mechanical vibrations - division of mechanics
TRANSCRIPT
LECTURENOTES ON
MECHANICAL VIBRATIONS
TRANSITIIISSIBILITY VS. FREOUENCY RATIOr00
0.t
20k0.010.1 t.0
Froquency Ratio
Per Lidström
Autumn2014
Division of MechanicsLund University
Faculty of Engineering, LTH
)
Index
active forcesadmittance matrixantiresonance frequencyassembly operationB-matrixcharacteristic polynomialconservativ e gener alize d forc e s
constitutive equationconstitutive equation, linear elastic plateconstraintconstrained systemsd'Alemberts principledamped natural frequencydamping matrixdifferential operatordisplacement, vibrating stringdissipation functiondouble Laplacianeigenvalueseigenvectorsequation of mechanical vibrationsequation of motion, bar in bendingequation of motion, bar in extensionequation of motion, free damped vibrationsequation of motion, free undamped vibrationsequation of motion, forced damped vibrationsequation of motion, forced undamped vibrationsequation of motion, Lagrangeequation of motion, membraneequation of motion, plateequation of motion, stringequilibriumequilibrium confi guration of non-disturbed systemEuler, master lawsexternal force densityexternal loadexternal moment densityforced response, particulate solutionforced vibrationforces of constraintsfree vibrationfrequency response matrixgeneral solutiongeneral solution, free undamped vibrationgeneral homogeneous solution, free damped vibrationgeneral combined homogeneous and particulate solution,Finite element methodFinite element method, longitudinal vibrations of a bar,
18
ll5,22l118
3587515646252,256331
l613619,3115275
8522854332949476,7925625213989178tt460323332232At4647,3244,303228244,303tl68718
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348357
element mass matrixFinite element method, longitudinal vibrations of a bar,element stiffness matrixFinite element method, beam in bending,element mass matrixFinite element method, beam in bending,element stiffness matrixgeneral particulate solution, forced damped vibrationgeneralized forcesgyroscopic inertia forcegyroscopic matriximpedance matrixinitial value problem solution, free undamped vibrationinitial homogeneous problem solution, free damped vibrationkinetic energyLagrangian equationLagrangian functionLaplacianlinearized equations of motionlocal equation of motion, stringmass matrixmaster laws, Eulermechanical energymodal dampingmodal massmodal matrixmodal stiffnessmode shape vectormodified potentialmomentmoment of inertia, areanon-conservative gener alized forcesnon-disturbed systemnormal axial forcenormal coordinatesnull-systemorthogonality relations, eigenvectorsorthogonality conditions, Fourier seriesparticulate solution, forced responseplate bending stiffnesspotential energyRayleigh dampingRayleigh-Ritzregular coordinate systemrelative damping, complex modesrelative damping, real modesresidual matrixresonance frequencyrigid body modescleronomous coordinate system
358
366
366
187
34,79427585, 155
105
Is433,40,7933,714732272-73230761,351
14699-1009999-100996630325653
602491004
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25, 47, 49,7914534130415150
227118
93
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sectional forcesectional momentshear forcespectral expansionstabilitystanding wave solutionstiffness matrixstring traction vectorsuperposition principlethickness parametertorsiontractiontransport inertia forcevariational derivativevirtual workwave propagation speed
244244249tt666-70907622986302328303
423515
234
Contents
IntroductionReferences
1. TheEulerequations2. The accelerating force3. Virtual work4. Lagrange's equation5. Struchre of the kinetic energy6. The conservative generultzed forces7. Non-conservative generalized forces8. Lagrange's equations in the general case9. The energybalance10. Equilibrium and stabilityll-Lineanzation1 2. Un-damped vibrations. Non-gyroscopic systems1 3. Eigenfr equency chm acteization14. Constrained systems1 5. Damped vibrations. Non-gyroscopic systems16. Damping mechanisms17. Forced vibrations18. Complex modes19. Admittance, tansfer functions and frequencyresponse20. The vibrating string21. One-dimensional bodies22. Thr e e- dimensional bodies23. Two-dimensional bodies24.Weak and variational formulations of the vibration
problem
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334
A1-45Appendix
-r.'/
Mechanical Vibrations
INTRODUCTION
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Mechanical Vibrations
LECTURE 2
EQUATIONS OF MOTIONLagrange's equations
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58
Mechanical Vibrations
LECTURE 3
EQUATIONS OF MOTIONDissipation
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Jean Le Rond d'Alembert
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1736 - 1813
Johann Peter Gustav Lejeune Dirichlet
1805 - 1859
John William Strutt Lord Rayleigh
1842 -1919
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Mechanical Vibrations
LECTURE 5
UNDAMPED VIBRATIONSNon-gyroscopic systerrui
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UN-DAMPED VIBRATIONSForced harmonic response
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LIN.DAMPED VIBRATIONSEigenfr equency chara ctenzatton
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LECTURE 12
DAMPED VIBRATIONSState space formulation
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LECTURE 15
CONTINUOUS SYSTEMSThe beam in bending
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-Qs1/- s)'
Jh /4e <rar4a /4/a7 y'l.l y'4a/;
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e/rzzszzy' dz.rp//r'ccfut-
lzza7)end /ha.z
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l.r'a az l.tZz'/,4 lha .r|-Zta /zra./ d,.4p./earzazz/
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7 = f h/o Ao /.r/, a/ , h/r-, ä,y-/ lUr €.v 7T
G+.rtJ
a, * ear, zeS
:t7 A^ = -2-Lz 4J:4e
eh/ 7.x-
d./ 15) - ,/ - 35'+ 2 53
x13 (5/: 5'1s- es/
x.* (S/ - -/. s'(S - /)
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Figtan 5.3.11
Shope functiow:Henition polgnomials of the thkd order
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åss
Mechanical vibrations
APPENDICES
Appendix A.l": Derivatives of vectors and matrices
We use the following notation. If we have a real-valued function
f=f(qt,q2,...,4,)elR
then
that is the ith component of this vector is given by
aföq,
€ R.J"'af afoq, oq,
af0q
0q
oq
0o,
0q,
0o,
öq, € lR'"'
(A.1.1)
(A.1.2)
(A.1.3)
(A.1.4)
(A.1.s)
(A.1.6)
({),={,oq oQi
I ,n
If we have the vector-valued function
where
then the so-called Jacobian is given by
a=a(8)=a(Q1,Q2,...,I)=fa, a2 an]t elR'"'
a, = ai(e) = ar(e1,e2,...,en), i = 1,...,n
oq, oqt
0o, 0o,
oq, oqt
öo, 6o,
öo,
0q,
öo, 0oo
oq, öq,
That is
(?),,oq"
_ 0o,
öq
Note that
A1
(A.1.7)
Mechanical vibrations
öq,
öq,
oqt
0q,
i.e. the unit matrix. This may be expressed as
With u =lu, r.t2 u,f .lRo'1 and o =fu, v2 v,l' elR"1 we have
6a1r' f 0o,
0qi0q
If we have the matrix-valued function
4,, 4,,
Arn,q = A@) = A(q 1, Q 2,..., Q o) : € IR"'
4,, t\,
where
then
öqt
oqtöq
öq
oq, oq,
oq, oq,
oq, öq,
oq' oqt
öq,
-1oqt oq
100001000010000I
0q,
(A.1.8)
(A.1.e)
(A.1.10)
(A.1.11)
(4.1.12)
(A.1.13)
(A.1.14)
ui! j
4,,
4,,
;,,
A2
4 = 4@)= A;.1(Qt,Qz,...,Qn)
or
,aA .(--v),r:oq
Note that if A is a symmetric matrix, i.e. Ar = A,then
u, (4*iw= ,,ä,#ul jwk
I'tiviwk = ,,f*,Hvil'tiwk
=" {ffD*
fa4"7, oqo '
u'(oAr\-= f a4'öq
,,,?1=, öQr
A2
(A.1.1s)
Mechanical vibrations
We have the followingderivation rules;
Proposition A.1.1 With a =lat a2 a,f' , u =lu, r.t2
matrix, according to (A. 1 . I 1), all depending on q. Then
u,l' and A an nxn-
A , - . -0a^ (a'u):u'
-oq oq+ar
0u
oq
(la,u,1
(A.1.16)
(4.1.17)
(A.1.18)
(4.1.1e)
öq, i=1
0u,
oqt
0u,
0qt
!ru'our=u'oAoq' öq
0o,
öq,
öot
oqt
AA 0u(Au)= ^ u*A-oq oq
,*u'AL+r'A'@oq oq
a
öq
Proof: A straightforward calculation gives
&r"',, = &rä',", =l* rä o,u,, * rä ",,,a n
,E#",n,#rrE#",*,h ,E*",*,#rf=)
f öo' u,
,tu )qn 'f oo'
u, * oo, u,
fröq, 'fr)q, ':I >
i=l
0u,
0q,
0u,
oq'
0u,
öqt
aiii=1
ör,
0q,aiI
i=1l.[ cli0u,
0q,,
öu,
0u,
öun
oqn
=l',', "' u,f
0o,
0q,
0o,
öq,
0o,
0q,
0o,
öq,
öo,
0q,
= 1,lT
)on don
oq, oqt
+la, a, ... a,l
oqn
0q,
öun ör,öq, oq,
0a -0u-+a' -oq öq
and this proves (A.1.16). Furhermore
4,, 4,,A,, 4,,
A,u
Arn
t[:lAnt Auz Arn
4,. a
-( Aul =-oq' öq
A3
Mechanical vibrations
ti=1
a1lA,,u,) a1lA,,u,) a1lA,,u,)i=1 i=1
0q, 0q, öq,
a1i.n,,u,) a1f ,t,,u,) a1f t,,u,)
n n
jujA1
n
a=-
öq
ZA,,u,
n
öq, 0q, 0q,
2A,,,,
!u,*A,,oQt
AA^"r u, + A",
öq, l zr
fiff",*a,,
äY",*u,,
0q,
äH",*o,,
äH,,*u,,
6u,+ 4,,;!" o{ln
öu,
0qt
öu,
,t n n
a1lA,,u,) a1l\u,) a1l/ru)
0q, Aq,
tt
0q,
0u.,
0q,
0u,
öqt
öu,
0qt
öu,
öqu
0u,
oqn
öuj
*du,t ,,uio öQn r{94.?' aq,
{94roo aq,
0u,u, * A,rC öu,
ur* A,rfi
f öA" uuio öQ,
f oA",E oq'
f oA" uuio
öQ,
, öA,, ,u1t 0Q,
j
J
j
*oA" u,,tu öq,, r
f oA" u,uio 0Q, r
:
ä*",
7'u'*Eo,*
E^,# fo,,
P,u'k fo,,+ öq,
uj
AA 0u:-It+A-oq oq
and this proves (A.1.17). Using (A.1.16) and (A.1 .17) it follows that
öu,
0qt
0r,öqt
0r,
ör,,
! @' or) = u' 0!u + (Av)r ! = u' (*-, +,q?) t, vr trr I =oq oq oq oq oq oq
Ze,ZArZA,94r0q,
uiIoq'f a4
(4.1.20)
(4.1.21)
1,t'AA ,,0v ,,76u_v+u'A_+v' A _0q öq 0q
and this proves (A.1 .18) tr
A4
Mechanical vibrations
Corollary 4.1.1
and if
then
(q' ,qd: n' #n + q' (A+ A')
4=o and Ar =Aöq
a
0q(A.r.22)
(A.r.23)
(A.r.24)
(A.r.26)
(A.1.27)
(4.1.28)
fio' ont = 2q' A
Proof: Taking Lt = v = q in (A.1.18) gives
!rn'uil= q' lq +q'A++q'A' +oq oq oq oq
where we have used (A.1.8). This proves (A.1.22).lf
4:o and Ar = A0q
then, from (A.1.22) it follows that
a
*@'Ad = 2q' A
and this proves (A.T.24). tr
Theorem A.1.1 The real-valued function
t-T =T(q,q) ==(f (q)+ a(q), q + q, A(q)q)
z
has the following derivatives
AT I .Af -.0a---(
" +a' -oq zoq 0q
., aAr .,+Q' ^ 4)oq
(A.r.2e)
aTI(ar + 4'(A(q)+ A'(q))442
Proof: Applying the derivation rules in proposition A.1 .l and Corollary A.1.1 one finds that
A5
Mechanical vibrations
and
where we have used
where
is the kinetic energy and
* o' ! * @@)il' * * A' aA\q,)4
r =0q oq dq
aT I ,Af .r 0a
-=-l-oq z oql oq
aT I .Af .,0a---(
- +a' -oq z oql oq
T2
+QaUq)
0q,=lrff ., aA .,+Q' ^ Q)
0q(A.1.30)
(A.2.2)
., 0a
oq
I(ar +(A(q)il'+4',4(q)) (ar + qr (A(q)+ A'(q)) (A.t.3t)
2
6ij (A.t.32)
and this proves the Theorem. n
Appendix A.2z Lagrange's equations and their linearization
This text may be seen as a complement to the pages 7l -76 in the Lecture Notes. It containsmore details on the linearization procedure.
AT(A.2.1)) Q,, i
!:o ana 1!1,,oq oq-
Lagrange's equations may be written
,aT(_oIi
Ir...rfld
dt öq,
T =T(t,q,q)=To+Tr+72
(A.2.3)
is the generalized force. We now divide the generalized forces into two parts, the conservativeand the non-conservative,
Q,=Qi"'+Q::o"
e,=lf .ffa*, i=1,...,n
and assume that
A6
(4.2.4)
Mechanical vibrations
where
Lagrange's equations may be written
Using the relative Lagrangian
Qfon=-%oq'"
(A.2.s)
(A.2.6)
(A.2.7)
(A.2.8)
(A.2.e)
(A.2.10)
(A.2.tt)
(A.2.t2)
(A.2.t3)
(4.2.14)
is the potential energy containing all conservative forces, such as the gravity and elasticforces. Introducing the Lagrangian
L = L(t,q,q) =T -V
V =Y(t,q)
ALQi", Id ,aL
-(-dt'0Q,) I )n
öq,
We may split the generalized non-conservative force Qi'on according to
Qi"'=Qi'+Qf'
i.e. into a sum of intemal and extemal forces. It may be assumed that
O:'' = -AD-t 0Q,
where
D = D(t,q,q)
is the so-called dissipationfunction. Lagrange's equations may then be written
AL AD--T--oq' oq,
d ,aL.dr( ad)
gr" i 1,...rfl
L =Tz -n =:E,aijqtq j -v
then Lagrange's equations take the form
ddt
(OL, OL,
öq'.#= -K! - Kf +e7', i = 1,...,n)
Aq
where the transport inertiaforce K! and the gtroscopic inertiaforce Kf are defined by
A7
Mechanical vibrations
K! =!ot
_!20t
*rk, k :äouu, * i,c*t,nr.n* d, *#
(#
fo,,i,*f c*u,ar*f*j=t j,k=t i=1 Ut.
,aT,(;+oQi
_Afo _0o0, 0aooand Kf =ä#q,-#:fr4,)
öq, öq,
respectively, and where
(4.2.1s)
(A.2.r6)
(A.2.r7)
(A.2.18)
(A.2.te)
o.6ij = a'7, _ atT, _oao, _öaoi0qi04, 0q,0qi 08i 0q,
Note that the matrix (Su) ir anti-symmetric
(E,)' = -(r,)
i.s. E,i = -8i,. A straightforward calculation gives the identity
where
cw %öq'
+oo^
Oq'), i, j,k = 1,...,fl
and then Lagrange's equations read
T
2
d, *#*ff:-Ki - Kf +er', i : 1,...,n
(4.2.20)
This is a set of n second order ordinary differential equations for the determination of themotion
Q,:Q,(t), i=1,...,n (A.2.2r)
once the external force
Q7 = Q:" Q) (A2.22)
and initial conditions
Q,(o)=q!, q,(o)=q! (A.2.23)
are given. Here q! , d! i = I,...,fr are given constants. The mechanical system is called non-
disturbed if there is no external force, i.e. if
A8
Mechanical vibrations
Lagrange's equations then read
where we have assume that
Qi'' Q)=0, t> o
(4.2.2s)
and this represents thefree motion of the mechanical system.
A fixed configuratior_ q = qo is an equilibrium configuration to the non-disturbed system if
q,(t) = q! , t' o (A.2.26)
is a solutionto (A.2.25). If we insert q,(t): q! , (q,(t) = 0, ti,Q) = 0) into (A.2.25) then we
obtain the equation
ffa,qo ,o) = -K:(t,qo ,o)
ffr,,no ,o) = o
Af. _ ö /ATr\_ öoo,
0q, 0t'0Q,' 0t
for4,*f c*od,*f+j=t j,k=] j=l ut'
(4.2.24)
(A.2.27)
(4.2.28)
(A.2.2e)
(4.2.30)
(A.2.31)
(4.2.32)
and used the fact that
Kf (t,qo ,0) = 0
The equilibrium equation(A.2.27) is equivalent to
where
V- =V -To
is the so-calledmodified potentiaL Now if
ooo,(t,qo) = 0
6t
which obviously is true if the coordinate system is scleronomous, then the equilibriumequation reads
A9
Mechanical vibrations
which means that the equilibrium configurations of the undisturbed system are theconfigurations for which the modified potential is stationary.
We take Lagrange's equations on the format
YQ,no)=ooQi
*r#, -# = ei'' + eI', i = 1,...,n
(A.2.33)
(A.2.34)
(4.2.3s)
(A.2.36)
and assume that we have an equilibrium state; Qi : Q! , 4i = 0. We introduce the matrices
qi
qi
,:
-0q:,l':),:)qq
Qt
4z
Q,,
an the deviationfrorn the equilibrium configuration
r1
r2
rn
A Taylor expansion of the Lagrangian gives
L(t,q,q) = L(t,80 +7,0 +i1 =
= L(t,q, ,0)+(*),, +(+); *+r'14!y,, *r'(%1,; *4;'<*oq oq z oq- oqloq z oq')oi +...
(A.2.37)
where higher order terms in F and i havebeen neglected, We use a notation where, forinstance,
(4.2.38),aL.( _),oq
0-0q(töL_oq
AL AL
oq, oq,
az)_toq, )o
)
and
410
Mechanical vibrations
a2L a2L a2L^')oq;
atL
öqtoqt
A,L
öq,öq,
atL.a2 L.(*)o=oq'
^)0q,6q, Aqi 0q,0q,
atl a2L atLöq,1q, öq,Oq, Aqt^
(A.2.3e)
(A.3.40)
(A.2.41)
(A.2.42)
(A.2.43)
(4.2.44)
0
Since (q0,0) corresponds to a equilibrium state
(+), = ({), - (%), = (%), - <%1, : - 1L1o = soq oq oq oq oq oq
where we assume that (A.2.38) is satisfied. Furthermore
and then
For the internal force
t*1, = <*), - r*r, = -(93)0, (*), = (%), ana 1!10 : (*),oq- oq- oq- oq- oqloqt oqloq oq- oq
q,q):L(t
L1t,r,i1= L(t,qo,0)*(#),i -i" r#),r +r'{ffi),i +qLo'q
aL ,aT,. _- . a'7, . :7 ,atT..^=
: ( ^= ), +r" (-)o - r" ( *,_, )o -.or oq oqoql oq-
aL _. .a2V*. :7 . a2T, ,':=-t" ( * )o +r'(-); +..or oq' oqoq
)oi +1,',
We now use coordinates (7,7) instead of (q,q) and calculate the partial derivatives
we obtain
All
(4.2.4s)
A"" (t, q, q) = A''' (t, qo + r,i) = A"" (t,do,o ) + r' 1V), +i' <ff), +
Mechanical vibrations
if we assume that
ddt
=r,eQ; ), +...
(A.2.46)
A*'Q,qo ,o): D and (aa''
)o:0Aq
-L=i'0r,
.a27".(#)o+o-q
(A.2.47)
(A.2.48)
(A.2.4e)
(4.2.s0)
Note that the matrix 'aö*' '(6), is, in general un-symmetric. It is defined by
,aö'" .
??-)o=oq
g:0q,
q0qt
W0q,
q: aQ:;'
öq, öq,
AQ:i, AQ:;,
oqt oqt
AOr,- Woq, oqn
0
Lagrange's equations may now be written
,aL aL d ,aLr- l- ^ ==t-rdei OQi dt O4
a2T, ./\^-^rr0oqoql
(!T+r'
Transposing and re-arranging this equation will leave us with
where we have used the symmetry
<*t,; *K*); -(%), -<9*r;ri *t*t,r = ef,' .t.., i = r,...,no- q oqoq oqoql oq o- q
*r'e:,':, -i'(:+l[ -i'1V)o : ef, +.., i = r,...,no q oqoq oq
<*t;:t*),o-q o-q
We now define the following matrices: The mass matrix
At2
(A.2.sr)
Mechanical vibrations
the'B-matrix'
an the stffiess matrix
is the so-called damping matrix and
is the gltroscopic matrix. We have here used the fact that
s : 1 @ - B' ) = - 1 K9+), - e+)h - (+-=>, - <*>h =z z oq oq oqoq oqloql
: -lu%r, - ru#r[> *(s,), e' = -e
w =tffi1,, M' =M
B-=(*);-(%>,-<4t[oqloq oqtoqt oq
E=(#),, K':K
(E+ B'):-iu#), *e#)D, e' = e
(*), - t*>; = (r,)' = -(eu )oqtoq oqtoq
(A.2.s2)
(A.2.53)
(A.2.58)
(A.2.se)
(A.2.s4)
If we assume that the q-coordinate system is regular then we may conclude that the massmatrix is positive definite, i.e.
u'Mi>0, V7;r0 eIR." (A.2.ss)
and then tr[ is invertible, i.e. M-1 exists. Furthermore, if the modified potential has a localminimum at the equilibrium state then the stiffness matrix is positive semi-definite, i.e.
i'6i>0, Yu+0 eF<' (4.2.s6)
The B-matrix is in general not symmetric. A decomposition of å into its syrnmetric and anti-syrnmetric parts gives
(4.2.s7)
where
B=C+G
c=!2
413
(A.2.60)
Mechanical vibrations
Note that if
a*'AD
Aq
where D = D(t,q,Q) is a dissipation function then
aa'' aD^: ^:)oq oq'
and this is a symmetric matrix and consequently
a (AD
-----:-=
oq'a=(sr)
The anti-symmetry of G has as a consequence
urGE=0, vzeiR'
The equations of motionfor mechanical vibrations may now be written
where
)o
(A.2.6r)
(A.2.62)
(A.2.64)
(A.2.6s)
(4.2.66)
(A.2.67)
(4.3.1)
>0
E7IIi+Ei+
7
P = Qf''
F
is the extemal force.
Appendix 4.3 Extremes of real valued functions
Let f = f (x,y) be a real-valued continuously differentiable function. The necessary
condition forf to have a stationary point (minimum. maximum or indifferent) is
öf (x,v) A-----;-=U,dx
of (*,v) A
Ay
Every solution x = x0, ! = lo to (A.2.1) is a stationary point of the functionl If
ö'f @o,yo) )0, O'f (xo,y) _ n A2 f (x0,y) Ot f (x*yr) ( a'f @o,y)\__T,u,G___@__l ata_ IOx-
then the stationary point is a strict local minimum.If
At4
(4.3.2)
Mechanical vibrations
then the stationary point is a strict local mqximum.If
O'f (xo_,!) <0. A2 f (x0=,y) .0. O'f (*o_,y) A2 f (xo=,yJ _( A'f @0,il\öx' --' oy' oxz ay' I aya* )
2
>0
(A.3.3)
( a'f @o,v)\I ur*- )
202f(x,y) ö2f (xs, !o) <0 (4.3.4)öx2 ay'
then the stationary point is a saddle point.
Appendix 4.4: On the definition of relative damping for complex modes
Given a mechanical system with the impedance matrix
Z(s): Msz +Cs + K (A'4'1)
Let s = o+ia eC, o,d)€ lR, o 10, at+ 0, be arootto the characteristic equation
p(s)=det(Z(s))=Q (A.4.2)
and let e be the corresponding mode shape vector, i.e. Z (s)z = 0 , thert the relative damping
f" corresponding to this mode (s,e) is defined by
rcr (A.4.3)
I+19'Yo
For a weakly damped system, which is diagonalizable using the classical normal modes, wehave (see Lecture Notes on p.150)
o=-(a)s, or=ora=rotfb( if o<(<I (A.4.4)
and then
I7C9= _, (A.4.s)
1
1
.CD-"+(-)-o +( 'oJi7
-(a\ )'
415
Mechanical vibrations
Appendix A.5: Partitioned matrices
Consider the matrix
= p(n+n)x(n+n) (A.s.1)
where I e iR"', 2 € ]Rn'n and Q€ ]R"'' and D e ]R"'. Calculate detÄ !
Case 1: Assume that det A+ 0 .Then
det.A,= det A.det(p - Q4-' E) (A.5.2)
Case 2: Assume that detD + 0. Then
det .A= det(A - ED-' Q) . det p (A.5.3)
Note that if all matrices are square, i.e. m=n then(A.5.2) and (A.5.3) maybe replaced by
detÄ: det(Ap- 4Q.4'E) (4.s.4)
detÄ= det(AD-ED'Q.D) (4.5.s)
respectively. Now if 4 and Q commute,i.e. AQ= g.A then (A.5.4) maybe written
detÄ= det(AD-AB_) (4.s.6)
and if Q and Q commute,i.e. Cp= De then (A.5.5) maybe written
det.4,= det(AQ -B_A) (A.s.7)
Now consider the matrix
(,q B\,4.=l - - |(.s D)
and
Ds+KMs
Ms(A.5.8)_M
where !f is the mass matrix, B the 'B-matrtx' and { the stiffrress matrix of a mechanical
system. Since det(-$) + 0 we obtain, by using expression (A.5.3)
det
= det(M s2 + ås + {) .det(-A!) = det Z!).det(-tt!) = (-I), det lttt p(s) (A.5.9)
B-p+E
MsWM = det(ås + K -Ms(-M)' Ms).det(-M) :
416