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Gyroscopic effects in micropolar elasticity
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Int. I. Engng Sri Vol. 17. pp. 433-439 0 Pergamon Press Ltd., 1979. Printed in Great Britain
GYROSCOPIC EFFECTS IN MICROPOLAR ELASTICITY JOSEPH PADOVAN University of Akron, Akron, OH 44325,U.S.A. Ah&act-The influence of gyroscopic fields on the dynamics of continuum systems modeled by micropolar elasticity is considered. In particular, the general effects of gyroscopic inertia on the properties of the micropolar frequency eigenvalues and their associated eigenfunctions are determined. For example, although the gyroscopic micropolar field equations are classically nonself-adjoint, through the use of the skew symmetry of the Coriolis operator, a modified form of orthogonality is developed. Additionally, through the use of a modified version of Rayleigh’s quotient, various of the properties of the eigenvalues are developed namely, potential bifurcations, realness, etc.
INTRODUCTION
much attention has been given to ascertaining the various properties of eigenvalue/vector relations arising from vibration problems in solid mechanics, little work has been directed toward understanding the potential effects of gyroscopic fields. This is an important shortcoming since numerous aerospace and commercial structures (satellites, gas/steam turbines, etc.) are subject to potentially important gyroscopic fields. Such fields generally give rise to both Coriolis and centripetal inertial force components and hence a skew symmetry is admitted in the governing equations of motion which induces a nonself-adjointness. In this context the series of works of Meirovitch[l-31 on lumped parameter systems subject to gyroscopic fields are notable exceptions. In these investigations Meirovitch[ l-3] developed orthogonality properties which ultimately lead to the development of a modal type solution procedure. More recently Padovan[41 has investigated the influence of gyroscopic fields on the dynamic characteristics of continuous systems modeled by nonpolar elasticity wherein the motions are represented by small deflections superposed on large. To further deepen our understanding of the effects of complex inertial fields, the current work will consider the influence of gyroscopic fields on the dynamics of continuum systems modelled by micropolar elasticity theory. In this context, in the sections to follow particular emphasis wilI be given to: (i) Ascertaining orthogonality conditions. (ii) Potential eigenvalue properties. (iii) Extension of results to multidomain situations [5]. ALTHOUGH
GOVERNING
FIELD EQUATIONS
In terms of the recent developments of Eringen[61, the governing linear micropolar elasticity field equations take the following form namely for V x E R
msp,s tsP
=
k&p
+
~~~~~~ +
(CL +
=
Iv2
Kksp
(2) +
Wps
msp = G&& + P& + &%., Ers= us,, + l&q
(3) (4) (5)
where Z,, and Ip2 are the inertial terms arising out of the dynamic equilibrium of forces and moments. Letting the surface of R be defined by
aR = aR,,+ aR,, 433
434
JOSEPH PADOVAN
where aR,, and aR,, are, respectively, those portions of aR wherein mrS, t,, and u, 4 are prescribed, it follows that the homogeneous boundary conditions associated with (l)-(5) arc given by (i) for V x E aR,, ;
(ii) for V x E aR,, (u,;(bJ=O.
(8)
Since the effects of gyroscopic fields are admitted, the inertial terms I,, and I,* take the form
such that
where the linear operators APSland APS2are, respectively, associated with the Coriolis and centripetal fields wherein fi, is the rotation vector. Due to the inclusion of Coriolis effects. the resulting field equations no longer retain their classical self-adjointness. In particular, the recently developed orthogonality principles associated with single [7] and multidomain [5] micropolar configurations do not apply to the current set of field equations. This is a direct outgrowth of the presence of the A,, operator. Although this operator obviously complicates the orthogonality process, as will be seen later, its skewness property will enable the development of the requisite orthogonality principle without resorting to adjoint operator procedures [S].
ORTHOGONALITY
PRINCIPLE
For the free vibration case, up. & t,, and m,, are all assumed separable in the form IS. 71 (Up; & ; t,, ; m,,)= ( Up,; O,, : Tspa; Mspu) e’“-*
(13)
where i = d- 1 and 0 is time. Hence, in terms of (13) (l)-(8) reduce to (i) forVxER;
(ii) for V x E dR,,; ( Tspn; Mspah
=0
(1%
(iii) for x E aR,: (Up,; @paI= 0.
(‘0)
Gyroscopic
effects
in micropolar
435
elasticity
As noted earlier, due to the appearance of the APl linear operator, (14)-(20) are not self-adjoint in the classical form. Rather, an alternative approach must be employed to establish the requisite orthogonality principle. Although an adjoint differential operator most probably could be associated with (14)-(20), the resulting set of field equations would require the generation of an additional set of eigenfunctions to develop the requisite biorthogonality condition. To circumvent the need to employ such an approach, the skew symmetric properties of APSlwill be utilized in the following development. In order to establish the requisite orthogonality conditions, (l)-(5), (7) and (8) are reduced to first order form. Before doing this, to take advantage of the skew symmetry of APSI,the following auxiliary identities are introduced, namely LP.9- Apszas= -$ (fsp.s- Apszus) mSP,S +l Ppq&- iASP2yS= 5 (m,,S + G,& - iA,&)
(21) (22)
where (23)
(a597s) = $ (US,A). Now recasting (l), (2), (21) and (22) in first order form yields
Aps2us
-
Ups24
Aps2us
tsp.9 -
+ Apse 0s -
mspqs -
~~~~~~
4p.s
(24)
In terms of (13) and (23), it follows that
where
I
up,
rqp.l=
i
0
0
0
@p= O
O
0
&a 0
0 0
0 apa
1 .
(25)
(26)
Employing (25), (24) reduces to iwa[A,]YY,
+
[B&Py, = 0
(27)
such that to derive benefit from the skew symmetry of BPS,,the form of the auxilary equation is so chosen that the off diagonal terms associated with [BP,] are negative symmetric. Hence, [A,] and [B,] take the following partitioned form
0” A;2Ao ; Pa3
0
0
0
Aw4
(28)
436
IOSEPH
PADOVAN
and B PaI
0
0
[B,,l=
Bpuz 0
BpoJ
0
BP,? 0 0 - BP&
-
Bpa4
0 0
(39)
0 0
where A Pal
=
PUPCI
(30a)
Apa2
=
Pi@,,
(30h)
A PO = b?U,a
i3Oc)
- Tspn.s
l30d)
A po4= jApr@sa - Kpa..~ - ~pqrTqro B pal = Aps, Us, Bpmz= 4,szUsz - La B pal = jApslQsa B pa4= jApdsa
- ~~~~~~~~ - Mspn.F.
Forming the inner product between @pa[6p,]; p# (Yand (27) yields the eigenvalue/function relationship central to the development of the orthogonal principle, that is
where the overbar denotes complex conjugation. Due to the choice of the auxilary identities, the matrices [C&l and [&I appearing in (32) have the following partitioned forms, namely
i
[Rsl=
-
Z.&l
0
Da04
0
0
0
- D+I
0
0
L&2
0
such that the various elements DaaI, . . and C+,. CapI
=
I
(34)
. are defined by
~U~oll?~,jdt:
i35a)
R
r35b)
cap,= Caa4=
I
R
IR
(ema+ Apsz~so~,d dv
(35C)
( Eap + iA,, IUs, ~pd do
(3Sd)
and (3%
Gyroscopic effects in micropolar elasticity
with
Although [&I is itself neither symmetric nor skew symmetric, after several extensive manipulations it can be shown that the bilinear form ~#&#Pn has the following conjugate antisymmetric property, that is _-
(y~D~~I~=) = - %fD.d’,.
(391
This is a direct outgrowth of: (i) The negative symmetry of the off diagonal terms. (ii) The skewness properties of the diagonal operators. In a similar context, the bilinear form I&~[C,,]‘Z’, can be shown to be conjugate symmetric, namely
Assuming that distinct frequency eigenvalues exist, that is w,# ws, then the eigenvalueeigenfunction relation associated with o,, UP@and OPs can be recast as
ioe[C,lY, t [D,lYo
= 0.
(41)
Now because of the conjugate symmetry and antisymmetry properties of the inner products described by (39) and (40), it follows that (32) and (41) can be manipulated to yield the following orthogonality conditions, that is
egc,BpP* = 0
(42)
+;[D,,]‘P,
(43)
and = 0.
Recast in scalar form, (42) is given by
For the case in which 0, + 0, it can be shown that .f(e=@+ E,,) du + 0 and thus (42) reduces to the orthogonality condition recently developed by Anderson[71. By also negfecting polar material behavior, (42) further reduces to the results of classical elasticity.
EIGENVALUEPROPERTIES
To determine various of the properties of the eigenvalues arising out of the foregoing development, a modified form of Rayleigh’s quotient must be developed. To do this, the sum of
438
JOSEPH PADOVAN
the inner products of (14) and (15) with (U,, aw) yields the following quadratic form namely
IR
(eqn(ll)
U,,+eqn(12)~,,)du~u,lw%+ia,2w,+ua3=0
(45)
where &I = RP( I
~,J&,
+ j@pn$,a)
dv
(46a)
Solving (45) for W, yields the following modified version of Rayleigh’s quotient, that is (47) As noted earlier, (47) can be used to establish certain of the properties of w,. In particular, since the operator APslis skew symmetric, it follows that aa is purely imaginary. Therefore, for gyroscopic fields for which aa > ~2,~/a~,, o, is itself purely real. Furthermore, due to the appearance of the aa2 term, (47) yields two distinct roots for each eigenfunction set. These correspond to backward and forward traveling waves. Hence, similar to classical nonpolar elasticity theory, gyroscopic inertial fields tend to cause a doubling of the frequency eigenvalues of micropolar elasticity. As the magnitude of the rotation vector is reduced, the bifurcated roots tend to merge back to the standard static natural frequency eigenvalue. MULTI-DOMAIN EXTENSION
For multi-domain situations, R takes the form
such that aR is given by
aRtr)= z aR(6t) + E
aR$i
+ aR(r) 4.
(49)
In terms of the domain definitions described by (48) and (49) the governing gyroscopic field equations take the form (i) for V x E R’? (50)
(53)
Gyroscopic effects in micropolar elasticity (ub”;
4:‘)
= 0.
439
(54)
Under the assumption that a common set of eigenvalues can be associated with the connected domain, (13) modifies to the form
In terms of (SO)-(SS), after extensive manipulations it can be shown that the multidomain extension of the orthogonality condition takes the form
(56) The modified Rayleigh quotient associated with (56) is given by (57) where here Aai =
2
I
a$‘;
j=
1,2,3.
Similar to the single domain case, (55) can be used to verify various of the properties of the foregoing eigenvalue problem. DISCUSSION
The main importance of orthogonality properties stems from the fact that they are central to the development of forced time dependent solutions. For this paper, the generation of such conditions was made possible by introducing a specialized set of auxiliary equations through which the governing gyroscopic polar field equations of motion are reduced to first order form. By subsequently employing the skewness property associated with Coriolis acceleration, various conjugate symmetriclantisymmetric identities were established. Through their use, the requisite orthogonality and generalized Rayleigh quotient are derived. The generality of the orthogonality conditions derived herein are such that for gyroscopic problems modeled by either plate, shell or 3-D polar elasticity continuum theories, the classical Sturm-Liouville approach can be directly applied in either its classical or finite element adaptions. Furthermore, by investigating the functional structure of the Rayleigh quotient, the bifurcating effects of Coriolis inertial fields are established wherein the various eigenvalue branches may be associated with backward and forward traveling waves. REFERENCES [I] L. MEIROVITCH, AIAA 1. 10, 1337 (1974). [Z] L. MEIROVITCH, 25th ht. Astronaut Congr. IAE Amsterdam (1974). [3] L. MEIROVITCH, J. Appl. Mech. 42,446 (1975). 141J. PADOVAN, ht. J. Engng. Sci. 16, 1061 (1978). [5] J. PADOVAN, ht. I. Engng. Sci. 14,273 (1976). [6] A. C. ERINGEN, Fracture: An Advanced Treutise (Edited by H. Liebovitz), Vol. Il. Academic Press, New York (1968). [7] G. L. ANDERSON, ht. 1. EngngSci. 11, 21 (1973). [B] J. PADOVAN, Int. J. Engng Sci. 14,819 (1976). (Received 31 August 1978)
ES Vol. 17, No. 4-C
Int. I. Engng Sri Vol. 17. pp. 433-439 0 Pergamon Press Ltd., 1979. Printed in Great Britain
GYROSCOPIC EFFECTS IN MICROPOLAR ELASTICITY JOSEPH PADOVAN University of Akron, Akron, OH 44325,U.S.A. Ah&act-The influence of gyroscopic fields on the dynamics of continuum systems modeled by micropolar elasticity is considered. In particular, the general effects of gyroscopic inertia on the properties of the micropolar frequency eigenvalues and their associated eigenfunctions are determined. For example, although the gyroscopic micropolar field equations are classically nonself-adjoint, through the use of the skew symmetry of the Coriolis operator, a modified form of orthogonality is developed. Additionally, through the use of a modified version of Rayleigh’s quotient, various of the properties of the eigenvalues are developed namely, potential bifurcations, realness, etc.
INTRODUCTION
much attention has been given to ascertaining the various properties of eigenvalue/vector relations arising from vibration problems in solid mechanics, little work has been directed toward understanding the potential effects of gyroscopic fields. This is an important shortcoming since numerous aerospace and commercial structures (satellites, gas/steam turbines, etc.) are subject to potentially important gyroscopic fields. Such fields generally give rise to both Coriolis and centripetal inertial force components and hence a skew symmetry is admitted in the governing equations of motion which induces a nonself-adjointness. In this context the series of works of Meirovitch[l-31 on lumped parameter systems subject to gyroscopic fields are notable exceptions. In these investigations Meirovitch[ l-3] developed orthogonality properties which ultimately lead to the development of a modal type solution procedure. More recently Padovan[41 has investigated the influence of gyroscopic fields on the dynamic characteristics of continuous systems modeled by nonpolar elasticity wherein the motions are represented by small deflections superposed on large. To further deepen our understanding of the effects of complex inertial fields, the current work will consider the influence of gyroscopic fields on the dynamics of continuum systems modelled by micropolar elasticity theory. In this context, in the sections to follow particular emphasis wilI be given to: (i) Ascertaining orthogonality conditions. (ii) Potential eigenvalue properties. (iii) Extension of results to multidomain situations [5]. ALTHOUGH
GOVERNING
FIELD EQUATIONS
In terms of the recent developments of Eringen[61, the governing linear micropolar elasticity field equations take the following form namely for V x E R
msp,s tsP
=
k&p
+
~~~~~~ +
(CL +
=
Iv2
Kksp
(2) +
Wps
msp = G&& + P& + &%., Ers= us,, + l&q
(3) (4) (5)
where Z,, and Ip2 are the inertial terms arising out of the dynamic equilibrium of forces and moments. Letting the surface of R be defined by
aR = aR,,+ aR,, 433
434
JOSEPH PADOVAN
where aR,, and aR,, are, respectively, those portions of aR wherein mrS, t,, and u, 4 are prescribed, it follows that the homogeneous boundary conditions associated with (l)-(5) arc given by (i) for V x E aR,, ;
(ii) for V x E aR,, (u,;(bJ=O.
(8)
Since the effects of gyroscopic fields are admitted, the inertial terms I,, and I,* take the form
such that
where the linear operators APSland APS2are, respectively, associated with the Coriolis and centripetal fields wherein fi, is the rotation vector. Due to the inclusion of Coriolis effects. the resulting field equations no longer retain their classical self-adjointness. In particular, the recently developed orthogonality principles associated with single [7] and multidomain [5] micropolar configurations do not apply to the current set of field equations. This is a direct outgrowth of the presence of the A,, operator. Although this operator obviously complicates the orthogonality process, as will be seen later, its skewness property will enable the development of the requisite orthogonality principle without resorting to adjoint operator procedures [S].
ORTHOGONALITY
PRINCIPLE
For the free vibration case, up. & t,, and m,, are all assumed separable in the form IS. 71 (Up; & ; t,, ; m,,)= ( Up,; O,, : Tspa; Mspu) e’“-*
(13)
where i = d- 1 and 0 is time. Hence, in terms of (13) (l)-(8) reduce to (i) forVxER;
(ii) for V x E dR,,; ( Tspn; Mspah
=0
(1%
(iii) for x E aR,: (Up,; @paI= 0.
(‘0)
Gyroscopic
effects
in micropolar
435
elasticity
As noted earlier, due to the appearance of the APl linear operator, (14)-(20) are not self-adjoint in the classical form. Rather, an alternative approach must be employed to establish the requisite orthogonality principle. Although an adjoint differential operator most probably could be associated with (14)-(20), the resulting set of field equations would require the generation of an additional set of eigenfunctions to develop the requisite biorthogonality condition. To circumvent the need to employ such an approach, the skew symmetric properties of APSlwill be utilized in the following development. In order to establish the requisite orthogonality conditions, (l)-(5), (7) and (8) are reduced to first order form. Before doing this, to take advantage of the skew symmetry of APSI,the following auxiliary identities are introduced, namely LP.9- Apszas= -$ (fsp.s- Apszus) mSP,S +l Ppq&- iASP2yS= 5 (m,,S + G,& - iA,&)
(21) (22)
where (23)
(a597s) = $ (US,A). Now recasting (l), (2), (21) and (22) in first order form yields
Aps2us
-
Ups24
Aps2us
tsp.9 -
+ Apse 0s -
mspqs -
~~~~~~
4p.s
(24)
In terms of (13) and (23), it follows that
where
I
up,
rqp.l=
i
0
0
0
@p= O
O
0
&a 0
0 0
0 apa
1 .
(25)
(26)
Employing (25), (24) reduces to iwa[A,]YY,
+
[B&Py, = 0
(27)
such that to derive benefit from the skew symmetry of BPS,,the form of the auxilary equation is so chosen that the off diagonal terms associated with [BP,] are negative symmetric. Hence, [A,] and [B,] take the following partitioned form
0” A;2Ao ; Pa3
0
0
0
Aw4
(28)
436
IOSEPH
PADOVAN
and B PaI
0
0
[B,,l=
Bpuz 0
BpoJ
0
BP,? 0 0 - BP&
-
Bpa4
0 0
(39)
0 0
where A Pal
=
PUPCI
(30a)
Apa2
=
Pi@,,
(30h)
A PO = b?U,a
i3Oc)
- Tspn.s
l30d)
A po4= jApr@sa - Kpa..~ - ~pqrTqro B pal = Aps, Us, Bpmz= 4,szUsz - La B pal = jApslQsa B pa4= jApdsa
- ~~~~~~~~ - Mspn.F.
Forming the inner product between @pa[6p,]; p# (Yand (27) yields the eigenvalue/function relationship central to the development of the orthogonal principle, that is
where the overbar denotes complex conjugation. Due to the choice of the auxilary identities, the matrices [C&l and [&I appearing in (32) have the following partitioned forms, namely
i
[Rsl=
-
Z.&l
0
Da04
0
0
0
- D+I
0
0
L&2
0
such that the various elements DaaI, . . and C+,. CapI
=
I
(34)
. are defined by
~U~oll?~,jdt:
i35a)
R
r35b)
cap,= Caa4=
I
R
IR
(ema+ Apsz~so~,d dv
(35C)
( Eap + iA,, IUs, ~pd do
(3Sd)
and (3%
Gyroscopic effects in micropolar elasticity
with
Although [&I is itself neither symmetric nor skew symmetric, after several extensive manipulations it can be shown that the bilinear form ~#&#Pn has the following conjugate antisymmetric property, that is _-
(y~D~~I~=) = - %fD.d’,.
(391
This is a direct outgrowth of: (i) The negative symmetry of the off diagonal terms. (ii) The skewness properties of the diagonal operators. In a similar context, the bilinear form I&~[C,,]‘Z’, can be shown to be conjugate symmetric, namely
Assuming that distinct frequency eigenvalues exist, that is w,# ws, then the eigenvalueeigenfunction relation associated with o,, UP@and OPs can be recast as
ioe[C,lY, t [D,lYo
= 0.
(41)
Now because of the conjugate symmetry and antisymmetry properties of the inner products described by (39) and (40), it follows that (32) and (41) can be manipulated to yield the following orthogonality conditions, that is
egc,BpP* = 0
(42)
+;[D,,]‘P,
(43)
and = 0.
Recast in scalar form, (42) is given by
For the case in which 0, + 0, it can be shown that .f(e=@+ E,,) du + 0 and thus (42) reduces to the orthogonality condition recently developed by Anderson[71. By also negfecting polar material behavior, (42) further reduces to the results of classical elasticity.
EIGENVALUEPROPERTIES
To determine various of the properties of the eigenvalues arising out of the foregoing development, a modified form of Rayleigh’s quotient must be developed. To do this, the sum of
438
JOSEPH PADOVAN
the inner products of (14) and (15) with (U,, aw) yields the following quadratic form namely
IR
(eqn(ll)
U,,+eqn(12)~,,)du~u,lw%+ia,2w,+ua3=0
(45)
where &I = RP( I
~,J&,
+ j@pn$,a)
dv
(46a)
Solving (45) for W, yields the following modified version of Rayleigh’s quotient, that is (47) As noted earlier, (47) can be used to establish certain of the properties of w,. In particular, since the operator APslis skew symmetric, it follows that aa is purely imaginary. Therefore, for gyroscopic fields for which aa > ~2,~/a~,, o, is itself purely real. Furthermore, due to the appearance of the aa2 term, (47) yields two distinct roots for each eigenfunction set. These correspond to backward and forward traveling waves. Hence, similar to classical nonpolar elasticity theory, gyroscopic inertial fields tend to cause a doubling of the frequency eigenvalues of micropolar elasticity. As the magnitude of the rotation vector is reduced, the bifurcated roots tend to merge back to the standard static natural frequency eigenvalue. MULTI-DOMAIN EXTENSION
For multi-domain situations, R takes the form
such that aR is given by
aRtr)= z aR(6t) + E
aR$i
+ aR(r) 4.
(49)
In terms of the domain definitions described by (48) and (49) the governing gyroscopic field equations take the form (i) for V x E R’? (50)
(53)
Gyroscopic effects in micropolar elasticity (ub”;
4:‘)
= 0.
439
(54)
Under the assumption that a common set of eigenvalues can be associated with the connected domain, (13) modifies to the form
In terms of (SO)-(SS), after extensive manipulations it can be shown that the multidomain extension of the orthogonality condition takes the form
(56) The modified Rayleigh quotient associated with (56) is given by (57) where here Aai =
2
I
a$‘;
j=
1,2,3.
Similar to the single domain case, (55) can be used to verify various of the properties of the foregoing eigenvalue problem. DISCUSSION
The main importance of orthogonality properties stems from the fact that they are central to the development of forced time dependent solutions. For this paper, the generation of such conditions was made possible by introducing a specialized set of auxiliary equations through which the governing gyroscopic polar field equations of motion are reduced to first order form. By subsequently employing the skewness property associated with Coriolis acceleration, various conjugate symmetriclantisymmetric identities were established. Through their use, the requisite orthogonality and generalized Rayleigh quotient are derived. The generality of the orthogonality conditions derived herein are such that for gyroscopic problems modeled by either plate, shell or 3-D polar elasticity continuum theories, the classical Sturm-Liouville approach can be directly applied in either its classical or finite element adaptions. Furthermore, by investigating the functional structure of the Rayleigh quotient, the bifurcating effects of Coriolis inertial fields are established wherein the various eigenvalue branches may be associated with backward and forward traveling waves. REFERENCES [I] L. MEIROVITCH, AIAA 1. 10, 1337 (1974). [Z] L. MEIROVITCH, 25th ht. Astronaut Congr. IAE Amsterdam (1974). [3] L. MEIROVITCH, J. Appl. Mech. 42,446 (1975). 141J. PADOVAN, ht. J. Engng. Sci. 16, 1061 (1978). [5] J. PADOVAN, ht. I. Engng. Sci. 14,273 (1976). [6] A. C. ERINGEN, Fracture: An Advanced Treutise (Edited by H. Liebovitz), Vol. Il. Academic Press, New York (1968). [7] G. L. ANDERSON, ht. 1. EngngSci. 11, 21 (1973). [B] J. PADOVAN, Int. J. Engng Sci. 14,819 (1976). (Received 31 August 1978)
ES Vol. 17, No. 4-C