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