1

Basics of Dynamics

Amit Prashant

Indian Institute of Technology Gandhinagar

Short Course on

Geotechnical Aspects of Earthquake Engineering

04 – 08 March, 2013

Our Dear Pendulum – Revisited

Force Equilibrium:

2

.sina g

g

.sing

sAcceleration,

Cord length, .s l

Velocity, .ds d

v ldt dt

.sindv

gdt

2 2

2 2. .sin

d s dl g

dt dt

.sin 0g

l

For small . 0g

l

l

2

Single Degree of Freedom Systems

3

Stiffness, k

Mass, m

Stiffness, k

Mass, m

Dampingc

Stiffness, k Mass, m

Stiffness, k Mass, m

Damping, c

Single Degree of Freedom Systems

Structures which have Most of their mass lumped at a single location

Only a single displacement as unknown

Elevated Water Tank

Bridges

Equivalent SDOF System

4

3

Dynamic Equilibrium

Three independent properties Mass, m

Stiffness, k

Damping, c

Disturbance External force f(t)

Response

Displacement,

Velocity,

Acceleration,

tu tu

tu

5

u(t)

f(t)

Column

Roof

Building

Internal forces

Inertia force

Damping force

Stiffness force tuktfS

tuctfD

tumtf I

6

tu

tf I

m

1 1

tu

c

tfD

1

tu

tfS

k

4

Force Equilibrium

Dynamic equilibrium

fI(t)+ fD(t) + fS(t) = f(t)

7

)(tfkuucum

f(t)

Inertia force

Stiffness force

Damping force

External force

Free Vibrations

8

Initial disturbance Pull and release : Initial displacement

Impact : Initial velocity

No external force

Divide by mass

Neutral position

Extreme position

0kuucum

0 um

ku

m

cu

m

kn

02 uuu nn

nmωc

2

Natural frequencyDamping Ratio

nn

T2Natural Period,

5

Free Vibration Response

9

v0

d0

0

u0

un

Dt

T

Dis

pla

cem

en

t u

(t)

Exponential decay

Time t

v0

d0

0

u0

un

TDis

pla

cem

en

t u

(t)

Time t

Undamped system

Damped system

Free Vibration Response of Damped Systems

10

d0

0

u0

u(t

)

t

Overdamped

d0

0

u0

u(t

)

t

Underdamped

In Civil Engineering Structures

6

Analogy of Swing Door with Dashpot Closing Mechanism

If the door oscillates through the closed position it is underdamped

If it creeps slowly to the closed position it is overdamped.

If it closes in the minimum possible time, with no overswing, it is critically damped. Critical Damping: the smallest amount of damping for

which no oscillation occurs

If it keeps on oscillating and does not stop, it is ??

11

Undamped System: Free vibrations

Equation of motion

Solution:

12

0 uu n

tutu

u non

n

o

cossin

Dis

pla

cem

en

t u

(t)

Time t0

v0 =

u0

ou

Initial velocity Initial

displacement

7

Undamped System: Free vibrations

13

Examplem = 5000 kg

k = 8000 kN/m

Hzscycles

nTn

f

s

nn

T

sradm

k

n

365.6/ 37.61571.011

1571.04022

/ 405000

10008000

m = 5000 kg

k = 8000 kN/m

Damped System: Free vibrations

Equation of motion

Solution:

14

onuω

DoD

D

oo

etωutωω

un

ωuu

cossin

Dis

pla

cem

en

t u

(t)

Time t0u0

v0 = ou

02 uuu nn

21 n

ωωD

Initial velocity

Initial displacement

8

Damped System: Free vibrations

15

Examplem = 5000 kg

c = 20 kN/(m/s)

k = 8000 kN/m

Hz =.

=

DT

=D

f

s.=.π=

Dωπ

DT

rad/s..n

ωD

ω

.

nmωc

366.61573011

157309539

22

9539205014021

0504050002

1000202

nD

m = 5000 kg

k = 8000 kN/m

c = 20 kN/(m/s)

Example: Damping

16

Time t (s)

From the given data,

a0 = 5.5 m,

a9 = 0.1 m

t9 - t0 = 4.5 s

Damped natural period

TD = (tN - t0)/N

= 4.5/9 = 0.5 s

Damping ratio

rad/s57.125.0

π2

T

π2ω

DD

07010

55e92

1

Na

0a

eN21 .

.

.loglog

9

Forced Vibrations

Apply a sinusoidal loading with frequency,

The equilibrium equations becomes

17

tfkuucum sin

tf sin

Dis

pla

cem

en

t u

(t)

Time t0

Forced Vibration Response

18

Sinusoidal Force

Constant Amplitude

Displacement

Frequency 0

n

tf sin

Static 1 2 3 4 5 6

ustatic u1 u2 u3 u4 u5 u6

10

Forced Vibration Response

19

Magnification Factor

=Normalised

Displacement umax/ustatic

Frequency 0

1

n

Resonance at natural frequency of structure

Critically dependant on damping

Undamped

Under-damped

Critically Damped

No

rmal

ised

Dis

pla

cem

ent

um

ax/

ust

ati

c

Frequency 0 A n B

1

Evaluation of Damping

Half-Power Method

20

X

2X

n

AB

2

11

Seismic Ground Motion

21From Earthquake Dynamics of Structures, Chopra (2005)

Seismic Ground Motion Response

Change of reference frame Rigid body motion causes no stiffness & damping forces

22

Mass m

tug

tum g

Moving-base StructureFixed-base Structure

0kuuc)uu(m g tumkuucum g

Absolute acceleration

Relative Velocity/displacement

12

Seismic Ground Motion Response

23

Time t0

tug

Time t0

tu

Deformation Response

24From Earthquake Dynamics of Structures, Chopra (2005)

13

Double Pendulum

25

1m

2m

1l

2l

1

2

Multi Degree of Freedom (MDOF) Systems

MDOF? Mass located at

multiple locations

More than one displacement as unknowns

Equilibrium equation in matrix form

Solution is found by Simultaneously solving the equation

Modal Analysis

26

)(t

Effff

SDI

Building

u1(t)

u2(t)

14

Summary

SDOF system Structures with SDOF

Internal Forces

Force Equilibrium

Free Vibration Response Undamped

Damped

Forced Vibration Response

Seismic Ground Motion Response

27

28

Thank You