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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