Spline approximation of thin shell dynamics - numerical examples

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SPLINE APPROXIMATION OF THIN SHELL DYNAMICS {

NUMERICAL EXAMPLES

1

R.C.H. del Rosario

Center for Research in Scientic Computation North Carolina State University

Raleigh, NC 26795 R.C. Smith

Department of Mathematics Iowa State University

Ames, IA 50011

Abstract

A spline-based method for approximating thin shell dynamics is presented here. While the method is developed in the context of the Donnell-Mushtari thin shell equations, it can be easily extended to the Byrne-Flugge-Lur'ye equations or other models for shells of revolution as warranted by applications. The primary requirements for the method include accuracy, exibility and eciency in smart material applications. To accomplish this, the method was designed to be exible with regard to boundary conditions, material nonhomogeneities due to sensors and actuators, and inputs from smart material actuators such as piezoceramic patches. The accuracy of the method was also of primary concern, both to guarantee full resolution of structural dynamics and to facilitate the development of PDE-based controllers which ultimately require real-time implementation. Several numerical examples provide initial evidence demonstrating the ecacy of the method.

1This research was supported in part by the National Aeronautics and Space Administration under NASA

Contract Numbers NAS1-18605 and NAS1-19480 while the authors were visiting scientists at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681. Additional support was also provided in part under NASA grant NAG-1-1600.

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1 Introduction

A spline-based method for numerically approximating thin shell dynamics is presented in this paper. This research is motivated by the growing number of applications involving shell models for which explicit solutions are typically unavailable. A specic example involves the development of control strategies for reducing vibrations, fatigue and structure-borne noise in aircraft fuselages. Typically, some form of cylindrical shell model is employed with the specic equations and boundary conditions dictated by the application or experimental setup. When numerically discretizing the models in such applications, the approximation method must be exible with respect to boundary conditions and adaptable with regard to nonhomogeneities in materials and geometries. For example, if piezoceramic patches bonded to or embedded in the shell are used as actuators or sensors, the numerical method must be suciently exible so as to permit extension to models incorporating these components. Also, the primary motion in many vibration and noise control applications is bending-dominated and the numerical method must accurately approximate such dynamics. Finally, from a control perspective, the approximation method must adequately preserve stability properties of the physical system as well as be suciently ecient so as to permit real-time implementation.

Current techniques for approximating shell dynamics include modal expansions [16, 17, 25], nite element discretizations [2, 3, 4, 11, 13, 14, 15, 19, 20, 21, 24, 27, 29] and nite dierence approximations [30, 31]. From a theoretical perspective, modal expansions arise naturally when separating variables in linear shell models. In models for which analytic expressions for the eigenfunctions or modes can be obtained, this provides a straightforward and often quite ecient method of approximation. The diculty, however, lies in the fact that analytic expressions describing mode shapes are known only for a restricted class of boundary conditions and for a limited set of models. For general boundary conditions and models, or systems involving coupling between a shell and an adjacent component (e.g., an acoustic eld or a piezoceramic actuator), the modes must rst be numerically approximated or experimentally determined before expansions can be formed.

Experimental determination of natural frequencies and modes is typically accomplished by exciting the structure using an acoustic source, shaker, magnetic driver, et cetera. At each natural frequency, the shape of the corresponding mode is determined and characterized for use in modal expansions. As detailed in [31, 32], however, diculties are encounted in this procedure in plate and shell-like structures due to the presence of multiple independent mode shapes which can occur at single experimental frequencies. In such cases, the experimentalist must excite the structure at various locations and complete an orthogonalization process to obtain a complete modal basis. The determination of which frequencies/modes to test for this behavior requires numerical analysis or extreme care when performing the experiments. Furthermore, as discussed in [31], internal or air damping will cause distortion of modes (e.g., modal lines that do not cross) which if not accounted for, will degrade modal expansions for approximating the structural dynamics.

In applications, modes determined for boundary conditions yielding analytic expressions (e.g., simply-supported end conditions) are occasionally used to approximate solutions to mod-els with more general boundary conditions or modmod-els incorporating additional components. In some cases, the inuence of various boundary conditions on modal characteristics has been investigated [16]; however, without any convergence analysis for such techniques, convergence

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of the numerical approximates to the true model solution cannot be guaranteed. Finally, it is dicult to apply modal techniques in applications modeled by nonlinear shell equations or nonlinear actuator contributions (e.g., electrostrictive actuators or piezoceramic actuators embedded in constrained damping layers).

Finite element approaches, on the other hand, are directly applicable for a variety of boundary and coupling conditions and in models including actuator contributions or nonlinear dynamics. Moreover, a large body of software exists for obtaining nite element solutions for various shell models (see [13, 19, 20, 21] for discussions of nite element methods and software for shells of revolution).

A diculty when developing and applying nite element methods for shell applications, however, concerns the manifestation of various locking phenomena. Shear locking, which has also been studied extensively in the context of Reissner-Mindlin plate models, is due to element incompatibility when enforcing the Kirchho-Love constraint of vanishing transverse shear strains as the shell thicknessh tends to zero [1, 10]. In shell applications, an even more serious problem leading to the failure of various nite element methods is the phenomenon of

membrane or shear-membrane locking. This phenomenon occurs when the total deformation energy is bending dominated, and is due to smoothness and asymptotic constraints in the shell model which are not appropriately represented by the approximation method (see [3, 4, 15, 24, 27]). If these constraints are not satised by approximating elements, the numericalsolution is often overly sti in the sense that the model exhibits bending dynamics which the approximate solution cannot match. As detailed in [24], mesh sizes must be chosen signicantly smaller than the shell thickness to ensure accurate approximations with high-order nite elements in such bending dominated regimes. These examples also illustrate that even with such mesh size restrictions, low-order nite element methods often fail in such regimes. The use of nite elements which exhibit locking is detrimental in simulations and will seriously degrade the performance of any controllers designed around such methods.

A second issue which must be considered when developing nite elementmethods for shells concerns the piecewise constant material parameters and inexact boundary conditions which often arise in smart materialapplications. For example, the use of piezoceramicactuators leads to piecewise discontinuities in the density, stiness, Poisson ratio and damping parameters. Consequently, nite elementmeshes must be aligned with the actuator boundaries to maintain accuracy (the reader is referred to [33] for a discussion of nite elements designed specically for piezoelectric applications). In experiments involving shells, slight energy loss must often be modeled into the boundary conditions to account for inexact boundary clamps. Consequently, elements must be provided with extra degrees of freedom to accommodate these boundary conditions. Standard commercialcodes not providing these capabilities will lead to potentially inaccurate results when applied to shells with smart material actuators and/or sensors.

The use of nite dierence discretizations for approximating thin shell dynamics is less common due to inherent diculties with the high-order equations and boundary conditions arising in the models. The reader is referred to [30] for further discussion of nite dierence methods for shell applications.

In this paper, we consider a Galerkin method for discretizing thin shell models with linear or cubic splines chosen as basis elements for approximating the longitudinal and circumferen-tial motion and cubic splines used to approximate the transverse component. These choices are motivated by smoothness and accuracy criteria, adaptability for a variety of boundary

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conditions, and suitability when including the contributions of actuators such as piezoceramic patches bonded to the shell [6]. For the discussion here, the shells are assumed to have lengths that are relatively short in relation to the radii, and the Donnell-Mushtari shell equations are employed. This facilitates the discussion while providing a framework which is easily extended to more accurate long shell models through the inclusion of Byrne-Flugge-Lur'ye components. In this study, the formulation of the method and numerical examples demonstrating the ac-curacy, eciency and exibility of the method are presented. Emphasis in these examples is placed on demonstrating that when hx denotes the axial mesh size, the expected O` (h2x)

and O`(h4x) accuracy of the method is maintained when approximating with linear and cubic splines, respectively. Convergence analysis and further analysis of the method with regards to membrane and shear locking will appear in a future companion paper.

In Section 2, the strong and weak forms of the modeling equations for a thin shell with surface-mounted piezoceramic patches are summarized. Modal analysis for the special case of an undamped shell with constant coecients and simply-supported boundary conditions is discussed in Section 3. This provides a framework for testing the convergence of the approxi-mate mass and stiness components in the system. The approximation method and resulting nite dimensional matrix system are detailed in Section 4. Finally, examples illustrating the method are presented in Section 5. These include modal approximations for various bound-ary conditions as well as results demonstrating convergence rates when approximating shell dynamics generated by various external inputs. To illustrate the shell dynamics generated by the piezoceramic patches, a modeled voltage spike to the patches is used as input in the nal example. The resulting frequencies are then compared with those obtained in the eigenvalue (modal) analysis to demonstrate the consistency of the method.

2 Donnell-Mushtari Equations

th

i patch

R h

x

θ

v w

u

i θ

(x , )i

Figure 1.

A cylindrical thin shell with surface mounted piezoceramic patches.

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We consider here a shell of length `, thickness h and radius R having mass density , Young's modulus E, Poisson ratio and Kelvin-Voigt damping coecient cD. As depicted

in Figure 1, the axial direction is taken along the x-axis. The displacements of the middle surface in the longitudinal, circumferential and transverse directions are denoted by u, v and w, respectively.

Bonded to the shell are s pairs of piezoceramic patches which can be employed as sensors and/or actuators in structural and structural acoustic applications [5, 18]. To simplify the exposition, the patches are all assumed to have thickness hpe, Young's modulus Epe, Poisson

ratio pe, and Kelvin-Voigt damping coecient cDpe. Furthermore, it is assumed that the

glue bonding layer provides negligible contribution to the structural dynamics. The reader is referred to to [6, 9] for details concerning the incorporation of diering patch characteristics and bonding layers in the ensuing models).

Throughout this discussion, it will be assumed that external inputs to the shell will be in the form of transverse, longitudinal and circumferential surface forces as well as line moments and forces generated by the patches. The surface forces can be used to model a variety of phenomena including coupling interactions with adjacent elds (e.g., acoustic elds { see [5]) and input from certain actuators. The patch moments and forces arise when the elements are used as actuators. While more general inputs can be considered, the above-mentioned cases demonstrate the exibility of the numerical method for typical smart materials applications.

2.1 Strong Form of Equations

As detailed in [6, 25], moment and force balancing yields the Donnell-Mushtari equations Rh@_@t2u₂ ?R@N

x

@x ? @Nx

@ = R^qx?R

s

X

i=1

@(Nx)pei @x Spe

i(x;) Rh@_@t2v₂ ?

@N

@ ?RN

x

@x = R^q?

s

X

i=1

@(N)pei @ Spe

i(x;) Rh@_@t2w₂ ?R@

2_M_x

@x2 ? 1 R@

2_M

@2 ?2@

2_M_x

@x@ + N =R^qn?

s

X

i=1

"

R@2(Mx)pei

@x2 + 1R@

2_(M₎_pe

i @2

#

(2.1)

as a model for the thin shell dynamics. Here Mx;M;Mx, and Mx are internal moments,

whileNx;N;Nx andNxdenote internal force resultants. External surface forces are denoted

by ^qx; ^q; ^qn whereas the external resultants (line moments and forces) generated by the ith

patch pair are designated by (Mx)pei;(M)pei;(Nx)pei and (N)pei. The indicator function

Spe i(x;)

S1;2(x)^S1;2() ; where

S1;2(x) =

8 > > < > > :

1 ; x < (x1i+x2i)=2

0 ; x = (x1i+x2i)=2

?1 ; x > (x1i+x2i)=2

; ^S1;2() =

8 > > < > > :

1 ; < (1i+2i)=2

0 ; = (1i+2i)=2

?1 ; > (1i+2i)=2 ;

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indicates the sense of the forces generated by theith _{patch pair having edges at}_x_1i_;x_2i_;_1i_;_2i_.

The symmetry of the function arises from the property that for homogeneous patches hav-ing uniform thickness, equal but opposite strains are generated about the point

xi; i

= ((x1i+x2i)=2;(1i+2i)=2) in the two coordinate directions. Similarly, the characteristic

function

pei(x;) = (

1; x1ixx2i ; 1i 2i 0; otherwise

will be used in ensuing discussion to isolate internal and external contributions due to theith

patch pair.

Under the assumption that stress is proportional to a linear combination of strain and strain rate, the internal moments and force resultants in regions of the shell not covered by patches are given by

Nx = Eh₍₁

?2) "

@u

@x + R @v@ + w !#

+ c₍₁ Dh ?2)

@ @t

" @u

@x + R @v@ + w !#

N = Eh₍₁

?2) "

1

R@v@ +R + w @u@x #

+ c₍₁ Dh ?2)

@ @t

" 1

R@v@ + R + w @u@x #

Nx =Nx = Eh_{2(1 +}₎

" @v

@x + R1 @u@ #

+ c_{2(1 +}Dh₎_@t@ "

@v

@x +R1@u@ #

Mx = ?Eh

3

12(1?2) "

@2_w

@x2 + R2@

2_w

@2

# ?

cDh3

12(1?2) @ @t

" @2_w

@x2 + R2@

2_w

@2

#

M= ?Eh3 12(1?2)

" 1 R2@

2_w

@2 +@

2_w

@x2

# ?

cDh3

12(1?2) @ @t

" 1 R2@

2_w

@2 +@

2_w

@x2

#

Mx =Mx = ?Eh3 12R(1 + ) @

2_w

@x@ ?

cDh3

12R(1 + )@t@ "

@2_w

@x@ #

(2.2)

(see [6]). The substitution of these equations in (2.1) yields the damped form of the Donnell-Mushtari equations for a uniform, homogeneous shell that is devoid of patches.

The bonding of piezoceramic patches to the shell produces contributions due to both internal (material) and external moments and forces. The internal contributions are due to the geometrical and material changes aorded by the patches. The external contributions result from the piezoelectric eect in the patches which is manifested as generated strains in response to applied voltages.

As detailed in [6, 9], the internal force resultant Nx is given by

Nx = Eh₁

?2 "

@u

@x + R @v@ + w !#

+Xs

i=1

2Epehpe

1?2pe "

@u

@x + Rpe @v@ + w !#

pei(x;)

+ c₁ Dh ?2

@ @t

" @u

@x +R @v@ + w !#

+Xs

i=1

2cDpehpe

1?2pe @ @t

" @u

@x +Rpe @v@ + w !#

pei(x;)

(2.3)

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for systems in which the patches have identical properties (compare with (2.2)). Analogous expressions are given in [6, 9] for the remaining resultants and the resultants when the patches have diering material characteristics.

To characterize the external contributions, it is typical to start with the assumption that the strains generated by a patch are proportional to the applied voltage [6]. Since diering voltages can be applied to the outer and inner patches in the pair, we will dierentiate between the two with Vi1(t) and Vi2(t) used to denote the voltages to the outer and inner patches in

the ith _{pair, respectively. The proportionality constant relating generated strain to the input}

voltage is designated byd31. As detailed in [6], the total external momentsand forces generated

by the patches are

(Mx)_pei = h

(Mx)_pei1 + (Mx)pei2 i

pei(x;) (M)_pei =

h

(M)_pei1 + (M)pei2 i

pei(x;) (Nx)_pei =

h

(Nx)_pei1 + (Nx)pei2 i

pei(x;) Spe

i(x;) (N)_pei =

h

(N)_pei1 + (N)pei2 i

pei(x;) Spe

i(x;)

(2.4)

where

(Mx)pei1 =

?Epe 1?pe

2 41

8 0 @4 h

2 + hpe !2

?h

2

1 A

+ 1R241 0 @8 h

2 + hpe !3 ?h 3 1 A 3 5d31

hpeVi1

(Mx)_pei2 = E

pe

1?pe 2 41

8 0 @4 h

2 + hpe !2 ?h 2 1 A ? 1 R241

0 @8 h

2 + hpe !3 ?h 3 1 A 3 5 d31

hpeVi2

(M)_pei1 =

?Epe 1?pe

2 41

8 0 @4 h

2 + hpe !2 ?h 2 1 A 3 5d31

hpeVi1

(M)pei2 = E

pe

1?pe 2 41

8 0 @4 h

2 + hpe !2 ?h 2 1 A 3 5d31

hpeVi2

(Nx)pei1 =

?Epe 1?pe

2 4h

pe+ 1R1₈

0 @4 h

2 + hpe !2 ?h 2 1 A 3 5d31

hpeVi1

(Nx)_pei2 =

?Epe 1?pe

2 4h

pe?

1 R18

0 @4 h

2 + hpe !2 ?h 2 1 A 3 5d31

hpeVi2

(N)_pei1 =

?Epe 1?pe

d31Vi1

(N)pei2 =

?Epe 1?pe

d31Vi2 :

(2.5)

When substituted into (2.1), the expressions (2.4) provide the input from the patches when voltages are applied.

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2.2 Boundary Conditions

Appropriate boundary conditions are dictated by the experimental setup or application under consideration. In many setups, such as the experimental shell apparatus at NASA Langley Research Center, the shell is supported by heavy endcaps. In such cases, the xed-edge conditions

u = v = w = @w_{@x = 0 ; x = 0;`} (2.6) may adequately model the end behavior of the shell.

If slight boundary movement or rotations are suspected, \almost xed" boundary condi-tions of the type discussed in [7, 8] can be employed. Such boundary condicondi-tions can be used to model the slight energy which result due to imperfect clamping of the structure.

A third type of commonly considered boundary conditions are the simply-supported or shear diaphragm edge conditions

v = w = Mx =Nx= 0 ; x = 0;` : (2.7)

These boundary conditions are theoretically attractive since they provide one of the few sit-uations in which explicit modal expansions can be calculated for the Donnell-Mushtari shell equations. They are of limited use when modeling experimental shells, however, since they are appropriate only for endcaps which prevent movement in the v and w directions but are suciently exible in the x-direction so that negligible moments Mx and forces Nx are

generated.

The Galerkin method of this work is equally ecient to implementfor models incorporating the xed-edge boundary conditions (2.6), \almost xed" boundary conditions, or the simple-supported edge conditions (2.7). The method is demonstrated for the xed-edge and simply-supported conditions while modications to adapt the method to \almost xed" can be found in [7, 8]. Later discussion will also illustrate the manner through which the method can be adapted to alternative boundary conditions which many arise in physical applications.

2.3 Weak Form of Equations

As noted in (2.1), the strong form of the modeling equations involves rst and second deriva-tives of both the moment and force resultants. For structures with surface-mounted or embed-ded actuators or sensors (e.g., piezoceramic patches), this dierentiation leads to diculties due to discontinuities in the resultants. The internal resultants contain piecewise discontinu-ities due to material changes introduced by the patches (see for example, (2.3)). The external patch contributions are also discontinuous since they are dened only over regions covered by patches (see (2.4)). As a result, formal analysis and approximation using the strong form of the modeling equations leads to derivatives of the Dirac delta \function."

To alleviate these diculties, we consider a weak form of the modeling equations. As detailed in [6], such equations can be derived from energy considerations and are equivalent to the strong form of the modeling equations under suitable smoothness conditions. This yields a form of the model which facilitates well-posedness analysis [5] and eliminates the diculties associated with the discontinuous resultants. Moreover, smoothness requirements on basis functions are reduced which proves advantageous when constructing system matrices.

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The weak form of (2.1), as derived from energy considerations in [6], is given by

Z 2

0

Z `

0

(

Rh@_@t2u₂1+RNx@_{@x + N}1 x@_@1 ?R^qx1?R

s

X

i=1(Nx)pei

@1

@x )

dxd = 0

Z 2

0

Z `

0

(

Rh@_@t2v₂2+N@_{@ + RN}2 x@_@x2 ?R^q2 ?

s

X

i=1(N)pei

@2

@ )

dxd = 0

Z 2

0

Z `

0

(

Rh@_@t2w₂ 3+N3?RMx @2₃

@x2 ? 1

RM@@223 ?2Mx @2₃

@x@

?R^qn3+

s

X

i=1

"

R(Mx)pei @2₃

@x2 + 1R(M)pei @2₃

@2

#)

dxd = 0

(2.8)

for all~ = [1;2;3]2V where V denotes the space of test functions (as detailed in [6], the indicator functions appear only in the denition of the external force resultants in the weak form of the equations). Specic choices for V depend upon the boundary conditions. For the xed-edge boundary conditions (2.6), the space of test functions is taken to be

V = H₀1₍₎

H

1

0()H

2

0()

where

H₀1_{() =}n 2H

1_{() :}_{(0) = (`) = 0}o H₀2_{() =}n

2H

2_{() :}_{(0) =}_x_{(0) =}_{(`) =}_x_{(`) = 0}o :

In the case of the simply-supported edge conditions (2.7), essential boundary conditions are imposed only on the circumferential and transverse functions and V is taken to be

V = H1₍₎

H

1

0()H2L()

with

H2L() =n 2H

2_{() :}_{(0) = (`) = 0}o :

In general,V is simply taken as the subset of the traditional Sobolev spaces satisfying essential boundary conditions (the reader is referred to [7] for modications to account for \almost xed" boundary conditions).

A comparison between (2.8) and (2.1) illustrates that in the weak form, derivatives are transferred from the discontinuous resultants onto suitably smooth test functions. This allevi-ates the diculties associated with the discontinuities and reduces smoothness requirements on approximate solutions.

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3 Modal Solution { Simply-Supported Boundary

Con-ditions, Constant Coecients, No Damping

As noted in the introduction, closed form modal expressions for shell models can be obtained only in a limited number of cases. One case in which analytic frequencies and modes can be calculated is the undamped (cD = 0) shell model with simply-supported edge conditions (2.7)

and constant parameters ; and E. Separation of variables in this setting is classical and can be found in numerous texts (e.g., [26]); we summarize the arguments here to facilitate numerical examples in later sections. While such conditions are not attained in typical shell applications, the consideration of the shell model in this form provides an excellent means of testing discretization techniques since approximate solutions can be compared with analytic values. The discretization techniques can then be used to approximate natural frequencies, modes and shell dynamics in models which incorporate piecewise constant parameters (in-cluding damping) and physically realistic boundary conditions.

Throughout this section, it is assumed that cD = 0 and ; and E are constant. Since

our interest here is restricted to the calculation of natural frequencies and modes, we will also consider the shell model to be unforced (no external patch contributions and ^qx = ^q = ^qn = 0).

In this case, the strong form of the Donnell-Mushtari thin shell equations (2.1) can be written in the operator format

h@_@t2~u₂ =L~u (3.1)

where ~u = 2 6 4 u v w 3 7 5 : The operator L here is given by

L= Eh 1? 2 2 6 6 6 6 6 6 6 6 6 4 @ 2 @x

2 + (1 ?) 2R 2 @ 2 @ 2

(1 +)

2R @ 2 @x@ R @ @x

(1 +)

2R @

2 @x@

(1?)

2 @

2 @x

2 + 1 R 2 @ 2 @ 2 1 R 2 @ @ ? R @ @x ? 1 R 2 @ @ ? 1 R 2 ?k @ 4 @x

4 + 2 R 2 @ 4 @x 2 @

2 + 1 R 4 @ 4 @ 4 3 7 7 7 7 7 7 7 7 7 5

where k = h2_{=12. Boundary conditions for the shell are denoted by}

B~u = ~0 : Due to linearity, the displacements are expressed as

u(t;x;) = T(t)U(x;) v(t;x;) = T(t)V (x;) w(t;x;) = T(t)W(x;) ;

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and spatial and temporal components are separated to yield the eigenvalue problem L~U + h!2~U =~0

B~U = ~0 (3.2)

along with the temporal equationT00+!2T = 0. Here ! is the circular frequency of vibration and ~U = [U;V;W]T _{contains the vibration modes in the axial, circumferential and transverse}

directions. We point out that separating variables to arrive at the eigenvalue problem (3.2) is equivalent to the assumption of a harmonic response in all components and one obtains the same eigenvalue problem in both cases.

The structure of the eigenfunctionsU;V and W is dependent upon the boundary conditions with closed form expressions attainable in only a few cases. To illustrate, we consider the eigenvalue problem (3.2) with the simply-supported boundary conditions (2.7). One form of the natural vibration modes for this case is

U(x;) = A1cos

nx

`

cos(m) V (x;) = B1sin

nx

`

sin(m) W(x;) = C1sin

nx

`

cos(m) :

(3.3)

The choice of the cosine representation for the x component of the axial vibration and sines for the circumferential and radial vibrations guarantees that the modes satisfy the boundary conditions. The relationship between the circumferential and axial/radial vibrations dictates that the former must be 90o _{out-of-phase from the latter two in} _{(see the moment and force}

expressions (2.2) or operator denition (3.1)).

To determine the relationship between the frequencies! and the wave numbers m and n, the expressions (3.3) are substituted into (3.2) to yield the system

2 6 6 6 4

2?H1 m

2 (1 +)

m

2 (1 +) 2?H2 ?m

?m 2?H3

3 7 7 7 5

2 6 6 6 4

A1

B1

C1

3 7 7 7 5=

2 6 6 6 4

0 0 0

3 7 7 7

5 (3.4)

where

H1 =2 + m_{2 (1}2 ?) H2 = _{2 (1}2 ?) + m

2

H3 = 1 + ~k

2₊_m22 and

2 _{= E(1}

?

2_)!2_R2

= nR_L ~k = kR2 = h

2

12R2 :

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A nontrivial solution is determined by setting the determinant to zero. This yields the cubic equation

6

?K2

4₊_K₁2

?K0 = 0 (3.5)

in 2_{. The coecients here are given by}

K2 = 1 + 12(3?)

m2 ₊2 + ~k

m2₊22 K1 = 12(1?)

(3 + 2)2 ₊_m2₊

m2₊22

+ 3? 1?

~k

m2 ₊23

K0 = 12(1?)

(1?

2₎4_{+ ~}_k

m2₊24

: The solutions to this cubic equation are then given by

2mn1=? 2 3

q

K22 ?3K1 cos

3

+ K₃2

2mn2=? 2 3

q

K22 ?3K1 cos

+ 2 3

+ K₃2

2mn3=? 2 3

q

K22 ?3K1 cos

+ 4 3

+ K₃2 where

= cos?1 0 @

?27K0?2K32+ 9K1K2 2q

(K22?3K1)

3

1 A :

Finally, the natural frequencies for the shell, with units of hertz, are given by fmnj = _2Rmnj

s E

(1?2) ; j = 1;2;3 :

(3.6) We point out that due to the cubic nature of the characteristic equation, three natural fre-quencies and mode shapes are obtained for each set of wave numbersm and n. This leads to a signicant interlacing of frequencies.

A second set of modes which satisfy the boundary conditions and compatibility criteria is

U(x;) = A2cos

nx

`

sin(m) V (x;) = B2sin

nx

`

cos(m) W(x;) = C2sin

nx ` sin(m) : (3.7)

The system which arises when using this form of the modes is 2

6 6 6 4

2?H1 ?m

2 (1 +)

?

m

2 (1 +) 2?H2 m

m 2 ?H3

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which diers in sign in the (1;2);(2;1);(2;3) and (3;2) elementsfrom (3.4) which was obtained using the rst set of modal expansions. The characteristic equations, however, are the same

in both cases and are given by (3.5). Hence both sets of expressions yield the same natural frequencies for the shell. This should be expected since the second set simply represents a phase shift in of the rst. For this reason, most authors consider only the rst set of vibration modes.

In some instances, however, the second set (3.7) contributes linearly independent modes and hence must be retained in order to obtain a complete basis for approximation. This is illustrated through consideration of the cases (i) m = 0 ; n 1: axisymmetric modes, (ii) M 1; n = 0: purely extensional modes, and (iii) m1 ; n1: general shell modes.

(i)

m = 0;n1

Axisymmetric Modes

To determine the modal structure and natural frequencies for the axisymmetric case, we substitute (3.3) and (3.7) into (3.2) and determine!. Consideration of the rst set (3.3) with m = 0 yields the relations

2 6 6 6 4

2?2 0

0 0 0

0 2?(1 + ~k4) 3 7 7 7 5

2 6 6 6 4

A1

B1

C1

3 7 7 7 5=

2 6 6 6 4

0 0 0

3 7 7 7

5 : (3.8)

Nontrivial magnitudes A1 and C1 are obtained by equating the determinant to zero and

solving for 2 _{in the manner previously discussed. We point out that for each xed value of}_n,

the solution of the resulting characteristic equation yieldstwo frequencies with corresponding modes given by

U(x;) = A1cos

nx

`

V (x;) = 0 W(x;) = C1sin

nx

`

:

(3.9)

Furthermore, consideration of the set (3.7) yields thethird frequency f0n= n2`

s E

2(1 + ) (3.10)

with the corresponding modes

U(x;) = 0 V (x;) = B2sin

nx

`

W(x;) = 0 :

(3.11)

Thus, three separate frequencies and sets of modes are associated with each xed value of n (which represents the number of axial half waves). For the axisymmetric case considered

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here, the axial/radial modes are uncoupled from the circumferential modes; this diers from the cases with m 1 where all three modes are coupled. Furthermore, one must consider both sets (3.3) and (3.7) in order to account for the complete sets of axisymmetric modes. Numerical examples illustrating both the analytic solutions and the approximation of the axisymmetric frequencies and modes are given in Section 5.1.1.

(ii)

m 1;n = 0

Purely Extensional Modes

A second case in which the modal structure is easily described occurs when m 1 and n = 0. The response in this case is asymmetric with m determining the number of circumfer-ential nodal lines. It is also purely translational in the longitudinal direction as noted by the analytic modal expressions

U(x;) = A1cos(m)

V (x;) = 0

W(x;) = 0 (3.12)

and

U(x;) = A2sin(m)

V (x;) = 0

W(x;) = 0 : (3.13)

The frequencies for this case are given by fm0= mR

s E

2(1 + ) : (3.14)

Numerical examples illustrating this case are presented in Section 5.1.2.

(iii)

m1;n 1

General Shell Modes

For the general case with m 1 and n1, analytic values of the natural frequencies can be determined from (3.6) where 2mnj is one of the three solutions to the cubic equation (3.5). Numerical examples comparing the analytic values obtained in this manner with numerical approximations computed using the Fourier-spline basis are reported in Section 5.1.3.

4 Approximation Method

To approximate the solutions u;v;w to the system (2.8) or eigenvalue problem (3.2), basis functions Bu

k(;x); Bv

k(;x) and Bw

k(;x) in V are chosen and used to form expansions uN_{(t;;x) =} Nu

X

k=1uk(t)

Bu k(;x) vN_{(t;;x) =} Nv

X

k=1vk(t)

Bv k(;x) wN_{(t;;x) =} Nw

X

k=1wk(t)

Bw

k(;x) :

(4.1)

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The time-dependent generalized Fourier coecientsuk(t);vk(t);wk(t) are then determined by

orthogonalizing the residual with respect to linearly independent test functions from the span of the bases.

To exploit the tensor nature of the cylindrical shell domain and periodicity in , the bases for the three displacements are constructed with Fourier components in and spline components inx. In terms of complexFourier expansions, this yieldsthe approximatesolutions

uN_{(t;;x) =} Mu X

m=?M

u

^Nu X

n=1umn(t)e

im_B_u

n(x) vN_{(t;;x) =} Mv

X

m=?M

v

^Nv X

n=1vmn(t)e

im_B_v

n(x) wN_{(t;;x) =} Mw

X

m=?M

w

^Nw X

n=1wmn(t)e

im_B_w

n(x)

(4.2)

where Bun(x);Bvn(x) and Bwn(x) denote the spline components of the longitudinal, circum-ferential and transverse approximates, respectively. If employing these complex expansions, complex inner products must be used when deriving the weak form (2.8). This yields the conjugates of the test functions and their derivatives in (2.8).

Through the Euler identities eim = cos(m)

isin(m), the complex expansions (4.2) are equivalent to real expansions of the form

uN_{(t;;x) =} ^Nu X

n=1u0n(t)Bun(x)

+ Mu X

m=1 ^Nu X

n=1umn(t)cos(m)Bun(x) +

Mu

X

m=1 ^Nu X

n=1u~mn(t)sin(m)Bun(x)

vN_{(t;;x) =} ^Nv X

n=1v0n(t)Bvn(x)

+ Mv X

m=1 ^Nv X

n=1vmn(t)cos(m)Bvn(x) +

Mv

X

m=1 ^Nv X

n=1v~mn(t)sin(m)Bvn(x)

(4.3)

wN_{(t;;x) =} ^Nw X

n=1w0n(t)Bwn(x)

+ Mw X

m=1 ^Nw X

n=1wmn(t)cos(m)Bwn(x) +

Mw

X

m=1 ^Nw X

n=1w~mn(t)sin(m)Bwn(x) :

In both cases, the total number of basis functions isNu = ^Nu(2Mu+1);Nv = ^Nv(2Mv+1), Nw = ^Nw(2Mw + 1) where Mu;Mv;Mw are the Fourier limits and ^Nu; ^Nv; ^Nw denote the number of splines used in each expansion.

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4.1 Axial Basis Functions

The choice of splines in the axial variable is motivated by the following criteria: (i) eciency;

(ii) exibility with regard to internal and external patch contributions; (iii) adaptability with respect to various boundary conditions;

(iv) accuracy.

For the longitudinal and circumferential displacements, both linear and cubic spline bases are considered. Due to dierentiability requirements, only cubic splines are used when approxi-mating the transverse displacement.

In all cases, a uniform partition along the x-axis is considered with gridpoints xn =

nhx;hx = `=N ;n = 1;;N. For n = ?1;0;1;;N + 1, standard cubic splines are de-ned by

bn(x) = 1h_3x

8 > > > > > > > > < > > > > > > > > :

(x?xn ?2)

3 _{; x}2[xn

?2;xn?1] h3x+ 3h2x(x?xn

?1) + 3hx(x ?xn

?1)

2 ?3(x?xn

?2)

3 _{; x}2[xn

?1;xn] h3x+ 3h2x(xn+1?x) + 3hx(xn+1?x)2 ?3(xn+1?x)3 ; x2[xn;xn+1]

(xn+2?x)3 ; x2[xn+1;xn+2]

0 ; otherwise

(4.4) (see [28]). The standard linear splines are dened by

cn(x) = 1h_x

8 > > < > > :

(x?xn

?1) ; x 2[xn

?1;xn] (xn+1?x) ; x2[xn;xn+1]

0 ; otherwise ; (4.5)

for n = 0;;N.

In both cases, the standard splines must be modied to satisfy the essential boundary conditions to ensure approximate solutions in V . These modications are summarized as follows.

(i) Fixed displacements at

x = 0;`

:

To construct basis functions satisfying xed displacements (but unspecied slopes) at x = 0;`, the modied cubic splines are taken to be

^bn(x) =

8 > > > > > > > > < > > > > > > > > :

b0(x)?4b

?1(x) ; n = 0 b1(x)?b

?1(x) ; n = 1

bn(x) ; n = 2;;N ?2 bN?1(x)

?bN+1 ; n = N?1 bN(x)?4bN+1(x) ; n = N

; (4.6)

for a total of N + 1 functions. To satisfy the same condition, the rst and last linear splines are omitted from the set to yield

^

cn(x) = cn(x) ; n = 1;;N ?1 : (4.7) It should be noted that with these denitions, ^bn(0) = ^bn(`) = 0 and ^cn(0) = ^cn(`) = 0.

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(ii) Fixed displacements and slope at

x = 0;`

:

Only the cubic splines are required to satisfy a condition of xed displacement and slope at the endpoints since this is a condition imposed only on the transverse displacements. In this case, the modied splines are taken to be

~bn(x) =

8 > > < > > :

b0(x)?2b ?1(x)

?2b1(x) ; n = 1

bn(x) ; n = 2;;N ?2

bN(x)?2bN ?1(x)

?2bN+1(x) ; n = N?1

; (4.8)

for a total ofN ?1 basis functions. Note that these functions satisfy ~bn(0) = ~b0

n(0) = ~bn(`) = ~b0

n(`) = 0 :

4.2 Bases for Specic Boundary Conditions

Appropriate bases for various boundary conditions are then constructed by considering mod-ied splines satisfying any essential boundary conditions. For example, the bases used for simply-supported shells (2.7) must satisfy the conditions Bvn(0) = Bvn(`) = 0, Bwn(0) = Bwn(`) = 0 (the moment and shear conditions are natural and hence do not need to be explic-itly enforced). Choices for shell models with xed-edge conditions are summarized in Table 1 while corresponding choices for simply-supported edge conditions are given in Table 2. The last column in each table summarizes the total number of axial functions in each expansion.

Once the axial bases are chosen, they are combined with the Fourier components in to form the total bases employed in (4.2) or (4.3). We reiterate that the basis limits in each case are given byNu = ^Nu(2Mu + 1);Nv = ^Nv (2Mv + 1) and Nw = ^Nw(2Mw+ 1):

We point out that the linear and cubic spline bases described here are but two choices from among many that can be made for the axial components. For the applications considered here, cubic splines provided a good balance between accuracy, eciency and adaptability with re-gards to patches and boundary conditions. If higher accuracy is desired, however (with slightly more expense when constructing system matrices), quintic splines can be employed. Similarly, spectral expansions employing Legendre, Chebyshev or sinc functions can be employed once modications have been made for boundary conditions.

Shell Component Axial Basis Functions Component Denition Axial Limit longitudinal { linear Bun(x) = ^cn(x) ^cn(x) dened in (4.7) ^Nu =Nu

?1 cubic Bun(x) = ^bn(x) ^bn(x) dened in (4.6) ^Nu =Nu+ 1 circumferential { linear Bvn(x) = ^cn(x) ^cn(x) dened in (4.7) ^Nv =Nv

?1 cubic Bvn(x) = ^bn(x) ^bn(x) dened in (4.6) ^Nv =Nv+ 1 transverse { cubic Bwn(x) = ~bn(x) ~bn(x) dened in (4.8) ^Nw =Nw

?1

Table 1.

Axial basis denitions for models with xed-edge boundary conditions (2.6).

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Shell Component Axial Basis Functions Component Denition Axial Limit longitudinal { linear Bun(x) = cn(x) cn(x) dened in (4.5) ^Nu =Nu+ 1 cubic Bun(x) = bn(x) bn(x) dened in (4.4) ^Nu =Nu+ 3 circumferential { linear Bvn(x) = ^cn(x) ^cn(x) dened in (4.7) ^Nv =Nv

?1 cubic Bvn(x) = ^bn(x) ^bn(x) dened in (4.6) ^Nv =Nv+ 1 transverse { cubic Bwn(x) = ^bn(x) ^bn(x) dened in (4.6) ^Nw =Nw+ 1

Table 2.

Axial basis denitions for models with simply-supported boundary conditions (2.7).

4.3 Components in the Finite Dimensional System

With bases thus dened, approximating subspaces are taken to beHNu = spanfBu k

g Nu

k=1;HNv =

spanfBv k

g N

v

k=1 andHNw = spanfBw k

g N

w

k=1. It should be noted thatHN =HNuHNvHNw V . The approximating system is then determined by restricting the weak form (2.8) to HN _with

basis functions used as test functions. This is equivalent to orthogonalizing the residual with respect to elements from HN_.

To illustrate, submatrices for the longitudinal, circumferential and transverse components withconstantmaterial properties are summarizedbelow. The matrices which arise when inter-nal patch contributions are incorporated are similar and details concerning their construction are left to the reader.

4.3.1 Approximate Longitudinal Displacement

The substitution of the force and moment resultants (2.2) into the rst equation of the weak form (2.8) and subsequent approximation gives

Z 2

0

Z `

0 h@

2_uN

@t2 Bu

jRdxd + Z 2

0

Z `

0 (1Eh?2)

(" @uN

@x + R @v@ + wN N !#

@Bu j @x + 12R(1?)

" @vN

@x + R1 @u@N #

@Bu j @

)

Rdxd

+Z 2

0

Z `

0 (1c?Dh2)

@ @t

(" @uN

@x + R @v@ + wN N !#

@Bu j @x + 12R(1?)

" @vN

@x + R1 @u@N #

@Bu j @

)

Rdxd

=Z 2

0

Z `

0

( R^qxBu

j +R

s

X

i=1(Nx)pei

@Bu j @x

) dxd for j = 1;;Nu .

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Consolidation yields the following mass and stiness submatrices:

(i) [UM]j;k= Z 2

0

Z `

0 hRBu

k Bu

jdxd

(ii) [U11]j;k = Z 2

0

Z `

0 1EhR?2

@Bu k

@x @

Bu j

@x dxd

(iii) [V11]j;k = Z 2

0

Z `

0 1Eh?2

@Bv k

@ @

Bu j

@x dxd

(iv) [W11]j;k = Z 2

0

Z `

0 1Eh?2

Bw k

@Bu j

@x dxd

(v) [V12]j;k = Z 2

0

Z `

0 2(1 +Eh)@

Bv k

@x @Bu j

@ dxd

(vi) [U12]j;k = Z 2

0

Z `

0 2R(1 +Eh )@

Bu k

@ @

Bu j

@ dxd

(vii) [Fu]j = Z 2

0

Z `

0

(

Rq^xBu j +R

s

X

i=1(Nx)pei

@Bu j

@x

)

dxd :

In a completely analogous manner, the matrices ~U11; ~V11; ~W11; ~V12 and ~U12 containing the

internal damping contributions can be formed by replacing E by cD in the denitions

(ii)

through

(vi)

above. Expressions containing the internal patch contributions are obtained in an analogous manner.

We point out that the orthogonality properties of the generalized Fourier/cubic spline bases give rise to matrices which are block diagonal with the block matrices composed of symmetric matrices.

4.3.2 Approximate Circumferential Displacement

Similarly, substitution of force and moment resultants and approximation of the second equa-tion of the weak form (2.8) yields the system

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Z 2

0

Z `

0 h@

2_vN

@t2 Bv

jRdxd + Z 2

0

Z `

0 (1Eh?2)

( 1 R

" 1

R@v@ +N wR + N @u@xN #

@Bv j @ +12(1?)

" @vN

@x + R1 @u@N #

@Bv j @x

)

Rdxd

+Z 2

0

Z `

0 (1c?Dh2)

@ @t ( 1 R " 1

@Bv j @ +12(1?)

" @vN

@x + R1 @u@N #

@Bv j @x

)

Rdxd

=Z 2

0

Z `

0

( R^qBv

j +

s

X

i=1(N)pei

@Bv j @

)

dxd :

From this equation, the mass and stiness components of the circumferential displacement are determined as follows:

(i) [VM]j;k = Z 2

0

Z `

0 hRBv

k Bv

jdxd

(ii) [V21]j;k = Z 2

0

Z `

0 R(1Eh?2)

@Bv k

@ @

Bv j

@ dxd

(iii) [W21]j;k = Z 2

0

Z `

0 R(1Eh?2)

Bw k

@Bv j

@ dxd

(iv) [U21]j;k = Z 2

0

Z `

0 1Eh?2

@Bu k

@x @

Bv j

@ dxd

(v) [V22]j;k = Z 2

0

Z `

0 2(1 +EhR)@

Bv k @x @ Bv j @x dxd

(vi) [U22]j;k = Z 2

0

Z `

0 2(1 +Eh)@

Bu k @ @ Bv j @x dxd

(vii) [Fv]j = Z 2

0

Z `

0

(

Rq^Bv j+

s

X

i=1(N)pei

@Bv j

@

)

dxd .

Similarly, the damping matrices ~V21; ~W21; ~U21; ~V22 and ~U21 can be formed by replacing the

coecientE by cD.

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4.3.3 Approximate Transverse Displacement

The substitution of (2.2) into the third equation of the weak form (2.8) gives

Z 2

0

Z `

0 h@

2_wN

@t2 Bw

jRdxd +Z 2

0

Z `

0 (1Eh?2)

( 1 R

" 1

Bw j+ h

2

12 "

@2_wN

@x2 + R2@

2_wN

@2

# @2Bw

j @x2

+ h_12R2₂ "

1 R2@

2_wN

@2 +@

2_wN

@x2

# @2

Bw j @2 + h

2

6R2(1?)@

2_wN

@x@@ 2 Bw j @x@ ) Rdxd

+Z 2

0

Z `

0 (1c?Dh2)

@ @t ( 1 R " 1

Bw j+ h

2

12 "

@2_wN

@x2 + R2@

2_wN

@2 # @2 Bw j @x2

+ h_12R2₂ "

1 R2@

2_wN

@2 +@

2_wN

@x2

# @2

Bw j @2 + h

2

6R2(1?)@

2_wN

@x@@ 2 Bw j @x@ ) Rdxd

=Z 2

0

Z `

0

( R^qnBw

j ? s X i=1 "

R(Mx)pei @2

Bw j @x2 ?

1

R(M)pei @2 Bw j @2 #) dxd :

The system components for the equation describing the transverse displacement can be de-scribed in the following manner:

(i) [WM]j;k = Z 2

0

Z `

0 hRBw

k Bw

jdxd

(ii) [V31]j;k = Z 2

0

Z `

0 R(1Eh?2)

@Bv k

@ Bw jdxd

(iii) [W31]j;k = Z 2

0

Z `

0 R(1Eh?2)

Bw k

Bw jdxd

(iv) [U31]j;k = Z 2

0

Z `

0 1Eh?2

@Bu k

@x Bw jdxd

(v) [W32]j;k = Z 2

0

Z `

0 Eh

3_R

12(1?2)

@2Bw k

@x2 @

2Bw

j

@x2 dxd

(vi) [W33]j;k = Z 2

0

Z `

0 Eh

3

12R(1?2)

@2Bw k

@2 @

2Bw

j

@x2 dxd

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(vii) [W34]j;k = Z 2 0 Z ` 0 Eh 3

12R3₍₁?2)

@2Bw k

@2 @

2Bw

j

@2 dxd

(viii) [W35]j;k= Z 2

0

Z `

0 Eh

3

12R(1?2)

@2Bw k

@x2 @

2Bw

j

@2 dxd

(ix) [W36]j;k = Z 2

0

Z `

0 Eh

3

6R(1 +)@

2Bw

k

@x@ @

2Bw

j

@x@ dxd

(x) [Fw]j = Z 2

0

Z `

0

(

Rq^nBw j ? s X i=1 "

R(Mx)pei

@2Bw j

@x2 + 1_R(M)pei

@2Bw j

@2

#)

dxd .

Again, the internal damping matrices ~W31; ~W32; ~W33; ~W34; ~W35; ~W36; ~U31 and ~V31 are obtained

by replacingE by cD.

4.4 Matrix System

With submatrices thus dened, we now form the complete matrix system. The generalized Fourier coecients for the three expansions (4.1) are consolidated in the vectors

U

Nu(t) = 2 6 6 4

u1(t)

... uN u(t) 3 7 7 5; V

Nv(t) = 2 6 6 4

v1(t)

... vN v(t) 3 7 7 5; W

Nw(t) = 2 6 6 4

w1(t)

... wN w(t) 3 7 7

5 : (4.9) The full set of coecients is then represented as #N(t) = [

U(t) Nu;

V Nv(t);

W

Nw(t)]T, where N =Nu+Nv +Nw.

The mass, stiness and damping matrices as well as the forcing vector for the full system are then given by

MN = 2 6 6 6 6 4 UM VM WM 3 7 7 7 7 5 ; KN E = 2 6 6 6 6 4

U11+U12 V11+V12 W11

U21+U22 V21+V22 W21

U31 V31 P

6k=1W3k

3 7 7 7 7 5 ; KN

cD =

2 6 6 6 6 4

~U11+ ~U12 ~V11+ ~V12 ~W11

~U21+ ~U22 ~V21+ ~V22 ~W21

~U31 ~V31 P

6k=1 ~W3k

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and

^FN(t) = 2 6 6 6 4

Fu

Fv

Fw

3 7 7 7

5 (4.11)

with components dened previously.

In rst-order form, this yields the system 2

4 KN

E 0

0 MN

3 5

2 4

_#N(t) #N(t)

3 5=

2 4

0 KN

E

?K N

E ?K

N

cD 3 5

2 4

#N(t) _#N(t)

3 5+

2 4

0 ^FN(t)

3 5

2 4

KN

E 0

0 MN

3 5

2 4

#N(0) _#N(0)

3 5=

2 4

gN

1

gN

2

3 5 :

(4.12)

Multiplication of the inverted mass matrix yields a Cauchy equation of the form _

yN_{(t) = A}N_yN_{(t) + g}N_(t)

yN_{(0) =} _y_N0 _; (4.13)

where yN 2 R

2N =

h

#N(t); _#N(t) iT

. This yields a form which is suitable for simulations, parameter estimation and control applications.

The discussion above illustrates the construction of the matrices in the context of constant material parameters. This was done solely to simplify the presentation. The case of piecewise constant parameters, which arises when considering many smart material systems, is handled analogously with the domains of integration specied by the characteristic functions. This exibility with regards to nonhomogeneities is one of the strengths of the method.

The system can also be adapted to alternative boundary conditions through modications of the rst and last basis functions. Flexibility in this regard is also a hallmark of the method.

4.5 Approximation of Natural Frequencies and Modes

Approximation of the eigenvalue problem (3.2) using the Fourier-Galerkin method yields the generalized matrix eigenvalue problem

KN

E#N =!2MN#N ; (4.14)

whereKN

E andMN are dened in (4.10). This can be solved using the

matlabcommandeig

as illustrated in the following section of code

[V,D] = eig(stiffness,mass); d = diag(D);

[lam,I] = sort(sqrt(d)); SV = V(:,I);

freq = (1/(2*pi))*lam; disp(freq(1:Nf)) .

In this code, Nfrefers to the number of displayed frequencies. Modes can be constructed by

multiplying the coecients inV by appropriate basis elements. In subsequent discussion, the

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5 Examples

To illustrate various facets of the Fourier-Galerkin method, two types of examples are pre-sented. In the rst set of examples, approximation of frequencies and modes is considered for simply-supported and xed-edge shells. This is accomplished by forming the mass and stiness matrices matrices and solving the generalized matrix eigenvalue problem (4.14). This serves two purposes. It illustrates the accuracy of the method in approximating the mass and stiness components in the system and demonstrates the method as an eective technique for obtaining modal information when analytic or experimental values are unavailable.

The second set of examples illustrates the approximation of static and dynamic shell responses to various forces. The initial steady state example illustrates that the expected

O` (h2x) and O`(h4x) convergence rates are obtained when approximating a static shell response (this example involves the solution of a matrix system with coecient matrixKN

E and vector

^FN). Analogous dynamic examples demonstrate that the same accuracy is obtained when the method is used to obtain the time-dependent system (4.13) which is then marched in time. The latter example also incorporates the internal Kelvin-Voigt damping. The dynamics of the system in response to patch inputs are considered in the nal example. Natural frequencies are compared to those obtained by solving the generalized matrix eigenvalue problem to illustrate consistency among techniques.

In these examples, the following shell characteristics were used: thickness h = :01 in, length` = 12 in, radius R = 3:0 in, mass density = :283 lb=in3_{, Young's modulus}_{E = 3:0} 107 _lb=in2_{, Poisson ratio}_{= 0:3 and Kelvin-Voigt damping coecient c}_D _{= 15:09936 lb}ins2. These values were chosen so as to permit comparison with Donnell-Mushtari shell results in [23, pp. 307-310]. We emphasize that the method is not restricted to such dimensions or ratios, however, and analogous convergence results have been obtained with a wide range of parameters.

In all examples, numerical approximation of the integrals was performed using a four point Gaussian quadrature rule of the form

Z `

0 f(x)dx

4Nq

X

k=1ckf(xk) :

HereNqwas taken to be 16 which impliesthat 64 quadrature points were used. The quadrature

weights ck and points xk, on the interval (0;hq), are given by

c1 = _{6(18 +}49p 30)

hq

2 ; x1 =hq 2 41

2 ? q

15 + 2p 30 2p

35 3 5

c2 = ₆₍₁₈49

? p

30) hq

2 ; x2 =hq 2 41

2 ? q

15?2 p

30 2p

35 3 5

c3 = ₆₍₁₈49

? p

30) hq

2 ; x3 =hq 2 41

2 + q

15?2 p

30 2p

35 3 5

c4 = _{6(18 +}49p 30)

hq

2 ; x4 =hq 2 41

2 + q

15 + 2p 30 2p

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where hq =`=Nq = 12=16.

In the following examples, three approximation limits in the axial direction were used, namely Nu = Nv = Nw = 4, Nu = Nv = Nw = 8, and Nu = Nv = Nw = 16. The

doubling in the axial discretization limitfacilitated numericalvalidation of the expectedO` (h2x) convergence rate when using linear splines and O` (h4x) rate when cubic splines were employed. The choice of Fourier limitsMu;Mv andMw in each example was determined by the nature of

the forcing function and the desired accuracy. Finally,Nx= 25 evaluation points in the axial

direction and N = 25 circumferential points were used when plotting true and approximate

solutions and evaluating errors.

5.1 Modal Examples: Simply-Supported Shell

In Section 3, separation of variables was used to obtain analytic expressions for modes and natural frequencies of shell models with simply-supportedboundary conditions. Here we com-pare the approximate solutions obtained by solving the matrix eigenvalue problem (4.14) with the analytic values. This provides a means of testing the accuracy of the method before using it to approximate in settings in which analytic solutions are unavailable. As in Section 3, we consider the cases of purely axisymmetric modes (m = 0;n 1), purely extensional modes (m1;n = 0) and general modes (m1;n1).

5.1.1

m = 0;n1

Axisymmetric Modes

Analytic frequencies for the axisymmetric case can be calculated via (3.8) or (3.10) while corresponding modes can be determined from (3.9) or (3.11). Corresponding approximate frequencies, obtained by solving (4.14) with Fourier limitM = 0, are compared with analytic values in Table 3. It was noted in Section 3 that three frequencies are obtained for each conguration of 1=2 wavelength (xed values of n), and this is reected in the analytic and approximate results in Table 3. Two of the frequencies correspond to coupled axial/radial modes while the remaining one corresponds to an uncoupled torsional (circumferential) mode. The frequencies of the torsional modes are boxed in Table 3 to facilitate comparison between the corresponding functional modal approximations.

The cubic spline approximates in Table 3 were obtained using N = 8 basis functions whereas N = 32 linear splines were used to obtain the corresponding results in columns 8-10. It is noted that accurate frequency approximates are obtained with the cubic splines with a maximum relative error (over the reported results) of 0:3% occurring for the frequency 2255:52 Hz. Due to the limitedaccuracy of the linear splines, signicantly more basis functions must be used to obtain accurate approximations, and a 1:25% error at 2255:52 Hz remains, even with N = 32 functions.

The quadratic rate of convergence of the linear spline approximates is illustrated by the results in Table 4. The ratios obtained by dividing the relative errors by the previous relative errors are almost four. Since the number of basis functions is doubled at each step, this indicates that the method is attaining the expected O` (h2x) convergence rate. On the other hand, the cubic spline approximates are converging at anO`(h4x) rate which leads to the highly accurate frequency approximations obtained with onlyN = 8 basis functions.

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Cross sections at = 0 of corresponding modes approximated using M = 0;N = 16 cubic splines are plotted in Figure 2. For n = 1, the coupling between the axial/radial components is readily veried for the 406:8 Hz and 603:83 Hz modes while the plot of the 266:05 Hz mode illustrates that for M = 0, the torsional mode is uncoupled (all three modes are coupled forM 1). Similarresults are illustrated forn = 2;;5 in the remaining plots of Figure 2. It should be noted that for eachn, all three frequencies and modes are automatically yielded by the approximation method whereas (3.9) and (3.11) must be employed to obtain analytic expressions for the axial/radial and torsional modes, respectively. In this aspect, the approximation method facilitates the calculation of a complete modal set.

n Analytic Frequencies Cubic Spline Approx Linear Spline Approx 1 266.05 406:80 603:83 266.06 406:80 603:83 266.19 406:99 603:87 2 531:53 532.11 924:29 531:53 532.11 924:29 531:71 533.18 925:95 3 541:03 798.16 1362:11 541:03 798.22 1362:19 541:25 801.79 1368:07 4 543:53 1064.22 1807:86 543:58 1064.82 1808:81 543:85 1072.81 1822:21 5 544:60 1330.27 2255:52 544:85 1333.95 2262:44 545:06 1347.10 2283:78

Table 3.

Analytic frequencies and approximate values obtained using N = 8 cubic splines

and N = 32 linear splines. The frequencies of the torsional modes are boxed.

Linear Spline Approx Approx Freq 267.77 409.40 604.24 N=8 Rel. Error .646-2 .639-2 .679-3 Approx Freq 266.48 407.45 603.93 N=16 Rel. Error .161-2 .160-2 .166-3

Rel: Err: N=16_{Rel: Err: N=8} _{4.012 3.993 3.994}

Approx Freq 266.19 406.99 603.87 N=32 Rel. Error .526-3 .467-3 .662-4

Rel: Err: N=32

Rel: Err: N=16 4.020 4.010 4.106

Table 4.

Linear spline approximates of the axisymmetric266:05;406:80, 603:83 Hz frequencies

which occur with n = 1. The observation that the ratios obtained by dividing the relative relative error by the previous error are approximately four illustrates the O` (h2x) convergence of the method.

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0 2 4 6 8 10 12 −1

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2.5 −2 −1.5 −1 −0.5 0

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

266.05 Hz

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X

406.8 Hz

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

X

603.83 Hz

UN(x;0) VN(x;0) WN(x;0)

Figure 2a.

Approximate axisymmetric modes UN(x;0);VN(x;0) and WN(x;0)

correspond-ing to the 266:05, 406:80 and 603:83 Hz natural frequencies.

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0 2 4 6 8 10 12 −0.6

−0.4 −0.2 0 0.2 0.4 0.6

X −10 2 4 6 8 10 12

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

531.53 Hz

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

532.11 Hz

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

X

924.29 Hz

UN(x;0) VN(x;0) WN(x;0)

Figure 2b.

Approximate axisymmetric modesUN(x;0);VN(x;0) and WN(x;0)

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0 2 4 6 8 10 12 −0.3

−0.2 −0.1 0 0.1 0.2 0.3

X −10 2 4 6 8 10 12

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

541.03 Hz

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

798.16 Hz

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

X

1362.11 Hz

UN(x;0) VN(x;0) WN(x;0)

Figure 2c.

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0 2 4 6 8 10 12 −0.2

−0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

543.53 Hz

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

1064.22 Hz

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

X

1807.86 Hz

UN(x;0) VN(x;0) WN(x;0)

Figure 2d.

Approximate axisymmetric modesUN(x;0);VN(x;0) and WN(x;0)

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0 2 4 6 8 10 12 −0.15

−0.1 −0.05 0 0.05 0.1 0.15

X −10 2 4 6 8 10 12

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

544.60 Hz

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

1330.27 Hz

0 2 4 6 8 10 12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

X

0 2 4 6 8 10 12

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 12

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

X

2255.52 Hz

UN(x;0) VN(x;0) WN(x;0)

Figure 2e.

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5.1.2

m1;n = 0

Purely Extensional Modes

A second case in which analytic expressions for the shell frequencies and modes are easily expressed occurs when purely extensional modes are present. The rst three analytic frequen-cies given by (3.14) are compared in Table 5 with approximate values obtained with N = 8 cubic splines. Due to the accuracy of the method, the approximate frequencies agree with analytic values to within at least two decimal places in all three cases. Corresponding modal plots are given in Figure 3. The numerical plots in the left column illustrate the two approx-imate solutions, for each frequency, obtained by solving the matrix eigenvalue problem (4.14) (again, the full solution set is automatically obtained by the approximation method). The corresponding analytic modes given by (3.12) and (3.13) are depicted in the right column of the gure. The accurate approximation of both frequencies and modes with N = 8 splines further adds to the eciency of the method.

Analtyic Galerkin m = 1;n = 0 338.75 338.75 m = 2;n = 0 677.50 677.50 m = 3;n = 0 1016.25 1016.25

Table 5.

Analytic and approximate frequencies for purely extensional modes obtained using

N = 8 cubic splines.

5.1.3

m1;n 1

General Shell Modes

For the general case with m 1;n 1, the axial, radial and torsional modes are all coupled with analytic values for the natural frequencies determined by (3.6) where 2mnj is one of the three solutions to the cubic equation (3.5). These analytic values are compared in Table 6 with approximate frequencies obtained using the cubic spline basis with N = 8;M = 3. For comparison sake, frequencies of the axisymmetric (m = 0) and purely extensional (n = 0) are also included in this table. The accuracy of the cubic spline discretization leads to approximate frequencies having relative errors less than 0:5% for the reported values (the largest relative error for the reported values occurs in the second frequency obtained withm = 3;n = 5). The convergence of the method and accuracy obtained using the cubic splines is further illustrated by the frequency results in Table 7 where approximates obtained with N = 8 and N = 16 are compared. A check of the relative errors shows that the method is converging more quickly than the expectedO` (h4x) rate.

The modes for this general case are obtained from (3.3) and (3.7). The coupling between the axial, radial and torsional modes can be seen in Figure 4 where the 147:09;508:59;873:61 Hz modes are plotted. In this gure, the modal displacement along the axial line ` = f(x;)j = =4;0x12g is depicted.

We reiterate that the full set of frequencies and modes can be approximated using the Fourier-Galerkin method by simply solving the matrix eigenvalue problem (4.14). Moreover, this techniquecan be directly extended to the problem of calculating frequencies and modes for shells with other boundary conditions by suitably modifying the spline basis. This is illustrated in the next section where the modal analysis of a shell with clamped ends is considered.

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2Π

θ 2Π

0 0 12

θ

-1.5 1.5 -1.5

1.5

X 338.75 Hz

0

0 θ 0

1.5

-1.5 1.5

-1.5

2Π 0

θ 2Π

X 677.50 Hz

12

-1.5 -1.5

1.5 1.5

0 2Π 0 X 12

1016.25 Hz

0

θ 2Π

θ

Figure 3.

Purely extensional modes in the longitudinal direction. For N = 8 cubic splines,

numerical values ofUN along the curve` =

f(x;)j02;x=0gare plotted in left column. Corresponding shell modes given by (3.12) and (3.13) are depicted in right column.

Spline approximation of thin shell dynamics - numerical examples