Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation

International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

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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 non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation O. Aghababaei, H. Nahvi ∗ , S. Ziaei-Rad Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

A R T I C L E

I N F O

Article history: Received 1 January 2008 Received in revised form 28 October 2008 Accepted 28 October 2008 Keywords: Geometric imperfection Inextensional beam Hamilton's principle

A B S T R A C T

The non-linear equations and boundary conditions of non-planar (two bending and one torsional) vibrations of inextensional isotropic geometrically imperfect beams (i.e. slightly curved and twisted beams) are derived using the extended Hamilton's principle. The assumptions of Euler–Bernoulli beam theory are used. The order of magnitude of the natural geometric imperfection is assumed to be the same as the first order of vibrations amplitude. Although the natural imperfection is small, in contrast to the case of straight beams (i.e. geometrically perfect beams), this study shows that the vibration equations are linearly coupled and have linear and quadratic terms in addition to cubic terms. Also, in the case of near-square or near-circular beams, coupling terms between lateral and torsional vibrations exist. Furthermore, a problem of parametric excitation in the case of perfect beams changes to a problem of mixed parametric and external excitation in the case of imperfect beams. The validity of the model is investigated using the existing experimental data. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The beam is one of the fundamental elements of an engineering structure. Structures like helicopter rotor blades, spacecraft antennae, flexible satellites, airplane wings, gun barrels, robot arms, highrise buildings, long-span bridges and subsystems of more complex structures can be modeled as a beam-like slender member. Beam modeling problems of non-rotating straight (i.e. geometrically perfect) and naturally curved and twisted beams and arches have been the subject of a majority of works over the past decades. In what follows, a review of some of the related works is presented. Woinowsky-Krieger [1] and Burgreen [2] considered free oscillations of a beam having hinged ends a fixed distance apart. Their equation of motion contained a non-linear term due to midplane stretching which results in non-linear strain–displacement relations. Burgreen [2] also studied, both theoretically and experimentally, the effects of a compressive axial load. Evensen [3] analyzed the effect of midplane stretching on the vibrations of a uniform beam with immovable ends for simply supported, clamped, and clamped-simply supported cases. Bolotin [4] showed that, for beams, inertia non-linearity effects are more significant than geometric nonlinearity effects. Ho et al. [5] accounted for the midplane stretching in the study of large-amplitude non-planar whirling motions

∗ Corresponding author. Tel.: +98 311 3915242; fax: +98 311 3912628. E-mail address: [email protected] (H. Nahvi). 0020-7462/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.10.006

of a simply supported beam. Epstein and Murray [6] formulated a theory for the three-dimensional large deformation analysis of thin-walled beams of arbitrary open cross-section. Crespo da Silva and Glynn [7,8] systematically derived the non-linear equations of motion and boundary conditions governing the flexural-flexuraltorsional motions of isotropic inextensional beams. They included non-linearities due to inertia and geometry up to order three and showed that the non-linearities arising from the curvature (geometry) are of the same order of magnitude as those due to inertia. Rosen et al. [9,10] derived equations for analyzing the non-linear coupled bending-torsion motions of pretwisted rods. Equations describing the non-linear flexural-flexural-torsional-extensional dynamics of beams were formulated by Crespo da Silva [11,12]. Non-linearities due to curvature, inertia, and extension were accounted for in a mathematically consistent manner. Gendy and Saleeb [13] presented an effective formulation on spatial free vibration of arbitrary thinwalled curved beam by including the shear deformation and rotary inertia. However, they partially considered the effect of thicknesscurvature and shear deformation. Oz et al. [14] investigated a simply supported slightly curved beam resting on a non-linear elastic foundation. Lee and Chao [15] derived the governing equations for out-of-plane vibrations of curved non-uniform beams of constant radius. Raveendranath et al. [16] studied the performance of a curved beam element with coupled polynomial distributions. €uczko [17] discussed a geometrically non-linear model of Timoshenko-type space-curved beams. The motion of the system was described by a non-linear matrix equation, which accounts for non-linearities up to

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Nomenclature E, G

Young's and shear modulus

m

mass per unit length of beam

, 

Lamé constants

Dii , i = , , 

stiffness constants

v(x, t), w(x, t), u(x, t)

beam neutral axis deflection along Y, Z and X axes

v0 (x), w0 (x)

beam neutral axis natural deflection along Y and Z axes beam neutral axis position in XYZ

¯ t), w(x, ¯ t) v(x,

(x, t), (x, t), (x, t)

beam neutral axis deflection rotation angles

0 (x), 0 (x), 0 (x)

beam neutral axis natural Euler rotation angles

¯ (x, t), ¯ (x, t), ¯ (x, t)

0 (x), (x, t) ci , i = , , 

beam neutral axis Euler rotation angles natural and deflection-induced twist angles

Fi , i = v, w M

distributed load per unit length of beam distributed torsional load per unit length of beam

(x, t)  (x, t)   0 (x), ¯ (x, t)

Lagrange multiplier

0 0 0

Initial principle axes coordinate system of beam cross-section



current principle axes coordinate system of beam cross-section

xyz

local coordinate system aligned with XYZ

XYZ eˆ i

inertial reference coordinate system unit vector along the i axis

[Ti ]

transformation matrix

[ ], [J]

Lagrange strain and moments of inertia tensors

damping coefficients

the second order. Kim and Kim [18] proposed an improved vibration theory for spatial free vibration of shear deformable circular curved beams with non-symmetric thin-walled cross-sections. Marur and Kant [19] formulated a higher-order refined model with seven degrees of freedom per node for the free vibration analysis of composite and sandwich arches accounting for warping shear effects. Mahmoodi and Jalili [20] investigated the non-linear vibrations of a piezoelectrically driven microcantilever beam considering the inextensibility condition and the coupling between electrical and mechanical properties in the piezoelectric materials which yielded cubic and quadratic non-linearities. In this paper, the kinematics of geometrically imperfect isotropic inextensional beam element is investigated. The components of curvature vector and strain tensor are derived using the Kirchohff's kinetic analog [21]. The kinetic and strain potential energy functions are formulated and using the extended Hamilton's principle [22] by introducing a Lagrange multiplier to enforce the inextensionality condition, three non-linear equations of non-planar vibrations, two bending and one torsional, and the associated boundary conditions are derived, a problem which is not covered by the previous works. Arafat [23] used such an approach to obtain the equations of motion and boundary conditions describing the non-linear vibrations of geometrically perfect metallic and symmetrically laminated composite beams. Then, by applying the assumptions of near-square and nearcircular beams, the three equations and the associated boundary conditions are reduced to two bending equations. The terms of the final equations are discussed and compared with the terms in the equations derived by Crespo da Silva and Glynn [7]. It is shown that the assumption of straightness for beams with very small imperfection may cause misleading results because even small imperfection leads to linear and quadratic terms in the equations of motion. By applying zero imperfection, the derived equations reduce to the equations of Crespo da Silva and Glynn [7]. Finally, the validity of the proposed model is investigated by the experimental data reported in [24].

2. Problem formulation In this section, the equations of motion and boundary conditions governing the non-planar non-linear vibrations of inextensional

angular velocity vector natural and the deformed beam curvature vectors

geometrically imperfect beams are derived. The assumptions are: (1) there is no transverse shear deformation, (2) there is no warping effect, and (3) the Poisson's effect is negligible. The same assumptions were made by Crespo da Silva and Glynn [7]. These simplifications are valid for slender beams. Also, the general natural geometric imperfection is assumed to be the combination of two small initial deflection functions of the neutral axis v0 (x) and w0 (x) and one rotation angle function 0 (x) as shown in Fig. 1. 2.1. Inextensional imperfect beam element Let us consider the deformation of an element of the imperfect beam with the initial length ds0 as shown in Fig. 1. The position of the beam element after deformation ds is presented in Fig. 2. From Figs. 1 and 2, the strain at point (x, v0 , w0 ) of the imperfect beam neutral axis along the elastic differential beam element ds0 can be calculated as e=

ds − ds0 ds0

(1)

For an inextensional imperfect beam, the beam element neutral axis elongation e ds0 is zero which yields the condition of inextensionality, i.e. ds = ds0

(2a)

Using Figs. 1 and 2, the length of imperfect beam element before and after deformation can be calculated as ds0 =



 ds =

2 1 + v2 0 + w0 dx

¯ 2 dx (1 + u )2 + v¯ 2 + w

(2b)

(2c)

¯ = w0 + w and assume where ( ) = (j/ jx)( ). Now let v¯ = v0 + v, w that v, w, v0 , and w0 are of O( ) and >1. Upon substituting v¯ and ¯ in Eqs. (2b) and (2c) and applying Eq. (2a) the inextensionality w constraint equation may be obtained as u2 + 2u + v2 + w2 + 2v0 v + 2w0 w = 0

(2d)

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

149

Fig. 1. Position and orientation of an imperfect undeformed beam element.

Fig. 3. Imperfect beam initial and deformed neutral axis.

Fig. 2. Position and orientation of the imperfect deformed beam element.

Solving for u yields u = − 12 (v2 + w2 + 2v0 v + 2w0 w )

(2e)

In the absence of large axial forces, the beams with either of boundary conditions fixed-free, fixed-fixed/sliding and fixed-pined/sliding can be reasonably modeled as inextensional beams [7]. Assuming the fixed end is located at x = 0 and integrating Eq. (2e) yields u(x, t) = −

1 2

 0

x

(v2 + w2 + 2v0 v + 2w0 w ) dx

(2f)

Therefore, for an inextensional imperfect beam, if v, w, v0 and w0 are of O( ), >1, then u is of O( 2 ). 2.2. Beam kinematics A schematic of the imperfect beam initial and deformed neutral axis, (C0 ) and (C), are shown in Fig. 3. Under the aforementioned assumptions in Section 2, a plane cross-section which is perpendicular to the neutral axis before deformation remains plane, intact and perpendicular to it after deformation. Considering Figs. 1–3, in the deformed configuration, the position of the beam at any point of its neutral axis with respect to the inertial ¯ of reference frame XYZ can be expressed by three functions u, v¯ and w each cross-section centroid O*. The orientation of each cross-section can be expressed by three successive rotation angles. It is assumed that the beam has uniform cross-section and material properties along its length. Therefore, the area centroid coincides with the mass centroid. To describe the orientation of the beam cross-section in

Fig. 4. Euler angle rotations, xyz: local coordinate system aligned with the inertial reference frame XYZ and attached to the cross-section centroid O* .

its deformed configuration, three successive counterclockwise Euler ¯ , ¯ and ¯ are angle rotations denoted in the order of rotation by  used. These rotation angles are functions of time t and the coordinate ¯ about the z x as shown in Figs. 2 and 4. The first rotation angle,      axis takes xyz to x y z , z = z. The second rotation angle, ¯ about y ¯ about axis takes x y z to x y z , y = y , and the last rotation angle,      x axis takes x y z to ,  = x . The unit vectors of the final and initial coordinate systems are related to each other as presented in Appendix A (Eq. (A.3)). From Fig. 4, the angular velocity of the  system may be obtained as ˙ ˙¯ ˆ ˙¯ ˆ  (x, t) =   ez + ¯ eˆ y + e ˙¯ ˙¯ ¯ + ˙¯ cos ¯ )eˆ + ( ¯ − ˙¯ sin ¯ )eˆ = ( cos ¯ sin cos ¯ cos   ˙¯ ˙¯ + ( − sin ¯ )eˆ  ≡  eˆ  +  eˆ  +  eˆ 

(3)

¯ (x, t) can simply be obThe curvature vector of the deformed beam tained by replacing the time derivatives in the expression of angular

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O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

velocity by spatial derivatives based on the Kirchhoff's kinetic analog [21]. Thus, using Eq. (3) one may obtain 







¯ (x, t) = (¯ cos ¯ sin ¯ +¯ cos ¯ )eˆ  +(¯ cos ¯ cos ¯ −¯ sin ¯ )eˆ  



¯ − ¯ sin ¯ )eˆ +(  ˆ ¯ ≡  e + ¯  eˆ  + ¯  eˆ 

(4)

¯ and ¯ can be calculated as Using Fig. 2, the rotation angles  Eqs. (A.4) and (A.5) in Appendix A. The sine, cosine and tangent of 0 and 0 can be calculated by replacing v¯ and w¯ by v0 and w0 and applying u = 0 in Eqs. (A.4) and (A.5). It is obvious that the only ¯ . As for v¯ and w ¯ in Section 2.1, we independent rotation angle is ¯ let = + 0 and assume that and 0 are of O( ), >1. The ex˙ ˙¯ ¯  , ¯  ,  and ¯ can be obtained by using the definitions pressions of  ¯ ¯ of tan  and tan  in Eqs. (A.4) and (A.5) and differentiating with respect to x and t (see Eqs. (A.6)–(A.9)). ˙¯ ˙¯ ¯  , ¯  ,  , , sin ¯ and cos ¯ from Eqs. (A.6) to (A.9) Substituting  and (A.5) into Eqs. (3) and (4), expanding the results in Taylor series and keeping the terms up to O( 3 ) yields the non-linear expressions of angular velocity and curvature vectors components (Eqs. (A.10)–(A.15)). ¯ in Eqs. (A.13)–(A.15) with v , w and , ¯ w ¯ and Replacing v, 0 0 0 respectively, and applying u = 0 yields the non-linear expressions of  0 (x) components the undeformed imperfect beam curvature vector (Eqs. (A.16)–(A.18)). 2.3. Strain tensor In this section, the components of strain tensor are derived using the definition of Lagrange strain tensor. Fig. 3 shows the initial and deformed configurations of the inextensional beam neutral axis with respect to the inertial coordinate system XYZ. In Fig. 3, P0 is a point of undeformed beam cross-section with center O∗0 located at (x, v0 , w0 ) with respect to the XYZ system, see Fig. 1. It is assumed that the position of P0 with respect to the point O∗0 in the central coordinate system 0 0 0 is (0 , 0 , 0). After deformation, the point O∗0 moves to ¯ w), ¯ see Fig. 2. In the central coordinate system O∗ located at (x + u, v, , the new position of the point P0 which moves to P with respect to the point O∗ is (, , 0). Since the shape of the cross-section remains intact, the positions of P0 and P with respect to O∗0 and O∗ are the same, that is (, )=(0 , 0 ). Using Figs. 1–3, the position vectors of P0 and P with respect to the XYZ coordinate system can be expressed as  O∗ + R  P = R ˆ x + v0 eˆ y + w0 eˆ z + eˆ 0 + eˆ 0 R P0 /O∗ = xe 0 0

(5a)

P = R  O∗ + R  ∗ = (x + u)eˆ x + v¯ eˆ y + w ¯ eˆ z + eˆ  + eˆ  R P/O

(5b)

0

Spatial differentiating Eqs. (5a) and (5b) and applying the condition of inextensionality, see Eq. (2a), using Kirchohff's kinetic analogue [21] to calculate the spatial derivative of unit vectors of the  system and the definition of Lagrange strain tensor [25] leads to  P · dR  P − dR  P · dR P dR 0 0 = 2(−(  − 0 ) + (  − 0 )) ds2 − 2(  − 0 ) ds d + 2(  − 0 ) ds d + H.O.T.  P · dR  P − dR  P · dR  P = 2[ds d d][ ][ds d d]t dR 0 0

Lagrange strain tensor [ ] can be expressed as

ss = −(  − 0 ) + (  − 0 ),

s = − 12 (  − 0 ),

s = 12 (  − 0 )

(7a)

 =  =  = 0

(7b)

The assumptions that are made in Section 2 result in the above components of strain. The shear strains in the plane of cross-section are due to torsion only. Since the warping, shear and Poisson's effects have been neglected, the other normal and shear strains are zero. 3. Equations of motion In this section, the non-linear equations governing the flexuralflexural-torsional vibrations of inextensional imperfect beams are derived using the extended Hamilton's principle. As a rigid body, the kinetic energy of a beam element consists of translational and rotational parts, T = Ttrans + Trot where   1 2 ˙¯ 2 ) ds = 1 ˙ 2 ) ds m(u˙ 2 + v˙¯ + w m(u˙ 2 + v˙ 2 + w (8a) Ttrans = 2 C 2 C  1 [   ][J][   ]t ds (8b) Trot = 2 C In Eq. (8b), the integration is over the length of the beam. For simplicity, it is assumed that the central coordinate system 0 0 0 is the principle one and consequently, the central coordinate system  is also the principle one, namely, the off-diagonal elements of the inertia matrix [J] vanishes. Therefore,  1 (J 2 + J 2 + J 2 ) ds (8c) Trot = 2 C The diagonal elements of the inertia matrix [J] in Eq. (8c) are defined in Appendix A, Eq. (A.19). The strain potential energy function can be calculated as [26]     U=  ij ij + 2kk dV, dV = d d ds (9a) 2 V In Eq. (9a),  and  are Lamé constants. Substituting the strain components form Eqs. (7a) and (7b) and the Lamé constants into Eq. (9a) and considering the assumption that  is the principle axes coordinate system yields  1 (D (  − 0 )2 + D (  − 0 )2 + D (  − 0 )2 ) ds (9b) U= 2 C The stiffness constants D , D and D are defined in Appendix A, Eq. (A.20). Since the kinetic and potential energy in Eqs. (8a) and (8b) and (9b) are functions of u, v, w and , the Lagrangian should be defined such that it accounts for the inextensionality condition. Therefore, the Lagrange multiplier (x, t) should be introduced to enforce the inextensionality constraint. Using Eq. (2d), the Lagrangian may be expressed as  1 (x, t)(u2 + 2u + v2 + w2 + 2v0 v + 2w0 w ) ds L=T −U+ 2 C  l ≡ (x, t) dx (10) 0

(6a) (6b)

where H.O.T. stands for higher order terms which can be neglected for slender beams with small natural imperfection undergoing small deflections. Comparing Eqs. (6a) and (6b), the components of the

In Eq. (10), (x, t) may be interpreted as Lagrangian density. The work of external excitations and damping forces for harmonic motion can be expressed as  W = (Fu u + Fv v + Fw w + M C

−

1 ˙ 2 )) ds ˙ 2 + c (cu u˙ 2 + cv v˙ 2 + cw w 2

(11)

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

In Eq. (11), Fi (i=u, v, w) and M are distributed forces and moments, respectively. Also, ci (i = u, v, w, ) are damping coefficients. Using Eqs. (2a)–(2c), for an inextensional beam, ds can be expressed as ds =



2 1 2 2 1 + v2 0 + w0 dx ≈ (1 + 2 (v0 + w0 )) dx

(12)

Substituting Eqs. (A.10)–(A.12) and (12) into Eqs. (8a) and (8c), ¯ ¯ w ¯ and Eqs. (A.13)–(A.15) and (12) into Eq. (9b) and replacing v, with v0 + v, w0 + w and + 0 , respectively, expanding the integrands of T and U and replacing them in Eq. (10), one obtains the Lagrangian (Lagrangian density) up to O( 4 ) (see Eq. (A.21)). Setting qi =u(x, t) and using Eqs. (A.22) and (A.23), the longitudinal equation of vibrations is obtained as

+ D ((w + w0 )(−w + (v + v0 ) + 0 v )) − D ((w + w0 )(−w + (v + v0 ) + 0 v ))

(13a)

and the boundary conditions are x=0

(13b)

(1 + u ) + D (v + v0 )(v + (w + w0 ) + 0 w ) − D (w 



+ (v

− D ((v



+ (w

+ w0 ) + 0 w ))

+ D ((w + w0 )(−w + (v + v0 ) + 0 v )) = 0 x = l

(13c)

It is assumed that the fixed end of the beam is located at x = 0. For a weakly damped beam, since u(x, t) is O( 2 ), see Eq. (2f), cu u˙ is O( 4 ) and hence can be dropped from Eq. (13a). Integrating both sides of Eq. (13a) form l to x and applying Eq. (13c) yields  l

x

   1 2 mu¨ − Fu 1 + (v2 + w ) d 0 2 0 





− D (v + v0 )(v + (w + w0 ) + 0 w ) + D (w + w0 )(−w + (v + v0 ) + 0 v ) + D ((v + v0 )(v + (w + w0 ) + 0 w )) − D ((w + w0 )(−w + (v + v0 ) + 0 v ))

(14)

Substituting u(x, t) from Eq. (2f) into Eq. (14), solving for (x, t) and keeping terms up to O( 3 ) yields 

 = − J (w w¨  ) − J (v v¨  ) + D (w w + w0 w )  ×

l

 +

x

x l



j2 jt 2

  0

2

2

(v + w

  1 2 Fu 1 + (v2 0 + w0 ) d 2

+ 2w0 w ) d

v = 0

(16c)

D v − J v¨   1 2 = D −v (v v ) − v (w w ) + v (−v2 0 + w0 ) 2 − (w0 ) − v0 v2 − 2(v0 v ) v − (v0 w w ) − v (w0 w )  1 2           − v 0 v − (v0 (w0 w ) ) − v (v0 v0 − w0 w0 ) 2 − (D − D )(v ( + 0 )2 − w ( + 0 ) + v0 0 )   1 2 − D − D ( v0 ) 2   j j ˙ ) + J v (v v˙  ) + v (w w jt jt

j ˙    ( (w + w0 ) + v˙  (w2 + w2 0 + 2w0 w )) jt j 2 ˙ ) + (J − J ) ( v˙  − w jt + J

(16d)

2

D v = D (w (w0 v + v0 w + v0 w0 ) − v )  1  2 1  2      v (v0 + w2 + D 0 ) − w0 + v0 − v (w0 w ) 2 2  1 −v0 w w − v0 (w0 w ) − v0 v2 2

 + 2v0 v

(16b)



¨ ) − J (v v ) = −(1 + u ) − J (w w

m 2

v=0

− D ((w + w0 )( + w v + v0 w + w0 v ))

 ¨

+ D (v v + v0 v ) +

  j ˙  ) ¨  − (v˙  2 − w − (J − J ) v w w jt     x 2   j m   2 2     − (v +w + 2v0 v +2w0 w ) d d (v +v0 ) 2 jt2 0 l    x − (v + v0 ) Fu d  (16a) The boundary conditions are

 ¨

+ v0 )(v

j ˙     ( (w + w0 ) + v˙  (w2 + w2 0 + 2w0 w )) jt

l

+ v0 ) + 0 v )

¨ ) + J (v v ) + J (w w 

      − (v0 (w w ) ) − ((v + v0 )(w 0 w + 2w0 w + w0 w ))  1 2  −(v0 w w ) − (w0 v0 w + w0 v0 w ) − (v v ) 2 0

+ J

+ D ((v + v0 )(v + (w + w0 ) + 0 w ))



+ (v (−v0 v0 + w0 w0 )) − (w0 ) − (v0 v2 ) − 2((v0 v ) v )

− (D − D )(v ( + 0 )2 − w ( + 0 ) + v0 0 )   1 2 ˙ 2 ) − D − D (v0 ) + J (v v˙ 2 + v w 2

− D ((v + v0 )(v + (w + w0 ) + 0 w ))

+ w0 )(−w

mv¨ + cv v˙ + D viv − J v¨    1 1 2 2 ¨ 2 = Fv 1 + (v2 0 + w0 ) − mv(v0 + w0 ) 2 2  1 2  + D −(v (v v ) + v (w w ) ) + (v (−v2 0 + w0 )) 2



¨  ) − J (v v¨  ) = (−(1 + u )) − J (w w



equation of bending vibrations for weakly damped inextensional imperfect beams as

− D ((w + w0 )( + w v + v0 w + w0 v ))

2 mu¨ + (cu u˙ − Fu )(1 + 12 (v2 0 + w0 ))

u = 0,

151

d (15)

Setting qi = v(x, t) and using Eqs. (A.22)–(A.24) and then substituting u and  from Eqs. (2f) and (15), respectively, yields the in-plane



− D (w + w0 )( + w v + v0 w + w0 v ) − (D − D )(( + 0 )2 v − w − 0 w + 0 v0 − v w w )

(16e)

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Setting qi = w(x, t) and using Eqs. (A.22)–(A.24) and then substituting u and  from Eqs. (2f) and (15), respectively, yields the out-of-plane equation of bending vibrations for weakly damped inextensional imperfect beams as iv



¨ + cw w ˙ + D w − J w ¨ mw   1 1 2 ¨ 2 = Fw 1 + (v2 + w2 0 ) − mw(v0 + w0 ) 2 0 2    1  2 w (v0 + w2 + D (−w (w0 w0 + w0 w )) + 0) 2         + (v0 ) + (v2 0 w + v0 v0 w + v0 v0 w )    1  2 − w0 w +w (w0 w ) +w (w w ) +w0 (2w w +w0 w ) 2

− D ((v (v + v0 )) (w + w0 )) + D ((v



 + v0 )(

 

+wv



+ v0 w 



m (w +w0 ) 2   − (w + w0 )

−

x



l x l

j2 jt 2

  0

(16b,c), (17b,c), (18b) (16b,c), (17b,c), (18b) (16b,c), (17b,c), (18b)

(16d,e), (17d,e), (18c) (16b,c), (17b,c), (18b) (16b,d), (17b,d), (18b)

Setting qi = (x, t) and using Eqs. (A.22)–(A.24) and then substituting u and  from Eqs. (2f) and (15), respectively, yields the equation of torsional vibrations for weakly damped inextensional imperfect beams as ¨ +c ˙ J − D



+ (D − D )(v w − v ( + 0 )(v + v0 ) + w ( + 0 )(w + w0 )) 2 ¨ − J (v¨  (w + w0 ) + v˙  w˙  + 12 (v2 0 + w0 ) )





(v2 +w2 +2v0 v +2w0 w ) d d

 Fu d 

˙ 2 − v˙  w ˙ ) + (J − J )( v˙ 2 − w

(18a)

The boundary conditions are

(17a)

=0

(18b)



(17b)

w = 0

(17c)

¨ D w − J w  1 2 2        = D ( v0 ) + w (v2 0 + w0 ) + v0 w + v0 v0 w + v0 v0 w 2 − (w (w0 w ) + w (w w ) + w0 (2w w + w0 w ))  1 2      − w 0 w − w (w0 w0 + w0 w ) 2 − D (v (v + v0 )) (w + w0 ) 

+ D (v + v0 )( + w v + v0 w + w0 v ) + (D − D )(w ( + 0 )2 + v ( + 0 ) + 0 w0 )   1 2 D − D (w0 ) + 2

D = − D  (17d)

−

0

x



x 0

  l

(w v + v0 w + w0 v ) d

(M + (D − D )v w

+ D (v0 w ) − D (w0 v )) d d

2      1  2 D w = D ( 12 (v2 0 + w0 )w + v0 − 2 w0 w + v0 v0 w )

+ (D − D )(w ( + 0 )2 + ( + 0 )v + 0 w0 ) 2

Eqs. (16a)–(16e), (17a)–(17e) and (18a)–(18c) are the governing equations and the associated boundary conditions of flexuralflexural-torsional vibrations of inextensional imperfect isotropic beams. Putting v0 = 0, w0 = 0 and 0 = 0, these equations reduce to the equations and boundary conditions for isotropic perfect beams reported in [7]. Table 1 shows the boundary conditions for different inextensional imperfect beams. Eqs. (16a), (17a) and (18a) are linearly coupled and contain quadratic and cubic non-linearities. For long slender beams, since the effect of rotary inertia terms (i.e. terms containing J and J ) is the same order as the effect of shear deformation [7], these terms may be neglected afterward in this paper. The fundamental torsional frequency of beams with near-square or near-circular cross-sections is much higher than that of the flexural modes. Then, the torsional inertia terms (i.e. terms containing J ) can be neglected in comparison with flexural inertia and stiffness ˙ can be neglected for terms [7]. Also, the torsional damping term c

˙ v˙  + v˙ 2 (w + w )) − J ( 0

+ ( 12 D − D )w0

(18c)

weakly damped beams [7]. Consequently, by twice integrating both sides of Eq. (18a) and using Eqs. (18b) and (18c) and keeping terms up to O( 2 ), one obtains

j   j ˙ ) + J w (v v˙  ) (w w jt jt j 2  ˙ + v˙  ) ( w jt



2 D = −D (w v + v0 w + w0 v ) − 12 D (v2 0 + w0 )

w=0

− (J − J )

Fixed-free Fixed-fixed/sliding Fixed-pined/sliding



The boundary conditions are

+ J w

x=L

 2 + D (w v + v0 w + w0 v + 12 (v2 0 + w0 ) )

˙ v˙  + v˙ 2 (w + w )) + J (w v˙ 2 ) − J ( 0



x=0

− D (( w w0 + w0 v + w2 0 ))

( + 0 ) + 0 w0 )

j  2 ˙ + v˙  ) − (J − J ) (w jt

Beam

2     2 = M (1 + 12 (v2 0 + w0 )) − D (( v v0 − v0 w + v0 ))

+ w0 v ))

+ (D − D )(w ( + 0 ) + v   1 2 ˙ 2 ) D − D (w0 ) + J (w w + 2 2

Table 1 Boundary conditions of inextensional imperfect beams

(17e)

(19)

It is clear from Eq. (19) that for slender beams, is O( 2 ). It should be noted that the angle of twist that is different from may be computed by integrating the difference between the third components of

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

curvature vectors in the deformed and undeformed configurations. Thus, using Eqs. (A.18) and (12) gives

=

 0

 =

x

0

x

(  − k0 ) ds   1  2 ( + w v + v0 w + w0 v ) 1 + (v2 0 + w0 ) dx 2

(20)

Expanding the integrand in Eq. (20) up to O( 2 ) and integrating yields the twist angle at an arbitrary x as

= +



x

0

(w v + v0 w + w0 v ) dx

(21)

1 D



x

  l

0

(Q + (D − D )v w + D (v0 w ) (22)

By putting v = 0, w = 0 and = 0 in Eq. (21), the initial twist angle may be computed as 0 = 0 . Substituting Eq. (19) into Eq. (16a), dropping all inertia terms and keeping terms up to O( 3 ) yields

−D [v (w0 w ) ] −

(D − D )2



D

  +(D − D ) w

x

l

  w

w

v w d − w

x

  l

0



 

x

l

0



x 0

 v w d d

 w v d

D x

0

−

D D

  w0

w v d x

0

 (−D v0 w + D w0 v ) d d

  l



0

  l





D

  + (D − D ) w0

0

l

0

x

M d d



 Fu d 

(23a)

− (w0 ) − v0 v − 2(v0 v ) v − (v0 w w ) − v (w0 w )  1 2           − v 0 v − (v0 (w0 w ) ) − v (v0 v0 − w0 w0 ) 2 2

− (D − D )(v 0 − w ( + 0 )) 2

(23b)

D v = D (w (w0 v + v0 w + v0 w0 ))  1  2      v (v0 + w2 + D 0 ) − w0 − v (w0 w ) 2  1 − v0 w w − v0 (w0 w ) − v0 v2 2 

2

− (D − D )( 0 v − ( + 0 )w − v w w ) x l

v w d

 (−D v0 w + D w0 v ) d d

   x + w0 (D v0 w − D w0 v ) d

w0

x 

− D (w + w0 )( + w v + v0 w + w0 v )



v w d d

  x  +D w0 (v0 w + w0 v ) d

l

l



 M d +



l

x

x



− D ((w + w0 )( + w v + v0 w + w0 v ))

0

  +D w0

  + (w + w0 ) 

l

0

M d d

 1 2 2 D v = D −v (v v ) − v (w w ) + v (−v 0 + w 0 ) 2

   x + w (D v0 w − D w0 v ) d   w0

D

 

x

By dropping all inertia terms in Eqs. (16d) and (16e) and keeping terms up to O( 3 ), the corresponding boundary conditions change to the following equations

   x −(D − D ) w (v0 w + w0 v ) d

D (D − D )

  w

(D − D )

l

  1 2         −D v0 v2 + v 0 v + 2v (v0 v ) + (v0 w w ) 2

+

      x j2  2 1 2 − m v0 (v + w ) d  d 2 jt 2 0 l

  − (v + v0 )

= −D [v (v v + w w ) ]  1 2       + D (v (−v2 0 + w0 )) − D (v0 (w0 w ) ) 2   2 −(D − D )( 0 v − 0 w )

D

      x j2    1    − m v0 (2v0 v + 2w0 w ) d d 2 jt 2 0 l

D

mv¨ + cv v˙ + D viv

(D − D )

      x j2    1    − m v (2v0 v + 2w0 w ) d d 2 jt 2 0 l

−

− D (w0 v )) d d

+

      x j2  2 1 2 − m v (v + w ) d  d 2 jt 2 0 l

 1 1 2 2 2 ¨ 2 − mv(v 0 + w0 )+Fv 1 + (v0 + w0 ) 2 2

Substituting Eq. (19) into Eq. (21) yields

= −

153



(23c)

Similarly, substituting Eq. (19) into Eq. (17a) and dropping all inertia terms and keeping terms up to O( 3 ) leads to ˙ + D wiv ¨ + cw w mw = −D [w (w w + v v ) ]  1 2 2      +D −w0 w0 w + w (v2 0 + w0 ) + v0 w + v0 (v0 w ) 2    D     D 2  ( 0 w + 0 v ) −w2 w0 (v0 v ) + 1 − 0w − D D

154

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

the following equations

 1 +D −w0 w w − w w2 − w (w0 w ) 2 0 D      − 2w0 w w − w0 (v v ) D −D [w (v v0 ) ] −   −(D − D ) v +

(D − D )



D

v

2

(D − D ) D x

v w d − v

l



v

x

  l

0



x

x

0

        + v0 v 0 w + v0 v0 w − (w (w0 w )

  l

0



2 2  D w = D (( v0 ) + 12 w (v2 0 + w0 ) + v0 w

+ w (w w ) + w0 (2w w + w0 w ))



2      − 12 w 0 w − w (w0 w0 + w0 w ))

v w d d

− D (v (v + v0 )) (w + w0 )





w v d

+ D (v + v0 )( + w v + v0 w + w0 v ) + (D − D )(w 0 + v ( + 0 )) 2



(−D v0 w + D w0 v ) d d

2      1  2 D w = D ( 12 (v2 0 + w0 )w + v0 − 2 w0 w + v0 v0 w )

   x −(D − D ) v (v0 w + w0 v ) d

2

+ (D − D )(w 0 + ( + 0 )v )

0

   x + v (−D v0 w + D w0 v ) d (D − D )



D

  −D v0

x

0

v0



x

  l

0



w v d

 v w d d

  −(D − D ) v0

x l

 v w d

  x  −D v0 (v0 w + w0 v ) d 0

    D  x  v0 + (−D v0 w + D w0 v ) d d D 0 l   x  + v0 (−D v0 w + D w0 v ) d l



1 − m w 2



x



l

j2 jt 2

  0

 2



2

(v + w ) d d

      x j2    1   − m w (2v v + 2w w ) d  d 0 0 2 jt 2 0 l       x j2    1   − m w0 (2v v + 2w w ) d  d 0 0 2 jt2 0 l       x j2  2 1 2 − m w0 (v + w ) d  d 2 jt2 0 l  1 1 2 2 2 ¨ 2 − mw(v + w )+F + w ) 1 + (v w 0 0 0 2 2 0 −

(D − D )



D

  − (v + v0 )

v

x l

  − (w + w0 )

x 0

  l

M d d



     D  x  v0 M d − M d d D 0 l

x l



 Fu d 

(24c)

In Eqs. (23b), (23c), (24b) and (24c), should be replaced with the expression in Eq. (19). Eqs. (23a)–(23c) and (24a)–(24c) are the governing equations and the associated boundary conditions of non-planar vibrations of inextensional imperfect beams with near-square or near-circular crosssections in which by substituting v0 = 0, w0 = 0 and 0 = 0, the equations and boundary conditions for perfect beams reported in [7] may be obtained. In Eqs. (23a) and (24a), the first once-underlined bracket represents cubic bending-bending geometric non-linearities which arise from deformation-induced curvature. The second and third one represent the imperfection-bending linear terms and the geometric quadratic non-linearities, respectively, which arise from deformation-induced curvature and natural imperfection and the last one represents the bending-bending-imperfection quadratic geometric non-linearity which is the additional coupling term between lateral vibrations along the Y- and Z-directions due to natural imperfection. The first and second twice-underlined brackets represent cubic bending-torsion geometric non-linearities and the third to fifth ones represent quadratic bending-torsion-imperfection geometric nonlinearities. Also, the sixth and seventh ones represent the quadratic imperfection-torsion geometric non-linearities and the eighth to tenth ones represent the imperfection-torsion linear coupling terms. The first three-time underlined bracket represents cubic bendingbending geometric inertia non-linearities and the second one represents the quadratic bending-bending-imperfection geometric inertia non-linearities. The third one represents the linear imperfection-bending inertia terms and the forth one represents the quadratic imperfection-bending geometric inertia terms. The last one represents the linear imperfection-bending inertia terms. The first four-time underlined bracket represents the forcing terms due to the distributed load along the Y- and Z-directions and their couplings with the natural imperfection. The second to forth ones are due to the distributed torsional load and the last one represents the forcing terms due to the distributed load along the X-direction and its coupling with the natural imperfection. The third and forth four-time underlined brackets show that there is coupling with torsional distributed load and the natural imperfection.

l

−

(24b)

(24a)

Also, by dropping all inertia terms in Eqs. (17d) and (17e) and keeping terms up to O( 3 ), the corresponding boundary conditions change to

4. Model validation In this section, the validity of the proposed model for the vibrations of geometrically imperfect cantilever beams is investigated by comparing the theoretical predictions of the model and the experimental data reported by Zaretzky and Crespo da Silva [24].

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

155

Table 2 Dimensions, measured natural frequencies and damping factors of the beams used in the experiments [24] Beam

Length (in)

Cross section (in×in)

Mode

meas (Hz)

c = 2 

1 1 2

70.5 70.5 45

1/4×1/4 1/4×1/4 1/4×1/4

2 3 1

9.77 27.4 3.79

0.09 0.11 0.035

Fig. 5. The base excited cantilever beam, deflections are shown exaggeratedly.

 The length of the beam L and the characteristic time L2 m/D are used to non-dimensionalize Eqs. (23a) and (24a). Expanding the in-plane and out-of-plane deflection variables (i.e. v(x,t) and w(x,t), respectively) in power series involving a bookkeeping perturbation parameter as (x, t; ) = 1 (x, T0 , T2 ) + 3 3 (x, T0 , T2 ) + · · · where  = v, w and (x, T0 , T2 ) = Xn (x)an (T2 ) cos[n T0 + n (T2 )], Xn (x) is the nth normalized flexural mode shape (i.e. single mode approximation of the response, it is assumed that the nth in-plane and out-of-plane flexural modes are interacting under the primary resonance  = n (1 + 2 )) and Ti = i t, i = 0, 2, the following equations are obtained from perturbation analyses of Eqs. (23a) and (24a) as the result of elimination of secular terms (for a detailed bifurcation analyses, we refer the reader to the part II of this paper [27]): − 2n avn + [C1 + C2 a2wn sin 2( vn − wn )]avn + C3 awn sin( vn − wn ) + f3 sin( wn ) = 0

(25a)

[2n vn + C4 + C5 a2vn + C6 a2wn + C2 a2wn cos 2( vn − wn )]avn + C3 awn cos( vn − wn ) + f3 cos( wn ) = 0

(25b)

− 2n awn + [C1 − C2 a2vn sin 2( vn − wn )]awn − C7 avn sin( vn − wn ) = 0

(25c)

[2n wn + C8 + C5 a2wn + C6 a2vn + C2 a2vn cos 2( vn − wn )]awn + C7 avn cos( vn − wn ) = 0

(25d)

In the above equations, f3 = 21 f 2n ( 3 f3 is called the excitation amplitude) where f denotes the scaled non-dimensional amplitude

1 of base excitation (i.e. 3 f = F, L 3 f = 3 f ∗ see Fig. 5), 21 = 0 Xn dx, C1 to C8 are defined in Appendix A of part II of this paper [27], and n = n T2 − n . It should be noted that the ratio D /D is ordered to 1 + 2  for near-square beams where  denotes the out-of-squareness of beam cross-section and appears in the constant C4 . The damping factors of both directions are assumed to be equal cw = cv = c and ordered to 2  which appears in the constant C1 . The equilibrium solutions to Eqs. (25a)–(25d) are obtained by letting an = n = 0. Zaretzky and Crespo da Silva [24] have performed an in-depth experimental investigation of one-to-one non-linear coupling phenomena in the dynamic response of cantilever beams whose torsional frequencies are much higher than the flexural frequencies. The experiments were conducted by harmonic in-plane transverse base excitation of two square cross-section aluminum beams of different

Fig. 6. The experimental [24] and perfect beam model frequency response of the third mode of beam 1 for 3 f3 = 0.081( 3 f ∗ = 0.15 mm), 2  = 0.11, 2  = 0 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

lengths, see Fig. 5. The dimensions and measured natural frequencies and damping factors of the beams are listed in Table 2. Zaretzky and Crespo da Silva [24] initially investigated the primary resonant excitation of the third mode of beam 1 for base excitation amplitude 3 f ∗ of 0.15 mm which corresponds to 3 f3 = 0.081. The theoretical predictions of the perfect beam model [7] and experimental results for the planar and non-planar components of the beam response at xaccel = 0.099 (i.e. the non-dimensional location of in-plane and out-of-plane accelerometers, referring to Eq. (12), for small imperfections one may assume that the arc-length s and x are equal for the sake of simplicity) are displayed in Fig. 6. In this figure, the normalized deflection amplitudes of the response defined as e = Xn (xaccel ) an ,  = v, w are plotted versus the base excitation frequency in Hz. The solid and dashed lines represent the stable and unstable predictions of the perfect beam model [7] amplitude–frequency response curves, respectively, while the dots represent the test results [24]. As Fig. 6 shows, the theoretical predictions underestimate the measured response amplitudes, even at excitation frequencies considerably far from the natural frequency when the response is nearly linear [24]. Also, the perfect beam model [7] predicts pitchfork bifurcations and consequently, non-zero-amplitude out-of-plane vibrations for a limited frequency band while the test results show the existence of whirling motions for approximately the whole range of excitation frequency. Zaretzky and Crespo da Silva [24] neglected the later discrepancy and assumed that the former discrepancy is exclusively due to an additional base motion other than a translation which would add to effective excitation on the beam. In order to incorporate the small additional motion of the base into the perfect beam model so as to assess its effect on the beam response, a torsional spring was added to the perfect beam model so as to exert a torsional moment along the Z axis at the base [24], see Fig. 5. A linear analysis was performed to show that the effective ex-

156

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

Fig. 7. The experimental [24] and imperfect beam model frequency response of the third mode of beam 1 for 3 f3∗ = 0.125( 3 f ∗ = 0.15 mm), 2  = 0.11, 2  = 0 and

ci = ci = 0.0015, (——) stable, (- - - - -) unstable: (a) ve and (b) we .

Fig. 8. The experimental [24] and perfect beam model frequency response of the third mode of beam 1 for 3 f3∗ = 0.125( 3 f ∗ = 0.15 mm), 2  = 0.11, 2  = 0 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

1

1 2 2 citation amplitude is [ 0 Xn dx +  /(2kz ) 0 xXn dx]F  ≡ 3 f3∗ rather 2 1 3 than F  0 Xn dx ≡ f3 for the resonant excitation of the nth mode [24], it is clear that if the torsional spring stiffness is infinite, the model reduces to the perfect beam model, then, the value of kz = 350 for the shaker was obtained from the linear range of the test results displayed in Fig. 6. (kz was varied until the theoretical curves generated for 3 f3∗ instead of 3 f3 closely agreed with the linear range of the test results shown in Fig. 6 [24]). It is believed that besides the additional base motion, small geometric imperfections should be taken into account to eliminate both discrepancies observed in Fig. 6 (i.e. underestimation of test results and prediction of pitchfork bifurcations by the perfect beam model). To do so, it is assumed that the non-dimensional geometric imperfections along Y and Z axis in Fig. 5 are of the forms v0 (x)= [c1 X1 +c2 X2 ] and w0 (x) = [c1 X1 + c2 X2 ], respectively. Care should be taken that ∗ by ordering the torsional imperfection 0 (x) to 0 (x); the effect of torsional imperfection does not appear in Eqs. (25a)–(25d). Then, by incorporating the base flexibility modeling proposed in [24] and

Fig. 9. The experimental [24] and imperfect beam model frequency response of the third mode of beam 1 for 3 f3∗ = 0.0833( 3 f ∗ = 0.1 mm), 2  = 0.11, 2  = 0 and

ci = ci = 0.0015, (——) stable, (- - - - -) unstable: (a) ve and (b) we .

Fig. 10. The experimental [24] and perfect beam model frequency response of the third mode of beam 1 for 3 f3∗ = 0.0833( 3 f ∗ = 0.1 mm), 2  = 0.11, 2  = 0 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

the assumed geometric imperfections, we tried to match the theoretical predictions of the imperfect beam model with the test results shown in Fig. 6 by varying the value of kz and the imperfection coefficients ci and ci . This yields kz = 449 (which gives 3 f3∗ = (968+235344/kz )(0.15 mm/1790.7 mm)=0.125) and ci = ci =0.0015, i = 1, 2. Fig. 7 shows the imperfect beam model response curves of the third mode generated for 3 f3∗ = 0.125( 3 f ∗ = 0.15 mm) instead of 3 f3 = 0.081, ci = ci = 0.0015 and 2  = 0 (i.e. near-square beam) and the same test results shown in Fig. 6. Fig. 7 demonstrates excellent agreement between imperfect beam model predictions and test results since both discrepancies observed in Fig. 6 are omitted. Fig. 8 shows the perfect beam model response curves of the third mode of beam 1 generated for 3 f3∗ = 0.125 and 2  = 0 and the same test results as in Fig. 6 that should be compared with Fig. 7. Fig. 8 reveals the importance of accounting for geometric imperfection. As Figs. 7 and 8 indicate, accounting for imperfection has a shifting effect to left on the theoretical predictions, furthermore, the out-of-plane vibrations (i.e. we ) is more precisely predicted since no pitchfork bifurcation occurs in the imperfect beam model predictions. The

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

Fig. 11. The experimental [24] and imperfect beam model frequency response of the second mode of beam 1 for 3 f3∗ = 0.0918( 3 f ∗ = 0.7 mm), 2  = 0.09, 2  = 0.024 and ci = ci = 0.0015, (——) stable, (- - - - -) unstable: (a) ve and (b) we .

Fig. 12. The experimental [24] and perfect beam model frequency response of the second mode of beam 1 for 3 f3∗ = 0.0918( 3 f ∗ = 0.7 mm), 2  = 0.09, 2  = 0.024 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

mathematical justification of this phenomenon is given in part II of this paper [27]. The imperfect and perfect beam model response curves of the third mode of beam 1 and the corresponding test results for a base excitation amplitude of 3 f ∗ = 0.1 mm ( 3 f3∗ = 0.0833) are shown in Figs. 9 and 10. Fig. 9 shows excellent agreement between theoretical and test results. Comparing Fig. 10 with Fig. 9 shows the importance of accounting for geometric imperfections. Figs. 11 and 12 and 13 show the imperfect beam model response curves of the second mode of beam 1 and the corresponding test results for the base excitation amplitudes of 3 f ∗ = 0.7 mm ( 3 f3∗ = 0.0918) and 3 f ∗ = 0.85 mm( 3 f3∗ = 0.1115), respectively. The value of

2 =0.024 yields the best correlation of theoretical and test results in Figs. 11 and 13. Figs. 12 and 14 should be compared with Figs. 11 and 13 to realize the effect of accounting for geometric imperfections. The theoretical and test results for the first mode resonant excitation of beam 2 are shown in Fig. 15. The non-dimensional location of the accelerometers in this case was xaccel = 0.267 [24]. The value of

157

Fig. 13. The experimental [24] and imperfect beam model frequency response of the second mode of beam 1 for 3 f3∗ = 0.1115( 3 f ∗ = 0.85 mm), 2  = 0.09, 2  = 0.024 and ci = ci = 0.0015, (——) stable, (- - - - -) unstable: (a) ve and (b) we .

Fig. 14. The experimental [24] and perfect beam model frequency response of the second mode of beam 1 for 3 f3∗ = 0.1115( 3 f ∗ = 0.85 mm), 2  = 0.09, 2  = 0.024 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

Fig. 15. The experimental [24] and imperfect beam model frequency response of the first mode of beam 2 for 3 f3∗ = 0.0171( 3 f ∗ = 2 mm), 2  = 0.035, 2  = 0.005 and

ci = ci = 0.0025, (——) stable, (- - - - -) unstable: (a) ve and (b) we .

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O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

parametric excitation one. The same situation holds for the case of base excitation along the X-direction. Although the natural imperfection is small, it is obvious that the dynamic behavior may be different from the straight beams (Figs. 7, 9, 11, and 13). Close agreement between imperfect beam model predictions and the experimental data reported in [24] certifies the validity of the proposed model. As a more complete model, Eqs. (23a) and (24a) can exhibit different non-linear behaviors such as resonant mechanisms, sub- and super-harmonic resonances and other non-linear phenomena such as jump, limit cycles and chaos which are to be investigated in the future. In the second part of this paper, the non-linear steady state responses of a cantilever beam with general imperfection under base excitations are analyzed and compared with the results of the same beam without imperfection, i.e. straight beam. Appendix A ⎡ Fig. 16. The experimental [24] and perfect beam model frequency response of the first mode of beam 2 for 3 f3∗ = 0.0171( 3 f ∗ = 2 mm), 2  = 0.035, 2  = 0.005 and

ci = ci = 0, (——) stable, (- - - - -) unstable, PF—Pitchfork bifurcation: (a) ve and (b) we .

kz = 449 was used and the imperfection coefficients were increased from zero to make the best correlation between theoretical and test results. The values ci = ci = 0.0025 and 2  = 0.005 were found to best correlate the theoretical and test results (see Fig. 15b),

1

0

⎤

0

¯ ¯ ⎥ cos sin ⎦ ¯ cos ¯ 0 − sin ⎤ ⎡ cos ¯ 0 − sin ¯ ⎥ ⎢ [T ] = ⎣ 0 1 0 ⎦ ¯ sin ¯ 0 cos ⎡ ⎤ ¯ ¯ cos  sin  0 ⎢ ¯ cos  ¯ 0⎥ [T ] = ⎣ − sin  ⎦

⎢ [T ] = ⎣ 0

(A.1)



0

0

1

⎤ ¯ ¯ cos ¯ sin  − sin ¯ cos ¯ cos  ⎢ ¯ sin  ¯ + sin ¯ sin ¯ cos  ¯ ¯ cos  ¯ + sin ¯ sin ¯ sin  ¯ ¯ cos ¯ ⎥ [T] = ⎣ − cos cos sin ⎦ ¯ sin  ¯ + cos ¯ sin ¯ cos  ¯ ¯ cos  ¯ + cos ¯ sin ¯ sin  ¯ cos ¯ cos ¯ sin − sin ⎡

meanwhile, this correlation is the worst one among other correlations (i.e. Figs. 7, 9, 11, and 13). Fig. 16 shows the importance of accounting for geometric imperfection. Underestimation of test results in Fig. 15b by the imperfect beam model predictions may be justified by a number of reasons. Since the amplitude of base excitation for the first mode was considerably larger than the second and third modes, small lateral variations in the “straightness” of the tracks on which the base assembly rides was known to have the most effect on the out-of-plane vibrations of the first mode [24]. Another source of discrepancy in Fig. 15b is the accelerometers. The lower operating frequency limit of the accelerometers is 1 Hz while the first natural frequency of beam 2 (i.e. 3.79 Hz) is still low enough to cast some doubt on the accuracy of the accelerometer readings [24]. 5. Conclusions In contrast to the equations derived by Crespo da Silva and Glynn [7] which only contain cubic non-linearities and are linearly uncoupled, Eqs. (23a) and (24a) contain linear, quadratic and cubic terms with the linear and quadratic terms coupled with the natural imperfection, furthermore, they are linearly coupled due to imperfection. Also, if D ≈ D (i.e. beams with near-square or near-circular crosssection), no coupling exists between the lateral and torsional vibrations in the equations of Crespo da Silva and Glynn [7] while there are two such terms in Eqs. (23a) and (24a) (the fifth twice-underlined brackets) which are the consequence of natural imperfection. The third and fifth four-time underlined brackets in Eqs. (23a) and (24a) show that when the excitation is distributed torsional or axial load, a mixed parametric and external excitation problem arises while in the case of geometrically perfect beams, the problem is a pure

⎧ ⎧ ⎫ ⎫ ⎧ ⎫ eˆ  eˆ  eˆ ⎪ ⎪ ⎪ ⎨ x ⎪ ⎨ x ⎪ ⎬ ⎬ ⎬ ⎨ ⎪ eˆ  = [T ¯ ] eˆ y = [T ¯ ][T¯ ] eˆ y ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎭ ⎭ ⎩ ⎪ eˆ  eˆ z eˆ z ⎧ ⎫ ⎧ ⎫ eˆ eˆ ⎪ ⎪ ⎨ x⎪ ⎨ x⎪ ⎬ ⎬ = [T ¯ ][T¯ ][T¯ ] eˆ y = [T] eˆ y ⎪ ⎪ ⎩ ⎪ ⎩ ⎪ ⎭ ⎭ eˆ z eˆ z v¯  ¯ = sin  , (1 + u )2 + v¯ 2 ¯ = tan 

¯ = cos 

(A.3)

1

,

(1 + u )2 + v¯ 2

v¯  1 + u

sin ¯ = 

(A.2)

(A.4) ¯ −w

, ¯ 2 (1 + u )2 + v¯ 2 + w  (1 + u )2 + v¯ 2 cos ¯ =  , ¯ 2 (1 + u )2 + v¯ 2 + w ¯ −w tan ¯ =  (1 + u )2 + v¯ 2 

¯ = 

¯ =

v¯  (1 + u ) − u v¯  (1 + u )2 + v¯ 2 ¯  (u (1 + u ) + v¯  v¯  ) ¯  ((1 + u )2 + v¯ 2 ) + w −w  ¯ 2 ) (1 + u )2 + v¯ 2 ((1 + u )2 + v¯ 2 + w

(A.5)

(A.6)

(A.7)

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 147 -- 160

˙¯ = ˙¯ =

 v˙¯ (1 + u ) − u˙  v¯  + v¯ 2 (1 + u )2

(A.8)

¯  (u˙ (1 + u ) + v¯˙ v¯  ) ¯˙  ((1 + u )2 + v¯  ) + w −w  2 ¯ 2 ) (1 + u )2 + v¯ 2 ((1 + u )2 + v¯  + w

(A.9)

       + w2 0 w + 2w0 w w + 2w0 w0 w )

+ 2 v (v0 + 0 v ) + 2v0 0 v ] 2

(A.10)

2     = v˙¯ + ¯ w˙¯ − v¯  u˙  − v˙¯ ( 12 ¯ + u + v¯ 2 + 12 w2 )

(A.11)

 = ˙¯ + w¯  v˙¯

(A.12)





+u +

1 ¯ 2 2v

− v u − v0 u −

− 0 v − 0 v0 − v2 v − v0 v2        − v2 0 v − 2v0 v v − 2v0 v0 v −

¯ ) +w

       + w2 0 v ) − w0 w v − w0 v0 w )



2 + 12 v2 0 + w0 )

1 2 + v2 0 + 2 w0 )

0 = 0 + w0 v0

J =

D = E

A

 D = G l

0

D = E

A

˙2

 2 d d (A.20)



2

2

2

2

2

˙ − 2u w ˙ −v w ˙ − 2w w ˙ ] − w ˙ 2

˙



2 2 ˙

˙  ˙

˙ − v − 2u v v + J [v + 2 v w

2 2 ˙ 2 + 2 ˙ v˙  (w + w ) ˙ − 2u v˙ 2 − v˙ 2 w2 − 2v2 v˙ 2 ] + J [ + w 0   + v˙ 2 (w2 + w2 0 + 2w0 w )] +

1 2 ˙2 ˙2 (v + w2 0 )(mv + mw 2 0

2

˙ ) ˙ 2 + J v˙ 2 + J + J w 2

2

2

2 − D [w2 + v2 + v2 0 + 0 v

− 2w ( v + v0 + 0 v + w0 u + w u + w0 u + w u + v w v + v0 v w + w0 v v + w0 v0 v + v0 w v + v0 v0 w + v0 w0 v + +

(A.21)

Since the Lagrangian density for the inextensional imperfect beams is expressed as  = (qi , q˙ i , qi , qi , q˙ i , x, t), equations of motion are [22]

j j − jqi jx





j j2 + 2  jqi jx





j j − jqi jt





j j2 + jq˙ i jxjt



0
j jq˙ i



(A.22)



j j − jqi jx 





j j − jqi jt



j jq˙ i





qi = 0,

x = 0, l

(A.23)



j qi = 0, jqi

x = 0, l

(A.24)

In Eqs. (A.22)–(A.24), qi (x, t) stands for the variables u(x, t), v(x, t), w(x, t) and (x, t).

2

˙ 2 − 2 v˙  w ˙  + 2 v˙ 2 − 2v v˙  w w ˙  − 2u˙  w w ˙ × [w 2

+ (2u + u2 + v2 + w2 + 2v0 v + 2w0 w )} dx



˙ ) + J {m(u + v + w

2

1 2 2 2 2 (v + w2 0 )(D w + D v + D ) 2 0

And the associated boundary conditions are

(2 +  ) d d

˙2

−

= −(Fi − ci q˙ i ),

2

A



(A.17)

(A.19) 

2 d d,

A

(A.16)

2 d d

(2 +  ) d d 

1 L= 2

 J =

2

A

+ 2w v (v0 w + w0 v ) + 2v0 w0 w v ]

(A.14)

(A.18)

2 d d,

A



+ 2 (w v + v0 w + w0 v )

(A.15)

0 = −w0 + 0 v0 + v0 w0 v0 + w0 ( 12 20



2 2 2 + D [ + w2 v2 + v2 0 w + w0 v

(A.13)

 = ¯ + w¯  v¯ 



1 2  (w v + v0 w2 2

2

2

2

J =

1 2  1  2 1 2  v − v0 − 0 v 2 2 2

+ 2 w (w0 + 0 w ) + 2w0 0 w )

 = v¯  + ¯ w¯  − v¯  u − v¯  ( 12 ¯ + u + v¯ 2 + 12 w¯ 2 )

0 = v0 + 0 w0 − v0 ( 12 20

2

+ 2v ( w + w0 + 0 w − v u − v0 u

 = − w¯  + ¯ v¯  + w¯  u + v¯  w¯  v¯  ¯2 ¯  ( 12 +w

2

2 − D [v2 + w2 + w2 0 + 0 w

2

2      = −w˙¯ + ¯ v¯˙ +w¯  u˙  +v¯  w¯  v¯˙ + w˙¯ ( 21 ¯ +u + 12 v¯ 2 + w¯ 2 )

159

1 2  2 2 ( w + w0 + 0 w ) + 0 w + 0 w0 2

1 2   (v w + w0 v2 + v2 0w ) 2

+ v0 v w + v0 w0 v + w2 w + w0 w2

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[20] S.N. Mahmoodi, N. Jalili, Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers, Int. J. Non-Linear Mech. 42 (2007) 577. [21] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications Inc., New York, 1944. [22] L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall, Upper Saddle River, NJ, 1997. [23] H.N. Arafat, Non-linear response of cantilever beams, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1999. [24] C.L. Zaretzky, M.R.M. Crespo da Silva, Experimental investigation of non-linear modal coupling in the response of cantilever beams, J. Sound Vib. 174 (2) (1994) 145. [25] G.E. Mase, Theory and Problems of Continuum Mechanics, McGraw-Hill, New York, 1970. [26] T.R. Tauchert, Energy Principles in Structural Mechanics, McGraw-Hill, New York, 1974. [27] O. Aghababaei, H. Nahvi, S. Ziaei-rad, Non-linear non-planar vibrations of geometrically imperfect inextensional beams, part II: base excitations, Int. J. Non-linear Mech. (2008), doi:10.1016/j.ijnonlinmec.2008.10.008.