Chapter 5 Two Degrees of Freedom
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Transcript of Chapter 5 Two Degrees of Freedom
Systems that require two independent coordinates to describe their motion are called Two Degree of Freedom Systems.
Systems that require two independent coordinates to describe their motion are called Two Degree of Freedom Systems.
The general rule for the computation of the number of freedom
Number of Degrees of
freedomof the system
Number of masses in the systemx=
number of possible types of motion of each mass
There are two equations of motion for a two degree of freedom system, one for each mass/DOF.
They are in the form of coupled differential equation- that is, each equation involves all coordinates
If harmonic solution is assumed for each coordinate, the equation lead to a frequency equation that gives two natural frequencies
Given a suitable initial excitation, the system vibrates at one of these natural frequencies
During free vibration at one of the natural frequencies, the amplitude of the TDOF are related in a specific manner and the configuration is called a normal mode, principal mode, or natural mode
If the system vibrates under the action of an external harmonic force , the resulting forced harmonic vibration takes place at the frequency of the applied force
Resonance occurs when the forcing frequency is equal to one of the natural frequencies of the system
The configuration of a system can be specified by a set of independent coordinates such as length, angle, or other physical parameter. Generalized coordinates
A set of coordinates which leads to an uncoupled equations of motions are called principal coordinates
1221212212111 Fxkxkkxcxccxm
2232122321222 Fxkkxkxccxcxm
tFtxktxctxm
2
1
0
0
m
mm
322
221
ccc
cccc
tFtxktxctxm
322
221
kkk
kkkk
tx
txtx
2
1
tF
tFtF
2
1
tFtxktxctxm
The solution involves four constant of integration. From the initial conditions;
00,00
00,00
2222
1111
xtxxtx
xtxxtx
00 32121 ccctFtF
02212111 txktxkktxm
02321222 txkktxktxm
tXtx cos11
tXtx cos22
0cos22121
2
1 tXkXkkm
0cos232
2
212 tXkkmXk
Equation 5.7
X1 and X2 are constants the maximum amplitude of x1 (t) and x2(t), φ is the phase angle.
022121
2
1 XkXkkm
0232
2
212 XkkmXk
0det
32
2
22
221
2
1
kkmk
kkkm
02
23221
2
131221
4
21
kkkkk
mkkmkkmm
nontrivial solution
Frequency or Characteristic equation
2/1
21
2
23221
2
21
132221
21
1322212
2
2
1
4
2
1,
mm
kkkkk
mm
mkkmkk
mm
mkkmkk
Natural frequencies of the system,
32
2
12
2
2
21
2
11
1
1
1
21
kkm
k
k
kkm
X
Xr
Frequencies ratios,
32
2
22
2
2
21
2
21
2
1
2
22
kkm
k
k
kkm
X
Xr
The normal modes of vibration (modal vectors),
1
11
1
1
1
2
1
11
Xr
X
X
XX
2
12
2
1
2
2
2
12
Xr
X
X
XX
The free vibration solution or the motion in time,
modefirst cos
cos
11
1
11
11
1
1
1
2
1
11
tXr
tX
tx
txtx
mode secondcos
cos
22
2
12
22
2
1
2
2
2
12
tXr
tX
tx
txtx
The unknown constant can be determine from the initial conditions,
2/1
2
1
2
2122
212
12
2/12
1
1
1
2
1
1
1
1
1
0000
1
sincos
xxrxxr
rr
XXtX
00,00
00,00
2222
1111
xtxxtx
xtxxtx
00
00tan
cos
sintan
2121
2121
1
1
1
1
1
11
1xxr
xxr
X
X
from the initial conditions,
2/1
2
2
2
2112
211
12
2/12
2
2
1
2
2
2
1
2
1
0000
1
sincos
xxrxxr
rr
XXX
00
00tan
cos
sintan
2112
2111
2
2
1
2
2
11
2xxr
xxr
X
X
Consider a torsional system consisting of two discs mounted on a shaft as shown below.
Parameters; k, J0 and Mt
11221111 ttt MkkJ
22312222 ttt MkkJ
12212111 tttt MkkkJ
22321222 tttt MkkkJ
Similar to the translational equations , but substituting θ → x, J → m, kt → k
0
0
Free Vibrations