Transcript Materialy/01/Applied Mechanics-Lectures/Applied Mechanics

Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 03
FUNDAMENTALS OF VIBRATIONS

The first remarkable work from filed of vibration was work
of Lord Rayleigh - Theory of Sound (published in 1887)

He introduced concept of oscillations of a linear system
and showed the existence of natural modes and natural
frequencies for discrete as well as continuous systems.

This work remains valuable in many ways even though
he was concerned with acoustics rather than with
structural mechanics.
FUNDAMENTALS OF VIBRATIONS

Vibration is in general a periodic motion in time and is
used to describe oscillation in mechanical systems.

In most cases, the general purpose is to prevent or
attenuate the vibrations, because of their detrimental
effects, such as fatigue failure of components and
generation of noise.

However, there are some applications where vibrations
are desirable and are usefully employed, as in vibration
conveyers, vibrating sieves, etc.
FUNDAMENTALS OF VIBRATIONS




Vibration - term describing oscillation in a mechanical
system - defined by the frequency (or frequencies) and
amplitude.
Either the motion of a physical object or structure or,
alternatively, an oscillating force applied to a mechanical
system is vibration in a generic sense.
The time-history of vibration may be considered to be
sinusoidal or simple harmonic in form.
The frequency is defined in terms of cycles per unit time,
and the magnitude in terms of amplitude (the maximum
value of a sinusoidal quantity).
FUNDAMENTALS OF VIBRATIONS

Vibration may be described as:
 deterministic
vibration - it follows an established
pattern so that the value of the vibration at any
designated future time is completely predictable from
the past history.
 random
vibration - its future value is unpredictable
except on the basis of probability - defined in
statistical terms wherein the probability of occurrence
of designated magnitudes and frequencies can be
indicated. The analysis involves certain physical
concepts that are different from those applied to the
analysis of deterministic vibration.
FUNDAMENTALS OF VIBRATIONS

The structures have the three fundamental properties which are
the inherent characteristics of a structure with which it will resist or
oppose vibration. The three fundamental properties are:

m - mass (kg) - represents the inertia of a body to remain in its
original state of rest or motion. A force tries to bring about a change in
this state of rest or motion, which is resisted by the mass.

k - stiffness (N/m) - there is a certain force required to bend or
deflect a structure with a certain distance. This measure of the force
required to obtain a certain deflection is called stiffness.

b - damping (Ns/m) - once a force sets a part or structure into
motion, the part or structure will have inherent mechanisms to slow
down the motion (velocity). This characteristic to reduce the velocity of
the motion is called damping.
FUNDAMENTALS OF VIBRATIONS
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The combined effects to restrain the effect of
forces due to mass, stiffness and damping
determine how a system will respond to the
given external force.
If the vibrations due to the external force are
much larger than the net sum of the three
restraining characteristics, the amount of the
resulting vibrations will be higher.
FUNDAMENTALS OF VIBRATIONS

Vibrations of mechanical systems may be generally
classified into three categories:

Free vibrations - occur only in conservative systems where there is
no friction, damping and exciting force. The total mechanical energy,
which is due to the initial conditions, is conserved and exchange can
take place between the kinetic and potential energies.

Forced vibrations - caused by the external forces, which excite the
system. Exciting forces supply energy continuously to compensate
for that dissipated by damping.

Self-excited vibrations - periodic oscillations of the limit cycle type
and are caused by some nonlinear phenomenon. Energy required to
maintain the vibrations is obtained from a non-alternating power
source. In this case, the vibrations themselves create the periodic
force.
VIBRATION OF SINGLE-DOF SYSTEM

Simplest dynamic systems - elastic, dissipating and inertia
forces interact.
Consists: mass m attached by means of a spring k and a damper b
to an immovable support
. ..
x, x, x
b
F(t)
k
m
a
a - translation system, b - torsional system
b
VIBRATION OF SINGLE-DOF SYSTEM
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The mass is constrained to translational motion in the
direction of the x axis - change of position from an initial
reference is described fully by the value of a single
quantity x - called a single DOF system.
Free vibration - mass m is displaced from its
equilibrium position and then allowed to vibrate free.
Forced vibrations - continuing force F acts upon the
mass or the foundation experiences a continuing
motion.
VIBRATION OF SINGLE-DOF SYSTEM
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The equation of motion - model of translation system:
mx  bx  kx  F (t )
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The equation of motion - model of torsional system the mass m is replaced by mass moment of inertia I,
the force F(t) by the moment M(t):
  bt   kt   M (t )
I
- torsional damper,
- torsional stiffness,
 - rotation, angular velocity and angular acceleration.
  
bt
kt
VIBRATION OF SINGLE-DOF SYSTEM
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The solution of the equation of motion is composed of two
parts:

solution of the homogenous equation – so-called free vibration

solution of the non-homogenous equation – with non-zero right
side of equation of motion – so-called forced vibration
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
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Mechanical model of a free undamped vibration
. ..
x, x, x
k
m
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The equation of motion
mx  kx  0
k
x  x  0
m
0 – natural angular frequency
x  02 x  0
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
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The characteristic equation
r 2  02  0
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Roots of the characteristic equation
r1,2    02   (1) 02  0

General solution
x  A sin( 0t )  B cos(0t )  C sin( 0t  )
where A, B
- constants,
C  A2  B 2
- amplitude,
  arctan( B A)
- phase angle
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
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Using the initial condition
 x  x0
t 0
 x  v0
and the derivative of solution x with respect to time,
x  A0 cos(0t )  B0 sin( 0t )
the constants are
v0
A
0
C
x02
B  x0

v02
02
x0 0
  arctan
v0
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
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The motion of the system
v
x  0 sin( 0t )  x0 cos(0t )
0
resp. in amplitude form
x

x02

x00 

 2 sin  0t  arctan
v0 
0

v02
The motion in this case is a harmonic vibration.
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Period of vibration
2
m
T0 
 2
0
k
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Linear natural frequency
f0 
1 0
1 k


T0 2 2 m
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
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The displacement x, velocity and acceleration for parameters:
x0  0 m
v0  1 m/s
0  0,5 rad/s
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING
SINGLE-DOF SYSTEM
FREE VIBRATION, WITHOUT DAMPING