Closed-form dynamic stiffness formulation for exact modal analysis of tapered and functionally graded beams and their assemblies

https://doi.org/10.1016/j.ijmecsci.2021.106887Get rights and content

Highlights

  • Closed-form DS for transverse vibration of tapered/functionally graded beams.

  • Applicable to multiform tapered and/or functionally graded beams and assemblies.

  • It fills the gap of and complete the closed-form DS element library of tapered/functionally graded beams.

  • Elegant and efficient J0 count of tapered element is proposed for the WW algorithm.

  • Effects of taper/functional gradient rate/index and BCs on vibration are investigated.

Abstract

The paper proposes a closed-form dynamic stiffness (DS) formulation for exact transverse free vibration analysis of tapered and/or functionally graded beams based on Euler–Bernoulli theory. The novelties lie in both the DS formulation and the solution technique. For the formulation, the developed DS is applicable to a wide range of non-uniform beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This fills a gap of existing closed-form DS element library which is generally limited to linearly tapered/functionally graded beams. For the solution technique, an elegant and efficient J0 count of tapered element is proposed to apply the Wittrick-Williams (WW) algorithm most effectively. The investigation sheds lights on the so-called J0 count challenge of the algorithm for other DS elements. The above two novelties make exact and highly efficient modal analysis possible for a wide range of tapered and/or functionally graded beams, without resorting to series solution, numerical integrations or refined mesh discretization. Results for a particular case show excellent agreement with published results. Moreover, we investigate the effects of the taper/functional gradient rate/index and boundary conditions on the free vibration behaviour. Benchmark solutions are provided for individual beams as well as beam assemblies.

Introduction

Beam structures are commonly used as load bearing structures in many engineering fields such as civil, aeronautical, mechanical and electronic engineering. Such structures are often optimized to improve the vibration and noise properties. For example, a tapered or functionally graded beam can be adopted for a light-weight design or specific wave propagation effects, such as acoustic black hole effect [1], wave propagation control [2], piezoelectric energy harvesting [3] amongst many others including architectural considerations.

Of course, in order to design tapered beams, the finite element method (FEM) is probably the most commonly used method in engineering. When modelling such beams with continuously varying cross-section or material, the FEM approximates the continuous beams by the so called finite elements. The shape functions for each element are assumed to be approximate polynomials, leading to separate and frequency-independent element mass and stiffness matrices. Finally the elemental matrices are assembled in the FEM resulting in global stiffness and mass matrices with frequency as the eigenvalue parameters in free vibration problem. This becomes a generalised eigenvalue problem which can be computed by usual linear algebra solvers. Based on the uniform Bernoulli–Euler theory and the Timoshenko theory respectively, Shahba et al. [4], [5] derived arbitrarily tapered or axially functionally graded beam elements and investigated their stability and free vibration characteristics. Both free vibration and wave propagation analysis of rotating Euler–Bernoulli tapered beams were also reported [6] by using spectrally formulated finite elements. Ramalingerswara and Ganesan [7] investigated the harmonic response of composite tapered beams by using FEM and a higher order shear deformation theory based on a similar principle of higher order plate theory. However, due to both the discretization of continuous function and the approximation of the shape functions, only the lower order eigenvalues could be accurately extracted from the FEM model. If one needs more accurate results especially in the high frequency range, a much finer mesh will be required, particularly for a tapered or functionally graded beam than that for a uniform beam.

Meanwhile, analytical methods can serve as useful alternatives whose advantages include accuracy, efficiency, convenience and physical-meaning clarity. For example, the Rayleigh–Ritz method is a common analytical method. By employing the Rayleigh–Ritz method, Abrate [8] performed the longitudinal vibration analysis of a variety of tapered rods with quadratic polynomials for the area along the length; Zhou and Cheung [9] established admissible functions representing the solution to analyse the free vibration of a type of rectangular tapered beams whereas the transverse vibration of rectangular Mindlin plates with variable thickness was investigated in [10]. On the other hand, the variational iteration method [11] was used to study the free vibration of a linearly tapered beam mounted on two-degrees of freedom spring-damper-mass subsystems. Huang and Lee [12] performed free vibration analysis of axially functionally graded beams with non-uniform cross-section based on the Bernoulli–Euler theory. The governing differential equation with variable coefficients was transformed into Fredholm integral equation, and the shape function was expanded in power series. Ashour [13] investigated the transverse vibration of orthotropic rectangular plates of variable thickness by combining the finite strip technique with the transfer matrix method. The asymmetric development method [14] was used for the free vibration of axially functionally graded beams based on the Bernoulli–Euler theory. The investigation in [14] obtained approximate analytical formulas for natural frequencies under several classical boundary conditions. Hein and Feklistova [15] studied the free vibration of non-uniform and axially functionally graded beams using the Haar wavelet approach and the Bernoulli–Euler theory. Based on the Rayleigh-Love theory, Banerjee et al. [16] analysed the axial vibration of a conical rod. By rewriting the governing differential equation into the Legendre’s equation, the shape function was obtained in series form, and the natural frequencies were computed by the authors by substituting the boundary conditions to eliminate the unknown constants. By contrast, the transfer matrix method was employed by Mahmoud [17] to determine the natural frequencies of axially functionally graded tapered cantilever Bernoulli–Euler beams with point masses at the tips. However, all of the above analytical methods are based on approximate series-form of shape functions, leading to approximate results. There are, of course, other research based on exact general solutions for some particular tapered beams. For instance, the authors of [8] obtained the exact natural frequency solutions of classical bar with quadratic cross-section area and quartic cross-section area variations. Ece et al. [18] gave analytical solution for the free vibration of tapered beam with exponential cross-sectional area and moment of inertia variations. Jong-Shyong Wu [19] summarized the analytical solutions in the form of Bessel functions for a number of linearly tapered beams, including axial vibration of conical rods, torsional vibration of conical shafts and bending vibration of single-tapered beams. Zhao et al. [20] solved the analytical shape function of a parabolically tapered annular Euler beam by using method of substitution. Eventually, the natural frequencies and mode shapes were solved by them using the Galerkin method. By applying the coupled placement field method, Rajesh and Saheb [21] performed the large deflection free vibration analysis of linearly tapered Timoshenko beam and obtained closed form expression of frequency ratio for hinged-hinged and clamped-clamped boundary conditions. Banerjee and Ananthapuvirajah [22] utilized Bessel functions to represent the free vibrational shape function of linearly tapered Bernoulli–Euler beam accurately, and computed the natural frequencies by imposing the boundary conditions. However, the above methods can only be applied to a single tapered beam under special boundary conditions, and cannot be applied to an assembly or combination of tapered beams or for complicated boundary conditions in engineering.

Different from the above analytical methods, the dynamic stiffness (DS) method uses the frequency-dependent shape function to derive the dynamic stiffness matrix of a structural element, which can be assembled directly to model complex built-up structures and importantly, any boundary conditions can be easily imposed. The DS method was first proposed by Kolousek [23]. Since then, many investigations have been carried out for the DS formulation in the free vibration and buckling analysis of the bars, beams, plates, shells, membranes and their assemblies. Another landmark in the DS method is the development of the solution technique namely the Wittrick-Williams (WW) algorithm [24], which facilitates an efficient and accurate eigenvalue analysis based on the DS matrices. In what follows, we summarize the existing research from both DS formulation aspect as well as from the solution technique aspect.

As for the DS formulation aspect, there are a series of work on the DS formulation of tapered and/or functionally graded bars and beams, which can be classified into four broad category of methods:

    Method 1

    The first category of method is described to be as an approximate model of a tapered element using a number of uniform elements with different cross-section parameters, such as reported in [25], [26]. By using this method, the natural frequencies and mode shapes of three types of linearly and parabolically tapered Bernoulli–Euler beams under axial force were computed and discussed in [25], [26]. This method was implemented into a program called BUNVIS-RG [27] for the free vibration and buckling analyses of space frame structures consisting of tapered Timoshenko beams under axial force. Later, Banerjee [28] modelled tapered rotating Bernoulli–Euler beams as an assembly of a large number of uniform beams, where the DS matrix of a uniform rotating Bernoulli–Euler beam was derived by Frobenius method of power series solution. However, the disadvantage of this type of approach is that it requires a large number of elements and therefore reduces computational efficiency. Moreover, a numerical convergence test is needed to determine the natural frequencies with required accuracy.

    Method 2

    As proposed by Yuan et al. [29], the general solutions and their derivatives of non-uniform beams with gradual or stepwise cross-section can be numerically solved by ODE solvers, which are then used to formulate the DS matrices and the mesh generation rules of element length. Then the free vibration analysis of arbitrarily tapered or axial functionally graded beam based on Timoshenko theory was performed by Yuan et al. [29]. Although the method is capable of providing highly accurate results, the number of degrees of freedom used is quite large which decreases the computation efficiency. This type of method is essentially a combination of analytical and numerical methods due to the fact that the general solutions are computed by using the numerical ODE solvers.

    Method 3

    The DS formulations are developed in this method based on series form or approximate polynomial shape functions. For example, Banerjee et al. [30], [31] used the Frobenius method to solve the differential equations and then established the dynamic stiffness matrix. Combined with the WW algorithm, natural frequencies of linearly tapered rotating Bernoulli–Euler beam [30] and Rayleigh-Love bar [31] were respectively computed by them. Leung and Zhou [32] applied the Frobenius method in the DS formulation of transverse vibration of tapered or axial functionally graded Timoshenko beams, to compute the natural frequencies. Similarly, Frobenius method was also used to formulate the transfer matrix [33] to analyse the free vibration of linearly tapered beam based on Bernoulli–Euler theory, and the natural frequencies were computed by the determinant of the transfer matrix. Kim et al. [34] adopted approximate polynomials as the shape function leading to the dynamic stiffness matrix, which was transformed into a state-vector form. However, it was found that in general, many series terms were required to converge upon results with acceptable accuracy.

    Method 4

    Dynamic stiffness formulation can also be achieved based on the closed-form exact solution of tapered beams, but this method is limited to the case when the governing differential equation has closed-form exact general solution. Kolousek [23] first proposed the dynamic stiffness matrix of a linearly tapered beam based on the exact shape function in the form of Bessel functions. Later, Banerjee and his coauthors [35] formulated the explicit expressions of DS matrices for torsion, axial and transverse free vibration of linearly tapered beam, based on the closed-form Bessel equation. Su et al. [36] formulated the DS for a functionally graded Bernoulli–Euler beam with material parameters varying in power function along the thickness direction, and the natural frequencies and mode shapes were computed by applying the WW algorithm. The explicit expressions of the DS matrix missing in [36] was later provided by Banerjee and Ananthapuvirajah [37]. Recently, Popov [38] stated the transfer matrix can be developed without deriving the explicit exact general solution as in the DS formulation [37]. In another research, Banerjee et al. [16] developed the DS matrix of a linearly tapered Rayleigh-Love rod, and natural frequencies for both individual rods and their assemblies were computed. To get more details, [39] summarized the historical development of the DS method.

Therefore, it can be easily seen from the above that Method 4 being a closed-form exact formulation is the most efficient and accurate DS formulation for tapered beams among the four methods. Nevertheless, all existing research has been confined to linearly tapered beams so far [16], [23], [35], [37]. Although nonlinear tapered beams are sometimes more commonly utilized in engineering applications, there appears to be no closed-form DS formulation available for it. A closed-form DS for tapered/functionally beams whose bending stiffness and linear density are assumed to be polynomial functions of position will no-doubt fill an important gap in the literature and add value to the closed-form DS element library of tapered/functionally graded beams with arbitrary cross sections.

Subsequently, once the analytically formulated matrices (such as dynamic stiffness matrix or transfer matrix) are developed, one needs to extract the natural frequencies and mode shapes from the analytically formulated matrices (based on either series-form solutions or closed-form solutions). There are generally two types of eigenvalue solution techniques available for this purpose.

    Technique 1

    One of the most commonly used eigenvalue solution technique is to determine the eigenvalue when the determinant of the system matrix becomes zero. This technique is used by most of the analytical methods (e.g., see [8], [19], [20]), including transfer matrix method [8], [17], [33], [34], [38], [40]. However, the determinant method needs the evaluation of the determinant numerically for a frequency range. Deciding the step size to determine the zeros of the frequency-determinant and avoid the poles is problematic and far from being trivial. The problem arises because a small step size leads to unnecessary computational cost whereas a large step size increases the possibility of missing some genuine natural frequencies. This is especially true for complex structures and also when computing higher order natural frequencies. The problem is further compounded by the fact that the frequency determinant often involves complex and irregular transcendental functions such as the hyperbolic functions. Nevertheless, the potential pitfalls and drawbacks of the determinant method as mentioned above, still exist and hard to overcome.

    Technique 2

    The WW algorithm [41], which has been used in many DS formulations, e.g., [28], [30], [31], [35], [36], [37], [42]. The WW algorithm is probably the most suitable solution technique for dynamic stiffness models with the following advantages

    • (i)

      Accuracy: Eigenvalues within any required precision can be computed;

    • (ii)

      High efficiency: It is highly efficient mainly due to the small-size matrix;

    • (iii)

      Analytical elegance: Infinite eigenvalues can be extracted from the finite dimensional matrix;

    • (iv)

      Certainty: The algorithm ensures that no eigenvalue will be missed.

    However, the advantages of (ii), (iii) and (iv) can be realized only when the key problem of the so called J0 count (the mode count of all fully clamped members) in the WW algorithm can be effectively solved; Otherwise, either some spurious modes will enter into the calculation or some true modes will be missed, so that the advantage of the above (iv) certainty cannot be fully realised. Thus, J0 count problem is an important key issue when applying the WW algorithm for the free vibration analysis. For the axial and transverse vibration of relatively simple uniform beam elements, the expression for J0 count can be derived analytically, e.g., [39], [41], [43], [44], [45], [46], [47], [48], [49], [50]. But for a tapered or functionally graded beam, the J0 count problem becomes more challenging. We found that the only work that mentioned the J0 problem of tapered beam is [35], which discussed the J0 problem of linearly tapered beam. Most of the existing dynamic stiffness of tapered beams usually discretize a structure into elements small enough so that each element will have J0=0 within the interested frequency range. However, as the number of elements is greatly increased, the above advantages of (ii) high efficiency and (iii) analytical elegance will be somehow sacrificed, let alone the fact that substructuring technique become less convenient to use for tapered beams compared to uniform beams. So it is very important to provide an efficient technique to compute the J0 count of the tapered beam member for the WW algorithm to be most effective.

This paper essentially aims to fill the gap which is the limitation of the existing research that closed-form DS formulation is confined to linearly tapered and functionally graded beams only. The novelty of this paper is two fold: (1) A closed-form DS formulation of tapered and/or functionally graded beams is developed based on the exact general solution of the governing differential equation; (2) An efficient solution for the most crucial issue of J0 count for the WW algorithm is proposed. By doing so, exact modal analysis can be efficiently performed for more general tapered and functionally graded beams and their assemblies. This undertaking is an important supplement to the existing non-uniform DS element library. A wide range of tapered and functionally graded beam members and their assemblies or when they are connected to uniform beams can be exactly modelled in the whole frequency range using a minimum number of degrees of freedom, and exact natural frequencies and mode shapes can be computed efficiently. This can be used for the efficient and accurate parametric studies and optimization of beam built-up structures.

The paper is organized as follows. Section 2 provides the development of DS formulation in explicit form based on the exact solution of tapered and/or functionally graded Euler–Bernoulli beam. Section 3 describes the modal analysis by using the WW algorithm, where the emphasis is placed on proposing an efficient and reliable J0 count procedure for a tapered and/or functionally graded beam member. In Section 4, the proposed method is validated by comparing current solutions with published results and those computed by commercial FEM software, to demonstrate the exactness, efficiency and wide application scope of the proposed theory and solution technique. Finally, Section 5 concludes the paper.

Section snippets

Governing differential equations and boundary conditions

The governing differential equation (GDE) for the flexural vibration of a tapered beam shown in Fig. 1 can be derived using Newton’s law or Hamilton’s principle to give [33] 2x2E(x)I(x)2wx2+ρ(x)A(x)2wt2=0where ρ(x)A(x)=ρ0A01cLxn,E(x)I(x)=E0I01cLxn+4where w=w(x,t) is the flexural displacement, t is time, the flexural rigidity E(x)I(x) and the mass of the beam per unit length ρ(x)A(x) are functions of the spatial variable x[0,L]. Apparently, E(x)I(x) depends on elasticity modulus E(x)

The Wittrick-Williams algorithm and modal analysis

Each entry of the global dynamic stiffness matrix of the final structure obtained above is a transcendental function of frequency, and the powerful solution technique of the Wittrick-Williams (WW) algorithm [41] will be applied to compute the natural frequencies. The WW algorithm uses the Sturm sequence property of the dynamic stiffness matrix to give the number of natural frequencies, below an arbitrarily chosen trial frequency J(ω#). It is essentially a counting method. Its basic principles

Results and discussions

The method described in this paper has been implemented in a MATLAB program to calculate the natural frequencies and mode shapes of non-uniform beams and their assemblies. First, Section 4.1 illustrates the high efficiency and exactness of the current method by comparing it with the FEM. Section 4.2 validates the present results with the existing results under different taper/functional gradient rates c. In Section 4.3, we explore the effect of the taper/functional gradient index n on the

Conclusions

Closed-form dynamic stiffness (DS) formulation for tapered/ functionally graded beams has been proposed. The formulation is based on the exact general solution of the governing differential equation of a non-uniform Euler–Bernoulli beam, and explicit expressions are derived. The developed DS is applicable to a wide range of tapered and/or functionally graded beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This is significant in the context

CRediT authorship contribution statement

Xiang Liu: Conceptualization, Methodology, Writing – review & editing, Supervision, Project administration, Funding acquisition. Le Chang: Investigation, Data curation, Writing – original draft, Writing – review & editing. J. Ranjan Banerjee: Supervision, Writing – review & editing. Han-Cheng Dan: Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors appreciate the supports from National Natural Science Foundation (Grant No. 11802345), State Key Laboratory of High Performance Complex Manufacturing, China (Grant No. ZZYJKT2019-07), Central South University (Grant No. 502045001) which made this research possible. The authors are grateful to the discussions on general solutions with X.W. Zhao from University of Shanghai for Science and Technology.

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