshapes Mode 1 Mode

9 General solution Assume the following initial conditions 17 Particular solution : projection of the solution in the modal basis Projection on the modal basis N independent equations of the type 18 9

10 Particular solution : projection of the solution in the modal basis The solution can be obtained by solving a set of n independent equations of the type This equation corresponds to the equation of motion of a sdof system with *mass (modal mass) *stiffness * angular eigenfrequency *excitation (modal excitation) 19 Particular solution : harmonic excitation 10

11 Harmonic excitation: modal basis solution or sdof oscillator solution 21 Harmonic excitation: modal basis solution The solution is the sum of sdof oscillators : 22 11

12 Example : two dofs system 23 Equations of motion : modal basis solution (m=1kg, k = 1N/m) 24 12

13 Equations of motion : modal basis solution (m=1kg, k = 1N/m) 25 Forced excitation videos 26 13

14 Forced excitation videos 27 Damped equations of motion Damping matrix 28 14

15 Damped equations of motion : general solution Non trivial solution if Complex roots of the characteristic equation -> Oscillatory functions with exponential envelope Complex eigen vectors = complex modeshapes -> Not often used in practice in vibrations 29 Equations of motion : modal basis solution real mode shapes Mode shapes of conservative system : Projection on the real modal basis: or In general is not diagonal and the equations remain coupled but 30 15

16 Equations of motion : modal basis solution real mode shapes Rayleigh damping: Often used as a simplifying assumption to decouple the equations but does not have a physical meaning Modal damping When damping is small, off-diagonal terms can be neglected leading to: = is the modal damping of mode i 31 Equations of motion : modal basis solution real mode shapes n independent equations of the type This equation corresponds to the equation of motion of a sdof system with *mass (modal mass) *stiffness *angular eigenfrequency * damping coefficient (modal damping) *excitation (modal excitation) 32 16

17 Link between Rayleigh damping and modal damping Rayleigh damping Modal damping Only two parameters to define the damping of all modes ->Overestimation at low and high frequencies ->Represents accurately the damping of two modes only 33 Particular solution : harmonic excitation 34 17

18 Harmonic excitation: modal basis solution Projection on modal basis Modal damping hypothesis (small damping) n decoupled equations Sum of damped sdof oscillators 35 Example : two dofs system 36 18

19 Example : two dofs system Damped resonances k=1 N/m, m=1kg, b=0.04 Ns/m Never goes to zero 37 Validity of modal damping hypothesis Neglect off-diagonal terms 38 19

20 Validity of modal damping hypothesis k=1 N/m, m=1kg, b=0.04 Ns/m Comparison of exact (coupled-dotted line) and approached (uncoupled) responses 39 Validity of modal damping hypothesis k=1 N/m, m=1kg, b=0.2 Ns/m The modal damping hypothesis is not valid for high values of damping 40 20

21 MDOF application : The tuned mass damper (TMD) 41 Application example : the tuned mass damper Equations of motion: 21

22 Application example : the tuned mass damper Undamped vibration absorber (b=0) for 43 Undamped tuned mass damper =0.03 =1 DVA (dynamic vibration absorber = TMD) tuned to eigenfrequency of primary system -> Reduces vibrations in a narrow band around eigenfrequency -> Amplification outside of this narrow band 44 22

23 Pendulum tuned mass damper * small neglected Inertial coupling of the two systems 45 Pendulum tuned mass damper for 46 23

24 Pendulum tuned mass damper Tuning of the PTMD based on the length of the pendulum Effect of the mass mainly on the spreading of the peaks =1 47 Undamped tuned mass damper The building is excited at its natural frequency (2.62 Hz) 48 24

25 Damped tuned mass damper =0.03 =1 -Reduction of vibration is lower around eigenfrequency with b increasing -Reduces the amplification outside of the narrow frequency band -Existence of P and Q : points where all curves cross 49 Optimal design of tuned mass dampers P and Q are at equal height for Optimum damping is given by 50 25

26 Steps to follow to design a TMD - The maximum mass of the device is decided fixing - Based on this value, the frequency of the TMD is tuned : - Which allows to compute the stiffnes of the TMD - And finally the optimal damping is computed 51 Tuned mass damper in action 52 26

27 Example of tuned mass dampers in structures Millenium bridge, London 53 Tuned mass damper in action on a bridge 54 27

28 John Hancock Tower (Boston-1976) Two TMDs of 2700 kn (approx 5.2x5.2x1m steel blocks) 55 City Corp Center (New York ) - 279m high - Fundamental period = 6.5s Tuned mass damper -400 Tons block installed at the top (2% of effective mass of first mode) 56 28

29 City Corp Center (New York ) 57 Chiba Port Tower (Japan ) Tuned mass damper : 15 tons Can slide in two directions 58 29

30 Damped pendulum tuned mass damper small 59 Damped pendulum tuned mass damper Let us define : TMD frequency 60 30

31 Damped PTMD : optimal parameters =0.03 with 61 Pendulum tuned mass damper example Dampers Tai Pei (Taiwan) 62 31

32 Pendulum tuned mass damper example Taipei Pendulum tuned mass damper example Taipei

33 Pendulum tuned mass damper example Pendulum motion during earthquake, May 12,

IJSRD - International Journal for Scientific Research & Development Vol. 4, Issue 05, 2016 ISSN (online):

IJSRD - International Journal for Scientific Research & Development Vol. 4, Issue 05, 2016 ISSN (online): 2321-0613 Experimental Investigation of Effect of Tuned Mass Damper in the Structure using Shake