Dr. S. K. Kudari, Professor, Department of Mechanical Engineering, B

Dr. S. K. Kudari,Professor,Department of Mechanical Engineering,B V B College of Engg. & Tech., HUBLI

email: [email protected]

Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

CHAPTER-6Systems with two degree of freedom

Recap

Mathematical modeling of two DOF system

Mathematical derivation of governing equations

Newton's method

Energy method (Lagrange’s method)

Solution to governing equations

Natural frequencies

Modal vectors and modal matrix

Mode shapes


Systems with two degree of freedom

Torsional systems

Definite and semi-definite systems

Equivalent shaft for a torsional system

A turbo-generator system

Recap



m

K

m

K

K

x1

x2K

K

Practice problems (Linear definite systems)

Steps(i) derive the equations of motion,(ii) setup the frequency equation and obtain the fundamental natural frequencies(iii) obtain the modal vectors and modal matrix(iv) draw mode shapes of the system.



m

K

m

K

K

x1

x2K

K

. Force equilibrium diagram

Governing equations Newton’s method

m1

Kx1

K(x2-x1)

11xm

Kx1

m2 22xm

Kx2Kx2

Practice problems


Systems with two degree of freedomPractice problems

m

K

m

K

K

x1

x2K

K

Lagrange’s method

22

21 xm

2

1xm

2

1T

22

22

2122

21

21 Kx

2

1Kx

2

1)x(xK

2

1Kx

2

1Kx

2

1U

2

1i x

xx

The Lagrange’s equation is :

iiii

Qx

U

x

T

x

T

dt

d


Systems with two degree of freedomPractice problems

m mK K K

x1 x2

m mK K K

x1 x2

K

For the systems shown in Figure(i) derive the equations of motion,(ii) setup the frequency equation and obtain the fundamental natural frequencies(iii) obtain the modal vectors and modal matrix(iv) draw mode shapes of the system.


Systems with two degree of freedomPractice problems (Torsional semi-definite systems)

J1J2

1 2K

2K

Turbine

Generator

Gears

Shaft-2

Shaft-1J1

J2

J3

J4

K1

K2

For the systems shown in figure, obtain natural frequencies and mode shapes

For the systems shown in figure, obtain natural frequencies and mode shapesNeglect inertia of gearsTake

2n and JJJ

KKK

21

21


Systems with two degree of freedomPendulum systems (double pendulum)

m1

m2

l1

l2

Obtain the natural frequencies of the double pendulum shown in the figure.For simplicity take m1=m2=m and l1=l2=l



Due to self weight of masses, the pendulum roads are in tension

m1g

m2g

l1

l2

T1

T2

T2

gmT 22

211 TgmT

)gm(mgmgmT 21211

Above equations holds good for small oscillations



Free body diagram

Pendulum systems (double pendulum)

l2

x1x2

2

l11

T1

T2

11xm

22xm

T2

1

111 l

xθsinθ

2

1222 l

xxθsinθ

0θTθTxm 221111

0l

xxT

l

xTxm

2

122

1

1111

11xm T1sin1= T11

T2sin2= T22

Mass 1

x1and x2 are generalized co-ordinates





0l

xxg)(m

l

x)gm(mxm

2

122

1

12111

m1=m2=m and l1=l2=l

0l

xx(mg)

l

x(2m)gxm 121

1

0xl

mgx

l

3mgxm 211

1 Eqn. of motion



Free body diagram


l2

x1 x2

2

l11

T1

T2

11xm

22xm

T2

1

111 l

xθsinθ

2

1222 l

xxθsinθ

22xm T2sin2= T22

0θTxm 2222

0l

xxTxm

2

12222

Mass 2



0l

xxTxm

2

12222

m1=m2=m and l1=l2=l

0l

xxgmxm

2

12222

0xl

mgx

l

mgxm 212

2nd Eqn. of motion



0xl

mgx

l

mgxm 212

2 Eqn. of motion

0xl

mgx

l

3mgxm 211

1 Eqn. of motion

Solution to governing eqns.:

φωtsinAx 11 φ)sin(ωAx 22 t

Assume SHM

The above equations have to satisfy the governing equations of motions

Governing equations of motions:



0φ)sin(ωAl

mgφ)sin(ωAmω

l

3mg21

2

0φ)sin(ωAmωl

mgφ)sin(ωA

l

mg2

21

0φ)sin(ω tIn above equations

The above equations reduces to: (characteristic equation)

0Al

mgAmω

l

3mg21

2

0Amωl

mgA

l

mg2

21



0

mωl

mgl

mg

........

l

mg

mωl

3mg

2

2

The above equation is referred as a characteristic determinant Solving, we get :

Frequency equation2

222242

l

g2m

l

gω4mωm




l

g0.27ω1

l

g3.73ω2 radians/sec

Natural frequencies of the system

Modal vectors and the mode shapes can be obtained by usual meaner



Systems with two degree of freedomString systems

Obtain the natural frequencies of the string system shown in the figure.For simplicity take m1=m2=m and l1=l2=l3=l

m2

l1 l2 l3

m1



m2

l1 l2 l3

m1

Free body diagram

x1 x21

2

3

TT

T

For small angular oscillations, it can be assumed that the tension in the string (T) do not change

l

xθ 1

1

l

xxθ 21

2

l

xθ 2

3

String systems



Free body diagram

m2

l1 l2 l3

x1 x21

2

3

TT

T

m1

m1

T TT 2T 1

1xm

m2

T 2

T 3

2xm

String systems



m1

T TT 2T 1

1xm

0TθTθxm 211

0l

xxT

l

xTxm 211

1

0xl

Tx

l

2Txm 211

0TθTθxm 322

0l

xT

l

xxTxm 221

2

0xl

2Tx

l

Txm 212

m2

T 2

T 3

2xm



0xl

Tx

l

2Txm 211

0xl

2Tx

l

Txm 212

Equations of motion

Solution to governing eqns.:

φωtsinAx 11 φ)sin(ωAx 22 t

Assume SHM

The above equations have to satisfy the governing equations of motions

2 Eqn. of motion

1 Eqn. of motion



0φ)sin(ωAl

Tφ)sin(ωAmω

l

2T21

2

0φ)sin(ωAmωl

2Tφ)sin(ωA

l

T2

21

0φ)sin(ω tIn above equations

The above equations reduces to: (characteristic equation)

0Al

TAmω

l

2T21

2

0Amωl

2TA

l

T2

21

String systems



0

mωl

2Tl

T

........

l

T

mωl

2T

2

2

The above equation is referred as a characteristic determinant Solving, we get :

Frequency equation2

2242

l

3T

l

4Tmωωm

String systems



ml

Tω1

ml

3Tω2 radians/sec

Natural frequencies of the system

String systems

As the system has two natural frequencies, under certain conditions it may vibrate with first or second frequency, which are referred as principal modes of vibration



Characteristic equations of the system are:

First principal mode of vibrationThe system vibrates with first fundamental natural frequency, i.e . 1ωω

String systems

0Al

TAmω

l

2T21

2

0Amωl

2TA

l

T2

21



For vibrations under Mode-I, consider

A11-amplitude of first mass (m1) due to frequency A21-amplitude of second mass (m2) due to frequency

Characteristic equations of the system changes to:

111

21 μAA

amplitude ratio Let

1ω

1ω

String systems

0Al

TAmω

l

2T2111

21

0Amωl

2TA

l

T21

2111



lT

mωl

2T

AA

μ

21

11

211

1

1

Aμ

A

A

AA

111

11

21

111

Substitute 1

First modal vector

String systems

0Al

TAmω

l

2T2111

21

1

lT

mlT

ml

2T

AA

μ11

211



For vibrations under Mode-II, consider

A12-amplitude of first mass(m1) due to frequency A22-amplitude of second mass (m2) due to frequency

Characteristic equations of the system changes to:

212

22 μAA

amplitude ratio Let

2ω

2ω

String systems

0Al

TAmω

l

2T2212

22

0Amωl

2TA

l

T22

2212



Substitute 2

Second modal vector

1

1

Aμ

A

A

AA

122

12

22

122

String systems

lT

mωl

2T

AA

μ

22

12

2210A

l

TAmω

l

2T2212

22

1

lT

ml3T

ml

2T

AA

μ11

211



11

11

122

12

111

1121 Aμ

A...

Aμ

AAAAModal matrix

String systems

m2

l1 l2 l3

m1

Mode shapes of the system

11

1-1

Mode I

Mode II


Summary


Torsional systems

String systems

Linear systems

Definite and semi-definite systems

Analysis of various 2DOF systems such as:

Dr. S. K. Kudari, Professor, Department of Mechanical Engineering, B

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Transcript of Dr. S. K. Kudari, Professor, Department of Mechanical Engineering, B