Stephen R. Borneman, Ph.D Candidate



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Chapter 4 Uniform Laminated Composite Wing Model

4.1    Introduction


In this chapter, the materially coupled bending-torsion vibration of laminated composite beam, based on an assembly of uniform beam elements, and using the Finite Element Method (FEM), Dynamic Stiffness Matrix (DSM) and Dynamic Finite Element (DFE) is presented. The DSM method is based on the exact solution to the governing differential equations of motion, as presented by Banerjee and Williams (1995). Therefore, for a uniform beam, the DSM needs only one element to achieve the exact natural frequencies. The DSM formulations can also be easily extended to approximate tapered geometry by using a piece-wise uniform stepped model. The FEM model is obtained using a Galerkin weighted residual method to formulate the element mass and stiffness matrices of the current uniform beam. In what follows, a Dynamic Finite Element method for the coupled vibration analysis of uniform and stepped composite beams is developed. The  comparison is then made between the DFE results and those obtained from the FEM and DSM formulations in order to validate the proposed methodology.


4.2    Wing Model


A cantilever composite beam with length L and a solid cross-section is the basis of the model (see Figure 4-1). All rigidities are assumed constant along y axis. The rigidities are: bending, EI, torsion, GJ, and coupled bending-torsion, K. The rigidities can be determined either experimentally or based on the theory presented in Chapter 3. The solid cross section is assumed to be symmetric with different fibre layer orientations (see Figure 4-1), where is the translational displacement associated with bending and  is the rotational twist associated with torsion.



Figure  4-1: Composite Beam on a Right Handed Coordinate System.



4.3    Assumptions

The simplest model of a composite wing is represented by a uniform Euler-Bernoulli beam, where the bending slope is the derivative of the bending displacement with respect to the span-wise direction (Lilico et al, 1997). Shear deformation and rotary inertia are neglected by assuming a long slender beam. Further simplifications have been made by applying the St. Venant assumptions, which is a pure torsion state, and neglecting all warping effects. The beam is assumed to be composed of composite material with unidirectional fibre lay-up. With any composite, material couplings between extensional-twist and bending-twist arise from ply orientation and stacking sequence. This research will focus on the bending-twist couplings as the other coupling behaviours are being investigated by other researchers (see, for example, Roach and Hashemi, 2003).


4.4    DFE Formulation

The governing differential equations of motion for the materially coupled vibration of a uniform composite beam are (Banerjee, 1998):




where denote the beam flexural displacement and  is the torsion angle. EI and GJ denote flexural and torsion rigidities respectively, m is the  mass per unit length and  represents the polar mass moment of inertia per unit length of the wing. The material bend-twist coupling rigidity is represented by K and primes denote differentiation with respect to span wise position y. Based on the simple harmonic motion assumption, the following separation of variables is applied on the flexural and torsional displacements  (sinusoidal variation with frequency).




     Then with substitutions of   (4.3) into   (4.1) and (4.2), the differential equations can be re-written in the following form:




     By implementing the Galerkin weighted residual method and integration by parts, the continuity requirements on the field variable are relaxed so that the integral weak form associated with equations  (4.4) and (4.5) can then be obtained as:








The above boundary terms can be associated with the Shear S(y), Moment M(y), and torque T(y) as:






     Boundary conditions associated with clamped-free (cantilever) beam are , and all force boundary terms are zero at the tip (y=L). The system is then discretized by 2-node 6-DOF beam elements (Figure 4-2).


Figure  4‑2: A 2-node 6-DOF beam element


Principle of Virtual Work (PVW) is also satisfied such that:



where,  represents element internal virtual work and  for free vibrations. After two integrations by parts on the differential equation governing the flexural motion, the element internal virtual work can be written in the following form:










The two above equations simply represent the bending and torsion contributions to the discretized internal virtual work for each element of length .

     The basis functions are then chosen based on the solutions to the differential equations of (*) and (**). For the first differential equation  (4.13) pertaining to bending, the following process is applied to formulate the trigonometric shape functions according to Hashemi and Richard (1999). The torsion interpolation functions are also evaluated in a similar way.


4.4.1 Frequency dependent trigonometric shape functions.

The non-nodal approximation for the flexural weighting function, , and the field variable,, can be written as:



Similarly for torsion:



where dw, w, are discretized over a single element (0  x  1). The basis functions of the approximation space are chosen as:



and, for torsion as:








     The basis functions are chosen as trigonometric terms based on the solution to the differential equations and were manipulated to reduce to Hermitian basis functions as . It is important to note that Hermitian basis functions have been used in beam finite elements for many years, since they satisfy the “Completeness” and “Compatibility” requirements. “Completeness” is satisfied by including the lowest order admissible term. The compatibility condition is also satisfied. With these conditions satisfied, the DFE with its Hermitian based Dynamic Trigonometric Shape Functions (DTSF’s) is guaranteed to converge to the exact solution. Classical basis functions of the standard “Hermite” beam element are [1, x, x2, x3]. The bending and torsion trigonometric basis functions lead to standard cubic and linear ones by taking the limit as , respectively. These variables are frequency dependent as seen above in equations (4.19) and (4.20). When the frequency approaches zero the DTSF’s reduce to polynomial basis functions which lead to satisfying the required conditions of compatibility and completeness.


 For the first bending basis function:



 The second basis function leads to:



The third basis function leads to:



The fourth basis function leads to:



     The coefficientshave no physical meaning and can be replaced by nodal variables for bending  and  and for torsionand . The derivation of bending shape functions are only considered in the following procedure, since, the torsion shape functions will follow the same development. Following the same systematic method as in FEM, one can write:









     The nodal approximations for element variables and can then be rewritten as:







     Expressions  (4.30) can then be rearranged as:  where  is the element displacements (i.e., degrees of freedom) and [N] represents the dynamic shape functions in matrix form.




     The four trigonometric shape functions pertaining to bending are (Hashemi and Richard 1999; Hashemi, 1998):








For torsion:

                    (4.37)                                    (4.38)



The six shape functions are plotted individually for first four natural frequencies of free coupled vibration of a uniform composite wing (see Figures 4-3 through 4-8). These shape functions are the approximations to the solution of the governing differential equations of motion.



     The above shape function plots display the inter-element continuity required to satisfy the  compatibility condition. With the frequency dependent trigonometric shape functions determined, the dynamic finite element matrix can be constructed from equations (4.13) and (4.14). The DFE matrix can be expressed in two matrices as:




     The uncoupled matrix is obtained from the boundary term expressions extracted from the integration by parts. The coupled matrix is formulated from the integral expressions representing the coupling terms in both equations  (4.13) and  (4.14). The symmetry of coupled matrix can be seen in the equivalence in both integral expressions.










      represents the dynamic finite element matrix which is now ready for assembly in the usual finite element way. The coupled matrix is integrated symbolically to ensure the final dynamic finite element matrix is purely algebraic. With all expressions in the DFE matrix symbolically computed, there is no need for a numerical integration which decreases the required computational time. The symbolic integrations for the coupled matrix in equation  (4.41) are carried using MAPLE©.


Numerical Results

Solution Algorithm (Wittrick-Williams Root Counting)