close

Вход

Забыли?

вход по аккаунту

?

STRUCTURAL MODIFICATIONS: HOW TO CIRCUMVENT THE

код для вставки
STRUCTURAL
MODIFICATIONS:
TO CIRCUMVENT
THE EFFECT
MODAL
TRUNCATION.
Techion
The paper outlines the basic theory, and then shows 2
examples demonstrating it’s implementation. An extended
presentation is available in [6a].
HOW
OF
2. Theory
I.sucher,
s BraIn
1srae1 Institute
of Technology
Haifa ISRAEL
2.1 Assigning Modes
A common design goti during design and uoublcshooting
stagesis to modify an existing vibrating structure in order to
obtain desired dynamic properties. The supplied knowledge
(model) about the vibrating structure, is limited due to
practical reesons.
This modification of the structurnl response may be
motivated, for example, by a desire to minimi7x the response
at specific locations.
The description here is suitable for systems with very low
damping or for systems having proportional damping (9;2].
In either case the natural vibrating frequencies and modeshapes are the solution for the following generalized
eigenvalue problem.
Abstract
The prediction of the effect of structural modifications is
problematic when the modal data is extracted via EMA, and
hcnnce truncated.
We show how to control a subset of the originally identified
eigenfrequencies and eigenvectors, and assign exact values to
them. Our method is based on choosing structural
modifications, such that the new eigenvectors are spanned by
the original modal base.Examplesare given for a discrete and
continous (FE modelled) system.
1. Introduction
(-o:M + K)x, = 0
r=l...n
(1)
Any structural or geometrical changes will yield appropriate
mass and stiffness changes. The new mass and stiffness
matriceswill be
The modal information available to engineers comes usually
from two possible sources:
2) An expeIimental test
I) A FE model
In both cases we only acquire a partial modal base, i.e. a
limited number of modes up to a maximum frequency. This
limitation is due to the desire to minimix the computational
effort ( in FE analysis)or experimental limitations.
The fact that only a subset of modal information is available
limits the use of structural modification techniques. Many
researchers have address the influence of such modal
truncation on the possible prediction of the effect of structural
modifications, when those are based on modal information
only (31, 321.
We address the design problem, whereas structural
modifications are undertaken in order to achieve ao exa~f
desired modal behavior.We show how to control a subset of
the originally identified eigenfrequencies and eigenvectors,
and assign exact values to them. The necessaryinformation
for such an implementation is still the truncated modal base,
as available from test? ( or a limited FE analysis1.
Our method is based on extracting both right and left
eigenvectors from the measured transfer functions (in FE
analysis both can obviously be extracted from the model 1.
The right eigenvectors are those extracted by classicalEMA,
the left eigenvectors are extracted by techniques similar to
those described by 1391. Our method is based on choosing
structural modifications, such that the new eigenvectors are
spanned by the original modal base
~?=M+AM
li?=~+~~
(2)
The modified structure will possesnew natural frequencies
and mode shapes.These will be the solution of:
(-rs:ti+qi,
=o
(3a)
Substituting Eq. (2) in Eq. (3a) and rearranging we obtain
(ti;M-K)i,=(-$AM+AK)i,
(3b)
Eq. (3b) is the modified structure’s eigenvalue problem. This
expressions includes the original mass and stiffness matrices
along with the new natural frequencies and mode-shapes.
In particular when the rth mode is unchanged the left hand
side of Eq. (3b) is zero. Any mass and stiffness changes
maintaining Eq. (3~) bellow, do not alter this mode.
i.e., iu.=w,
2,=x,,
(-o:AM + AK)x, = 0
(3C)
The mass and stiffness increments, in this case, cause inertia
and potential forces that balance each other at the harmonic
excitation frequency wr.
1188
2.2 Structural
Changes Based on Truncated
Data
This section describes how to circumvent the incomplete
knowledge due to truncation (incomplete number of modes
due to finite measureme”t hand-width). The fix1 that the mass
and stiffness matrices are not known can also be
circumvented, using measured FRFs a~ will he shown.
Solving Eq. (3h) for mass and stiffness modification
requires knowledge of the original mass and stiffness
matrices. This information, although it can be dctcrmined
using finite element analysis,will be practically inaccurate due
to modeling errors. In any case the number of extracted
modes will be smaller than the number of degrees of freedom
(i.e., a truncated model). This section describesthe constraints
and a numerical process by which Eq. (3a) can he solved
using strictly realistic experimental data. A proof for the
validity of the above statementsis also given.
Defme the rth Left Modal Vector (LMV), I,, [39] for an
2.3 Main theorems
The method is based on 2 thalitimc ~ihichcan be found in
1W.
Tlzeorem 1: Structural modil~icaiions maintaining
(A,%+
whose steady state response vccto~.x(r)is a linear comhiaation
of the truncated modal b&seI; ...�.,
modified structure’s modal vector, i, can he expressed
exactly by the original modes, i.e., r ’ c spm[ XI .q,, ]
2.4 Circumventing the knowledge of Mass and
Stiffness matrices
The above proofs allow us to use Eq.(3h). If we use the
LMV definition and Eq.(4), we may write:
nommlized modal vectors throughout the paper). We also
define L=[l,...l,,]- a matrix who’s columns are the (truncated
model’s) left modal vectors.
In a recent paper 1391, it is shown that using the same
information required for the extraction of the rth mode, x,,
Mi, = MXc, = C,,@,[i]
x”[, ,... I.,]
(i.e., the frequency hand which includes this mode) the LMV,
I,. can he extracted, as well. This is true for a generally
C,EYP x,Em
= Lc,
Lip /... /“,I
ml
Similarly
damped system, where the vibratory modes are complex.
However, for the sake of simplicity we deal here only with
proportional (or undamped) systems, which simplifies the
presentation of the procedure. An improved procedure for
extracting the LMV is given in [ 6a]. The given equations do
not involve the extracted modal vectors and use only the
measured FRFs.
We choose that any assigned mode will he a linear
combination of the original set of known modal vectors. We
next show that any excitation wctor, spanned by the .sctof left
modal vectors. yields a spatial motion which can he described
strictly as a combination of the truncated modal set. An
equivalent condition is given for AM, AK to yield similar
results. Since we prescribe the modes, WCcan always describe
them as the required linear combination.
We write the new ti mode as:
XE%-
2: If (-C,?m + AK)-, c Sl,un[l, .I,,,], then the
Theorem
undamped (or proportionally damped) system as /, LA%. The
LMV is orthogonal to all vibratory modes, I,, hut one, i.e..
ljTX, = s,,, 6% is the kroneckcr delta (assuming mass
i,=xc,
c .SPUU[~
. ..l..]. yield a modified structure
AKx)
where W: is the rtb original natural frequency and c&j the ith
element of cr
Substituting (%I and (Sh) into the L.H.S. of (3b)
(“:M-
K):,
= c;,(fi;
- m,?)/,c,[i] = L.d;pg{W;
-o;}.c,
(6)
Finally, we can write the eigcnpair assignment equation (~q.
(3a)) in terms of the experimentally obtained values and the
unknown massand stiffness modifications.
(Ak5i:M4)xcr
= L’rliog{Wj
-c$}.c,
(7)
Here the modifications AM(h) and AK(h) are functions of a
design vector, h, while cr is a vector “1� the linear combination
ten”.% c,,i,
(4)
Eq. (7) has a solution if and only if there exists a non zero
vector C, 01 equivalently
1189
The last equation gives the needed discrete mass
modifications to obtain a new mode shape (%,fi,). The only
required data for this calcolation is. as mentioned before: the
truncated modal base (natural frequencies, right and left
modes).
equivalently:
de+%)
=0
[
Ee (Af-r3~AM)X-L~diog{+o~}
i
1
m
4. Examples
equations
The importance of Eq. (7&b) is in those applications where
only modal frequencies are to be assigned, still the resultant
mode will be spanned by the original ones. Note that the new
mode shape does not appear in this equation.
The LMV c;m be extracted directly from measured FRFs.A
description of the numerical procedurer can be found in [6x
by means
using
the modification
We proceed with some example. ranging from simple
discrete spring mass systems towards discretized continuous
(more realistic) systems.
391.
3. Assignment
of modes
mass modifications
for
4.1 Example 1: Non-uniqueness of the solution
Using the redundant d.o.1. for optimization
of point
We now present a solution imvolving point mass changes
only. Such modifications, can be approximated with relative
ease in practical situations. Another reason for our interest in
such casesis that we are able to obtain closed form solutions.
The following example casts the modilication problem into
an optimization problem by introducing an appropriate cost
function. The cost function can be chosen to obtain additional
desired properties.
Consider the spring-masssystem described bellow :
3k
k
Consider mass modification at the exact nodal points at
which the mode shape is measured (EMA) or calculated (FE).
In this case the mars modification matrix is diagonal.
Fig.1 Three degrees of freedom vibrating system
AM = diog@“+}, i=l...n.
Under these terms, the L.H.S of
Eq.(7) become:
A?
-iiifAAC, = -c$ ~dipg{Z,[il]~ “7
Assuming we have a truncated model consisting of the first
i14
(8)
Using the left modal vectors with Eq.(7) as before, we
have a set of linear equations in the point massmodifications:
= (L)d;pg{ri,
hvo modes:
I1
11
-oi}.c,
(9)
Solving for the massmodifications:
We wish to assigna different first mode, �t, (from SpanXX
il=[l=xcf)=ii
3~)
and a matching nat”ra,
Again we “se the modification Eq.(7). This can be written in
the form Aq=B. A more detailed description of such a
formulation will be given in the next example:
1190
Aq=B
outlines
the desired
freedom
at measured
by a spetial
locations
is descibed
procedure
allowed
mass
Each
mode
in
changes
have
the
curve,
the degrees
to bc calculated.
appendix.
In
our
are to lie on the exact
modification
will
have
tl-anslatory
cdse,the
nodal
mass
of
This
points.
and
wary
inertia.
Consider
next
g ={~,,AIw~.A~+KA~~~}
The
general
solution
for Eq.(l3)
Where
A’
is the pseudo
columns
span
inverse
of A
the Null-space
[3],
N is a matrix
of A, /3 is a vector
0
0
solution
function
have
only
solve
the next
Q
Fig.2
Clamped
0
can
-I
0
still
For
changes
standard
(IS)
be found
to be minimized.
positive
The
1
while
if we
example
adding
wish
third
a
if we wish
tnass.
with
two
FE elements
desired
8
to
to assign
a vibrating
mode
specifying
the
demands:
beam
[Rad#
introduce
tbe least
Linear-programing
beam
of
We
!
cost
in the
figure
following
A particular
depicted
@I
coefficients.
=(a
beam
(14)
whose
(A’B)’
cantilever
is:
q=A’B+NP
arbitrary
a the vibrating
should
while
mode
mass
have
having
a naturitl
the same
shape.
We
(and
inertia)
apply
frequency
mode
directly
shape
Eq.(ll)
at
to;=10
as the original
to obtain
the
changes.
We
6
problem:
o-Desire&Mode
,
4.
Find q such that
c~q-afnin
2.
Subject to :
where
Aq=B
c’=(l
(If4
l
l
0
0
O),A.B.q
0
are given
in
-4
Eq.(l3)
0
The
solution
01-3
I - / 2 (oblaincd
for
exactly
Fig.3
) is:
10
First
four
modes
describing
Am,
Any
choice
Eq.(46).
of p. will
If
we
realizable
restrict
say
still
all mainttdning
F.q.(13).
4.2
2:
Example
without
the
mode
the
we have
Since
span
desired
without
here
one).
is simple
In
mode
the
changes
number
a Natural
demonstrates
one.
specifications
ourselves
an intinitc
the
the assigned
original
allowed
Changing
frequency
third
the rcquircd
restrict
changing
example
vibrating
mode.
the
solution,
modifications,
This
meet
positive
the ability
to change
affecting
the
from
cast
point
Desired
MC&S
20
30
a
Shape
the
identical
the
the third
where
0
il natural
accompanied
is to be exactly
choose
and The
Mode
frequency
mode
( WC just
Beam
50
of solutions,
matching
selection
the general
oblain
to
of a Cantilever
the desired
40
/
in
to
30
4,
A,,,, = Am, = 0 Ak, = I Akk, = 2 Akk, = 0 Ak, = 3
=
20
the
Fig.4
to
original
mode
as
designer
1191
10
Desired
and accepted
Mod?
(FE
40
c;IIalxi,wl
so
s
Ori
inal natural
Modified
natural
Conclusions
We have presented a theory enabling to assign specific
eigenfrcqucncies and eigcnvcctors to existing StNctuTcS.
The necessary information is extracted via classical modal
analysis. Our results arc thus of interest both to analyst
using PE models,and cxpcrimenttelists who use EMA
results.
One important contribution of our work lies in the
possibility of obtaining exact results cvcn when the original
unmodified model is incomplete. i.c., includes only a partial
range of it’s eigcnsolutions. This is always the case when
experimental test results arc used for the design of
moditications~
fre uencies fRad/s]2
fre uencies [Rad/s12
m
4
Mass modifications
A%
AI,
2.65
0.098
0.0667
AI,
0.0181
Table 1 modification results
The results show that the goal was obtained exactly. It is
also obvious that the other natural frequency where
decreased. The actual shape of the added masses can be
chosen so as to satisfy Am1 and AI1 for each location.
*Acknowledgement: This researchwt supportid by Ulc
Bcrstcin Fund for the Promotion of Research
REFERENCES
5. Discussion
Example number 1 demonstrates the fact that multiple
solutions exist, enables to add additional design
requirement. In this example we then add the requirement of
minimum addition of mass.
Example 2 refers to continuous structures. A
relatively simple requirement posed, namely a shifting of
frequencies. The fact that one modal vector (the third one) is
required to stay unchanged, automatically retains it’s
position in the original modes span. It seems important to
notice that this demonstrates the possibility of shifting a
natural frequency without affecting the associated modal
deflection.
Ox additional aspect needs to be mentioned. This
concerns the fact that the solution could include negative
mass changes which might be impossible to implement (in
some cases). A more complicated approach based on the
same basic equations is required. This approach will seekan
approximate solution still complying with the exact
mcdclling constraints. Such an approach is not introduced in
this paper.
Another aspect not covered by us is that of sensitivity
with respect to the structural changes. In our view it is
possible to develop further the presented equations, in order
to calculate the senstttvtttes with respect to mass and
stiffness changes, using the same truncated modal data.
Low senstttvthes could imply the possibility of relaxing
some requirements.
1192
Appendix: Selecting
spatial curw
When
points
mode
(discrete)
least
squares
least
sense)
squwcs
It is worth
natural
[2;25].
M, K - Mass and Stiffness
AM. AK- Mass and Stiffness
c,
h
vector
of constants.
vector of s!~ctural
At-Pseudo
inverse
(i running
matrices
modification
Determines
index)
matrices
the modified
and geometrical design
of the mstix
ilh mode.
pxamertes
A
1193
choice
The
to
extracting
the
discrete
corvc
noting
that
would
be the
formulation
and is described
for
for
the
in reference
from
closest
above
[6a].
to a
(in
lcast
set of modes.
The
the disactc
a Finite-element
element
the
resort
uses a set of interpolating
spatial
points
at which
one must
by II given
a
These
points
case.
fit
some
arc given.
the grid
In this
spanned
procedure
entries.
standard
x.d
procedure
the continuous
curve
with
are given.
fomting
the
o
coincide
mode.
coordinates
is to be specified,
the desired
do not
squares
..,
shape
describing
usually
lo
a mode
modal
basis
problem
functions,
modal
model,
functions
is fairly
Документ
Категория
Без категории
Просмотров
9
Размер файла
346 Кб
Теги
1/--страниц
Пожаловаться на содержимое документа