WEEK-2: Free Vibration of SDOF systems - DEUkisi.deu.edu.tr/saide.sarigul/MVib2_Chp3.1.pdf · A...
Transcript of WEEK-2: Free Vibration of SDOF systems - DEUkisi.deu.edu.tr/saide.sarigul/MVib2_Chp3.1.pdf · A...
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WEEK-2:
Free Vibration of SDOF systems
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Free vibration of SDOF systems
•Natural frequency
•Damping Ratio
•Equation of motion
•Standart form of equation of motion
•Assume following initial conditions
•Assume a harmonic solution 0eq eq eqm x c x k x
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Free vibration of SDOF systems
•Equation of motion •Harmonic solution
• Types of Free Vibrations
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Free Vibration of SDOF Systems (Undamped Case)
•General harmonic solution
•Applying initial conditions
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Example:
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Example:
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Damped Free Vibrations with Viscous Damping
1.Underdamped vibration
•By applying initial conditions and expanding with cos and sine terms gives
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Underdamped vibration
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Underdamped vibration
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Example: Governing equation of motion of an underdamped single degree of
freedom system is given as
a) Find the natural frequency
b) Find the damping ratio
c) Find the damped natural frequency of the system
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Example: Governing equation of motion of an underdamped single degree of
freedom system shown below is given as
Find all vibration parameters
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Solution:
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2.Critically damped vibrations
•By applying initial conditions:
1 2( ) ntx t e B B t
If the initial conditions are
opposite and
0
0 0
0n
x
x x
then the response overshoots the
equilibrium position before
eventually decaying and
Approaching equilibrium from the
direction opposite that of the initial
position.
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Example:
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M=0
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3.Overdamped vibrations
•By applying initial conditions:
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The response of a system that is
overdamped is similar to a critically
damped system.
An overdamped system has more
resistance to the motion than critically
damped systems.
Therefore, it takes longer to reach a
maximum than a critically damped system,
but the maximum is smaller. An
overdamped system also takes longer than
a critically damped system to return to
equilibrium.
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Example:
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b)
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(3.29)
or 3.29
(3.48)
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(3.53)
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Coulomb damping
Equation of motion:
Initial conditions:
Response:
b) 0t /n
a) /nt 2/n
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Coulomb damping
/mg kWhen motion ceases a constant displacement from equilibrium of is maintained.
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Example:
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Hysteretic (Structural) Damping
A system undergoing periodic vibration has the following load-displacement diagram.
Energy dissipated per cycle is independent of frequency and proportional to the
amplitude
Hysteris loop
2E khX
h: Hysteretic damping coefficient
2
h
1
2
ln ln(1 )X
hX
Logarithmic decrement for hysteritic damping
For small h: h
Equivalent viscous damping coefficient: eq
n
kc h
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Problem
The area under the hysteresis curve is approximated by
counting the squares inside the hysteresis loop. Each square
represents (1 104 N)(0.002 m)=20 N m of dissipated energy.
There are approximately 38.5 squares inside the hysteresis
loop resulting in 770 N m dissipated over one cycle of motion
with an amplitude of 20 mm.
keq= the slope of the force deflection curve=5 106 N/m.
=0.123
=0.0613
h =0.385
2
h
100 /n
kr s
m
2 6 3 2
770
(5 10 )(20 10 )
Eh
kX
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The response of this structure with hysteretic damping is approximately the same
as the response of a simple mass-spring-dashpot system with a damping ratio of
0.0613 and a natural frequency of 100 rad/s. Then using underdamped free
vibration response equation, with