mechanical vibrations. introduction 1 examples
Post on 20-Jan-2016
265 Views
Preview:
TRANSCRIPT
Introduction
1
Examples
Examples
Equation of motion
Newton's 2nd law of motion
The rate of change of
momentum of a mass is equal to the force acting
on it.
Newton's 2nd law of motion
Energy conservation
0
constant
UTdt
d
UT
Potential energy
2
2
1kxU
Kinetic energy 2.
2
1xmT
0..
kxxm
0....
..2
2
kxxmxmkxtF
xmdt
txdm
dt
txdm
dt
dtF
Vertical system
0..
; ..
kxxm
kmgmgxkxm stst
Solution to the equation of motion:
0
or
0 0)(
:yields (1)eqation intosolution thesubstitute
solution. of roots theare andconstant is B where)(
: type theofsolution thehasmay (1)equation osolution t
frequency natural theasknown theis / where
0
as, written becan (1)Equation equation. aldifferentiordinary
order 2nd homogenuos ait (1), 0
22
22
2
n
ststn
st
n
n
s
eBes
SBetx
mk
xx
kxxm
(2)
Then
)( and Put
sin)(cos)()(
,sincos
: thatNote
BB)(
:as written be nowcan (1)equation osolution t The
.
212211
2121
21
tsinωAtcosωAx(t) n2n1
BBjABBA
tBBjtBBtx
Therefore
tjte
eetx
jsei
nn
nntj
tjtj
n
n
nn
motion. theof applying
from obtained becan ) and (or and constants The :
.frequancy and amplitudean of
motion harmonic a of consists system theof vibrationfree hesay that tcan we
angle. phase theis tan
andmotion of amplitude theis Where
(3)
:form in the written be alsocan (1)equation osolution t The
21
n
1
21
22
21
nditionsinitial co
AAANote
A
A
A
AAA
φ)tAcos(ωx(t) n
. any timeat ion consideratunder system
theofnt displaceme of valueobtain the toused becan (4)Equation
tan and
:become angle phase theand amplitude The
)........(4 sincos)(
:becomessolution general then the and
:yields (3)or (2)solution general theinto conditions
thesegSustitutin . )( and )( ; 0at Let
:conditions Inatial
1
2
20
0
0
.
201
0
..
0
t
XA
ttXtx
XAXA
XtxXtxt
nn
n
n0
0
.
n
0
.
n
0
.
ωXX
ωX
φ
ωX
A
Graphical representation x(t ) = A cos(ωnt – ϕ )
Torsional vibration
rad/sec.
as obtained becan frequency natural which thefrom and
motion ofequation theis 0
0
:law 2 Newtons Applying
ineria. ofmoment polar theis 32
where
as; calculated isshaft theof
stiffness that theNote shown. as ineria ofmoment of
disk a and K stiffness torsionalofshaft a has systemA
n
nd
4
t
D
t
tDDt
D
pp
t
D
J
K
KJJK
JM
dI
L
GIK
J
Natural Frequency (ωn) :
It is a system property. It depends, mainly, on the stiffness and the mass of vibrating system.
It has the units rad./sec, or cycles/sec. (Hz)
It is related to the natural period of oscillation (τn) such that, τn = 2π/ωn
and ωn = 2 π fn where fn is the natural frequency in Hz.
Example 2.1 solution:
Initial assumptions:
1. the water tank is a point mass
2. the column has a uniform cross section
3. the mass of the column is negligible
Example 2.1 solution:
a. Calculation of natural frequency:
1. Stiffness: , But:
So:
2. Natural frequency :
3
3
l
EIk 44444 3475.24.23
6464mddI io
mNxxx
k / 812,28990
3475.2103033
9
srxm
kn /ad 9829.0
103
812,2895
Example 2.1 solution:
b. Finding the response:
1. x(t ) = A cos (ωn t - ϕ )
So, x(t ) = 0.3 cos (0.9829 t )
mXX
XA on
o
o 3.0
. 2
2
0
0tan
.
tan 11
nono
o
XX
X
Example 2.1 solution:
c. Finding the max. velocity:
Finding the max. acceleration :
0.9829)*/(2then t
/20.9829t when maximum is )(
9829.0sin9829.03.0.
tx
ttx
smx /2949.09829.03.0.
max
9829.0/then t
9829.0 when maximum is ..
9829.0cos9829.03.0..
2
ttx
ttx
22max /2898.09829.03.0
..smx
Example 2.2 :
mm
kk
kk
m
m
eqneq
21
21eq
eqeq
2121
2211
21
k isfrequency natural theand 0,xkx
thenismotion ofequation the,4
1k
:becomes system theof stiffness equivalent for x the
ngsubstitutiby then x,k wsuch that k stiffness and
mass of system spring mass a toequivalent is system The
k
2w
k
2w2or x 2x2x x
thatNote g. w where xk 2w and xk 2w
: thatshow
pulleys theof diagramsbody free The ly.respective xand xare
2 and 1 pulleys of centres theofnt displaceme Let the
Example 2.2 : solution
Simple pendulum
Governing equation:
Assume θ is very small
Natural frequency (ωn)
.. OOO JJM
0sin..
mglJO
sin
0..
mglJO
mgl
J
J
mgl O
On 2
Solution
n
oon
o
AAt
At
.
,.
0.
0
22
1
ttt nn
o
no sin
.
cos
tAtAt nn sincos 21
0sin..
WdJO
0..
WdJO
OOn J
mgd
J
Wd
Example 2.5: Q2.45
Draw the free-body diagram and derive the equation of motion using Newton s second law of motion for each of the systems shown in Fig
top related