Linear modal analysis of L-shaped beam structures

Mechanical Systems and Signal Processing 38 (2013) 312–332

Contents lists available at SciVerse ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Linear modal analysis of L-shaped beam structures Fotios Georgiades a,n, Jerzy Warminski a, Matthew P. Cartmell b a b

Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland Department of Mechanical Engineering, University of Sheffield, Sheffield, UK

a r t i c l e in f o Article history: Received 10 May 2012 Received in revised form 29 October 2012 Accepted 17 December 2012 Available online 13 March 2013 Keywords: L-shaped beam structure Modal analysis Elastic continua dynamics Euler–Bernoulli beams Autoparametric structure

Abstract: In this article a theoretical linear modal analysis of Euler–Bernoulli L-shaped beam structures is performed by solving two sets of coupled partial differential equations of motion. The first set, with two equations, corresponds to in-plane bending motions whilst the second set with four equations corresponds to out-of-plane motions with bending and torsion. The case is also shown of a single cantilever beam taking into account rotary inertia terms. At first for the case of examination of the results for the L-shaped beam structure, an individual modal analysis is presented for four selected beams which will be used for modelling an L-shaped beam structure; in order to investigate the influence of rotary inertia terms and shear effects. Then, a theoretical and numerical modal analysis is performed for four models of the L-shaped beam structure consisting of two sets of beams, in order to examine the effect of the orientation of the secondary beam (oriented in two ways) and also shear effects. The comparison of theoretical and finite element simulations shows a good agreement for both in-plane and out-of-plane motions, which validates the theoretical analysis. This work is essential to make progress with new investigations into the nonlinear equations for the L-shaped beam structures within Nonlinear Normal Mode theory. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction There has been continuous interest in the dynamics of L-shaped coupled structures exhibiting nonlinearities and autoparametric coupling since the 1960s. Bolotin in [1] studied principal parametric resonance in a simply supported beam subjected to periodic end moments. In [2], Bux and Roberts studied nonlinear vibratory interactions in L-shaped beam structures. Roberts and Cartmell in [3] studied theoretically and certified experimentally nonlinear modal interactions between a pair of beam modes in forced vibration of an L-shaped beam structure. Cartmell and Roberts in [4] studied simultaneous combination resonances in a parametrically excited nonlinear cantilever beam using first and second order approximation. Balachandran and Nayfeh performed nonlinear modal analysis and studied nonlinear resonances considering only in-plane motions of an L-shaped beam structure with tip masses at some spots of the structure [5]. Warminski et al. in [6] formulated the third order partial differential nonlinear equations for an L-shaped beam structure with different flexibilities in the two orthogonal directions, without taking into account rotary inertia effects. Ozonato et al. studied post-buckled chaotic vibrations of an L-shaped beam structure considering only in-plane bending nonlinear motions [7]. Nayfeh and Pai [8], studied many cases of autoparametric excitations of beams. In Georgiades et al. [9] using Lagrange formulation, the linear equations of motion have been derived for an L-shaped beam structure, considering the inextensionality conditions and rotary inertia terms. That study demonstrated well separated

n

Corresponding author. Tel.: þ 48 81 53 84 144; fax: þ 48 81 53 84 205. E-mail addresses: [email protected] (F. Georgiades), [email protected] (J. Warminski), m.cartmell@sheffield.ac.uk (M.P. Cartmell).

0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.12.006

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

313

in-plane and out-of-plane motions, which has also been shown with Abaqus Finite Element (FE) simulations. Also the necessity for rotary inertia terms in out-of-plane bending was confirmed. In the case that the rotary inertia terms are neglected then the torsional equation of the secondary beam becomes uncoupled from the rest of the out-of-plane motions, and modal analysis for torsion of the secondary beam leads to very different results from the numerical simulations. In this article we start from the equations of motion and then a theoretical linear modal analysis of the L-shaped Euler–Bernoulli beam structure is undertaken considering in-plane and out-of-plane motions. Using two sets of dimensions four models are built in the Abaqus software in order to compare the theoretical with the numerical results. Initially FE models are built and theoretical and numerical modal analyses are undertaken for each one of the individual beams to examine shear and rotary inertia effects in the dimensions of the beams that are considered. Finally, the results for the L-shaped beam structure of the analytical model are compared with the numerical results to validate the theoretical linear modal analysis. 2. Theory 2.1. Equations of motion Beams constructed of isotropic material are considered, with a constant cross section with respect to the longitudinal direction X1, X2 for the primary and secondary beams respectively (see Fig. 1). The equations of motion were derived in [9] and are given by, In-plane motions (bending in the X1Y1 plane for the primary beam and the X2Z2 plane for the secondary beam), € 00 m1 v€ 1 Dz1 vIV 1 þ d4 Iz1 v 1 ¼ 0,

ð1Þ

0

00

€ 2 s2 v€ 1C þ u€ 1C ÞDZ2 wIV € m2 ðw 2 þ d5 IZ2 w 2 ¼ 0,

ð2Þ

where subscript indices 1 and 2 refer to the primary and secondary beams respectively, also m1 and m2 are the masses, Dz1 and DZ2 are the bending stiffnesses of in-plane bending, Iz1 and IZ2 are the rotary inertia terms in in-plane bending. All these parameters are defined explicitly in Section 3, with the following boundary conditions: v01 ð0,t Þ ¼ 0, 0 € d4 Iz1 v€ 1 ðl1 ,t Þ þDz1 v000 1 ðl1 ,t Þl2 m2 v 1 ðl1 ,t Þ ¼ 0, 0 € 2 ðl2 ,t Þ Dz1 v001 ðl1 ,t ÞDZ2 w002 ð0,t Þ þIZ2 d4 d5 l2 v€ 1 ðl1 ,t ÞIZ2 d5 w

v1 ð0,t Þ ¼ 0,

þ

    IZ2 IV € 00 d4 d5 DZ2 wIV 2 ðl2 ,t Þw2 ð0,t Þ IZ2 d5 w 2 ð0,t Þ ¼ 0, m2

w02 ð0,tÞ ¼ 0, 000 € 02 þ v€ 01 ðl1 ,tÞÞ DZ2 w2 ðl2 ,tÞ þ d5 IZ2 ðw

ð3a2dÞ

w2 ð0,tÞ ¼ 0,

¼ 0,

w002 ðl2 ,tÞ ¼ 0,

ð4a2dÞ

Out-of-plane motionsbending, € 1 þ d1 IZ1 w € 001 DZ1 wIV m1 w 1 ¼ 0,

ð5Þ 00

€ 1C Þ þ d2 Iz2 v€ 2 Dz2 vIV € 1C þ w m2 ðv€ 2 þs2 j 2 ¼ 0,

ð6Þ

φ2

w1

ζ2

η2

ξ1

s1+u1

X2 s2+u2

v1 ζ1 Z1

ξ2

η1

Y1

X1

φ1

v2

w2

C

Z2 Y2

Fig. 1. Indcation of axis orientation and displacements for (a) the primary beam and (b) the secondary beam.

314

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

torsion € 1 þ Dx1 j001 ¼ 0, Ix1 j

ð7Þ

€ 01C Þ þ Dx2 j002 ¼ 0, € 2 d1 d3 w Ix2 ðj

ð8Þ

where IZ1 and Iz2 are rotary inertia terms for out of plane bending, DZ1 and Dz2 are the bending stiffnesses for out of plane bending, Ix1 and Ix2 are the rotary inertia terms in torsion, and Dx1 and Dx2 are the torsional rigidities. All these parameters are defined explicitly in Sction 3,with these boundary conditions w1 ð0,tÞ ¼ 0, w01 ð0,tÞ ¼ 0, 000 € 01 ðl1 ,tÞ þ DZ1 w000 € d1 IZ1 w 1 ðl1 ,tÞ þ d2 I z2 j 1 ðl1 ,tÞDz2 v2 ð0,tÞ ¼ 0, DZ1 w001 ðl1 ,tÞd1 d3 Dx2 j02 ð0,tÞ ¼ 0

ð9a2dÞ

j1 ð0,tÞ ¼ 0, € 1 ðl1 ,t Þ þ d2 Iz2 v€ 2 ðl2 ,t Þ þ Dz2 v002 ð0,t Þ þ Dx1 j01 ðl1 ,t Þ þ d2 l2 Iz2 j v2 ð0,tÞ ¼ 0,

I2z2 m2

d2 v€ 002 ð0,tÞ þ d2

 Iz2 Dz2  IV v2 ðl2 ,t ÞvIV 2 ð0,t Þ ¼ 0, m2

ð10a; bÞ

v02 ð0,tÞ ¼ 0,

0 € 1 ðl1 ,tÞÞ þDz2 v000 d2 Iz2 ðv€ 2 ðl2 ,tÞ þ j 2 ðl2 ,tÞ ¼ 0,

v002 ðl2 ,tÞ ¼ 0:

j2 ð0,tÞ ¼ 0, j

ð11a2dÞ 0 2 ðl2 ,tÞ ¼ 0:

ð12a; bÞ

The coordinates u  1, v1, w1, denote the displacements of a selected arbitrary point of the primary (horizontal) beam in the absolute X1,Y1,Z1 coordinate set, and u  2, v2, w2, are the displacements of the secondary (vertical) beam in the local coordinate set X2,Y2,Z2 which is fixed to the centre of a cross-section at point C (see Fig. 1), and j1, j2 are angles of rotation about the x1 and x2 axes respectively. The dot denote derivatives with respect to time. The prime in a variable with subscript 1(2) means derivative with respect to space variable s1(s2). The Kronecker delta function is used in order to model the effect of the rotary inertia terms in the final equations. Considering the boundary conditions, Eqs. (3a,b), (4a,b), (9a,b), (10a), (11a,b), and (12a), are strong boundary conditions which are coming from the system configuration (see Fig. 1). Weak boundary conditions are coming from equilibrium in the shear forces, and the torques and the equations in detail are:  Eq. (3c) corresponds to the shear force equilibrium at point C (see Fig. 1) with respect to the Z1-axis. In this equation the first inertia term is present due to the consideration of the rotary inertia terms (since there is a delta coefficient) and the second inertia term is due to the shear force from the existence of the secondary beam at the free end of the primary beam (it also exists in cases where the secondary beam is considered without flexibility, just as a tip mass).  Eq. (3d) corresponds to the equilibrium of torques at point C with respect to the z1-axis. In this equation the terms with the fourth order space derivative are coming from the replacement of the acceleration terms using the main equation, and since there is a delta function coefficient they appear due to the consideration of rotary inertia terms.  Eq. (4c) corresponds to shear forces equilibrium at the free end of the secondary beam with respect to the z2-axis.  Eq. (4d) corresponds to equilibrium of torques at the free end of the secondary beam with respect to the Z2-axis.  Eq. (9c) corresponds to shear force equilibrium at point C with respect to the z1-axis. The inertia terms in this equation, are coming from consideration of the rotary inertia terms, because they have a delta function coefficient.  Eq. (9d) correspond to torque equilibrium at point C with respect to the Z1-axis.  Eq. (10b) corresponds to equilibrium of torques at point C with respect to the x1-axis. The terms with fourth order derivatives in this equation are coming from the replacement of acceleration using the main equation, and exist due to the consideration of the rotary inertia terms (due to the existence of a coefficient with the delta function).  Eq. (11c), corresponds to equilibrium of shear forces at the free end of the secondary beam with respect to the Z2-axis.  Eq. (11d) corresponds to equilibrium of torques at the free end of the secondary beam with respect to the z2-axis. The inertia terms exist due to the consideration of rotary inertia terms (as indicated by the coefficient of the delta function).  Eq. (12b) corresponds to equilibrium of torques at the free end of the secondary beam with respect to the x2-axis. The following transformations, are applied W 2 ¼ w2 s2 v01C þ u1C ,

V 2 ¼ v2 þs2 j1C þw1C ,

f2 ¼ j2 w01C

ð13a2cÞ

to express the local displacements and rotations of the secondary beam in terms of global displacements and rotations. More precisely regarding Eq. (13a), we re-express the displacement of the secondary beam along the Z2-axis (with local displacement w2) in global coordinates (using system (X1,Y1,Z1)) to give motion along the X1-axis. The local displacement w2 should be considered with the rotation of the primary beam at point C with respect to the z1-axis multiplied by the distance of the considered point on the secondary beam, and also considering the axial displacement of the primary beam at point C. In the case of Eq. (13b), we express the displacement of the secondary beam along the Y2-axis (with local

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

315

displacement v2) in global coordinates (again using system (X1,Y1,Z1)) to produce motion along the Z1-axis. Now the local displacement v2 should be considered, the rotation of the primary beam with respect to the x1-axis multiplied by the distance of the considered point on the secondary beam, and also the displacement w1 at point C of the primary beam in the direction of Z1. Finally, taking Eq. (13c) one expresses the rotation of the secondary beam with respect to the X2-axis in global coordinates (noting that the rotation is with respect to the Y1-axis) and it is necessary to consider the local rotation of the secondary beam j2 and also the rotation of the primary beam at point C, both with respect to the Y1-axis. From Eq. (13a–c) it follows that € 2¼w € 2 s2 v€ 01C þ u€ 1C , W w02

¼ W 02 þ v01C ,

w002

w2 ¼ W 2 þ s2 v01C u1C ,

¼ W 002

€ 1C þ.’ V€ 2 ¼ v€ 2 þs2 j w1C ,

ð14a2dÞ

v02 ¼ V 02 j1C ,

v002 ¼ V 002 ,

ð15a2cÞ

€ 01C , f02 ¼ j02 € 2 w f€ 2 ¼ j

ð16a; bÞ

For in-plane bending the rotary inertia terms have a very small influence and can be neglected. Therefore using

d4 ¼ d5 ¼ 0, and considering the inertia terms for out-of-plane bending d1 ¼ d2 ¼ d3 ¼ 1, and also using Eqs. (13a–c), (14a–d), (15a–c), (16a,b) in Eqs. (1)–(12), then the equations of motion and the boundary conditions are given by a) In-plane motions, m1 v€ 1 þ Dz1 vIV 1 ¼ 0,

€ 2 þ DZ W IV ¼ 0, m2 W 2 2

with boundary conditions v01 ð0,t Þ ¼ 0,

v1 ð0,t Þ ¼ 0,

 € Dz1 v000 1 ðl1 ,t Þ ¼ l2 m2 v 1 ðl1 ,t ,

0

W 2 ð0,tÞ ¼ v01 ðl1 ,tÞ,

W 2 ð0,tÞ ¼ 0,

ð17a; bÞ

W 002 ðl2 ,tÞ ¼ 0,

  Dz1 v001 ðl1 ,t ¼ DZ2 W 002 ð0,t ,

W 000 2 ðl2 ,tÞ ¼ 0

ð18a2hÞ

b) Out-of-plane motions1 00 € 1 IZ1 w € 001 þDZ1 wIV m1 w m2 V€ 2 Iz2 V€ 2 þDz2 V IV 1 ¼ 0, 2 ¼ 0, € D f00 ¼ 0, € D j00 ¼ 0, I f I j

x1

x1

1

x2

1

2

x2

2

ð19a; dÞ

with boundary conditions, w1 ð0,tÞ ¼ 0,

w01 ð0,tÞ ¼ 0,

0

000 € 1 ðl1 ,tÞ þ DZ1 w000 € IZ1 w 1 ðl1 ,tÞ þIz2 j 1 ðl1 ,tÞDz2 V 2 ð0,tÞ ¼ 0, 0

DZ1 w001 ðl1 ,tÞ þ Dx2 f2 ð0,tÞ ¼ 0,

ð20a2dÞ

j1 ð0,tÞ ¼ 0, € 1 ðl1 ,t ÞIz2 V€ 2 ðl2 ,t ÞDz2 V 002 ð0,t Þ Dx1 j01 ðl1 ,t Þ þ Iz2 w V 2 ð0,tÞ ¼ w1 ðl1 ,tÞ,

 I2z2 00 I z Dz IV V€ 2 ð0,t Þ 2 2 V IV 2 ðl2 ,t ÞV 2 ð0,t Þ ¼ 0, m2 m2

ð21a; bÞ

V 02 ð0,tÞ ¼ j1 ðl1 ,tÞ,

0 Iz2 V€ 2 ðl2 ,tÞ þ Dz2 V 000 2 ðl2 ,tÞ ¼ 0,

V 002 ðl2 ,tÞ ¼ 0,

f2 ð0,tÞ ¼ w01 ðl1 ,tÞ, f02 ðl2 ,tÞ ¼ 0,

ð22a2dÞ ð23a; bÞ

It should be noted that the equations of motion and boundary conditions which corresponds to in-plane bending Eqs. (17) and (18), are the same as those ones used in [5] by Balachadran and Nayfeh. In the next section a theoretical modal analysis of the equations of motion is performed (Eqs. (17)–(23)). 2.2. Modal analysis 2.2.1. In-plane motion Using the method of separation of variables in space and time, by means of v1 ðs1 ,tÞ ¼ Y v ðs1 Þav ðtÞ,

W 2 ðs2 ,tÞ ¼ Y W ðs2 ÞaW ðtÞ,

ð24a; bÞ

1 In this boundary condition, W2(0,t)= u1c, but the axial displacement is of second order due to the inextensionality condition, therefore it is neglected for the first order linear problem.

316

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

the equations of in-plane motion (Eq. (17)) take the form 4 Y IV v ðs1 Þmv Y v ðs1 Þ ¼ 0,

4 Y IV W ðs1 ÞmW Y W ðs1 Þ ¼ 0,

DZ Dz a€ v ðtÞ a€ W ðtÞ ¼ ¼ o2in ¼  1 m4v ¼  2 m4W , av ðtÞ aW ðtÞ m1 m2

ð25a2cÞ

The equality on the left hand side of Eq. (25c) arises from the fact that for linear modes the two beams must execute synchronous motion, therefore they must have the same period. Using Eqs. (24), (25) and also Eq. (17a), the boundary conditions of Eq. (18) take the form Y 0v ð0Þ ¼ 0,

Y v ð0Þ ¼ 0,

Dz1 Y 00v ðl1 ,t Þ ¼ DZ2 Y 00W ð0Þ, Y 0W ð0Þ ¼ Y 0v ðl1 Þ,

l2 m2 IV Y ðl1 Þ, m1 v Y W ð0Þ ¼ 0,

Y 000 v ðl1 Þ ¼ 

Y 00W ðl2 Þ ¼ 0,

Y 000 W ðl2 Þ ¼ 0:

ð26a2hÞ

The mode shapes can be determined from the solution of the ordinary differential equations of (25a,b), which is a trivial problem and those solutions are given by Y v ðs1 Þ ¼ C 1 sinðmv s1 Þ þC 2 cosðmv s1 Þ þ C 3 sinhðmv s1 Þ þ C 4 cosðmv s1 Þ  Y W ðs1 Þ ¼ D1 sinðmW s2 þ D2 cosðmW s2 Þ þ D3 sinhðmW s2 Þ þD4 coshðmW s2 Þ,

ð27a; bÞ

The constants in these equations are determined as functions of C 1 and are given in Appendix A by means of some appropriate algebraic manipulations and using the boundary conditions Eq. (26a–h) and Eq. (27a,b). These constants are given by C 3 ¼ C 1 ,

ð28aÞ

C2 ¼ C1CL,

ð28bÞ

C 4 ¼ C 1 C L ,

ð28cÞ

D1 ¼ 

mv fcosðmv l1 Þcoshðmv l1 Þ þ C L ½sinðmv l1 Þsinhðmv l1 Þg C1, mW ð1þ C g Þ

ð29aÞ

by using Eqs. (29a) and (5),(7),(8) the other constants for Eq. (27b) are given as

mv fcosðmv l1 Þcoshðmv l1 Þ þ C L ½sinðmv l1 Þsinhðmv l1 Þg C1, mW ð1 þC g Þ

ð29bÞ

mv fcosðmv l1 Þcoshðmv l1 Þ þ C L ½sinðmv l1 Þsinhðmv l1 Þg C1Cg , mW ð1þ C g Þ

ð29cÞ

mv fcosðmv l1 Þcoshðmv l1 Þ þ C L ½sinðmv l1 Þsinhðmv l1 Þg C1, mW ð1 þC g Þ

ð29dÞ

D2 ¼ C a

D3 ¼ 

D4 ¼ C a

The natural frequencies can be determined through Eq. (26d), and mv which are defined by the roots of a transcedental equation. This equation originates from the boundary condition equation (26d) and Eq. (27a,b) and is given by h Dz1 m2v C 1 sinðmv l1 ÞC 2 cosðmv l1 Þ  ð30Þ þ C 3 sinhðmv l1 Þ þC 4 coshðmv l1 Þ ¼ DZ2 m2W ½D2 D4 , by also using Eqs. (28) and (29) and some manipulations, it emerges that vffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 u          4 m1 Dz  1 t sin mv l1 sinh mv l1  cos mv l1 þ cosh mv l1 3 m2 DZ2 "          #) cos mv l1 cosh mv l1 þ m2 l2 mv =m1 sin mv l1 sinh mv l1            sinh mv l1 sin mv l1 þ m2 l2 mv =m1 cosh mv l1 cos mv l1 (               ) sin mv l2 =g cos mv l2 =g sinh mv l2 =g sin mv l2 =g cosh mv l2 =g cos mv l2 =g þ cosh mv l2 =g  þ             1þ sinh mv l2 =g sinh mv l2 =g sin mv l2 =g sinh mv l2 =g þ cos mv l2 =g cosh mv l2 =g þ1            sin mv l2 =g sinh mv l2 =g þcos mv l2 =g cosh mv l2 =g þ 1          þ 2 sinh mv l2 =g cos mv l2 =g sin mv l2 =g cosh mv l2 =g            cos mv l1 cosh mv l1  sin mv l1 þ sinh mv l1 "          #) cos mv l1 cosh mv l1 þ m2 l2 mv =m1 sin mv l1 sinh mv l1            ¼0 sinh mv l1 sin mv l1 þ m2 l2 mv =m1 cosh mv l1 cos mv l1

ð31Þ

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

317

with

m g¼ v ¼ mW

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 m1 DZ2 : m2 Dz1

ð32Þ

in the literature there are only few simple cases e.g. axial motion and torsion of cantilever beam etc., whereas the natural frequencies are given explicitly by a formula considering the stiffness and dimensions of the system. It should be noted that in this case it is very difficult to obtain such explicit formulae for the roots of the transcedental equation (31) which are giving explicitly the natural frequencies in-plane motion considering the dimensions and stifnesses. Considering parameters for a given L-shaped beam structure, then the natural frequencies can be determined by the roots of this equation and can be found easily by examining the zero crossings of the plot of this equation with respect to mv . 2.2.2. Out-of-plane motion Using the method of separation of variables in space and time for out-of-plane motion, by means of w1 ðs1 ,tÞ ¼ Y w ðs1 Þhw ðtÞ,

V 2 ðs2 ,tÞ ¼ Y V ðs2 ÞhV ðtÞ

j1 ðs1 ,tÞ ¼ Y j ðs1 Þhj ðtÞ, f2 ðs2 ,tÞ ¼ Y F ðs2 ÞhF ðtÞ Eq. (19) then take the form  IZ1 2 Y 00w Z2w Y w ¼ 0, Y IV w þ Zw m1 Y 00j þ Z2j Y j ¼ 0,

2 Y IV V þ ZV

ð33a2dÞ



Iz2 Y 00V Z2V Y V ¼ 0, m2

Y 00F þ Z2F Y F ¼ 0,

hw ðt Þ ¼ hV ðt Þ ¼ hj ðt Þ ¼ hF ðt Þ ¼ hðt Þ, € DZ Dz Dx Dx hðtÞ ¼ o2out ¼  1 Z2w ¼  2 Z2V ¼  1 Z2j ¼  2 Z2F : hðtÞ m1 m2 Ix1 Ix2

ð34a2fÞ

Similarly to the case of the in-plane motions, Eq. (34e) arises from the fact that for linear modes the two beams must execute synchronous motions with the same period. Consequently the solution of the differential equation in the time domain will be the same for all the variables. Therefore, using Eqs. (33) and (34), the boundary conditions (20)–(23) take the form Y w ð0Þ ¼ 0,

Y 0w ð0Þ ¼ 0,

o2out IZ1 Y 0w ðl1 Þ þDZ1 Y 000w ðl1 ,tÞIz2 o2out Y j ðl1 ÞDz2 Y 000V ð0Þ ¼ 0, DZ1 Y 00w ðl1 Þ þDx2 Y 0F ð0Þ ¼ 0, Y j ð0Þ ¼ 0, 

Dx1 Y 0j ðl1 Þo2out Iz2 Y w ðl1 Þ þ Iz2 o2out Y V ðl2 ÞDz2 Y 00V ð0Þ þ o2out

 Iz2 Dz2 IV Y V ðl2 ÞY IV V ð0Þ ¼ 0, m2

Y V ð0Þ ¼ Y w ðl1 Þ,

ð35a2dÞ I2z2 00 Y ð0Þ m2 V

Y 0V ð0Þ ¼ Y j ðl1 Þ,

o2out Iz2 Y 0V ðl2 Þ þDz2 Y 000V ðl2 Þ ¼ 0, Y 00V ðl2 Þ ¼ 0, Y F ð0Þ ¼ Y 0w ðl1 Þ,

ð36a; bÞ

Y 0F ðl2 Þ ¼ 0:

ð37a2dÞ ð38a; bÞ

As shown in Appendix B the general solutions of equations (34a,b) are given by Y w ðs1 Þ ¼ E1 sinðd1 s1 Þ þ E2 cosðd1 s1 Þ þ E3 sinhðd2 s1 Þ þ E4 coshðd2 s1 Þ         Y V ðs2 Þ ¼ F 1 sin f 1 s2 þ F 2 cos f 1 s2 þ F 3 sinh f 2 s2 þ F 4 cosh f 2 s2 , with

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 u 2 tZw IZ1 =m1 þ Z4w IZ1 =m1 þ4Z2w , d2 ¼ , d1 ¼ 4 4 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  u  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 u 2 u 2 tZV Iz2 =m2 þ Z4V Iz2 =m2 þ 4Z2V tZV Iz2 =m2 þ Z4V Iz2 =m2 þ 4Z2V , f2 ¼ , f1 ¼ 4 4

ð39a; bÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 u 2 tZw IZ1 =m1 þ Z4w IZ1 =m1 þ 4Z2w

ð40a; bÞ

The general solutions of equations (34c,d) are given by Y j ðs1 Þ ¼ K 1 sinðZj s1 Þ þ K 2 cosðZj s1 Þ,     Y F ðs2 Þ ¼ M1 sin ZF s2 þ M2 cos ZF s2 :

ð41a; bÞ

318

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

The mode shapes for out-of-plane motion are defined by Eqs. (39) and (41), and the general solution constants have been determined with respect to E1 through the boundary conditions (35)–(38), as explained in Appendix C. These constants are given by E1 ¼ E2 q3 ,

E3 ¼ 

d1 E2 q3 , d2

F 1 ¼ E2 ðq10 q6 þ q7 Þ,

E4 ¼ E2 ,

F 2 ¼ q10 E2 ,

K 1 ¼ E2 ðq10 q2 þq3 q4 þ q5 Þ,

ð42a2cÞ

F 3 ¼ E2 ðq10 q8 þ q9 Þ,

F 4 ¼ E2 ðq1 q10 Þ,

K 2 ¼ 0,

ð44a; bÞ

( M 1 ¼ E2

ð43a2dÞ

i q3 DZ1 h 2 DZ1 h 2 2 d sinðd1 l1 Þ þd1 d2 sinhðd2 l1 Þ þ d cosðd1 l1 Þ þ d2 coshðd2 l1 Þ Dx2 mj2 1 Dx2 mj2 1

M 2 ¼ E2 fq3 ½d1 cosðd1 l1 Þ þd1 coshðd2 l1 Þ þ d1 sinðd1 l1 Þ þ d2 sinhðd2 l1 Þg:

) i ð45a; bÞ

The natural frequencies of the linear modes for out-of-plane motion are given by the roots of the transcendental equation. This arises from boundary condition (36b) using also Eqs. (42)–(45)  Dx1 q10 q2 þ q3 q4 þ q5 ÞZj cos Zj l1

 d1 o2out Iz2 q3 sinðd1 l1 Þ þcosðd1 l1 Þ q3 sinhðd2 l1 Þcoshðd2 l1 Þ d2   2 þ Iz2 oout ðq10 q6 þ q7 Þsinðf 1 l2 Þ þq10 cosðf 1 l2 Þ þ ðq10 q8 þq9 Þsinhðf 2 l2 Þ þ ðq1 q10 Þcoshðf 2 l2 Þ ! h i I D h I2z  2  4     z z 2 4 þ o2out 2 Dz2 q1 q10 f 2 q10 f 1  2 2 q10 q6 þ q7 f 1 sin f 1 l2 þq10 f 1 cos f 1 l2 m2 m2 4

4

2

2

þ ðq10 q8 þ q9 Þf 2 sinhðf 2 l2 Þ þðq1 q10 Þf 2 coshðf 2 l2 Þq10 f 1 ðq1 q10 Þf 2  ¼ 0,

ð46Þ

It should be noted that the parameters qi (i ¼1...10) are frequency dependent, and are used to simplify the expressions. They are defined explicitly in Appendix C. Similarly with in-plane motion, due to complexity of the transcedental Eq. (46) an explicit formulae to determine the natural frequencies of the system which corresponds to the roots of this equation are very difficult to obtain. In the case of an L-shaped beam structure with specific dimensions, the natural frequencies are the roots of this transcendental equation (46) and can be found easily by examining the zero crossings of the plot of this equation with respect to oout . 2.2.3. Cantilever beam In the case of a single cantilever beam in which rotary inertia terms are considered, the equation of motion and boundary conditions arising (e.g. by Eqs. (19a or b), (20), or (22) by eliminating terms which corresponds to the connection with the other beam) are given by, 00 mV€ IV€ þ DV IV ¼ 0,

ð47Þ

with boundary conditions Vð0,tÞ ¼ 0,

V 0 ð0,tÞ ¼ 0,

0

IV€ ðL,tÞ þ DV 00 ðL,tÞ ¼ 0,

V 00 ðL,tÞ ¼ 0

ð48a2dÞ

using the separation of variables by means of Vðs,tÞ ¼ YðsÞaðtÞ, in Eq. (47), leads to  I Y 00 Z2 Y ¼ 0, Y IV þ Z2 m

ð49Þ

€ aðtÞ D ¼ o2rot ¼  Z2 , aðtÞ m

ð50a; bÞ

with boundary conditions Yð0Þ ¼ 0,

Y 0 ð0Þ ¼ 0,

o2rot IY 0 ðLÞ þ DYðLÞ ¼ 0, Y 000 ðLÞ ¼ 0:

ð51a2dÞ

The solution of Eq. (50a) with the necessary boundary condition (Eq. (51c)) and Eqs. (B.9)–(B.11), as defined in Appendix B, leads to a transcendental equation. The roots of this transcendental equation define the natural frequencies of the beam in bending, considering also the inertia terms. This transcendental equation is given by D½s31,s cosðs1 LÞDL s31,s sinðs1 LÞs1,s s22,s coshðs2,s LÞ þ DL s32,s sinhðs2,s LÞ þ o2rot I½s1,s cosðs1,s LÞ þ DL s1,s sinðs1,s LÞs1,s coshðs2,s LÞ þDL s2,s sinhðs2,s LÞ ¼ 0,

ð52Þ

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

319

In the case where rotary inertia terms are neglected the transcendental equation (52) when Eqs. (50b) and (B.12) are considered, takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi       4 4 4 ðmo20 Þ=D L cos ðmo20 Þ=D L DL,0 sin ðmo20 Þ=D L cosh qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4 þ DL,0 sinh ðmo20 Þ=D LÞ ¼ 0

ð53Þ

for which we also have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sin 4 ðmo20 Þ=D L þ sinh 4 ðmo20 Þ=D L DL,0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    cos 4 ðmo20 Þ=D LÞ þ cosh 4 ðmo20 Þ=D L

ð54Þ

The roots of Eq. (53) with respect to o0 denote the natural frequencies of a cantilever beam, when the rotary inertia terms are neglected.

3. Models In order to validate the theoretical results of modal analysis, four FE models were constructed in the Abaqus programme consisting of two sets of beams (with indices a,b respectively). The material for all the models was aluminium of density, r ¼2800 kg/m3, Young0 s modulus E¼70 MPa, Poisson0 s ratio n ¼0.33, and shear modulus, G12 ¼ 26.32 MPa. The dimensions of the beams for each set are l1a  b1a  h1a ¼ 0:2 m  0:004 m  0:003 m, l2a  b2a  h2a ¼ 0:15 m  0:004 m  0:003 m, l1b  b1b  h1b ¼ 0:18 m  0:01295 m  0:00216 m l2b  b2b  h2b ¼ 0:21 m  0:01295 m  0:00216 m, For each set of dimensions two different models were constructed with two different orientations of the secondary beam (Fig. 2a and c). In the first orientation the secondary beam is oriented in such a way that its width is in the same direction as the width of the primary beam (Fig. 2a). In the second model the secondary beam has a transversal orientation (Fig. 2c). The parameters used in Eqs. (17) and (19) are defined as follows: a) For model 1 with dimensions given for the first set of beams with the first orientation of the secondary beam, and also considering the warping coefficient for torsional rigidity [8] 3

h1a b1a ¼ 4:480  108 kg m m1,a ¼ rb1a h1a ¼ 0:034 kg=m, IZ1,a ¼ r 12 ! 3 3 3 h1a b1a þb1a h1a b1a h1a ¼ 0:630 Pa m4 , Ix1,a ¼ r ¼ 7:000  108 kg m, Dz1,a ¼ E 12 12 3

h1a b1a ¼ 1:120 Pa m4 , 12   ! 1 1 192h1a X 1 npb1a 3 ¼ G12 b1a h1a 1 5 tanh ¼ 0:513 Pa m4 , 3 p b1a n ¼ 1,3,... n5 2h1a

DZ1,a ¼ E Dx1,a

3

h2a b2a ¼ 4:480  108 kg m, m2,a ¼ rb2a h2a ¼ 0:034 kg=m, Iz2,a ¼ r 12 ! 3 3 3 h2a b2a þb2a h2a h2a b2a ¼ 1:120 Pa m4 , Ix2,a ¼ r ¼ 7:000  108 kg m, Dz2,a ¼ E 12 12 3

b2a h2a ¼ 0:630 Pa m4 , 12   ! 1 1 192h2a X 1 npb2a 3 ¼ G12 b2a h2a 1 5 tanh ¼ 0:513 Pa m4 , 3 p b2a n ¼ 1,3,... n5 2h2a

DZ2,a ¼ E Dx2,a

ð55a2lÞ

320

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

Fig. 2. Characteristic mode shapes of two motions for first set of dimensions: (a) in-plane bending motion for the first orientation of the secondary beam, (b) out-of-plane motion for the first orientation of the secondary beam, (c) in-plane bending motion for the transversal orientation of the secondary beam, and (d) out-of-plane motion for the transversal orientation of the secondary beam.

b) For model 2 and using the first set of dimensions but with the transversal orientation of the secondary beam, 3

b2a h2a ¼ 2:520  108 kg m, 12 3 h2a b2a ¼ 1:120 Pa m4 , ¼E 12

Iz2,a,t ¼ r DZ2,a,t

3

Dz2,a,t ¼ E

b2a h2a ¼ 0:630 Pa m4 , 12 ð56a2cÞ

the other parameters are the same as those for the first orientation of the secondary beam (model 1). c) For model 3 the dimensions are given by the second set of beams and they take the first orientation of the secondary beam, being defined by 3

h b m1,b ¼ rb1b h1b ¼ 0:078 kg=m, IZ1b ¼ r 1b 1b ¼ 1:095  106 kg m, 12 ! 3 3 3 h1b b1b þb1b h1b b h Ix1,b ¼ r ¼ 1:125  106 kg m, Dz1,b ¼ E 1b 1b ¼ 0:761 Pa m4 , 12 12   ! 3 1 X h b 1 192h 1 npb1b 3 ¼ 1:024 Pa m4 , tanh DZ1,b ¼ E 1b 1b ¼ 27:364 Pa m4 , Dx1,b ¼ G12 b1b h1b 1 5 1b 3 12 p b1b n ¼ 1,3,... n5 2h1b ! 3 3 3 h b h b þ b2b h2b m2,b ¼ rb2b h2b ¼ 0:078 kg=m, Iz2,b ¼ r 2b 2b ¼ 1:095  106 kg m, Ix2,b ¼ r 2b 2b 12 12 ¼ 1:125  106 kg m, 3

3

h2b b2b b h ¼ 27:364 Pa m4 , DZ2,b ¼ E 2b 2b ¼ 0:761 Pa m4 , 12 12   ! 1 1 192h2b X 1 npb2b 3 ¼ G12 b2b h2b 1 5 tanh ¼ 1:024 Pa m4 , 3 2h2b p b2b n ¼ 1,3,... n5

Dz2,b ¼ E Dx2,b

ð57a2lÞ

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

321

d) Finally for model 4, using the second set of dimensions but with the secondary beam oriented in the transversal manner, they are given by 3

b2a h2a ¼ 3:045  108 kg m, 12 3 b2a h2a ¼ 0:761 Pa m4 , Dz2,a,t ¼ E 12 3 h2a b2a ¼ 27:364 Pa m4 , DZ2,a,t ¼ E 12 Iz2,a,t ¼ r

ð58a2cÞ

The other parameters are the same as those for the first orientation of the secondary beam (model 3). According to Vlasov0 s theory the shear effects are dominant only in cases where the dimensions with respect to the length have a ratio higher than 0.1 [11], and for the second set of dimensions (models 3 and 4) we expect to observe some shear effects for the higher modes. This is due to the fact that the widths in the second set of beams are one order of magnitude higher than those of the first set of beams, whereas the lengths are of the same order. Before modelling the L-shaped beams a modal analysis of a single cantilever beam was performed in Abaqus, with the dimensions used for the L-shaped beam structure. We modelled them using two kinds of elements, denoted as wire (B31) and shell (S4) elements, and also considering shear effects. The difference in the results obtained by using these two kinds of elements, with examination restricted to the first 5 bending modes for each one motion (in-plane and out-of-plane bending), was less than 1%. Therefore only the results of the model with wire elements are presented here noting that 150 elements were used for all cases. In order to perform a comparison between the mode shapes we selected specific points on each beam which corresponded to a division of the beam length into 30 elements for all cases. Furthermore the four FE models of the L-shaped beam structures were constructed in Abaqus in order to perform numerical modal analysis. For each model, and to ensure correct numerical results, two kinds of elements were used in the form of wire (B31) and shell (S4) elements, for which shear effects were also considered. In the case of wire modelling, 150 elements were used for each beam. In the case of shell modelling and (a) model 1, 150 elements in the longitudinal direction  30 elements for each beam, were used and (b) models 2 and 4, 151 elements in the longitudinal direction  16 elements for the primary beam and 150 elements in the longitudinal direction  16 elements for the secondary beam, were used and (c) in model 3, 150 elements in the longitudinal direction  16 elements for all beams were used. In order to compare mode shapes between the theory and the FE simulations we selected specific points in each beam. In the case of models 1 and 3, the selection of points corresponds to division of the beam lengths into 30 elements. In the case of models 2 and 4 the primary beam was divided into 31 elements and the secondary beam was divided into 30 elements. 4. Results and discussion In order to validate the theoretical modal analysis the theoretical results were compared with the FE simulations in terms of the natural frequencies and mode shapes. The comparison of mode shapes was carried out using the Modal Assurance Criterion (MAC). The MAC comparison between two mode shape vectors YMAC,1 and YMAC,2 can be estimated using the following formula [10]:   MAC Y MAC,1 ,Y MAC,2 ¼

9fY MAC,1 gT fY MAC,2 g9 T

2

T

ð59Þ

ðfY MAC,1 g fY MAC,1 gÞðfY MAC,2 g fY MAC,2 gÞ

noting that the superscript T defines the transpose. In the case of in-plane motion of the L-shaped beam structure the mode shape is given by fY MAC g ¼ ½fY v g,fY w gT

ð60aÞ

for which the mode shape vector is determined at the aforementioned selected points. In the case of out-of-plane motion for the structure then the mode shape is given by fY MAC g ¼ ½fY V g,fY w g,fY F g,fY j gT

ð60bÞ

Fig. 2a,b (c,d) depict the mode shapes using Abaqus for the 10th modes of in-plane and out-of-plane motion, respectively, for model 1 (model 2). 4.1. Results for single beams Theoretical modal analysis was performed for a single cantilever beam in bending by finding the roots of transcendental equation (52) considering the inertia terms and (53) neglecting the inertia terms for all the beams used in the modelling of the L-shaped structure. The dimensions are defined in the previous section and the inertia and stiffness terms are defined by Eqs. (56)–(59). For both sets of beam dimensions the results for the primary and the secondary beam are

322

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

Table 1 In-plane bending of beam 2 model 1. Mode Freq. FE (Hz)

Freq., theory without inertia terms (Hz)

Freq., theory, with inertia terms (Hz)

% Rel. dif. with inertia effect (theory)

% Rel. dif. FE with theory (inertia)

MAC FE with theory (inertia)

1 2 3 4 5

107.7 674.9 1889.7 3703.2 6121.6

107.7 67.5 1887.3 3694.4 6098.4

0.0  0.1  0.1  0.2  0.4

0.0 0.2 0.4 0.7 1.2

1.000 1.000 1.000 1.000 1.000

107.7 673.4 1879.6 3667.0 6027.3

Table 2 Out-of-plane bending of beam 2 model 1. Mode Freq. FE (Hz)

Freq., theory without inertia terms (Hz)

Freq., theory, with inertia terms (Hz)

% Rel. dif. with inertia effect (theory)

% Rel. dif. FE with theory (inertia)

MAC FE with theory (inertia)

1 2 3 4 5

143.6 899.9 2519.7 4937.5 8162.1

143.6 899.0 2513.9 4916.8 8107.5

0.0  0.1  0.2  0.4  0.7

0.1 0.3 0.7 1.3 2.0

1.000 1.000 1.000 1.000 1.000

143.5 896.3 2496.0 4853.2 7943.9

Table 3 In-plane bending of beam 2 model 2. Mode Freq. FE (Hz)

Freq., theory without inertia terms (Hz)

Freq., theory, with inertia terms (Hz)

% Rel. dif. with inertia effect (theory)

% Rel. dif. FE with theory (inertia)

MAC FE with theory (inertia)

1 2 3 4 5

39.6 247.9 694.2 1360.3 2248.7

39.6 247.9 694.0 1359.5 2246.5

0.0 0.0 0.0  0.1  0.1

0.0 0.0 0.1 0.2 0.3

1.000 1.000 1.000 1.000 1.000

39.6 247.8 693.2 1356.7 2239.2

Table 4 Out-of-plane bending of beam 2 model 2. Mode Freq. FE (Hz)

Freq., theory without inertia terms (Hz)

Freq., theory, with inertia terms (Hz)

% Rel. dif. with inertia effect (theory)

% Rel. dif. FE with theory (inertia)

MAC FE with theory (inertia)

1 2 3 4 5

237.2 1486.4 4161.9 8155.7 13,482.0

237.0 1478.8 4111.9 7977.1 13,019.7

 0.1  0.5  1.2  2.2  3.6

0.2 1.6 3.5 6.0 8.8

1.000 1.000 0.999 0.997 0.993

236.5 1455.8 3967.8 7497.0 11,874.0

similar, therefore only an analysis for the secondary beam is presented in this paper. In Tables 1–4) the results of the examination of the modal analysis of the secondary beam of model 1 (model 2) in bending are shown. The frequencies which resulted from theoretical analysis, with and without considering the inertia terms, are presented there, and also the frequencies from the FE simulations. There is also a comparison of the natural frequencies from the theoretical results which demonstrates the effect of the rotary inertia terms, and a comparison of the theoretical with the FE simulation results in terms of frequencies and mode shapes. In the case of the in-plane bending of model 1 (Table 1) the inertia terms clearly have no significant effect. The comparison of theoretical results with FE simulations, shown in Table 1, shows that they are in very good agreement in terms of frequencies and MAC. Therefore it may be concluded that the shear effect for the first five modes is not significant. In the case of out-of-plane bending of the secondary beam of model 1, the results are shown in Table 2.The inertia and shear effects are also insignificant there. The results for the secondary beam of model 2 are shown in Tables 3 and 4 for in-plane bending (out-of-plane bending). In the case of in-plane bending the shear and rotary inertia effects are insignificant but in out-of-plane bending of the secondary beam (model 2) the shear and rotary inertia effects are more dominant, as shown in Table 4.

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

323

4.2. In-plane motions of L-shaped beam structure Considering the in-plane motions in this structure then a comparison of the theoretical and numerical results for the four models, taking into account just the first 10 modes, is presented in Tables 5–7. In Table 5 the theoretical results for model 1 are in a very good agreement with those from FE simulations. There is no significant difference between the results of the wire and the shell element models, and the maximum relative difference between the numerical and theoretical results is 2%, whereas the MAC diagonal values are 1.00. Fig. 3a depicts the theoretical and numerical (wire model) results for the 10th mode shape for both beams (the same scale coefficient is used for both beams undergoing theoretical mode shapes) with a good agreement and this confirms the high MAC values. In the case of model 2 for which the secondary beam is placed in the transversal configuration, the results are shown in Table 6. In this table there is a good agreement between the natural frequencies for both models, with less than a 4% difference, for the first nine modes. Therefore, the shear effect influence is minimal. In the case of the wire model for the 1st mode and then in the case of the shell model for the 9th mode there is a low MAC value due to numerical errors. It should be highlighted that in the 10th mode there is greater discrepancy in the natural frequencies for both models, and also there are low MAC values due to the imposed inextensionality conditions. More precisely, the axial displacement of Table 5 In-plane motion, model 1. Mode FE mode

Freq.,FE-a, B31 Freq., FE-b, S4 Freq., theory % Rel. diff. theory (Hz) (Hz) (Hz) with FE-a

% Rel. diff. theory with FE-b

MAC theory with FE-a

MAC theory with FE-b

1 2 3 4 5 6 7 8 9 10

25.8 77.9 337.2 573.5 998.2 1628.0 2032.4 3102.7 3529.2 4828.8

0.0 0.1 0.1 0.3 0.3 0.7 0.8 1.0 1.9 1.4

0.0 0.0 0.0 0.2 0.2 0.6 0.6 0.9 1.7 1.2

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1 3 5 7 9 11 12 15 16 20

25.8 78.0 337.5 573.8 999.3 1629.0 2035.2 3106.8 3534.1 4839.2

25.8 78.0 337.6 575.2 1001.7 1639.1 2048.5 3133.7 3596.6 4896.6

Table 6 In-plane motion, model 2. Mode FE mode

Freq.,FE-c, B31 Freq.,FE-d, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-c

% Rel. diff. theory with FE-d

MAC theory with FE-c

MAC theory with FE-d

1 2 3 4 5 6 7 8 9 10

25.8 84.8 365.4 697.7 1053.9 1894.7 2275.7 3299.4 4284.6 4997.5

0.4 0.0 0.1 0.4 0.4 0.6 1.4 1.1 2.9 2.5

0.4 2.9 1.8 1.1 2.7 0.7 3.4 2.1 3.2 3.8

0.96 0.99 0.99 1.00 1.00 1.00 1.00 0.99 0.99 0.98

1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.97 0.95

1 4 5 8 9 12 13 15 18 20

25.8 82.3 359.4 692.4 1029.5 1892.7 2228.1 3264.0 4268.9 4930.8

25.9 84.8 365.9 700.4 1058.6 1906.8 2307.0 3335.0 4412.1 5126.0

Table 7 In-plane motion, model 3. Mode FE mode

Freq., FE-e, B31 Freq., FE-f, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-e

% Rel. diff. theory with FE-f

MAC theory with FE-e

MAC theory with FE-f

1 2 3 4 5 6 7 8 9 10

15.2 41.4 192.2 321.1 612.2 878.9 1258.6 1722.6 2132.2 2842.5

0.0 0.0 0.1 0.1 0.2 0.2 0.4 0.4 0.6 0.6

 0.7  0.5  0.1  0.6  0.1  0.4 0.0  0.3 0.0  0.2

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1 3 4 6 7 8 11 12 14 16

15.3 41.6 192.6 323.2 613.8 884.6 1263.8 1734.7 2145.0 2864.8

15.2 41.4 192.4 321.4 613.2 881.0 1263.3 1729.5 2146.0 2860.2

324

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

In-plane bending, 1st Model, 10th Mode

In-plane bending, 2nd Model, 10thMode

0.5

0

-0.5 FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

-1

0

0.05

0.1

0.15

0.2

Displacement-scaled

Displacement-scaled

1

0.25

1 0.8 0.6 0.4 0.2 0 -0.2 FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

-0.4 -0.6 -0.8 -1 0

0.05

In-plane bending, 3rd Model, 10th Mode

0.5 0 -0.5

0.05

0.1

0.15

0.25

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-1 0

0.2

In-plane bending, 4th Model, 10th Mode

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

1

0.15

1

Displacement-scaled

Displacement-scaled

1.5

0.1

x-position

x-position

0.2

0.25

-1

0

0.05

x-position

0.1

0.15

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2 0.2 0.25

x-position

Fig. 3. 10th Mode shape in in-plane bending for: (a) the 1st model, (b) the 2nd model, (c) the 3rd model, and (d) the 4th model.

Table 8 In-plane motion, model 4. Mode FE Mode

Freq., FE-h, B31 Freq., FE-g, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-g

% Rel. diff. theory with FE-h

MAC theory with FE-g

MAC theory with FE-h

1 2 3 4 5 6 7 8 9 10

15.9 55.4 360.1 927.4 1070.6 1857.1 3005.0 3278.5 4526.9 5965.4

0.0 0.0 0.1 0.4 1.3 0.5 1.1 4.4 1.3 6.2

 0.6  2.9  3.5  1.6  0.5  3.4  1.2 2.6  2.9 6.1

0.98 0.98 0.99 0.98 1.00 0.96 0.94 0.99 0.91 0.12

1.00 1.00 1.00 0.98 0.97 0.99 0.90 0.88 0.99 0.11

1 3 6 8 9 13 16 18 21 25

16.0 57.0 373.2 946.2 1090.4 1930.3 3074.7 3339.7 4718.6 5968.4

15.9 55.4 360.5 931.3 1085.0 1867.2 3037.5 3430.6 4586.9 6356.6

the primary beam due to the inextensionality condition, is a second order nonlinear term, and this has been neglected from the formulation. Fig. 3b depicts the theoretical and numerical wire model 10th mode shape for the 2nd model, using the same scale coefficient for both beams for the theoretical mode shape. Examining the FE mode shape of the secondary beam in Fig. 3b it is possible to notice a displacement at the clamped end (x ¼0) which is due to the axial displacement of the primary beam, this was neglected in the theoretical modal analysis. Therefore the coupling between bending in the secondary beam and the axial motion of the primary beam has been neglected, and so this axial displacement makes a significant difference to this mode shape. The mode shape from the shell model in Abaqus is presented in Fig. 2b, in which the axial displacement or shortening of the primary beam is visible. The theoretical and numerical results for model 3, presented in Table 7, are in a good agreement, with a minimal shear effect. This is justified further on from examination of Fig. 3c. In this figure (Fig. 3c), the theoretical and numerical results for the wire model of the 10th mode shape are depicted with the same scale coefficient for both beams in the theoretical mode shapes, and they are in a good agreement.

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

325

In model 4 the secondary beam is oriented in the transversal direction, therefore in the higher modes it is expected that the influence of shear effects will be apparent. The results of modal analysis for model 4 are shown in Table 8, in which the theoretical results are in a relatively good agreement with the finite element simulations results. The differences in the natural frequencies of the theoretical results with those of the corresponding finite element simulations are higher than the differences shown by the other models e.g. Tables 5–7. Detailed examination of the mode shapes for this model (4) indicates that the low MAC values for the 6th and 9th modes of the B31 wire element model in Table 8 are due to numerical errors. Similarly, the low MAC values for the 5th and 8th modes in the case of the S4 shell element model are also due to numerical errors. For both the B31 and S4 element models the low MAC values in Table 8 for the 7th mode seem to be due to shear effects. Examination of the 10th mode shapes of the finite element models shows, apart from the influence of some shear effects, that there is also significant axial displacement in the primary beam, and this is similarly for model 2, and it explains the low MAC values and the relative differences in the natural frequencies (Table 8) for this mode. Fig. 3d depicts the theoretical and numerical results for the wire element model specifically the 10th mode. By considering the mode shape of the secondary beam from the finite element simulation in Fig. 3d, it is clear that at the clamped end (x ¼0) there is a significant displacement caused by the axial displacement of the primary beam, and this explains the low MAC values for this mode. Also it is expected that shear effects will be more dominant in this mode due to significant displacements of the secondary beam. Table 9 Out-of-plane motion, model 1. Mode FE Mode

Freq., FE-a, B31 Freq., FE-b, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-a

% Rel. diff. theory with FE-b

MAC theory with FE-a

MAC theory with FE-b

1 2 3 4 5 6 7 8 9 10

30.2 79.4 407.1 787.3 1285.5 2176.8 2643.2 3981.1 4384.5 4791.2

0.0 0.0 0.1 0.1 0.4 0.5 0.6 0.7 0.5 0.9

 0.3  0.1  0.1  0.1 0.0 0.2 0.3 0.5 0.5 0.6

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2 4 6 8 10 13 14 17 18 19

30.3 79.5 408.3 788.8 1289.8 2182.2 2652.1 3990.6 4385.5 4802.0

30.2 79.4 407.7 788.4 1290.2 2187.4 2660.0 4011.0 4406.1 4833.0

Table 10 Out-of-plane motion, model 2. Mode FE Mode

Freq., FE-c, B31 Freq., FE-d, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-c

% Rel. diff. theory with FE-d

MAC theory with FE-c

MAC theory with FE-d

1 2 3 4 5 6 7 8 9 10

29.8 77.7 393.3 629.7 1246.8 1733.3 2538.6 3390.3 4056.8 4638.3

0.0 0.0 0.2 0.1 0.3 0.3 0.6 0.6 0.5 0.6

 0.3  0.3  0.1 0.0 0.1 0.2 0.4 0.4 2.1 1.8

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2 3 6 7 10 11 14 16 17 19

29.9 77.9 394.4 630.4 1250.3 1735.1 2545.4 3394.6 3993.3 4583.8

29.8 77.7 393.9 630.4 1251.1 1738.4 2555.2 3409.5 4079.2 4667.8

Table 11 Out-of-plane motion, model 3. Mode FE Mode

Freq., FE-e, B31 Freq., FE-f, S4 Freq., theory (Hz) (Hz) (Hz)

% Rel. diff. theory with FE-e

% Rel. diff. theory with FE-f

MAC theory with FE-e

MAC theory with FE-f

1 2 3 4 5 6 7 8 9 10

23.9 217.6 1111.1 1160.3 1782.8 2621.1 3377.7 3553.7 4871.7 5263.6

0.4 0.3 0.5 0.7 1.7 0.4 0.5 2.3 3.9 0.5

 1.3 0.1  0.4 0.5 1.5  2.4  0.9 2.1 3.6  2.6

1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 1.00

1.00 1.00 1.00 0.99 1.00 1.00 1.00 0.98 0.92 0.99

2 5 9 10 13 15 18 19 22 23

24.3 217.9 1121.4 1163.3 1785.7 2696.3 3422.4 3560.6 4886.4 5429.7

24.0 218.2 1116.6 1168.8 1812.9 2632.4 3393.0 3636.2 5068.8 5291.5

326

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

In conclusion, for in-plane motion the comparison of theoretical with numerical results shows a relatively good mutual agreement, which tends to validate the analytical work undertaken for in-plane motion. 4.3. Out-of-plane motions of the L-shaped beam The theoretical and FE simulations results for all the models and the comparison considering the out-of-plane motions are all shown in Tables 9–12. Similarly to the case of in-plane motion, these tables show the theoretical and numerical wire and shell model natural frequencies, the % relative difference between the theoretical and numerical results and also the MAC analysis for the comparison of the theoretical mode shapes with each one of the mode shapes from the FE simulations. Table 9 presents the results for model 1. Both FE models and theory are in a very good agreement with 1% of Table 12 Out-of-plane motion, model 4. Mode FE mode

Freq., FE-g, B31 Freq., FE-h, S4 Freq., theory % Rel. diff. theory (Hz) (Hz) (Hz) with FE-g

% Rel. diff. theory with FE-h

MAC theory with FE-g

MAC theory with FE-h

1 2 3 4 5 6 7 8 9 10

20.6 157.5 297.4 624.5 1115.2 1197.0 1760.4 2053.9 2528.8 3208.8

 0.5 0.1 0.1  0.1 1.6  0.1 1.1 0.0  0.8  0.1

1.00 1.00 1.00 1.00 1.00 1.00 0.97 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 0.99 0.99 1.00 0.99 0.67

2 4 5 7 10 11 12 14 15 17

20.8 157.6 297.8 625.8 1102.6 1200.8 1764.4 2066.4 2559.2 3229.7

20.7 157.7 298.0 625.4 1120.0 1200.0 1784.9 2067.3 2538.8 3226.34

0.5 0.1 0.2 0.1 0.4 0.3 1.4 0.6 0.4 0.5

x 10-3Out-of-plane: bending, 1st Model, 10th Mode

Out-of-plane: torsion, 1st Model, 10th Mode 1 0.8

1 0.5 0 -0.5 -1 FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

-1.5 -2 -2.5

Displacement-scaled

0

0.05

0.1

0.15

0.2

Rotation-scaled

2 1.5

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

0.6 0.4 0.2 0 -0.2 -0.4

0.25

0

0.05

0.1

0.15

0.2

0.25

x-position

x-position

x 10-3Out-of-plane: bending, 2nd Model, 10th Mode

Out-of-plane: torsion, 2nd Model, 10th Mode

1.5

1

1

0.8

0.5 0 -0.5 FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

-1 -1.5 0

0.05

0.1

0.15

x-position

0.2

0.25

Rotation-scaled

Displacement-scaled

2.5

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

0.6 0.4 0.2 0 -0.2

0

0.05

0.1

0.15

0.2

0.25

x-position

Fig. 4. .10th Mode shape for out-of-plane motions: (a) 1st model bending, (b) 1st model torsion, (c) 2nd model bending, and (d) 2nd model torsion.

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

327

maximum relative difference in the natural frequencies, and the MAC diagonal is 1.00, all across the modes. Fig. 4a and b depict the theoretical and FE wire model simulation mode shape for the 10th mode of the 1st model, noting that all the theoretical motions for bending and torsion have the same scale coefficient. More precisely, Fig. 4a(b) depicts the mode shape in bending (torsion), and there is a good agreement between the theoretical and FE simulations which confirms the MAC value. The results for model 2 in Table 10 present again a good agreement between the FE simulations with the theoretical values. The maximum relative difference is 2% in the natural frequencies, and the MAC diagonal is equal to 1.00 for all the mode shapes. Fig. 4c,d depict the theoretical and FE wire model simulations for mode 10 for the 2nd model. In Fig. 4c (d) the theoretical and FE simulation mode shape in bending (torsion) are in a good agreement, which validates the high MAC value. Table 11 shows the results for model 3, in which the theory is seen to be in good accordance with the FE simulations. The highest relative difference in natural frequencies is 4% for the 9th mode noting that there is clear evidence of shear effect influences, and that all the others have a maximum of 3% difference. For all the mode shapes the MAC diagonals show high values, apart from mode 9 which is just acceptable. Fig. 5a, b depict the theoretical and FE wire model mode shapes for mode 10 for model no. 3. It should be noted that in bending there is some difference in the mode shapes (Fig. 5a) but the displacement values are very small, due to numerical errors. The dominant motions in this mode are the rotations (Fig. 5b). There is a very good agreement between the theoretical and numerical simulation mode shapes (Fig. 5a and b), which is confirmed by the MAC value in Table 11. The results for model 4 show a very good agreement between the theory and the FE simulation results as shown in Table 12. The maximum relative difference in the natural frequencies is 1% and the MAC values are between 0.99 and 1.00, neglecting the 7th mode for the wire elements model which is again lower due to numerical error. Fig. 5c and d depict the theoretical and FE wire model simulation mode shape for mode 10 for the 4th model. Also in Fig. 5c (d) the mode shapes in bending (torsion) suggest very good agreement between the theoretical and the FE simulations, and this justifies the high MAC value in Table 12. In conclusion, there is very a good agreement in the natural frequencies with a maximum relative difference of 4% only in one case, and between 0.99 and 1.00 MAC values, apart from one case which is due to shear effect influence. These results validate the theoretical analysis for out-of-plane motion, and also the comparison shows that shear effect influences are negligible for the modes examined.

x 10-4Out-of-plane:

bending, 3rd Model, 10th Mode

Out-of-plane: torsion, 3rd Model, 10th Mode

8

Displacement-scaled

4 2 0 -2 -4

1 0.8 0.6

Rotation-scaled

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

6

0.2 0 -0.2 FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

-0.4 -0.6 -0.8

-6 -8

0.4

-1 0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

0.15

Out-of-plane: bending, 4th Model, 10th Mode

Out-of-plane: torsion, 4th Model, 10th Mode 100

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2

80

Rotation-scaled

Displacement-scaled

0.25

x-position

1

0.5 0 -0.5

-1 0

0.2

x-position

0.05

0.1

0.15

x-position

FE-beam 1 Theory-beam 1 FE-beam 2 Theory-beam 2 0.2 0.25

60 40 20 0 -20 -40 -60 -80

0

0.05

0.1

0.15

0.2

0.25

x-position

Fig. 5. 10th Mode shape for out-of-plane motions: (a) 3rd model bending, (b) 3rd model torsion, (c) 4th model bending, and (d) 4th model torsion.

328

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

5. Conclusions In this article we performed a theoretical modal analysis of an L-shaped beam structure, by solving two sets of coupled equations of motion for in-plane bending and out-of-plane bending and torsion. Two sets of dimensions were considered for the FE models. For each beam a theoretical and numerical modal analysis was performed to examine the inertia and shear effects. It was shown for both models that for in-plane bending the shear and inertia effects are insignificant and for out-of-plane bending the shear and inertia effects are more significant especially for the second beam. Then, four models of the L-shaped beam structure were constructed with two configurations of the secondary beam and the second one of these was oriented in the transversal direction. Both theoretical and numerical modal analyses were performed for the four models, and the results were compared. For both solutions, for in-plane motion and out-of-plane motion there is a good agreement between the theory and the FE simulations, which tends to validate the theoretical analysis. In the case of in-plane motion for the two models, in the 10th mode there is a high discrepancy due to the coupling of axial motion of the primary beam with the in-plane bending motion of the secondary beam, which was neglected in the theoretical approach. Also in some of the higher modes some discrepancies occurred due to shear effects. Considering the out-of-plane motions, a comparison of the theoretical with numerical results shows smaller differences and in these cases the shear effects are negligible. It should be noted that although in the case of a simple cantilever beam the shear effect is more dominant even in the lower modes, in the L-shaped beam structure the situation is different. In in-plane motion of the L-shaped beam the first mode of each beam is coupled together, which results in 2 modes. Therefore, a shifting of the corresponding second mode of the cantilever beam to a higher mode in the L-shaped beam structure is expected. Considering this phenomenon the shear effects attributed, generally speaking, to higher modes are shifted to even higher modes in the case of the L-shaped beam structure. Finally, the theoretical modal analysis performed in this article for an L-shaped beam structure can give relatively accurate results for the first modes, and will be used to discretise the nonlinear equations for the second order approximation, with a projection to the linear mode shapes in order to continue with nonlinear modal analysis of the L-shaped beam structure.

Acknowledgement The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013), FP7-REGPOT-2009-1, under grant agreement no. 245479. The support by the Polish Ministry of Science and Higher Education—Grant no. 1471-1/7.PR UE/2010/7 is also acknowledged by the second author.

Appendix A In this appendix the coefficients of the mode shapes (Eq. (27a,b)) are determined for in-plane bending, through the boundary conditions (Eq. (26a-c,e–h)). All the coefficients are stated as functions of constant C 1 . Starting from Eq. (26a) and using Eq. (27a), we get C 4 ¼ C 2 ,

ðA:1Þ

also, by using Eq. (26b), and then considering Eq. (27a), it is shown that C 3 ¼ C 1 , Applying Eq. (26c) with Eq. (27a) and then taking Eq. (28a,b), we get 8         9 <cos m l1 cosh m l1 þ m2 l2 mv sin m l1 sinh m l1 = v v v v m1 C2 ¼ C1 ¼ C1CL          : sinh m l1 sin m l1 þ m2 l2 mv cosh m l1 cos m l1 ; v v v v m1 Therefore the constants in Eq. (27a) are given by Eq. (28a,b) and also by (          ) cos mv l1 cosh mv l1 þ m2 l2 mv =m1 ½sin mv l1 sinh mv l1            ¼ C 1 C L , C 4 ¼ C 1 sinh mv l1 sin mv l1 þ m2 l2 mv =m1 ½cosh mv l1 cos mv l1 

ð28aÞ

ð28bÞ

ð28cÞ

Eq (26e) with Eq. (27b) leads to D4 ¼ D2 , then making use of Eq. (26g), and Eq. (27b), and then taking into account (A.5), we have   sinðmW l2 Þ cosðmW l2 Þ þcoshðmW l2 Þ þ D2 D3 ¼ D1 sinhðmW l2 Þ sinhðmW l2 Þ

ðA5Þ

ðA:6Þ

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

Eq. (26h), and Eq. (27b), (A.5) and (A.6), leads to   cosðmW l2 ÞsinhðmW l2 ÞsinðmW l2 ÞcoshðmW l2 Þ ¼ D1 C a , D2 ¼ D1 sinðmW l2 ÞsinhðmW l2 Þ þ cosðmW l2 ÞcoshðmW l2 Þ þ 1 Therefore, using (A.7) in (A.6) we obtain   sinðmW l2 Þ ½cosðmW l2 ÞsinhðmW l2 ÞsinðmW l2 ÞcoshðmW l2 Þ½cosðmW l2 Þ þ coshðmW l2 Þ þ ¼ D1 C g , D3 ¼ D1 sinhðmW l2 Þ sinhðmW l2 Þ½sinðmW l2 ÞsinhðmW l2 Þ þcosðmW l2 ÞcoshðmW l2 Þ þ1

329

ðA:7Þ

ðA:8Þ

Finally, using (26f), with (27), (28a,b), (A.5), (A.7), (A.8), gives the following: D1 ¼ 

mv fcosðmv l1 Þcoshðmv l1 Þ þC L ½sinðmv l1 Þsinhðmv l1 Þg C1, mW ð1 þ C g Þ

ð29aÞ

Appendix B The general solution of the ODE which arises from the out-of-plane bending and takes the inertia terms into account is not trivial and is presented within this appendix. In both cases of out-of-plane bending the ODEs (Eq. (34a,b)) are structurally the same but with different coefficients, and in general can be written as follows: 2

2

Y IV þ b dY 00 b Y ¼ 0:

ðB:1Þ

The characteristic polynomial for this form of ODE is 4

2

2

2

PðkÞ ¼ k þ b dk b ¼ 0

ðB:2Þ

with d 40. This equation is trivial to solve since it is a bi-squared polynomial, therefore the roots are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 2 2 2 4 2 2 ðb d þ b d þ 4b Þ b d þ b d þ4b ¼ 7i ¼ 7 is1 ðimaginaryÞ k1,2 ¼ 7  2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 2 2 2 4 2 2 b d þ b d þ 4b b d þ b d þ4b ¼7 ¼ 7 s2 ðrealÞ: k3,4 ¼ 7 2 2

ðB:3Þ

Considering the roots (B.3) of the characteristic polynomial (B.2) the general solution of Eq. (B.1) is given by YðsÞ ¼ B1 expðis1 sÞ þB2 expðis1 sÞ þ B3 expðs2 sÞ þ B4 expðs2 sÞ,

ðB:4Þ

Using the hyperbolic and trigonometric identities expðxÞexpðxÞ expðxÞ þ expðxÞ , coshðxÞ ¼ 2 2 expðixÞexpðixÞ expðixÞ þexpðixÞ , cosðxÞ ¼ , sinðxÞ ¼ 2i 2

sinhðxÞ ¼

ðB:5a2dÞ

we have expðxÞ ¼ sinhðxÞ þ coshðxÞ, expðixÞ ¼ i sinðxÞ þ cosðxÞ,

expðxÞ ¼ coshðxÞsinhðxÞ expðixÞ ¼ cosðxÞi sinðxÞ

ðB:6a2dÞ

Therefore substituting relations (B.6a–d) in Eq. (B.4), and after some algebraic manipulations, we get the final form of the general solution of Eq. (B.1) YðsÞ ¼ L1 sinðs1 sÞ þ L2 cosðs1 sÞ þ L3 sinhðs2 sÞ þ L4 coshðs2 sÞ, where the constants Li (i¼1,y,4) are determined through the boundary conditions. Eq. (50a) is in the form of Eq. (B.1) therefore the mode shapes are as given by Eq. (B.7), with vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi u   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 u 2 tZ I=m þ Z4 I=m þ4Z2 s1,s ¼ 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi u  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 u 2 tZ I=m þ Z4 I=m þ 4Z2 s2,s ¼ 2

ðB:7Þ

ðB:8a; bÞ

Considering Eq. (51a) and Eq. (B.7) we get, L4 ¼ L2

ðB:9Þ

330

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

Also, Eq. (51b) and Eqs. (B.7), (B.9), give L3 ¼ 

s1,s L , s2,s 1

Then, taking Eq. (51d), and Eqs. (B.7), (B.9), (B.10), yields " # s21,s sinðs1,s LÞ þ s1,s s2,s sinhðs2,s LÞ L2 ¼ L1 ¼ L1 DL , s21,s cosðs1,s LÞ þ s22,s coshðs2,s LÞ In the case when the inertia terms are neglected (I¼0) Eq. (B.3) takes the form pffiffiffi s1,0 ¼ s2,0 ¼ Z:

ðB:10Þ

ðB:11Þ

ðB:12Þ

Appendix C In this appendix, the coefficients of the mode shapes for out-of-plane motion are obtained (Eqs. (39) and (41)). Starting from Eq. (34a), and using Eq. (39a), we get K 2 ¼ 0,

ðC:1Þ

Also, taking Eq. (35a), and Eq. (39a), gives E4 ¼ E2 :

ðC:2Þ

Then applying Eq. (35b), and Eqs. (39a), (C.2), shows that E3 ¼ ðd1 =d2 ÞE1 :

ðC:3Þ

Eq. (37a), and Eqs. (39a,b), (C.2) and (C.3), combine to give F 4 ¼ E1 ½sinðd1 l1 Þðd1 =d2 Þsinhðd2 l1 Þ þ E2 ½cosðd1 l1 Þcoshðd2 l1 ÞF 2 ¼ E1 p1 þ E2 p2 F 2

ðC:4Þ

Then Eq. (37b), and Eqs. (39b), (41a), together with Eq.(C.1), develop to show that K1 ¼

f 1F1 þ f 2F3 : sinðZj l1 Þ

ðC:5Þ

Applying Eq. (38b), and Eq. (41b), gives M 1 ¼ M 2 tanðZF l2 Þ Using Eq. (37d), with Eq. (39b), leads to " #  2  2       1 f f F1 ¼ F 2 cos f 1 l2 þ F 3 2 sinh f 2 l2 þF 4 2 cosh f 2 l2 sinðf 1 l2 Þ f1 f1 Therefore substituting (C.4), into Eq. (C.7), provides " # " #  2 2 cosðf 1 l2 Þ þ f 2 =f 1 coshðf 2 l2 Þ f 2 =f 1 sinhðf 2 l2 Þ F 1 ¼ F 2 þF 3 sinðf 1 l2 Þ sinðf 1 l2 Þ   coshðf l Þsinðd l Þðd =d Þsinhðd l 2 1 1 1 2 2 1 Þcoshðf 2 l2 Þ 2 2 þ E1 ðf 2 =f 1 Þ sinðf 1 l2 Þ   2 coshðf 2 l2 Þcosðd1 l1 Þcoshðd2 l1 Þcoshðf 2 l2 Þ þ E2 ðf 2 =f 1 Þ sinðf 1 l2 Þ ¼ F 2 p3 þ F 3 p4 þ E1 p5 þ E2 p6

ðC:6Þ

ðC:7Þ

ðC:8Þ

Then using Eq. (37c), together with Eq. (39a), we obtain the following: 3

F 1 ðDz2 f 1 þ o2out Iz2 f 1 Þcosðf 1 l2 Þ 3

þ F 2 ðDz2 f 1 o2out Iz2 f 1 Þsinðf 1 l2 Þ 3

þ F 3 ðDz2 f 2 þ o2out Iz2 f 2 Þcoshðf 2 l2 Þ 3

þ F 4 ðDz2 f 2 þ o2out Iz2 f 2 Þsinhðf 2 l2 Þ ¼ F 1 p7 þ F 2 p8 þ F 3 p9 þ F 4 p10 ¼ 0 Eqs. (C.9) and (C.4), (C.8 provide    p p p p p p þ p5 p7 p p þp6 p7 E2 2 10 F 3 ¼ F 2 8 3 7 10 E1 1 10 p4 p7 þ p9 p4 p7 þp9 p4 p7 þ p9 ¼ F 2 q8 þ E1 p11 þE2 p12 ,

ðC:9Þ

ðC:10Þ

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332

331

Also, using (C.8), (C.10) in Eq. (C.5) we get K1 ¼

F 2 ½f 1 ðq8 p4 p3 Þ þ f 2 q8  þ E1 ½f 1 ðp11 p4 þp5 Þ þf 2 p11  þ E2 ½f 1 ðp12 p4 þ p6 Þ þ f 2 p12  ¼ F 2 q2 þE1 q4 þ E2 q5 , sinðZj l1 Þ

ðC:11Þ

Eq. (38a) and Eqs. (39a), (41b), (C.2), (C.3), gives M 2 ¼ E1 ½d1 coshðd2 l1 Þd1 cosðd1 l1 Þ þE2 ½d1 sinðd1 l1 Þ þ d2 sinhðd2 l1 Þ ¼ E1 p13 þ E2 p14 ,

ðC:12Þ

Also, taking into account Eq. (C.12) and so Eq. (C.6) takes the final form     M 1 ¼ E1 p13 tan ZF l2 þ E2 p14 tan ZF l2

ðC:13Þ

Eqs. (38a) and Eqs. (41b), (39a), and also considering Eqs. (C.2) and (C.3), provides the following: i i DZ1 h 2 DZ1 h 2 2 d1 sinðd1 l1 Þ þ d1 d2 sinhðd2 l1 Þ þE2 d1 cosðd1 l1 Þ þ d2 E2 coshðd2 l1 Þ M 1 ¼ E1 Dx2 ZF Dx2 ZF ¼ E1 p15 þ E2 p16

ðC:14Þ

Therefore from Eqs. (C.13) and (C.14) it arises that  ! p p tan ZF l2  ¼ E2 q3 E1 ¼ E2 16 14 p13 tan ZF l2 p15

ðC:15Þ

So Eq. (C.15), with Eq. (C.14), gives M 1 ¼ E2 ðq3 p15 þ p16 Þ

ðC:16Þ

By considering Eqs. (C.15) (C.4) we obtain the form F 4 ¼ E2 ðq3 p1 þp2 ÞF 2 ¼ q1 E2 F 2

ðC:17Þ

From Eqs. (C.15) and (C.10) results F 3 ¼ F 2 q8 þ E2 ðq3 p11 þ p12 Þ ¼ F 2 q8 þ E2 q9

ðC:18Þ

Also, using Eqs. (C.15) and (C.18) Eq. (C.8) becomes F 1 ¼ F 2 ðq8 p4 p3 Þ þE2 ðq9 p4 þq3 p5 þ p6 Þ ¼ F 2 q6 þ E2 q7 :

ðC:19Þ

It is shown that by utilising (C.15) Eq. (C.11) becomes K 1 ¼ F 2 q2 þ E2 ðq3 q4 þq5 Þ

ðC:20Þ

Eq. (35c) together with Eqs. (39a), (C.2), (C.3), (C.15), (41a), (C.1), (C.20), (39b), (C.17)–(C.19), generates the following:     Iz2 o2out F 2 q2 þ E2 q3 q4 þ q5 sin Zj l1 h  i 3 3 3 3 Dz2 F 2 q6 f 1 þ f 2 q8 þ E2 q7 f 1 þ q9 f 2 h i 3 3 2 3 þ DZ1 E2 q3 d1 cosðd1 l1 Þ þd1 sinðd1 l1 Þq3 d1 d2 coshðd2 l1 Þd2 sinhðd2 l1 Þ   ðC:21Þ þ o2out IZ1 E2 q3 d1 cosðd1 l1 Þq3 d1 coshðd2 l1 Þd1 sinðd1 l1 Þd2 sinhðd2 l1 Þ ¼ 0 which, with some simple algebraic manipulation, leads to

The parameters pi (i¼1,y,16) are used in order to simplify the expressions and make the algebraic manipulations simpler and they are given by p1 ¼ ½sinðd1 l1 Þðd1 =d2 Þsinhðd2 l1 Þ,

p2 ¼ ½cosðd1 l1 Þcoshðd2 l1 Þ

"

#  2 cosðf 1 l2 Þ þ f 2 =f 1 coshðf 2 l2 Þ p3 ¼ , sinðf 1 l2 Þ

" p4 ¼

2

f 2 =f 1 sinhðf 2 l2 Þ sinðf 1 l2 Þ

ðC23a; bÞ # ðC:24a; bÞ

1 =d2 Þsinhðd2 l1 Þcoshðf 2 l2 Þ p5 ¼ ðf 2 =f 1 Þ2 ½coshðf 2 l2 Þsinðd1 l1 Þðd sinðf l2 Þ 1

½coshðf 2 l2 Þcosðd1 l1 Þcoshðd2 l1 Þcoshðf 2 l2 Þ , p6 ¼ ðf 2 =f 1 Þ sinðf 1 l2 Þ 2

ðC:25a; bÞ

332

F. Georgiades et al. / Mechanical Systems and Signal Processing 38 (2013) 312–332 3

p7 ¼ ðDz2 f 1 þ o2out Iz2 f 1 Þcosðf 1 l2 Þ, 3

p9 ¼ ðDz2 f 2 þ o2out Iz2 f 2 Þcoshðf 2 l2 Þ,  p p þ p5 p7 , p11 ¼  1 10 p4 p7 þ p9

3

p10 ¼ ðDz2 f 2 þ o2out Iz2 f 2 Þsinhðf 2 l2 Þ

 p p þp6 p7 p12 ¼  2 10 , p4 p7 þ p9

  p14 ¼ d1 sinðd1 l1 Þ þd2 sinhðd2 l1 Þ , p16 ¼

3

p8 ¼ ðDz2 f 1 o2out Iz2 f 1 Þsinðf 1 l2 Þ

p15 ¼

i DZ 1 h 2 2 d cosðd1 l1 Þ þ d2 coshðd2 l1 Þ Dx2 ZF 1

  p13 ¼ d1 coshðd2 l1 Þd1 cosðd1 l1 Þ

ðC:26a; bÞ ðC:27a; bÞ ðC:28a; b; cÞ

i DZ1 h 2 d1 sinðd1 l1 Þ þd1 d2 sinhðd2 l1 Þ , Dx2 ZF ðC:29a; b; cÞ

The final definition of the constants for the mode shape Eqs. (39) and (41) for the simplified expressions are defined through a set of parameters qi (with i¼1,y,10) which are given by q1 ¼ ðq3 p1 þ p2 Þ

ðC:30Þ

q2 ¼

½q8 ðf 1 p4 þf 2 Þp3 f 1  sinðZj l1 Þ

ðC:31Þ

q3 ¼

p16 p14 tanðZF l2 Þ , p13 tanðZF l2 Þp15

ðC:32Þ

q4 ¼

½p11 ðf 1 p4 þf 2 Þ þ f 1 p5  sinðZj l1 Þ

ðC:33Þ

q5 ¼

½p12 ðp4 f 1 þf 2 Þ þ p6 f 1  sinðZj l1 Þ

ðC:34Þ

q6 ¼ q8 p4 p3

ðC:35Þ

q7 ¼ q9 p4 þ q3 p5 þ p6 ,

ðC:36Þ

 p p p p q8 ¼  8 3 7 10 , p4 p7 þ p9

ðC:37Þ

q9 ¼ q3 p11 þ p12

ðC:38Þ

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

V.V. Bolotin, The Dynamic Stability of Elastic Systems, San Fransisco, Holden-Day, 1964. S.L. Bux, J.W. Roberts, Non-linear vibratory interactions in systems of coupled beams, J. Sound Vib. 104 (1986) 497–520. J.W. Roberts, M.P. Cartmell, Forced vibration of a beam system with autoparametric coupling effects, Strain 20 (1984) 123–131. M.P. Cartmell, J.W. Roberts, Simultaneous combination resonances in a parametrically excited cantilever beam, Strain 23 (1987) 117–126. B. Balachandran, A.H. Nayfeh, Nonlinear motion of beam-mass structure, Nonlinear Dyn. 1 (1990) 39–61. J. Warminski, M.P. Cartmell, M. Bochenski, I. Ivanov, Analytical and experimental investigations of an autoparametric beam structure, J. Sound Vib. 315 (2008) 486–508. N. Ozonato, K.-I. Nagai, S. Maruyama, T. Yamaguchi, Chaotic vibrations of a post-buckled L-shape beam with an axial constraint, Nonlinear Dyn. 67 (2012) 2363–2379. A.H. Nayfeh, F. Pai, Linear and Nonlinear Structural Mechanics, first ed, John Wiley and sons, New Jersey, 2004. F. Georgiades, J. Warminski, P. Cartmell, M. Towards, Linear modal analysis for an L-shaped beam: equations of motion, Mech. Res. Commun., in preparation 47 (2013) 50–60. D.J. Ewins, Modal Testing (Theory, Practice and Application), second ed, Research Studies Press Ltd, 2000. L. Librescu, O. Song, Thin-Walled Composte Beams (Theory and Application), Springer, 2006.