Basics of Dynamics. Amit Prashant. Indian Institute of Technology Gandhinagar. Short Course on. Geotechnical Aspects of Earthquake Engineering

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1 Basics of yamics Amit Prashat Idia Istitute of Techology Gadhiagar Short Course o Geotechical Aspects of Earthquake Egieerig 4 8 March, 213 Our ear Pedulum Revisited g.si g l s Force Equilibrium: Cord legth, s. l ds d Velocity, v l. dt dt Acceleratio, ag.si dv g.si dt 2 2 d s d l. g.si 2 2 dt dt g.si l For small g. l 2 1

2 Sigle egree of Freedom Systems Stiffess, k Mass, m Stiffess, k Mass, m Stiffess, k Mass, m Stiffess, k ampig c ampig, c Mass, m 3 Sigle egree of Freedom Systems Structures which have Most of their mass lumped at a sigle locatio Oly a sigle displacemet as ukow Bridges Elevated Water Tak Equivalet SOF System 4 2

3 yamic Equilibrium Three idepedet properties Mass, m Stiffess, k ampig, c Buildig isturbace Exteral force f(t) Respose isplacemet, Velocity, Acceleratio, ut u t u t Colum Roof u(t) f(t) 5 Iteral forces Iertia force ampig force f Stiffess force t kut f I f S t t mu t c u t f I t u t f t m 1 1 c u t f S t 1 k ut 6 3

4 Force Equilibrium yamic equilibrium f I (t)+ f (t) + f S (t) = f(t) mu cu ku f (t) Stiffess force Iertia force ampig force Exteral force f(t) 7 Free Vibratios Neutral positio Iitial disturbace Pull ad release : Iitial displacemet Impact : Iitial velocity No exteral force mu cu ku ivide by mass c k u u u m m Natural frequecy ampig Ratio k m u 2 u u c 2mω Natural Period, T 2 Extreme positio 8 4

5 u(t) u(t) isplacemet u(t) isplacemet u(t) Free Vibratio Respose d v u u Udamped system Time t T v d u Expoetial decay u amped system Time t T t 9 Free Vibratio Respose of amped Systems d u Overdamped t d u I Civil Egieerig Structures Uderdamped t 1 5

6 isplacemet u(t) Aalogy of Swig oor with ashpot Closig Mechaism If the door oscillates through the closed positio it is uderdamped If it creeps slowly to the closed positio it is overdamped. If it closes i the miimum possible time, with o overswig, it is critically damped. Critical ampig: the smallest amout of dampig for which o oscillatio occurs If it keeps o oscillatig ad does ot stop, it is?? 11 Udamped System: Free vibratios Equatio of motio Solutio: u u uo u si t u v = u o Iitial velocity o Iitial displacemet cos t u Time t 12 6

7 isplacemet u(t) Udamped System: Free vibratios Example m = 5 kg k = 8 kn/m m = 5 kg k = 8 kn/m k m rad / s T s 4 f 1 T cycles / s Hz 13 amped System: Free vibratios Equatio of motio Solutio: u u o v = u 2 u u ω u ω u o Iitial velocity o si ωt uo cos ωt e ω 1 2 ω Iitial displacemet ω u o u Time t 14 7

8 amped System: Free vibratios Example m = 5 kg c = 2 kn/(m/s) k = 8 kn/m k = 8 kn/m m = 5 kg c = 2 kn/(m/s) c mω 25 4 ω T f = ω 1 4 2π = 2π = s ω = 1 = Hz T rad/s 15 Example: ampig From the give data, a = 5.5 m, a 9 =.1 m t 9 - t = 4.5 s amped atural period T = (t N - t )/N = 4.5/9 =.5 s 2π T 2π.5 ω rad/s ampig ratio a 5 5 e N e 2 1. log log. a 9. 1 N Time t (s) 16 8

9 isplacemet u(t) Forced Vibratios f si t Apply a siusoidal loadig with frequecy, The equilibrium equatios becomes mu cu ku f sit Time t 17 Forced Vibratio Respose f si t Siusoidal Force Costat Amplitude Static u static u 1 u 2 u 3 u 4 u 5 u 6 isplacemet Frequecy 18 9

10 Forced Vibratio Respose Resoace at atural frequecy of structure Critically depedat o dampig Magificatio Factor = Normalised isplacemet u max /u static Uderdamped Udamped 1 Critically amped Frequecy 19 Evaluatio of ampig Half-Power Method X B A 2 Normalised isplacemet u max /u static 1 X 2 A B Frequecy 2 1

11 Seismic Groud Motio From Earthquake yamics of Structures, Chopra (25) 21 Seismic Groud Motio Respose Chage of referece frame Rigid body motio causes o stiffess & dampig forces Movig-base Structure Mass m mu g Fixed-base Structure t u g t m(u u g ) cu ku mu cu ku mu g t Absolute acceleratio Relative Velocity/displacemet 22 11

12 Seismic Groud Motio Respose u g t Time t ut Time t 23 eformatio Respose From Earthquake yamics of Structures, Chopra (25) 24 12

13 ouble Pedulum 1 l 1 m 1 2 l 2 m 2 25 Multi egree of Freedom (MOF) Systems MOF? Mass located at multiple locatios More tha oe displacemet as ukows Equilibrium equatio i matrix form f f f f (t) Solutio is foud by I Simultaeously solvig the equatio Modal Aalysis S E u 1 (t) u 2 (t) Buildig 26 13

14 Summary SOF system Structures with SOF Iteral Forces Force Equilibrium Free Vibratio Respose Udamped amped Forced Vibratio Respose Seismic Groud Motio Respose 27 Thak You 28 14