Bda31103 Lect02 -1 Dof Part1
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Transcript of Bda31103 Lect02 -1 Dof Part1
BDA 31103 – VIBRATION
CHAPTER 2 –
SINGLE DEGREE
OF FREEDOM
SYSTEM
MR MOHD AMRAN BIN HJ. MADLAN
Faculty of Mechanical & Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia
2
Free Vibration of Single Degree of
Freedom (SDOF)
� A system is said to undergo free vibration when it oscillates only under an initial disturbance with no external forces acting after initial disturbance
3
Introduction - SDOF
� One coordinate (x) is sufficient to specify the position of the mass at any time
� There is no external force applied to the mass
� Since there is no element that cause dissipation of energy during the motion of the mass, the amplitude of the motion remains constant with time, undamped system
4
Introduction - SDOF
� If the amplitude of the free vibration diminished gradually over time due to the resistance offered by the surrounding medium, the system are said to be damped
� Examples of free vibration: oscillations of the pendulum of grandfather clock, the vertical oscillatory motion felt by a bicyclist after hitting a road bump, and the swing of a child on a swing under an initial push.
5
6
7
Free Vibration of an Undamped
Translation System
� Equation of Motion using Newton’s Second Law
� Select a suitable coordinate to describe the position of the mass or rigid body
� Determine the static equilibrium configuration of the system and measure the displacement of the mass or rigid body
� Draw the free body diagram of the mass or rigid body when a positive displacement and velocity are given
� Apply Newton’s second law of motion
8
Free Vibration of an Undamped
Translation System
� Newton’s second law
� Applied to undamped SDOF system
A spring-mass system in horizontal position
xmtF &&rr
=)( θ&&rrJtM =)(
For rigid body undergoing
rotational motion
9
Free Vibration of an Undamped
Translation System
A spring-mass system in horizontal position
xmkxtF &&rr
=−=)(
0=+ kxxm &&
10
Free Vibration of an Undamped
Translation System
� Equation of Motion using other methods
� D’Alembert’s Principle
� Principle of Virtual Displacements
� Principle of Conservation of Energy
0=+ kxxm &&
� Spring-Mass System in Vertical Position
For static equilibrium
11
stkmgW δ==
Wxkxm st ++−= )( δ&&
0=+kxxm&&
� The solution can be found assuming,
substituting
characteristic equation eigenvalues12
stCetx =)(
0)()(2
2
=+ stst CekCedt
dm 0)( 2 =+ kmsC
02 =+ kmsni
m
ks ω±=
−±=2
12
1
=m
knω
� The general solution,
where C1 and C2 are constants
using
where A1 and A2 are new constants and can be determine
from the initial conditions13
titi nn eCeCtxωω −+= 21)(
tAtAtx nn ωω sincos)( 21 +=
tite ti ααω sincos ±=±
� The initial conditions at t = 0
14
02
01
)0(
)0(
xAtx
xAtx
n&& ===
===
ω
Hence, . Thus the solution
subject to the initial conditions is given by
nxAxA ω/ and 0201&==
tx
txtx n
n
n ωω
ω sincos)( 00
&+=
� Free vibration of an undamped: Harmonic
Motion
15
)sin()( 00 φω += tAtx n
where A0 and are new constants, amplitude
and phase angle respectively: 0φ
2/12
02
00
+==
n
xxAA
ω&
= −
0
01
0 tanx
x n
&
ωφ
amplitude
phase angle
The nature
of harmonic
oscillation
can be
represented
graphically
in the figure
16
Example 1:
Consider a small spring about 30 mm long,
welded to a stationary table (ground) so that
it is fixed at the point of contact, with 12 mm
bolt welded to the other end, which is free to
move. The mass of the system is 49.2 x 10^-
3 kg. The spring constant, k = 857.8 N/m.
Calculate the natural frequency and period
of system.
17
Example 1: Solution
18
srad
xm
kn 132
102.49
8.8573=== −ω
Natural frequency:
In hertz:
The period:
Hzf nn 21
2==
πω
sf
Tnn
0476.012===
ωπ
Example 2: Harmonic
Obtain the free response of
in the form
Initial condition are and
19
)(1282 tfxx =+&&
tAtAtx nn ωω sincos)( 21 +=
mx 05.0)0( =
smx /3.0)0( −=&
Example 2: Solution
20
tt
ttx
8sin0375.08cos05.0
8sin8
3.08cos05.0
−=
−+=