Non-linear dynamics of curved beams. Part 1: Formulation

Non-linear dynamics of curved beams. Part 1: Formulation

International Journal of Non-Linear Mechanics 44 (2009) 623 -- 629 Contents lists available at ScienceDirect International Journal of Non-Linear Mec...

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International Journal of Non-Linear Mechanics 44 (2009) 623 -- 629

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / n l m

Non-linear dynamics of curved beams. Part 1: Formulation D. Zulli, R. Alaggio, F. Benedettini ∗ DISAT, University of L'Aquila, 67040 Monteluco di Roio (AQ), Italy

A R T I C L E

I N F O

Article history: Received 7 June 2007 Received in revised form 3 February 2009 Accepted 11 February 2009

Keywords: Beams Non-linear dynamics Initial curvature

A B S T R A C T

The non-linear dynamics of elastic beams with uniform initial curvature and double-symmetric crosssection are considered in this work. In particular, the work is divided into two parts. In Part 1, the interest is oriented to the formulation of an accurate model, able to describe the finite dynamics of initially curved beams as to obtain a parameterization of the initial configuration and the weak expression of the equations of motion. To this end, an explicit description of the deformation field and inertia terms is presented. The equations of motion can be used, with slight modifications, for extensible and inextensible, or shear-deformable and shear-indeformable, beams. A description of the free dynamics, of the possible classes of motion under a sinusoidally varying shear tip force, and of bifurcation phenomena is presented in Part 2 for a case-study, together with the results of experimental tests on an aluminum prototype. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The analysis of finite dynamics, even for simple structural systems, is of great interest in different engineering fields. Often a rich and varied response is due to the presence of non-linear terms. In the case of beams, arches and cables, a key role is played by the initial configuration (null, small, or large initial curvature) and by the simplifying assumptions (e.g., internal constraints). Works devoted to model formulation and to non-linear dynamic analysis of beams are widely present in literature. Non-linear dynamics of straight beams, subjected to time-periodic forces, were studied in [1], where particular attention was given to flexural–torsional dynamic coupling in the case of thin-walled beams. In [2–5], the formulation of approximate models of Euler–Bernoulli beams, considering or neglecting extensional deformation, was presented. It was proved that, when one end of the beam is free to move, the beam behaves essentially as inextensional. In [6], an accurate formulation of the non-linear model of an elastic beam, as descending from a corresponding three-dimensional continuum, was presented. In [7], the steady-state solutions of reduced-order models for planar beams, having different constraint conditions, were analyzed via a perturbation approach. In [8], the formulation of non-linear models and various stability and post-critical analyses were provided for onedimensional bodies using Cosserat rod theory. In [9], the sensitivity to imperfections for thin-walled beams, where warping is considered, were studied in the case of buckling. In [10], a non-linear model of a thin-walled, non-symmetric open-section beam were

∗ Corresponding author. Tel.: +39 862 434513; fax: +39 862 434548. E-mail address: [email protected] (F. Benedettini). 0020-7462/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2009.02.014

studied including both non-linear warping and torsional elongation effects. It was also shown that, in beams with double-symmetric sections, these two last contributions are negligible. In [11], the exact non-linear equations for the statics of beams were obtained and numerical computations were performed considering various configurations. In [12], the equations describing the kinematics and statics of initially curved and twisted beams, in case of very large rotations and displacements, were formulated and then solved using multiple shooting method. More recently, in [13], the stability of thin-walled cantilever beams under static non-conservative forces was studied, using the non-linear Cosserat rod model. This work is composed of two parts. In Part 1, an accurate geometrically non-linear one-dimensional model of initially curved beams is formulated. A double-symmetric, uniform, cross-section is considered. The initial curvature terms, assumed as uniform, are considered as explicit parameters characterizing the initial configuration. The deformation field, the inertia terms, and then the equations of motion are explicitly derived as function of such parameters, without considering the contributions due to the warping [10]. These equations describe, with slight modifications, extensional and inextensional, as well as shear-deformable and shearindeformable, beams. In particular, the model is developed for an extensional, shear-deformable, and rigid-section beam. Then, internal constraints for inextensible and shear-indeformable beam are added directly in weak form, considering the corresponding stress terms as reactive forces. As an example, a cantilever with distributed and tip mass, and time-periodic shear tip force, is considered. In Part 2, a description of the results obtained by using the model in a case-study is presented. The free dynamic problem is solved for different initial configurations, in order to discuss the variation

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a¯ 3 (1 )} are bases for V. In analogy with Eq. (1), the parameterization ¯ is for R

a2

¯ 1 , 2 , 3 ) = x¯ 0 (1 ) + 2 a¯ 2 (1 ) + 3 a¯ 3 (1 ) x(

a1

(2)

It can be assumed that the sections, in the reference configuration, are orthogonal to the tangent vector of the axis curve, i.e.,

x0

x¯ 0 = a¯ 1

a3 Fig. 1. Geometry of the beam in a generic configuration: (a) the cylinder, spanned by the axis curve and the cross-sections; (b) the axis curve and the directors.

(3)

where ( ) stands for differentiation with respect to 1 . 2.2. Initial curvature

of the spectral properties as the initial configuration is varied, enlightening on the possible internal resonance conditions as well as possible veering phenomena among different couples of adjacent modes. The non-linear forced problem is then studied, after use of a Galerkin projection of the continuous problem on a reduced basis. Direct integrations and path-following procedures in the excitation control-parameters plane are applied to the discretized equations. An analysis of the flexural–torsional dynamic instability phenomenon, of bifurcation boundaries, and of post-critical classes of motion for different initial configurations are considered as well. The results are compared with experimental observations on a straight beam characterized by geometrical imperfections on the initial configuration, in order to evaluate if imperfections, however, present in experimental prototypes and in real full-scale cases, could justify the use of a model with some initial curvature to obtain better agreement between analytical results and experimental evidence.

¯  )a¯ i ( ), a¯ i (1 ) = K( 1 1

i = 1, 2, 3

(4)

Its representation with respect to the basis {a¯ i } is ⎡

0 −k¯ 3 (1 ) ⎢ ¯ ¯ = ⎢ k¯ 3 ( ) [K] 0 ai 1 ⎣ ¯ ¯ −k2 (1 ) k1 (1 )

⎤ k¯ 2 (1 ) ⎥ −k¯ 1 (1 ) ⎥ ⎦

(5)

0

In the following, the initial curvature is assumed to be uniform, i.e., k¯ i do not depend on 1 , for i = 1, 2, 3. This provides a limitation to the possible classes of initial shapes; anyhow the obtained set of possible shapes permits one to study several interesting cases. Under such hypothesis, Eq. (4) furnishes a¯ 1 (1 ) = k¯ 3 a¯ 2 (1 ) − k¯ 2 a¯ 3 (1 ) a¯ 2 (1 ) = −k¯ 3 a¯ 1 (1 ) + k¯ 1 a¯ 3 (1 )

2. The model

a¯ 3 (1 ) = k¯ 2 a¯ 1 (1 ) − k¯ 1 a¯ 2 (1 )

2.1. Geometry and configurations The shape of the beam is a cylinder, occupying the space spanned by the axis line and the cross-sections (Fig. 1a). The axis line is a curve in the three-dimensional Euclidean configuration space E. Let I be a closed set of R, and 1 ∈ I is chosen as a length abscissa for the axis curve, whose position, in the current configuration, is x0 (1 ). The time-dependent positions of the points of the curve are oriented, i.e., in correspondence to every position x0 (1 ), three orthonormal vectors {a1 (1 ), a2 (1 ), a3 (1 )}, called directors, are considered (Fig. 1b). They belong to the vector translation space V of E. These vectors describe the attitude of the rigid plane sections of the beam, which are images of a closed set S of R2 . The vectors a2 and a3 are supposed to lay on the sections; therefore, if (2 , 3 ) ∈ S, the position x of a point of the beam in the current configuration can be written as x(1 , 2 , 3 ) = x0 (1 ) + 2 a2 (1 ) + 3 a3 (1 )

The orientation of the directors in the reference configuration is ¯  ), assumed provided by the skew tensor of the initial curvature K( 1 as known and defined as

(1)

(6)

which is a linear ordinary differential equation system with constant coefficients. The necessary initial conditions read a¯ 1 (0) = e1 a¯ 2 (0) = e2 a¯ 3 (0) = e3

(7)

asserting, without loss of generality, that the initial section lays on a plane parallel to the natural base {e1 , e2 , e3 } of V. The initial value problem (6)–(7) can be solved to obtain the expressions of a¯ i (see Appendix A). To get information on the axis curve in the reference configuration, Eq. (3) can be integrated, considering x¯ 0 (0) = 0, i.e., the axis curve starts from the zero position of E (Appendix A). In ¯ (Eq. (2)) is completely written this way, the parameterization of R in terms of the three initial curvature components k¯ i (i = 1, 2, 3). 2.3. Deformation A bijective and smooth transformation

Eq. (1) represents a parameterization of the shape of the beam, denoted as R, a subset of E, in the current configuration. In particular, the parameters that identify the position of a point in R are {1 , 2 , 3 }. A particular configuration, corresponding to the initial time t¯, is ¯, chosen as reference. In this configuration, the shape is denoted by R the axis curve by x¯ 0 (1 ), and the directors by {a¯ 1 (1 ), a¯ 2 (1 ), a¯ 3 (1 )}. The overbar indicates time-independent terms related to the reference configuration. Both {a1 (1 ), a2 (1 ), a3 (1 )} and {a¯ 1 (1 ), a¯ 2 (1 ),

¯ →R :R

(8)

is assumed to deform the initial shape of the beam. In particular this transformation allows one to express the current position as a function of the initial one ¯ 1 , 2 , 3 )) x(1 , 2 , 3 ) = (x( x0 (1 ) = (x¯ 0 (1 ))

(9)

D. Zulli et al. / International Journal of Non-Linear Mechanics 44 (2009) 623 -- 629

To take into account the rigidity of the section, the directors {a¯ 1 , a¯ 2 , a¯ 3 } can only perform a rotation. If R is a rotation of V, the evolution of the directors, from the reference configuration, is described by the following relation: ai (1 ) = R(1 )a¯ i (1 ),

i = 1, 2, 3

(10)

Hence, Eq. (1) becomes (omitting the independent variables) x = x0 + 2 Ra¯ 2 + 3 Ra¯ 3

(11)

The rotation R can be written in terms of three functions {1 (1 ), 2 (1 ), 3 (1 )} describing the finite angles carrying a¯ i (1 ) on ai (1 ). A possible expression1 for R with respect to a¯ i is ⎡

cos 2 cos 3 [R] = ⎣ cos 2 sin 3 − sin 2

sin 1 sin 2 cos 3 − cos 1 sin 3 cos 1 cos 3 + sin 1 sin 2 sin 3 sin 1 cos 2

¯ 1 + h, 2 , 3 ) c¯ 1 (h) := x( ¯ 1 , 2 + h, 3 ), c¯ 2 (h) := x(

and substituting Eqs. (14) and (16) in Eq. (17), one obtains Fa¯ 1 =

1



[x0 + R (2 a¯ 2 + 3 a¯ 3 ) + (−2 k¯ 3 + 3 k¯ 2 )Ra¯ 1 ]

Fa¯ 2 = Ra¯ 2 Fa¯ 3 = Ra¯ 3

(18)

where ¯ := 1 − 2 k¯ 3 + 3 k¯ 2 . Now the objective is to evaluate the determinant of F. It represents the current volume of a parallelepiped having unit volume in the initial configuration. To this aim, it is necessary to write in a different form Eq. (18)1 ; the comparison of Eqs. (16)1 with (18)1 , leads one to the following equation: 1 (19) Fa¯ 1 = [x − (−3 k¯ 1 Ra¯ 2 + 2 k¯ 1 Ra¯ 3 )] ¯

⎤ cos 1 sin 2 cos 3 + sin 1 sin 3 cos 1 sin 2 sin 3 − sin 1 cos 3 ⎦ cos 1 cos 2

It is useful to describe the deformation of the beam by means of its gradient. This is defined as the tensor transforming the tangent ¯ 1 , 2 , 3 ) to the tangent vectors of the curves passing through x( vectors of the corresponding curves passing through x(1 , 2 , 3 ). In this way, choosing three curves passing through the position ¯ 1 , 2 , 3 ) as x(

¯ 1 , 2 , 3 + h) c¯ 3 (h) := x(

(13)

and using Eqs. (2), (3) and (6), the tangent vectors in h = 0 are  dc¯ 1  = x¯  = a¯ 1 + 2 (−k¯ 3 a¯ 1 + k¯ 1 a¯ 3 ) + 3 (k¯ 2 a¯ 1 − k¯ 1 a¯ 2 ) dh h=0 = (1 − 2 k¯ 3 + 3 k¯ 2 )a¯ 1 − 3 k¯ 1 a¯ 2 + 2 k¯ 1 a¯ 3   dc¯ 2  = a¯ 2 dh h=0  dc¯ 3  = a¯ 3 dh h=0

(12)

Calling , 2 , 3 the components of x with respect to the basis {ai } (i.e., x = Ra¯ 1 + 2 Ra¯ 2 + 3 Ra¯ 3 ), Eq. (19) becomes Fa¯ 1 =

1



[Ra¯ 1 + (2 + 3 k¯ 1 )Ra¯ 2 + (3 − 2 k¯ 1 )Ra¯ 3 ]

(20)

Hence, using Eqs. (20) and (18)2,3 , the representation of F on the basis {a¯ i } is ⎡

h∈R

625

 ¯ ⎢  ⎢ ⎢  + 3 k¯ 1 [F]a¯ i = [R] ⎢ 2 ⎢ ¯ ⎣ 3 − 2 k¯ 1 ¯

0 1 0

0



⎥ ⎥ ⎥ 0⎥ ⎥ ⎦ 1

(21)

and so det F =

(14)

 ¯

(22)

A more explicit expression for Eq. (22) will be provided in the following paragraphs.

The corresponding curves passing through x(1 , 2 , 3 ) are

2.4. Velocity field

c1 (h) := x(1 + h, 2 , 3 )

The velocity field is also needed to describe the kinematics of the beam. Eqs. (2) and (10) can be combined to obtain

c2 (h) := x(1 , 2 + h, 3 ),

h∈R

c3 (h) := x(1 , 2 , 3 + h)

(15)

˙ T (x − x0 ) x˙ = x˙ 0 + RR

w = w0 + W(x − x0 ) (16)

Therefore, defining the deformation gradient F as

1

i = 1, 2, 3

(24)

Hence the velocity field is written as

h=0

  dc¯ i  dci  = F , dh h=0 dh h=0

(23)

Considering that the terms with overbar are time-independent, time differentiation of Eq. (23) provides, using the orthogonality of R, the expression of the velocity of a generic point of the beam

and, using Eqs. (11) and (6), the tangent vectors in h = 0 are  dc1  = x = x0 + R (2 a¯ 2 + 3 a¯ 3 ) dh h=0 + R[2 (−k¯ 3 a¯ 1 + k¯ 1 a¯ 3 ) + 3 (k¯ 2 a¯ 1 − k¯ 1 a¯ 2 )]  dc2  = Ra¯ 2 dh h=0  dc3  = Ra¯ 3 dh 

x − x0 = R(x¯ − x¯ 0 )

(25)

˙ T is the skew angular velocity tensor, w(x) := x˙ and where W := RR w0 := w(x0 ) = x˙ 0 . 2.5. Internal power; stress and strain measures

(17)

This expression for R describes the composition of a rotation of amplitude

3 around a¯ 3 , followed by a rotation of amplitude 2 around the new a2 , and then by a rotation of amplitude 1 around the final a1 .

The expression of the internal power (i.e., the power expended by the internal stress for the gradient of the velocity field) is deduced, in the case of the beam, from the corresponding well-known expression for the three-dimensional Cauchy continuum, applying the kinematic description of the beam developed in the previous paragraphs.

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For a three-dimensional Cauchy continuum of shape R, the internal power reads [14] Wint = − T · G dV (26) R

where T is the Cauchy stress tensor, G := ∇w is the gradient of the velocity field, the dot represents the tensor inner product (T · G := tr(TT G)) and dV is a measure of R. In the spirit of the referential description, the integral in Eq. (26) ¯ . Hence, it becomes must be evaluated in the reference set R Wint = − T · G det F d V¯ (27) ¯ R

and applying the inner product in both the members of Eq. (35) to a¯ 1 , a¯ 2 and a¯ 3 , alternatively, it follows that

 = RT a¯ 1 · a¯ 1 +

3

¯ a¯ i · a¯ 1 − 1 (ui RiT + ui RT K)

i=1

2 = RT a¯ 1 · a¯ 2 +

3

¯ a¯ i · a¯ 2 (ui RiT + ui RT K)

i=1

3 = RT a¯ 1 · a¯ 3 +

3

¯ a¯ i · a¯ 3 (ui RiT + ui RT K)

(37)

i=1

Using Eqs. (5) and (12), Eq. (37) eventually becomes

¯. where dV¯ is a measure of R ¯ of the beam can be parameterized As seen before, the shape R on I × S and, accordingly, Eq. (27) can be written as Wint = − T · G det F dA d1 (28)

 = (k¯ 2 s2 + c2 k¯ 3 s3 )u1 + (−c2 c3 k¯ 3 − k¯ 1 s2 )u2 + (c2 c3 k¯ 2 − c2 k¯ 1 s3 )u3

where dA is a measure of the area of S. It can be proved [15] that, stating the principle of material frame-indifference, Eq. (28) is equivalent to Wint = − [t · Rc˙ + m · Rv˙ ] d1 (29)

+ (c3 s1 s2 − c1 s3 )(1 + u1 ) + (c1 c3 + s1 s2 s3 )u2 + c2 s1 u3

I

+ c2 c3 (1 + u1 ) + c2 s3 u2 − s2 u3 − 1

2 = (k¯ 3 (c1 c3 + s1 s2 s3 ) − c2 k¯ 2 s1 )u1 + (c2 k¯ 1 s1 − k¯ 3 (c3 s1 s2 − c1 s3 ))u2 + (k¯ 2 (c3 s1 s2 − c1 s3 ) − k¯ 1 (c1 c3 + s1 s2 s3 ))u3

S

3 = (k¯ 3 (c1 s2 s3 − c3 s1 ) − c1 c2 k¯ 2 )u1 + (c1 c2 k¯ 1 − k¯ 3 (c1 c3 s2 + s1 s3 ))u2 + (k¯ 2 (c1 c3 s2 + s1 s3 ) − k¯ 1 (c1 s2 s3 − c3 s1 ))u3

+ (c1 c3 s2 + s1 s3 )(1 + u1 ) + (−c3 s1 + c1 s2 s3 )u2 + c1 c2 u3

I

where the dot, here, represents the inner product in V, and Ta1 dA t := S ¯ × Ta1 dA m := (x − x) S

in which ci := cos i and si := sin i , i = 1, 2, 3. In Eq. (38), the components of the vector c are expressed in terms of u := {u1 , u2 , u3 },  := {1 , 2 , 3 } and k¯ := {k¯ 1 , k¯ 2 , k¯ 3 }. The components of v (30)

are, respectively, the internal force and couple (the symbol × stands for vector product in V), and

c := RT x0 − x¯ 0 v := ax(RT R )

(31)

are the right strain measures (ax( ) is the axial vector of a skew tensor). To express the components of c

c = a¯ 1 + 2 a¯ 2 + 3 a¯ 3

(32)

it is useful to introduce the vector u ∈ V describing the translations of the axis curve from the reference configuration and defined such that x0 = x¯ 0 + u

(33)

v = 1 a¯ 1 + 2 a¯ 2 + 3 a¯ 3

(39)

can be expressed in an explicit way from Eq. (31)2 . To this end, the representation of the tensor RT R with respect to the vectors a¯ i (i = 1, 2, 3) is needed. In particular, the rotation R can be written as 3

R=

rij a¯ j ⊗ a¯ i

(40)

i,j=1

where rij is the (i, j)-th component of the matrix [R] (Eq. (12)) and ⊗ is the tensor product (∀v ∈ V, (a¯ j ⊗ a¯ i )v := (a¯ j · v)a¯ i ). Using Eq. (4) and the properties of the tensor product, the differentiation of both the members of Eq. (40) with respect to 1 provides R =

3

¯ a¯ j ⊗ a¯ i )) ¯ T + rij K( (rij a¯ j ⊗ a¯ i + rij (a¯ j ⊗ a¯ i )K

¯ + [R]T [K][R] ¯ [RT R ]a¯ i = [R]T [R] − [K]

x0 = a¯ 1 + u

1 = −k¯ 1 + c2 c3 k¯ 1 − k¯ 3 s2 + c2 k¯ 2 s3 + 1 − s2 3

(34)

Using Eqs. (32) and (34), Eq. (31)1 becomes 

a¯ 1 + 2 a¯ 2 + 3 a¯ 3 = R a1 + R u − a¯ 1 If u = u =

3

3

i=1

T

¯ it follows that ¯ T = −K, hence, considering that K (42)

Thus, the expressions of i are obtained from the matrix on the righthand side of Eq. (42), extracting the axial vector

2 = − k¯ 2 + c1 c3 k¯ 2 + c2 k¯ 3 s1 + c3 k¯ 1 s1 s2 − c1 k¯ 1 s3 (35)

¯ the expression for u is (see Eq. (4))

i=1 ui ai ,

¯ a¯ i ) (ui a¯ i + ui K

(41)

i,j=1

The objective is to express all the kinematic terms as functions of the three components of u with respect to {a¯ i } and the three angles {1 , 2 , 3 }. These six scalar functions are the Lagrangian parameters describing the deformation of the beam. By differentiating Eq. (33) with respect to 1 and using Eq. (3), one obtains



(38)

+ k¯ 2 s1 s2 s3 + c1 2 + c2 s1 3 



3 = − k¯ 3 + c1 c2 k¯ 3 − c3 k¯ 2 s1 + c1 c3 k¯ 1 s2 + k¯ 1 s1 s3   + c1 k¯ 2 s2 s3 − s1 2 + c1 c2 3 (36)

(43)

These components are explicitly expressed in terms of  and k¯ .

D. Zulli et al. / International Journal of Non-Linear Mechanics 44 (2009) 623 -- 629

Starting from Eq. (22), it can be useful to express the determinant of F in terms of the strain components. Combining Eqs. (16)1 and (31)1 , it follows that x0 = (1 + )a1 + 2 a2 + 3 a3

(44)

¯ as parameterized on I×S and the axis curve crossing Considering R  the sections in their centroid ( S (x − x0 ) dA = 0), if ¯ is considered as uniform, Eq. (54) becomes

¨ T Wext =− ¯ A x¨ 0 · w0 d1 + RR (x−x0 ) ⊗ (x−x0 ) dA · W d1 I

Hence, evaluating  = x · a1 one gets ¯ a¯ 2 · a¯ 1 +  (RT R + K) ¯ a¯ 3 · a¯ 1  = (1 + ) + 2 (RT R + K) 3

I

S

(55) (45)

and after some tensor algebra

 = (1 + ) − 2 ( 3 + k¯ 3 ) + 3 ( 2 + k¯ 2 )

627

(46)

where A is the area of the generic cross-section. If a¯ 2 and a¯ 3 are principal axes for the section and I2 , I3 are the respective principal moments of inertia, Eq. (55) becomes

¨ 3 (a¯ 2 ⊗ a¯ 2 )+I2 (a¯ 3 ⊗ a¯ 3 )]RT · W d Wext =− ¯ A u¨ · w0 d1 + R[I 1 I

I

(56)

Accordingly, an explicit expression for det F is derived: det F =

1 +  − 2 ( 3 + k¯ 3 ) + 3 ( 2 + k¯ 2 ) 1 −  k¯ +  k¯ 2 3

(47)

3 2

It is worth noticing that, in correspondence to the axis curve, i.e., when considering 2 = 0 and 3 = 0, Eq. (47) reduces to det F = 1 + , therefore the effect of the initial curvature is present only in the definition of .

As for the description of the internal stress and strain, the expression of the inertia forces is deduced from the corresponding terms valid for the three-dimensional Cauchy continuum. In particular, the expression of the (external) power expended by the inertia forces for the velocity field, for a three-dimensional Cauchy continuum of shape R, is Wext = − x¨ · w dV (48) R

where is the volume mass density in the current configuration, x¨ is the acceleration of a point of the body occupying the position x, and w is the velocity field. The integration in Eq. (48) must be evaluated in the reference ¯ , so shape R Wext = − x¨ · w det F dV¯ (49) ¯ R

Moreover, considering the principle of conservation of mass (50)

¯ being the volume mass density in the initial configuration, Eq. (49) becomes

Wext = −

¯ R

¯ x¨ · w dV¯

I

where b := − ¯ Au¨ ¨ 3 (a¯ 2 ⊗ a¯ 2 ) + I2 (a¯ 3 ⊗ a¯ 3 )]RT )) c := 2 ¯ ax(skw(R[I

2.6. Inertia forces

¯ = det F

being x¨ 0 = u¨ from Eq. (33). Thus, the inertia forces are described, by means of Eq. (56), in terms of the six components u and . Eq. (56) can be written in a more compact form as ˙ ) d 1 Wext = − (b · u˙ + c · x (57)

(51)

and where u˙ = w, x := ax(W). Therefore, when expressing the inertia forces and couples in terms of volume mass density in the initial configuration ¯ , they do not depend on the initial curvatures. 3. Equations of motion The objective is to obtain the equations of motion for a beam having double-symmetric cross-section and initial length , constituted by linear elastic isotropic material, with mass density ¯ in the reference configuration, carrying possible tip masses m0 and m and subjected to tip forces f0 and f , and couples 0 and  . The subscripts 0 and  indicate terms relevant to 1 = 0 and 1 = , respectively. The equations of motion are obtained using the principle of virtual power. The internal power has the form of Eq. (29) and can be written, in scalar form, as Wint = − (N˙ + T2 ˙ 2 + T3 ˙ 3 + M3 ˙ 1 + M2 ˙ 2 + M3 ˙ ) d1 (59) I

˙ := where the act of motion is described by u˙ := {u˙ 1 , u˙ 2 , u˙ 3 } and  ˙ , ˙ } after using Eqs. (38) and (43). ˙ , { 1 2 3 The external power contains terms due to distributed and concentrated tip masses and terms due to tip forces and couples ˙0 Wext = − (b · u˙ + c · x)d1 + b0 · u˙ 0 + c0 · x I

From Eq. (23) and by using the orthogonality of R, the expression for the acceleration is ¨ T (x − x0 ) x¨ = x¨ 0 + RR

(52)

Using Eqs. (25) and (52), the external power reads ¨ T (x − x0 )) · (w0 + W(x − x0 )) dV¯ Wext = − ¯ (x¨ 0 + RR

(53)

¯ R

Using the properties of the tensor product, Eq. (53) can be written as ¨ T (x − x0 )) · w0 Wext = − ¯ [(x¨ 0 + RR ¯ R

¨ T ((x − x0 ) ⊗ (x − x0 ))] · W] dV¯ + [(x − x0 ) ⊗ x¨ 0 + RR

(54)

(58)

˙  + f0 · u˙ 0 + f · u˙  + 0 · x ˙ 0 +  · x ˙ + b · u˙  + c · x

(60)

In the following, the case of a clamped–free beam is reported as ex˙ 0 = 0 are considered. Moreample. In this case m0 = 0, u˙ 0 = 0, and x over, a tip force f is assumed parallel to the axis a2 (), sinusoidally varying in time and having an amplitude f f := f sin( t)a2

(61)

The internal constraints, to describe vanishing of the extension and of the shear deformations, are

=0 2 = 0 3 = 0

(62)

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D. Zulli et al. / International Journal of Non-Linear Mechanics 44 (2009) 623 -- 629

and the corresponding stress quantities T := {N, T2 , T3 } are considered as reactive forces. A linear elastic uncoupled relation is considered to express {M1 , M2 , M3 } in terms of the strain components M1 = GJ 1 M2 = EI2 2 M3 = EI3 3

(63)

For a subsequent Galerkin projection, it is useful to express the internal constraint in weak form by introducing a constraint functional Vc := (N˜ + 2 T˜ 2 + 3 T˜ 3 ) d1 (64) I

In order to consider the internal constraint (62), this term needs to ˜ T˜ 2 , T˜ 3 . If the internal constraint is vanish for every trial function N, not used, this term is not considered and constitutive relations for T must be added. Hence, the boundary value problem governing the dynamics of the beam can be obtained by the vanishing of both the total power and constraint term

W := W Vc = 0

int

+W

ext

=0 (65)

for every act of motion, described by the nine-independent param˙ , T˜ := {N, ˜ T˜ 2 , T˜ 3 }, defined in I, and the six-independent eters u˙ ,  boundary parameters u ,  . Moreover, the six boundary conditions at the clamped end must be added

u(0) = (0) = 0

(66)

The expressions of the total power and constraint term (65) are manipulated by means of the software Mathematica䉸 [16] to obtain ˙ , T˜ and of u˙ ,  ˙ the coefficients of u˙ ,    . The separate vanishing of these coefficients provides the equations of motions and the boundary conditions on the tip section, respectively. The equations of motion appear as

Ji (u, u˙ , u¨ , , ˙ , ¨ ) + Hi (u, u , ,  ,  , T, T , k¯ ) = 0,

i = 1, . . . , 6

(u, u , , k¯ ) = 0 2 (u, u , , k¯ ) = 0 3 (u, u , , k¯ ) = 0

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i=1, . . . , 6

Appendix A. Expressions of the terms describing the initial configuration Defining   := k¯ 21 + k¯ 22 + k¯ 23

(A.1)

the components of a¯ 1 (1 ) on {e1 , e2 , e3 } are a¯ 11 (1 ) = a¯ 12 (1 ) = a¯ 13 (1 ) =

1 ¯2 [k1 + (k¯ 22 + k¯ 23 ) cos(1 )]

2 1

3 1

3

[k¯ 1 k¯ 2 (1 − cos(1 )) + 2 k¯ 3 sin(1 )] [k¯ 1 k¯ 3 (1 − cos(1 )) − 2 k¯ 2 sin(1 )]

(A.2)

The components of a¯ 2 (1 ) on {e1 , e2 , e3 } are a¯ 21 (1 ) = a¯ 22 (1 ) = a¯ 23 (1 ) =

1

3

[k¯ 1 k¯ 2 (1 − cos(1 )) − 2 k¯ 3 sin(1 )]

1 ¯2 [k2 + (k¯ 21 + k¯ 23 ) cos(1 )]

2 1

3

[k¯ 2 k¯ 3 (1 − cos(1 )) + 2 k¯ 1 sin(1 )]

(A.3)

The components of a¯ 3 (1 ) on {e1 , e2 , e3 } are a¯ 31 (1 ) = a¯ 32 (1 ) = a¯ 33 (1 ) =

1

3 1

3

[k¯ 1 k¯ 3 (1 − cos(1 )) + 2 k¯ 2 sin(1 )] [k¯ 2 k¯ 3 (1 − cos(1 )) − 2 k¯ 1 sin(1 )]

1 ¯2 [k3 + (k¯ 21 + k¯ 22 ) cos(1 )]

2

(A.4)

Eventually, the components of x¯ 0 (1 ) on {e1 , e2 , e3 } are

where Ji contain inertia terms while Hi contain elastic terms. The boundary conditions in 1 = 0,  are of the form

u(0)=(0)=0 Ji (u , u˙  , u¨  ,  , ˙  , ¨  )+Ki (u ,  ,  , T , f , k¯ )=Pi ,

a relevant power term in the expression of the total power and considering the internal forces as reactive. Investigations on free and forced dynamics for a case-study, and how these are affected by initial curvatures, are described in Part 2 of this work.

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where Ki contain elastic and parametric-exciting boundary terms while Pi contain direct-exciting terms. The full expressions of the equations of motion, omitted here for sake of space, are reported in [15]. 4. Conclusions In this work a geometrically non-linear elastic beam, with doublesymmetric, rigid planar cross-sections and uniform initial curvature was considered. In Part 1 of this work, the geometry of the beam was defined in terms of the axis curve and directors. The initial curvature parameters were introduced to specify the initial configuration. The description of the deformation and of inertia terms was deduced from a corresponding three-dimensional Cauchy continuum. The weak form of the equations of motion was obtained. These equations are valid, with slight modifications, both for extensible and inextensible beams, and both for shear-deformable and shearindeformable beams. The internal constraints were introduced with

x¯ 01 (1 ) = x¯ 02 (1 ) = x¯ 03 (1 ) =

1 ¯2 [k1 1 + (k¯ 22 + k¯ 23 ) sin(1 )]

3 1

3 1

3

[k¯ 3 (1 − cos(1 ) + k¯ 1 k¯ 2 (1 − sin(1 )))] [−k¯ 2 (1 − cos(1 ) + k¯ 1 k¯ 3 (1 − sin(1 )))]

(A.5)

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[11] W.M. Smolenski, Statically and kinematically exact nonlinear theory of rods  and its numerical verification, Comput. Methods Appl. Mech. Eng. 178 (1999) 89–113. [12] P.F. Pai, A.N. Palazotto, Large-deformation analysis of flexible beams, Int. J. Solids Struct. 33 (9) (1996) 1335–1353. [13] A. Paolone, M. Vasta, A. Luongo, Flexural–torsional bifurcations of a cantilever beam under potential and circulatory forces 1. Non-linear model and stability analysis, Int. J. Non-linear Mech. 41 (2006) 586–594.

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[14] M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981. [15] D. Zulli, Dynamic instability of an initially curved elastic beam (Orig. title: Instabilità dinamica di una trave elastica a curvatura iniziale non nulla), Ph.D. Thesis, University of L'Aquila, Italy, 2005 (in Italian). [16] S. Wolfram, The Mathematica Book, fifth ed., Wolfram Media, Inc., Champaign, IL USA, 2005.