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SDOF systems- a recap
CIVE 405
Instructor: Dr. Budhaditya Hazra
Department of Civil & Environmental
Engineering
SDOF system
• A single geometrical co-ordinate often
called a degree-of-freedom is adequate to
represent the motion of the system
• A handy approximation for most of the
complex engineering systems
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SDOF system: A physical model
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c c
SYSTEM LUMPED MASS
MODEL
A mass-spring model
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Spring-mass-damper model
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• Inertia force
• Spring force
• Damping force
Equation of motion
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• Natural frequency
• Damping
Writing equation of motion
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• Force equilibrium
• Moment equilibrium
• Principal of virtual work
• Lagrange’s equations
Example: Writing equation of
motion
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Force type Force Moment about O
Inertia
Spring-1
Spring-2
Tor spring
Example: Moment equilibrium
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𝐽𝑜𝜃
𝑘1𝑎𝜃 (𝑘1𝑎𝜃)𝑎
𝑘2𝑙𝜃 (𝑘2𝑙𝜃)𝑙
𝑘𝑡𝜃
Example….
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Equation of motion (∑M = 0 about the pt O)
ml2
3θ + kt + k1a
2 + k2l2 θ =0
𝐽𝑜 =ml2
12+m(
𝑙
2)2=
ml2
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Free vibration
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Assume a solution of the form:
For un-damped case the general solution is :
Free vibration…
• The constants A1 & A2 are found using
initial conditions
•
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Complete solution:
Damped free vibration
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For the under-damped case the general solution is :
Initial conditions:
Complete solution:
Logarithmic decrement
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Logarithmic decrement
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(ζ < 0.3)
Reasonably accurate for most
practical structures
Forced vibration under
harmonic excitation
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Assume the steady state solution :
EQN of motion
harmonic excitation…..
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=>
&
Amplification factor
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Applications of harmonic
excitation
• Force transmissibility
• Base motion
(base isolation)
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Forced vibration under arbitrary
excitation
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Step load or a constant load
Pulse load
SDOF system under step input
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SDOF system subjected to a step input
Initial conditions
SDOF system under impulse
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Dirac delta function
SDOF system under impulse
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Solution:
Duhamel Integral
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Frequency response Function
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Approach the solution to the equation
using Fourier transforms
Therefore,
Frequency response Function
• The quantity
is called frequency response function
•
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Frequency & impulse response
function •
•
•
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Frequency & impulse response
function
•
• Then
•
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