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Response of MDOF systems Degree of freedom (DOF): The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time. Two DOF systems Three DOF systems

Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

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Page 1: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Response of MDOF systems

Degree of freedom (DOF): The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time.

Two DOF systems Three DOF systems

Page 2: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

The normal mode analysis (EOM-1)

Example: Response of 2 DOF system

m 2mk k k

x1 x2

FBD m 2mkx1 k(x1-x2) kx2

EOM 1211 )( xmxxkkx &&=−−−

2221 2)( xmkxxxk &&=−−

In matrix form, EOM is ⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡00

22

200

2

1

2

1

xx

kkkk

xx

mm

&&

&&

Page 3: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

EOM -2 (example)

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡00

22

200

2

1

2

1

xx

kkkk

xx

mm

&&

&&

x

EOM

M K Fx

)()()()( tttt FKxxCxM =++ &&&In general form

M is the inertia of mass matrix (n x n)C is the damping matrix (n x n)K is the stiffness matrix (n x n)F is the external force vector (n x 1)x is the position vector (n x 1)

Page 4: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Synchronous motion

From observations, free vibration of undamped MDOF system is a synchronous motion.

• All coordinates pass the equilibrium points at the same time

• All coordinates reach extreme positions at the same time

• Relative shape does not change with time

=21 xx constant

time

x1 x2 x1x2

No phase diff. between x1 and x2

)sin(11 φω += tAx)sin(22 φω += tAx

)(1

φω += tjeA)(

2φω += tjeA

oror

Page 5: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Response of 2DOF system (example-1)

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡00

22

200

2

1

2

1

xx

kkkk

xx

mm

&&

&&EOM

Synchronous motion

Sub. into EOM

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

ω−ω−

00

22

200

2

1

2

12

2

xx

kkkk

xx

mm

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

−−−−

00

222

2

12

2

AA

mkkkmkω

ω

0KxMx =+ω− )()(2 tt

0xMK =ω− )()( 2 t

022

22

2

=ω−−

−ω−mkk

kmk 0)det( 2 =ω− MK Characteristic equation (CHE)

)sin(11 φω += tAx)sin(22 φω += tAx

)(1

φω += tjeA)(

2φω += tjeA

oror

Page 6: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Response of 2DOF system (example-2)

022

22

2

=ω−−

−ω−mkk

kmk0

233

224 =⎟

⎠⎞

⎜⎝⎛+ω⎟

⎠⎞

⎜⎝⎛−ω

mk

mk

mk634.01 =ω

mk366.22 =ω

CHE

Solve the CHE Natural frequencies of the system;

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

ω−−−ω−

00

222

2

12

2

AA

mkkkmk

kmk

mkk

AA 2

22

1 222

ω−=

ω−=

1ω=ω

731.0)634.0(2

)1(

2

1 =−

=⎟⎟⎠

⎞⎜⎜⎝

mmkk

kAA

2ω=ω

73.2)366.2(2

)2(

2

1 −=−

=⎟⎟⎠

⎞⎜⎜⎝

mmkk

kAA

From

Page 7: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Response of 2DOF system (example-3)

1ω=ω

731.0)1(

2

1 =⎟⎟⎠

⎞⎜⎜⎝

⎛AA 73.2

)2(

2

1 −=⎟⎟⎠

⎞⎜⎜⎝

⎛AA

⎭⎬⎫

⎩⎨⎧

=φ1731.0

)(1 x⎭⎬⎫

⎩⎨⎧−

=φ1

73.2)(2 x

Amp. ratio Amp. ratio

The first mode shape The second mode shape

0.731 1

-2.73

1

2ω=ω

same directionOpposite direction

Page 8: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Response of 2DOF system (example-4)

In general, the free vibration contains both modes simultaneously (vibrate at both frequencies simultaneously)

)sin(1

73.2)sin(

1732.0

2221112

1 ψ+ω⎭⎬⎫

⎩⎨⎧−

+ψ+ω⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

tctcxx

2121 ,,, ψψcc are constants (depended on initial conditions)

Page 9: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Initial conditions (1)

)sin(1

73.2)sin(

1732.0

2221112

1 ψ+ω⎭⎬⎫

⎩⎨⎧−

+ψ+ω⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

tctcxx

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

42

)0()0(

2

1

xx

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

00

)0()0(

2

1

xx&

&Initial conditions and

)cos(1

73.2)cos(

1732.0

222211112

1 ψ+ω⎭⎬⎫

⎩⎨⎧−

ω+ψ+ω⎭⎬⎫

⎩⎨⎧

ω=⎭⎬⎫

⎩⎨⎧

tctcxx&

&

Velocity response

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

42

)0()0(

2

1

xx

2211 sin1

73.2sin

1732.0

42

ψ⎭⎬⎫

⎩⎨⎧−

+ψ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

cc

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

00

)0()0(

2

1

xx&

&222111 cos

173.2

cos1732.0

00

ψ⎭⎬⎫

⎩⎨⎧−

ω+ψ⎭⎬⎫

⎩⎨⎧

ω=⎭⎬⎫

⎩⎨⎧

cc

Page 10: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Initial conditions (2)

2211 sin1

73.2sin

1732.0

42

ψ⎭⎬⎫

⎩⎨⎧−

+ψ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

cc

222111 cos1

73.2cos

1732.0

00

ψ⎭⎬⎫

⎩⎨⎧−

ω+ψ⎭⎬⎫

⎩⎨⎧

ω=⎭⎬⎫

⎩⎨⎧

cc

4 Eqs., 4 unknowns

Solve for four unknowns,732.31 =c ,268.02 =c 2/21 π=ψ=ψ

)2

sin(1

73.2268.0)

2sin(

1732.0

732.3 212

1 π+ω

⎭⎬⎫

⎩⎨⎧−

+ω⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

ttxx

The response is

ttxx

212

1 cos268.0732.0

cos732.3732.2

ω⎭⎬⎫

⎩⎨⎧−

+ω⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

Page 11: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Initial conditions (3)

)sin(1

73.2)sin(

1732.0

2221112

1 ψ+ω⎭⎬⎫

⎩⎨⎧−

+ψ+ω⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

tctcxx

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

2464.1

)0()0(

2

1

xx

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

00

)0()0(

2

1

xx&

&(a) Initial conditions and

⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

173.2

)0()0(

2

1

xx

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

00

)0()0(

2

1

xx&

&(b) Initial conditions and

Try to do

Page 12: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Summary (Free-undamped) (1)

0KxxM =+ )()( tt&&

The motion is synchronous: constant ω and φ

0KxMx =+ω− )()(2 tt

0xMK =ω− )()( 2 t

0)det( 2 =ω− MKCharacteristics equation

2nω Eigen value

nNnn ω

Eigen value problem

ωω ,,, 21 K N natural freq.

0xMK =ω− ini )( 2

ix Eigen vector

)sin( φ+ω= tAx )( φ+ω= tjeAor

N mode shapesNxxx ,,, 21 K

EOM1

2

3

4 5

Direct Method

Page 13: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Summary (Free-undamped) (2)

Free-undamped response

)sin()sin()sin()( 22221111 NNNN tAtAtAt φ+ω+φ+ω+φ+ω= xxxx K

∑=

φ+ω=N

iiiii tAt

1)sin()( xx

6

where A and φ are from initial condition x(0) and v(0)

Direct Method

Page 14: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Example

1k2k

θx

2l1l

Determine the normal modes of vibration of an automobile simulated by simplified 2-dof system with the following numerical values

lb3220=W

ft5.41 =l lb/ft24001 =k

ft5.52 =l lb/ft26002 =k

ft4=r2rgWJC =

Page 15: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Forced harmonic vibration (1)

Example

EOMt

Fxx

kkkk

xx

mm

ω⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡sin

000 1

2

1

2221

1211

2

1

2

1

&&

&&

System is undamped, the solution can be assumed as

tXX

xx

ω⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡sin

2

1

2

1

Sub. into EOM⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

ω−ω−

01

2

12

22221

122

111 FXX

mkkkmk

[ ] ⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡ω

0)( 1

2

1 FXX

ZSimpler notation, [ ] ⎥⎦

⎤⎢⎣

⎡ω=⎥

⎤⎢⎣

⎡ −

0)( 11

2

1 FZ

XX

Page 16: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Forced harmonic vibration (2)

[ ] [ ]⎥⎦

⎤⎢⎣

⎡ωω

=⎥⎦

⎤⎢⎣

⎡ω=⎥

⎤⎢⎣

⎡ −

0)()(adj

0)( 111

2

1 FZ

ZFZ

XX

))(()( 222

22121 ω−ωω−ω=ω mmZWhere

ω1 and ω2 are natural frequencies

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

ω−−−ω−

ω=⎥

⎤⎢⎣

⎡0)(

1 12

11121

122

222

2

1 Fmkkkmk

ZXX

The amplitudes are))((

)(22

222

121

12

2221 ω−ωω−ω

ω−=

mmFmkX

))(( 222

22121

1212 ω−ωω−ω

−=

mmFkX

Page 17: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Forced harmonic vibration (3)

))(()2(

222

221

21

2

1 ω−ωω−ωω−

=m

FmkX

))(( 222

221

21

2 ω−ωω−ω=

mkFX

m mk k k

x1 x2

F1sinωt

tF

xx

kkkk

xx

mm

ω⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡sin

022

00 1

2

1

2

1

&&

&&

mk

mk 3, 21 =ω=ω

EOM

Force response of a 2 DOF system

0 1 2 3

012

3

45

-1

-2-3

-4

-5

FXk

1ωω

1

2

FkX

1

1

FkX

1ω=ω 2ω=ω

Same direction

Opposite direction

Page 18: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Solving methods

Page 19: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis

• is a method for solving for both transient and steady state responses of free and forced MDOF systems through analytical approaches.

• Uses the orthogonality property of the modes to “decouple” the EOM breaking EOM into independent SDOF equations, which can be solved for response separately.

Introduction

Page 20: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Coordinate coupling

1k 2k

2l1l

mg

θ

xRef.

)( 11 θ− lxk)( 22 θ− lxk

1k 2k1l mg

θ1x

Ref.

11xk)( 12 θ+ lxk

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡+−−+

+⎥⎦

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡00

00

222

2111122

112221 xlklklklklklkkkx

Jm

&&

&&⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡ ++⎥

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡001

222

2211

11

1 xlklklkkkx

Jmlmlm

&&

&&

Page 21: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Concept of modal analysis

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡+−−+

+⎥⎦

⎤⎢⎣

⎡θ⎥

⎤⎢⎣

⎡0

)(0

0222

2111122

112221 tFxlklklklklklkkkx

Jm

&&

&&)()()( ttt FKxxM =+&&

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

ωω

+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡)()(

00

1001

2

1

2

12

2

21

2

1

tNtN

rr

rr

n

n

&&

&& )()()( ttt NΛrr =+&&

EOM in modal coordinate (Independent SDOF equations)

EOM in physical coordinate (Coordinates are coupled)

Solve for )(tr

Transform r(t) back to x(t)

Page 22: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Orthogonality

x = eigen vector (vector of mode shape)

iiiTi M=Mxx

jiiTj ≠= ,0Mxx jii

Tj ≠= ,0Kxx

iiiTi K=Kxx

If M and K are symmetric and then xi and xj are said to be “orthogonal” to each other.

njni ω≠ω

Page 23: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Normalization

u = normalized eigen vector (respect to mass matrix)

1=iTi Muu

jiiTj ≠= ,0Muu

constant is , CC ii xu

0xMK =ω− )()( 2 tFrom eigen value problem

=

0uMK =ω− )()( 2 t

or iii MuKu 2ω=

22ii

Tiii

Ti ω=ω= MuuKuu

Page 24: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal matrix

Modal matrix is the matrix that its columns are the mode shape of the system

[ ]nuuuU K21=

Then

⎥⎥⎥⎥

⎢⎢⎢⎢

==

100

010001

K

MOMM

K

K

IMUUT

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

ω

ωω

=

2

22

21

00

0000

nN

n

n

T

K

MOMM

K

K

KUU

Λ (Spectral matrix)

Page 25: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis (undamped systems)-1

1. Draw FBD, apply Newton’s law to obtain EOM2. Solve for natural frequencies through CHE3. Determine mode shapes through EVP4. Construct modal matrix (normalized)

Procedures)()()( ttt FKxxM =+&&

0xMK =ω− )()( 2 t0)det( 2 =ω− MK

[ ]nuuuU K21=

IMUU =T

ΛKUU =T

5. Perform a coordinate transformation )()( tt Urx =

)()()( ttt FKxxM =+&& )()()( ttt FKUrrMU =+&&

)()()( ttt TTT FUKUrUrMUU =+&&

)()()( ttt TFUΛrr =+&&

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Modal analysis (undamped systems)-2

)()()( ttt TFUΛrr =+&&

⎥⎥⎥⎥

⎢⎢⎢⎢

=

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥

⎢⎢⎢⎢

=

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

ω

ωω

+

⎥⎥⎥⎥

⎢⎢⎢⎢

)()()()(

)()()()(

)(

)()(

00

0000

)(

)()(

4

3

2

1

4

3

2

1

21

22221

11211

2

1

2

22

21

2

1

tNtNtNtN

tFtFtFtF

uuu

uuuuuu

tr

trtr

tr

trtr T

NNNN

N

N

NnN

n

n

N K

MOMM

K

K

M

K

MOMM

K

K

&&

M

&&

&&

Independent SDOF equations, can be solve for r(t)

6. Transform the initial conditions to modal coordinates

)()( tt Urx =

IMUU =T

)0()0( Urx =

)0()0( MUrUMxU TT =

)0()0( MxUr T=

From

and )0()0( xMUr && T=

Page 27: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis (undamped systems)-3

7. Find the response in modal coordinates8. Transform the response in modal coordinate

back to that in original coordinate )()( tt Urx =

K=)(tr)(tr

)(tx

Page 28: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Example (Modal analysis) -1

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡00

33327

1009

2

1

2

1

xx

xx&&

&&

EOM

⎥⎦

⎤⎢⎣

⎡=

01

0x ⎥⎦

⎤⎢⎣

⎡=

00

0vInitial conditions

Page 29: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Example (Modal analysis) -2

2dof string-bead system

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−+⎥

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡00

2112

00

2

1

2

1

xx

aT

xx

mm

&&

&&

EOM

⎥⎦

⎤⎢⎣

⎡=

00

0x ⎥⎦

⎤⎢⎣

⎡=

10

0vInitial conditions

Page 30: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Example (Modal analysis) -2_2

Page 31: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Rigid body mode

• Rigid body mode is the mode that the system moves as a rigid body.

• The system moves as a whole without any relative motion among masses.

• There is no oscillation. 0=ωn

Page 32: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Rigid-body modes

Compute the solution of the system. Let m1 = 1 kg, m2 = 4 kg and k = 400 N/m.

Initial condition ⎥⎦

⎤⎢⎣

⎡=

001.0

0x ⎥⎦

⎤⎢⎣

⎡=

00

0v

Page 33: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

More than two degrees of freedom

Calculate the solution of the n-degree-of-freedom system in the figure for n = 3 by modal analysis. Use the values m1 = m2 = m3 = 4 kg and k1 = k2 = k4 = 4 N/m, and the initial condition x1(0) = 1 m with all other initial displacements and velocities zero.

Page 34: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis on damped systems (1)

)()()()( tttt FKxxCxM =++ &&&

CKMKCM 11 −− =

The original modal analysis can be applied to MDOF damped system if and only if

KMC β+α=

α β

However, there are subsets of the above systems where C can be written as a linear combination of M and K.

and are constants. Such system is called “proportionally damped.”

Necessary and sufficient condition

Sufficient but not necessary condition

Such system is called “classically damped”.

EOM

Page 35: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis on damped systems (2)

For proportionally damped )()()()( tttt FKxxCxM =++ &&&

)()()()()( tttt FKxxKMxM =+β+α+ &&&

)()( tt Urx = )()()()()( tttt FKUrrUKMrMU =+β+α+ &&&

TU )()()()()( TTTT tttt FUKUrUrUKMUrMUU =+β+α+ &&&

)()()()()()( T ttttt NFUΛrrΛIr ==+β+α+ &&&

)()()(2)( 1121111 tNtrtrtr nn =ω+ζω+ &&&

M2112 nn βω+α=ζω

Let

Premultiply by

Thus, the system when it is written in modal coordinates r(t) can be decoupled into k sets of SDOF equations

where

Page 36: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis on damped systems (Ex.)

A belt-driven lathe•bearings are modeled as providing viscous damping •shafts provide stiffness •belt drive provides and applied torque.

/radkg.m10 2321 === JJJ

N.m/rad10321 == kkN.m.s/rad2=c

• Zero initial conditions• Applied moment M(t) is a unit

impulse function

Page 37: Response of MDOF systems - Chulapioneer.netserv.chula.ac.th/~rchanat/2103433 Intro Mech Vib/Ch5... · Synchronous motion From observations, free vibration of undamped MDOF system

Modal analysis on damped systems (Ex.)