Lecture 03
of 16
description
Mechanical Vibrations Lecture 3
Transcript of Lecture 03
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Lecture 2:
Spring and Stiffness
Mass or Inertia Elements
Damping
Simple Harmonic Motion
Free Vibrations of SDOF
Introduction -1
With many figures and models from Mechanical Vibrations, S. S. Rao
Lecture 3:
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tAAx sinsin
tAdt
dx cos
xtAdt
xd 222
2
sin
Harmonic Motion
Periodic Motion: motion repeated after equal
intervals of time
Harmonic Motion: simplest type of periodic motion
Displacement (x): (on horizontal axis)
Velocity:
Acceleration:
Introduction -2
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Harmonic Motion
Harmonic motion as
the projection of the
end of a rotating
vector. This requires
the description of both
the horizontal and
vertical components.
It is more convenient
to represent harmonic
motion using complex
number representation
Introduction -3
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Relationship between Displacement, Velocity and
Acceleration
a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
0
1
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20
0
20
v
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200
0
200
Time (sec)
A=1, n=12
x(t) Asin(nt )
x(t) nAcos(nt )
x(t) n2Asin(nt )
Note how the relative magnitude of each increases for n>1
Displacement
Velocity
Acceleration
Harmonic Motion
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Harmonic Motion
Introduction -5
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)2sin()( ftAtx
Harmonic Motion
Time (s)
Displacement x(t)
Period T
0
Amplitude Adelay t
Introduction -6
tt fphase 2
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Free Vibration of SDOF Systems
Free Vibration occurs when a system oscillates only
under an initial disturbance with no external forces
acting after the initial disturbance
Undamped vibrations result when amplitude of motion
remains constant with time (e.g. in a vacuum)
Damped vibrations occur when the amplitude of free
vibration diminishes gradually overtime, due to
resistance offered by the surrounding medium (e.g. air)Free vibration -7
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Free Vibration of SDOF Systems
Undamped vs. damped vibration
Free vibration -8
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Free Vibration of SDOF Systems
Several mechanical and structural systems can be
idealized as single degree of freedom systems, for
example, the mass and stiffness of a system
Free vibration -9
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Free Vibration of SDOF Systems
To study the free-vibration of the mass, we
need to derive the governing equation, known
as the equation of motion.
The equation of motion of the undamped
translational system is derived using four
methods:
Newtons second law of motion
DAlemberts principle
The principle of virtual displacements
The principle of conservation of energy
Free vibration -10
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Free Vibration of SDOF Systems
The equation of motion using Newtons second law:
Select a suitable coordinate to describe the position of the mass or rigid body in the system.
Determine the static equilibrium configuration of the system and measure the displacement of the mass
from its static equilibrium
Draw the free-body diagram of the mass when a positive displacement is given to it
Apply Newtons second law of motion to the mass
Resultant force on the mass = mass acceleration
F = maFree vibration -11
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Free Vibration of an Undamped System
For undamped single degree of freedom system,
the application of Newtons second law to mass m yields the equation of motion:
Free vibration -12
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Does gravity matter in spring problems?
Let be the deflection caused by
hanging a mass on a spring
( = x1-x0 in the figure)
Then from static equilibrium:
mg k
Next use Newtons law in the vertical direction for some point x > x1 measured
from
So no, gravity does not have an effect on the vibration
(note that this is not the case if the spring is nonlinear)Introduction -13
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Free Vibration of an Undamped System
Does it matter if the mass and the spring are
hanged vertically?
M
x(t)
No, gravity does not have
an effect on the vibration
Free vibration -14
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Principle of Conservation of Energy.
A system is said to be conservative if no energy
is lost due to friction or energy-dissipating
nonelastic members.
If no work is done on the conservative system by
external forces, the total energy of the system
remains constant. Thus the principle of
conservation of energy can be expressed as:
0)(
constant
UTdt
d
UT
or
Free Vibration of an Undamped System
Free vibration -15
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The kinetic energy is stored in the mass and the
potential energy is stored in the spring and are
given by:2
2
1xmT
2
2
1kxU
Substitution yields the desired equation:
0 kxxm
Free Vibration of an Undamped System
0)( UTdt
dRecall;
Free vibration -16
