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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