APPLIED MECHANICS

Lecture 06

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

VIBRATION OF MULTI-DOF SYSTEM

Many real structures can be represented by a single degree of freedom model.

Most actual structures have several bodies and several restraints and therefore several degrees of freedom.

The number of DOF that a structure possesses is equal to the number of independent coordinates necessary to describe the motion of the system.

Since no body is completely rigid, and no spring is without mass, every real structure has more than one DOF, and sometimes it is not sufficiently realistic to approximate a structure by a single DOF model.

VIBRATION OF MULTI-DOF SYSTEM

The deployment of the structure at its lowest or first natural frequency is called its first mode, at the next highest or second natural frequency it is called the second mode, and so on.

Two DOF structure will be considered initially. This is because the addition of more DOF increases the labour of the solution procedure but does not introduce any new analytical principles.

Equations of motion for 2-DOF model, natural frequencies and corresponding mode shapes will be obtained.

VIBRATION OF MULTI-DOF SYSTEM Examples of 2-DOF models of vibrating structures

a)

b) c)

d)e)

f)

a) horizontal motion - x1, x2; b) shear frame - x1, x2; c) combined translation and rotation - x , ; d) rotation plus translation - y, ; e) torsional system - 1, 2;

f) coupled pendula - 1, 2.

TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING

Model of vibrating structures with 2-DOF

m1 m2

k1 k2 k3

x1, x1, x1 . ..

x2, x2, x2 . ..

m1 m2

k1x1 k2(x1 - x2)

x1, x1, x1 . ..

x2, x2, x2 . ..

k2(x1 - x2) k3x2

The equations of motion for x1 x2

),( 2121111 xxkxkxm 1body for

.)( 2321222 xkxxkxm 2body for

The equations of motion for x1 x2

),( 2121111 xxkxkxm 1body for

Equations can be solved for the natural frequencies and mode shapes by assuming a solution of the form

)sin( 011 tAx )sin( 022 tAx

TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING

Substituting these solutions into the equations of motion

System algebraic equations

),sin()()sin()sin(- 021201102011 tAAktAktAm

),sin()sin()()sin(- 023021202022 tAktAAktAm

.0)(

,0)( 20223221

22201121

mkkAkA

kAmkkA

.0202232

220112

mkkk

kmkk

A1 and A2 can be eliminated

20112

2

2

1

mkk

k

A

A

2

20223

2

1

k

mkk

A

A

TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING

The frequency equation

Considering the case: k1 = k 2 = k3 = k, m1 = m2 = m.

Frequency equation:

The frequencies 01 and 02 and the corresponding mode shapes

0))(( 22

20223

20112 kmkkmkk

0)2( 2220 kmk 034 22

040

2 kmkm 0)3)(( 20

20 kmkm

m

k01

m

k302

rad/s

rad/s.

1 2

1010

A

A

m

k

1 3

2

1020

A

A

m

k

TWO-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING

The first mode and second mode of free vibration

TWO-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The mechanical model

),sin()( 02121111 tFxxkxkxm

.)( 2321222 xkxxkxm

The equations of motion

TWO-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The solution is supposed in the form

Substituting the solutions into the equations of motion

)sin(11 tAx )sin(22 tAx

0222

1211 )()( FkAmkkA

0)()( 2223221 mkkAkA

)(

2

12101

mkkFA

20

2 kF

A

22

2121

2223 ))(( kmkkmkk The frequency equation

N-DOF SYSTEM

The mechanical model

F2

k1

x1, x1, x1 . ..

m1b1

k2

b2m2

ki

bimi

kn+1

bn+1

x2, x2, x2 . ..

ki+1

bi+1

mn

xi, xi, xi . .. xn, xn, xn

. ..

F1 Fi Fn

).()()(

),()()(

),()()(

1111

23323212332321222

1221212212111

tFxkkxkxbbxbxm

tFxkxkkxkxbxbbxbxm

tFxkxkkxbxbbxm

nnnnnnnnnnnnn

N-DOF SYSTEM

The equations of motion in matrix notation

)(tFKxxBxM

xxx ,, - displacement, velocity, acceleration

],...,,[)( 21 nT FFFt F - vector of time depending exciting forces

- mass matrix,

nm

m

m

m

000

000

000

000

3

2

1

M

N-DOF SYSTEM

1

433

3322

221

000

00

0

00

nn bb

bbb

bbbb

bbb

B

1

433

3322

221

000

00

0

00

nn kk

kkk

kkkk

kkk

K

- matrix of damping

- stiffness matrix

N-DOF SYSTEM UNDAMPED FREE VIBRATION

0KxxM Free undamped vibrations are described by equation

tie ux

.

The solution

- vector of amplitudes of harmonic motion ],...,,[ 21 nT uuuu

Ω - circular frequency

Equation of motion for the assumption of harmonic motion

0uMK )( 2

N-DOF SYSTEM UNDAMPED FREE VIBRATION

For non-trivial solution - the determinant must be equal to zero

.

The determinant is called the frequency determinant.

Developing the determinant - the frequency equation of n order for

While the matrices are positive and definite the roots of this equations are real values

0)det( 20 MK

0... 0201

)2(202

)1(201

20

aaaaa nn

nn

nn

20

n00201 ...0

N-DOF SYSTEM UNDAMPED FREE VIBRATION

Substituting the natural frequency, the set of homogenous equations is obtained. Therefore it is necessary to divide each equation by one element of the amplitude vector uri. We get for example

11

2

1

1 ,...,,r

rn

r

r

r

rTr u

u

u

u

u

uv

The vectors vr gives the shape of the vibrating system but not the

absolute value of the displacements of its members. Therefore these vectors are called modal vectors. This process is called normalization. The normalization is possible to carry out by using one of the following procedure

1rTr vv 1r

Tr Mvv 1r

Tr Kvv

N-DOF SYSTEM UNDAMPED FREE VIBRATION

The displacements that belong to r mode are given

tirr

re 0~ vx )sin( 0 rrrr t vxresp.

The general solution is given by linear combination of all modes

n

r

tirr

reC1

0~~ vx

rC~

are complex integration constants

n

rrrrr tC

10 )sin(vx

n

rrrrrr tBtA

100 )]sin()cos([vx

The integration constants Cr, r or Ar, Br for r = 1,2,…,n are determined from initial conditions.

N-DOF SYSTEM UNDAMPED FREE VIBRATION

The modal vectors is possible arrange in modal matrix

nnnn

n

n

n

vvv

vvv

vvv

21

22221

11211

21 ],...,,[ vvvV

The natural circular frequencies are arranged in spectral matrix

20

202

201

00

00

00

n

S

N-DOF SYSTEM UNDAMPED FREE VIBRATION

Orthogonality of vibration modes. For sr 00

.)(

,)(20

20

0vMK

0vMK

ss

rr

Multiply the first equation by the vector Tsv ,the second one by T

rv

.)(

,)(20

20

0vMKv

0vMKv

ssTr

rrTs

The second of these equations will be transposed

0vMKv rsTs )( 2

0

After arrangement

0Mvv rTssr )( 2

020

N-DOF SYSTEM UNDAMPED FREE VIBRATION

Because it has been supposed that sr 00

These equations are the orthogonality relationships between natural modes of distinct natural frequencies.

0Mvv rTs and 0Kvv r

Ts for sr

It is also possible to say:

The mode vectors belonging to various natural frequencies are

orthogonal with respect to the mass matrix as well the stiffness

matrix.

The quadratic forms

yrrTr mMvv yrr

Tr kKvv

are called generalized stiffness and generalized mass of mode r.

N-DOF SYSTEM UNDAMPED FREE VIBRATION

The orthogonality relationships - in more complex form

where V is called the modal matrix. The matrices My and Ky are

diagonal.

We notice that the mass matrix is positive definite. Therefore all

generalized masses are positive. The modal matrix is possible to

use to define the main or normal coordinates.

.][][

,][][

yyrrTr

T

yyrrTr

T

k

m

KKvvKVV

MMvvMVV

The normal coordinates y we obtain by modal transformation

xVy 1 or Vyx

N-DOF SYSTEM UNDAMPED FORCED VIBRATION

Let us consider the undamped system

after substituting for x

)(tFKxxM

)(tFKVyyMV

Multiplying this equation from left by modal transformed matrix VT

)(tTTT FVKVyVyMVV or

)(tyyy FyKyM

N-DOF SYSTEM UNDAMPED FORCED VIBRATION

The matrices My and Ky are diagonal - n independent equations

If the modal vectors have been normalized

)(tFykym yrryrryr for r = 1,2, ... , n.

the equation of motion

EMVVM Ty

)(tyy FyKy

and .SK y