EFFECTS OF NON-CLASSICAL BOUNDARY CONDITIONS ON THE FREE VIBRATION RESPONSE OF A CANTILEVER EULER-BERNOULLI BEAMS

In this article, the problem of the free vibration behavior of a cantilever Euler-Bernoulli beam with various non-classical boundary conditions, such as rotational, translational spring, and attached mass is investigated. For describing the differential equation of the system. An analytical procedure is proposed firstly, and a numerical method based on the differential transform method DTM is developed in order to validate the obtained results. A parametric study for various degenerate cases is presented with the aim to analyze the influence of rotational stiffness, vertical stiffness, and mass ratio on the free vibration response of the beam, particularly on its modal characteristics. The results show that the non-classical boundary conditions significantly affect the natural frequency and mode shapes of the studied beam system in comparison to the case of a classical boundary conditions such as Simply supported, clamped-clamped, etc. The comparison between the obtained results based on the proposed analytical solution and numerical scheme, and those available in the literature shows an excellent agreement.


INTRODUCTION
Recently, a new class of boundary conditions known as non-classical mechanics systems has drawn considerable attention in the field of engineering, these systems can be used to produce excellent and optimization structural elements in various engineering structures and technologies such as robotic structure, aircraft, vehicles, building, and bridges. The structural response of beams with linear and non-linear elastic boundary conditions has been a topic of many investigations. Paupitz et al. [1] proposed a new method for calculating the natural frequencies and mode shape by using the terms of dynamic stiffness presented by boundary condition, this method is valid for any linear boundary conditions. Raimondo Luciano et al. [2] studied the free flexural vibrations of nanobeams constrained by non-rigid supports, modelled by transversal and rotational springs. Laila chalah et al. [3] using the finite element method (FEM) for determining the transverse free vibration of a cantilever beam with torsional and translational springs attached at the end, identifying the immediate effects of the elastic restraints on the dynamic behavior of the system. Ding et al. [4] studied the free vibration of an axially moving beam supported by torsional and vertical springs at both ends. the critical speed of the axially moving beam does not change with the vertical spring stiffness. Sayed Mojtaba [5] analysed the free vibration response of a cantilever beam with exponentially varying width and joined by a mass-spring system at the free end. Wang et al. [6] studied the derivation of the frequency equation of flexural vibrating cantilever beam considering the bending moment generated by 2 DIAGNOSTYKA, Vol. 24, No. 1 (2023) Afras A, El Ghoulbzouri A.: Effects of non-classical boundary conditions on the free vibration response of … a mass attached at the free end of the beam The results show that the inertial moment of the mass has the significant effect on the natural frequency and the shape mode. Nuttawit and Arisara Chaikittiratana. [7] applied the differential transform method (DTM) to investigate linear and nonlinear vibration problems of elastically end restrained of functionally graded beams. Suddoung et al. [8] by using the DTM analysed the free vibration stepped response of beams with arbitrary boundary conditions. Dongyan et al. [9] investigated an accurate solution method for free vibration of the Timoshenko beam with general elastic restraints at the end points. Sayed Mojtaba et al. [10] investigated the free vibration analysis of beam with an intermediate sliding connection and joined by a mass-spring system and elastically supported. Banerjee J. R [11]. assembling the dynamic stiffness matrix of the beam and spring-mass system, the Wittrick-Williams algorithm employed to derive natural frequencies and mode shapes of the combined system. The Fourier series has also been used to investigate the free vibration of beams with general restrained boundaries Kim, H. K et al. [12]. Lau, J. H [13]. dealt the vibration Frequencies and mode shapes for a constrained cantilever at some point. Darabi et al. [14] analysed the free vibrations of a beam with a mass-spring system with different boundary conditions both numerically and analytically.
In this study, according to a literature survey and as known that the free vibration's amplitude is the important phenomenon governing the level of vibrations produced in a bridge-beam like structure, the objective is to analyse the free vibration response of a cantilever E-B beam with various non-classical boundary conditions, a problem which can be viewed as a generalization of some cases study presented in the literature [1], [14] in which a developed analytical solution and a numerical method based on the DTM method are used, in order to understand the effect of various boundary conditions on behaviors of the studied beam. The use of the differential transforms method (DTM) in the present analysis is justified by the fact that this has the advantages of rapid convergence and of its effectiveness, to solving nonlinear equations which there is no analytical solution.

ANALYTICAL SOLUTION
Consider the configuration represented schematically in Fig. 1, a uniform cantilever Euler-Bernoulli beam attached by mass M and supported at the tip by various boundary conditions, these linear boundary conditions include the translation spring with stiffness KT, rotational spring with stiffness KR, the beam is assumed to be of length L and uniform cross-sections. In order to study the variation of the natural frequency and mode shape by taking into account the influence of the boundary conditions, as shown in Fig. 1, seven configuration of boundary condition study are presented. From these examples, some of them have been studied previously and presented here for validation, and we note that the first case is the general cases which combine all the defined non-classical boundary conditions.
Determining the characteristic frequency equation described the general system. The equation of motion for free vibration of a Euler-Bernoulli beam is: is the transverse displacement response at the coordinate measured along the axis of the beam with its origin at its left extremity and at time t, EI, A, ρ and L are respectively the flexural rigidity, the cross-sectional area, the mass density, and the length of the beam.
For any mode of vibration, the transverse displacement y (x, t) may be written in the form: y(X, t) = W(X)e ω (2) Where ω is the circular frequency and w(x) is the mode shape, substituting Eq. (2) into Eq. (1) leads to: ( ) − W(X) = 0 (3) Eq. (3) can be cast into the dimensionless form as: ( ) − β w(x) = 0 (4) Where x = , w(x) = ( ) and β = ω ρ As a function of x, the solution of Eq. (4) can be given by: With σ1, σ2, σ3 and σ4 are shape coefficients which can be found by using the boundary conditions. The beam is clamped at the left end, hence the deflection and slope, Eq. (5) reduce to: w(x) == σ 1 (cos(βx) − cosh(βx)) + σ (sin(βx) − sinh(βx) By using the new approach proposed by Paupitz et al. [1], the boundary conditions are DIAGNOSTYKA, Vol. 24, No. 1 (2023) Afras A, El Ghoulbzouri A.: Effects of non-classical boundary conditions on the free vibration response of … 3 described in terms of dynamic stiffness. The force and moments acting on the beam at the tip are given respectively by: Where ZW, and Zθ represent the dynamic stiffness for lateral and rotator displacement. At the tip, Shear force and bending moment are given by: The Boundary condition of the mechanical system in the dimensionless form at the end of the cantilever beam shown in Fig. 1 can be expressed as: Eqs (9) -(6) can be combined to give: Where ax and aθ are dimensionless stiffness parameters translational and rotational respectively and R is the ratio between the additional mass M and the cantilever beam mass which. The frequency equation can be obtained by equating the determinant of Eq.(10a) the coefficients in the above set to zero: Then, the expression for the mode shape of the ℎ natural frequencies is given by expression: (sin( β x) − sinh( β x))) (12a) the normalized mode shape is defined as: (12b)

DTM SOLUTION
In order to understand the influence of various boundary conditions as shown in Fig. 1, in the free vibration of the cantilever beam with length L such as variation of natural frequencies and mode shapes, and in addition with the proposed analytical solution presented below, it is of great interest to effectuate an analysis with the Differential Transforms Method (DTM). With its advantage of rapid convergence and easier implantation, the principle of the DTM which is based on the Taylor series expansion, is to transform the governing differential and boundary condition equations into a set of algebraic equations using transformation rules. Tables 1-2 respectively show the basic operation required in differential transformation for the governing differential and boundaries conditions.  Original The general function f(x) in Tables 1-2 is considered as the transversal displacement w(x), we apply the basic operation of DTM presented in Eq. (4) we obtained the recurrence equation as: DIAGNOSTYKA, Vol. 24, No. 1 (2023) Afras A, El Ghoulbzouri A.: Effects of non-classical boundary conditions on the free vibration response of … Where β = ω ρAL EI Let the non-zero values of shear force and bending moment indicates by Ω1 and Ω2, with applying the basic operations of DTM for the boundary condition at x=0, and by using the Table. 2, once we obtain: Substituting Eq. (14) into the recurrence Eq. (13) leads to [ ] for all values of k as follows: For the boundary conditions x=1 as presented in Eq. (9), by applying the basic operations of DTM presented in Table. 2, and by using the terms of dynamic stiffness Eq. (7), one obtains: Substituting the expression w[k] from Eq. (16) into Eq. (14) leads to two polynomial equations which can be arranged into the matrix form: Where the elements in the matrix are: The frequency equation is: Solving the frequency equation. Eqs. (11) and (18) by using the algorithm of Newton Raphson programmed in MATLAB environment, and by solving the Eq. (16), one can obtain the frequency values in the flowing form β = βk [ϵ] , where k=1,2,3,…,ϵ, in which βk [ϵ] is the k th estimated frequency corresponding to ϵ, hence, an appropriate value of ϵ is obtained by convergence analysis with the following equation βk [ϵ] -βk [ϵ-1] ≤ λ where λ is a given error tolerance. The mode shape function can be obtained by using the expression: Where ψ2 is defined in the Eq. (10b).

NUMERICAL RESULTS AND DISCUSSION
In this section, with the aim of analysing the effect of various boundary conditions in the free vibration response of a cantilever beam as shown in Fig.1, and to show the versatility of the aforementioned theory, like the analytical and numerical solution, six examples are given to illustrate this, and are presented as a simplification referring to the example 1. In addition, most parameters affect the frequency and mode shape of the specified case, are considered in term of non-dimensional ratios, such as non-dimensional translational stiffness ax, rotational stiffness aθ, and non-dimensional mass R, these ratios are defined in Eq. (10b). DIAGNOSTYKA, Vol. 24, No. 1 (2023) Afras A, El Ghoulbzouri A.: Effects of non-classical boundary conditions on the free vibration response of … 5 Example 1: cantilever beam with rotational, translation spring and mass restraint at the tip By solving Eq. (11) and (18), we can obtain the values of three first dimensionless natural frequencies, which are listed in Table3  (Appendix 1). Fig. 2, shows the variation of the first three dimensionless frequencies as a function of the k th eigenvalues defined by the Eq. (18) corresponding to the values R=ax=aθ=100. From this figure, we can observe that dimensionless natural frequencies determined by the differential transform method converge very rapidly, in which for the first non-dimensional frequency the convergence is attained for k=1, and for the second and third non-dimensional frequency convergence is attained for k=2, and for k=4 respectively. The convergence is assured in terms of the third non-dimensional frequency, consequently this is one of the reasons why the DTM is used in this work. Examining Fig. 3, which represents the first three mode shapes of the beam with fixed values R=ax=aθ=100 and a comparison between the analytical solution and DTM solution, it can be seen that there is an excellent agreement between the two solutions, which shows the accuracy of the proposed solution. Fig. 4 shows the first three dimensionless natural frequencies βi=1,2,3 of a beam with various parameters of R, ax and aθ. It is observed that the effect of the value of R, ax and aθ on the lower mode is more significant, and can be drawn, particularly for the fundamental mode, that when dimensionless stiffness ax=aθ keep constant and the mass ratios increase, the dimensionless natural frequencies βi of system decreases, when dimensionless stiffness ax=aθ increase and the mass ratios R . keep constant, the dimensionless natural frequencies βi increase, at the lower modes are more sensitive to the boundary restraints then the natural frequencies at the higher modes, stiffness parameter, and mass ratios give significant change on the natural frequencies. In the present case, and referring to the general case, the rotational spring is neglected aθ, which a variation of the natural frequencies can be produced. Table 4 (Appendix 1) shows the variation of the first three dimensionless natural frequencies as a function of the non-dimensional mass coefficient R and the non-dimensional stiffness coefficient ax, which are obtained from Eqs. (11) and (18) and includes a comparison between the results proved by Kim, H. K et al. [12], which they, in their paper, treated the same problem by using the Fourier series. From this table, one can notice clearly that the obtained results are in good agreement with those obtained by using the Fourier transform, with a slight error, the think which explain the accuracy of the presented methods. In addition, other conclusions also can be drowned for a fixed value of R, as the non-dimensional stiffness coefficient ax increases the dimensionless natural frequencies βi increase, Inversely, for a fixed value of ax, as the nondimensional mass coefficient R increases, the dimensionless natural frequencies βi decreases.

Example 3: Cantilever beam joined by mass and rotational spring at the tip
In this case, comparably with the case 2, only the linear rotational spring and the mass are considered, and the vertical spring is neglected ax =0, as there are no results in term of frequency in the literature. Table 5 (Appendix 1). presents the values of dimensionless natural frequencies βi=1,2,3 for a various value of aθ, and R. From this table, especially for the first natural frequency, can be drown that for a fixed value of R, as the nondimensional stiffness coefficient aθ increases the dimensionless natural frequencies βi increase, Inversely, for a fixed value of aθ, as the nondimensional mass coefficient R increases, the dimensionless natural frequencies βi decreases. In this case, beam with translational, rotational spring, and no mass attached spring boundary condition R=0. The dimensionless natural frequencies, in this case, are provided in Table. 6 (Appendix 1) By comparing the present results with the available results given by Lau et al [13], the good agreement demonstrate clearly the accuracy of the present solutions. From Fig. 7, which represents the variation of the first three dimensionless frequencies as a function of ax, aθ the most conclusions that can be drawn are when dimensionless stiffness aθ keep constant and the ax increase, the natural frequency of system increases, also when dimensionless stiffness aθ increase and the ax keep constant, the natural frequency of system increases. If ax and aθ approach infinity, the beam becomes a clampedclamped.
Example 5: Cantilever beam joined by translational spring at the tip We consider the beam with translational spring. The dimensionless natural frequencies in this case are provided in Table. 7 (Appendix 1). Comparing the present's results with those available given by Banerjee. [11], in which in his work the dynamic stiffness matrix and the Wittrick -Williams algorithm to derive the natural frequencies of a similar system, demonstrate the accuracy of the proposed analytical solution. From  Fig. 8, which illustrates the variation of the first three dimensionless natural frequencies as a function of ax, important conclusions can be drawn that when dimensionless stiffness ax increase, the natural frequency of system increases too, Fig. 9 shows the variation of the first three mode shape with various value of ax. It can be seen from the results that the non dimensionless parameter ax has a significant effect on the natural frequencies and mode shape, if ax approaches to infinity the beam becomes clamped-pinned system.

Example 6: Cantilever beam joined by rotation spring at the tip
In this case, the cantilever beam is connected only by a rotational spring at the end, the effect of the vertical spring and mass are neglected, to show the influence of the rotational spring with nondimensional stiffness aθ, Table 8 (Appendix 1) represents the variation of the first three dimensionless natural frequency as a function of aθ and a comparison between the present results and the results presented by Lau et al. [13]. From this table and fig.12, the accuracy of the proposed solution proposed has been proved, and one can conclude that there is a slow effect in dimension-   fig. 12, represents the variation of the first dimensionless natural frequency as a function of the non-dimensional stiffness ratio aθ, ax, one can notice that for a specified value of these parameters, with the present calculation and examples, can be obtained for any value of aθ, ax and i.e. for pinned, clamped and free condition and one can say that the powerful of the presented solutions reside in its higher ability to give an idea about different system of boundary conditions in reality. 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.