Post on 12-Feb-2017
transcript
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
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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
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3
Dynamic Equilibrium
Three independent properties Mass, m
Stiffness, k
Damping, c
Disturbance External force f(t)
Response
Displacement,
Velocity,
Acceleration,
tu tu
tu
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u(t)
f(t)
Column
Roof
Building
Internal forces
Inertia force
Damping force
Stiffness force tuktfS
tuctfD
tumtf I
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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)
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)(tfkuucum
f(t)
Inertia force
Stiffness force
Damping force
External force
Free Vibrations
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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
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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
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d0
0
u0
u(t
)
t
Overdamped
d0
0
u0
u(t
)
t
Underdamped
In Civil Engineering Structures
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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 ??
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Undamped System: Free vibrations
Equation of motion
Solution:
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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
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Undamped System: Free vibrations
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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:
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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
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Damped System: Free vibrations
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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
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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
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tfkuucum sin
tf sin
Dis
pla
cem
en
t u
(t)
Time t0
Forced Vibration Response
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Sinusoidal Force
Constant Amplitude
Displacement
Frequency 0
n
tf sin
Static 1 2 3 4 5 6
ustatic u1 u2 u3 u4 u5 u6
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Forced Vibration Response
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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
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X
2X
n
AB
2
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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
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Mass m
tug
tum g
Moving-base StructureFixed-base Structure
0kuuc)uu(m g tumkuucum g
Absolute acceleration
Relative Velocity/displacement
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Seismic Ground Motion Response
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Time t0
tug
Time t0
tu
Deformation Response
24From Earthquake Dynamics of Structures, Chopra (2005)
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Double Pendulum
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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
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)(t
Effff
SDI
Building
u1(t)
u2(t)
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Summary
SDOF system Structures with SDOF
Internal Forces
Force Equilibrium
Free Vibration Response Undamped
Damped
Forced Vibration Response
Seismic Ground Motion Response
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Thank You