Research article

A pair of dual Hopf algebras on permutations

  • Received: 24 November 2020 Accepted: 18 February 2021 Published: 09 March 2021
  • MSC : 05A05, 08C20, 16T05

  • Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra.

    Citation: Mingze Zhao, Huilan Li. A pair of dual Hopf algebras on permutations[J]. AIMS Mathematics, 2021, 6(5): 5106-5123. doi: 10.3934/math.2021302

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  • Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra.



    L: Length; E: Young's modulus; I: a moment of inertia of cross section; A: cross section area; x: longitudinal axis; ρ: density; s: Displacement; K1, K2: linear stiffness of the springs at right and left ends, respectively; S(x): normal function; ω: eigenfequencey; c: wave speed; η: eigenvalues; Ci(i=1,2,3,4): cnstant coefficient; Q(x): weighted functions; Ni(i=1,2,3,4): shape functions; sj: function of space and time component; ¯sj: amplitude of vibration varying with time; kϵ: stiffness matrix; mϵ: mass matrix; N: number of element

    The beam, a fundamental structural element, is typically characterized by having one dimension significantly larger than its other dimensions. This elongated geometry allows beams to efficiently carry loads and distribute them along their length. Euler-Bernoulli beam (EBB) theory stands as one of the foundational frameworks used to describe the behavior of beams under certain conditions [1]. The development of beam equations has a rich historical context, with contributions from notable figures such as Vinci and Galilei. During the eighteenth century, significant advancements in beam theory were made by eminent mathematicians including Euler, Jacob, and Daniel, culminating in the establishment of comprehensive analytical tools for understanding beam behavior. The application of beam theory extends across a diverse range of engineering disciplines, reflecting its essential role in various practical applications. In transportation engineering, beams are integral components of bridges, roadways, and railway tracks, where they provide structural support and facilitate the safe passage of vehicles and pedestrians. Structural engineering heavily relies on beam theory to design and analyze buildings, bridges, and other infrastructure projects, ensuring their stability and resilience against external forces such as wind and seismic loads. In aerospace engineering, beams play a crucial role in the design and construction of aircraft, spacecraft, and aerospace structures, where they must withstand complex loading conditions and extreme environments. Scholarly research in the field of continuous structural beam systems has yielded valuable insights and methodologies that contribute to advancements in engineering practice. Works by [2,3] exemplify the breadth of studies focused on beam theory, covering topics ranging from structural analysis and design optimization to materials research and performance evaluation. These studies underscore the interdisciplinary nature of beam theory and its significance in addressing real-world engineering challenges across multiple sectors.

    Numerous researchers have delved into the intricate dynamics of transverse vibrations exhibited by beam-spring systems, conducting both exact and approximate analyses to ascertain the natural frequencies governing their behavior. These investigations have explored the nuanced interplay between beam vibrations and various factors, including the characteristics of the elastic foundation supporting the beam. The effects of springs, rotary inertia, and mass distribution have been subjects of intensive scrutiny across a spectrum of studies, underscoring the multidimensional nature of beam-spring dynamics. The significance of springs in modulating beam vibrations has been a focal point of investigation, with studies such as [4,5] shedding light on their role in altering the structural response. Additionally, the influence of rotary inertia, mass distribution, and other dynamic parameters has been rigorously examined in works [6], among others. These investigations have contributed valuable insights into the intricate dynamics of beam-spring systems and their implications for diverse engineering applications. ¨Oz [7] and ¨Ozkaya [8] determined beam frequencies, incorporating mass and using both analytic and finite element methods. Grossi and Arenas [9] employed Rayleigh-Ritz and optimized Rayleigh-Schmidt methods to investigate frequency variations with changes in height and width. Smith et al. [10] introduced the fully Sinc-Galerkin method in space and time to solve beams with cantilever and fixed boundary conditions. Moaaz et al. [11] formulated mathematical expressions for transverse resonance in simply supported, axially compressed thermoelastic nanobeams, employing nonlocal elasticity theory and the dual-phase-lag heat transfer model to explore the impact of length scale and axial velocity on system responses. Baccouch [12] applied the Galerkin method to solve the Euler-Bernoulli beam equation, while Xie and Zhang [13] investigated difference methods for nonlinear equations with damping. Additionally, Shi et al. [14] explored the mixed finite element method for solving such equations.

    The variational iteration technique was used by Liu and Gurran [15] to find the natural frequencies and mode shapes of the beam under different boundary conditions. Galerkin finite element method (FEM), Rayleigh-Ritz, and exact solutions were compared by Hamdan and Latif [16], and the exhibited FEM was preferable because of good accuracy. Jafari et al. [17] derived the equation of motion of beam by applying Hamilton's principle and obtained the eigenfrequencies and mode shapes for cantilever beam connected with linear spring at the tip by using the FEM.

    Unlike boundary element method, finite difference method, and finite volume method, which are widely used in acoustic and fluid mechanics (see for example, [18,19]), the study of FEM is considered as the standard approach in presenting solutions to structural problems, see for instance, [20,21]. Many industries use FEM software such as ABAQUS (based on Abundant Beads Addition Calculation Utility System tool), NASA structure analysis, and analysis system for the commercial purpose. It has been widely discussed on static problems where numerous codes existed while having less validation and verification for standard problems with respect to dynamics of determining the eigenfrequencies. In the study of beam dynamics using FEM, consistent mass (CM) and lumped mass (LM) matrices are critical in accurately representing the system's behavior. The choice between these two matrices has an impact on the accuracy and computing efficiency of the analysis. The consistent mass matrix accurately preserves the system's mass distribution by taking into account the mass contributions of all finite constituents. This matrix is constructed directly from the discretized governing equations, yielding more accurate results, particularly for bigger models with variable element sizes. However, its bigger size necessitates more computing effort and storage. The lumped mass matrix, on the other hand, combines the mass contributions into nodal points, making the computing process easier. While this matrix decreases computational effort and storage needs, it may produce less accurate results, especially in systems with large mass variations or irregular geometries. Hence, the primary aim of this study is to present a comprehensive analysis of the vibration behavior of beam-spring systems using FEM with a focus on the comparative performance of CM and LM matrices. Our work bridges the gap between mathematical analysis and engineering applications by addressing two primary objectives:

    (1) Determining the preferred matrix type (CM or LM) for such analyses from a mathematical perspective;

    (2) Examining the deflection behavior while adjusting the stiffness of the supports at each end of the beam from an engineering perspective.

    The novelty of our study lies in its detailed comparison of CM and LM matrices and providing insights into their respective accuracies in calculating eigenfrequencies and mode shapes, which has not been extensively covered in existing literature. By varying the stiffness of the beam supports, we analyze the dynamic behavior of the system, offering practical insights valuable for both theoretical and applied contexts. Additionally, we provide a numerical results section to include a deeper analysis of the outcomes prior to yielding a detailed introduction to the FEM procedure, including the assembly of global stiffness and mass matrices and the incorporation of boundary conditions. Therefore, the underlying study is crucial for both mathematical analysis and practical engineering applications, making our study a valuable contribution to the field of applied mathematics.

    This article is categorized as follows: the governing problem is formulated in Section 2. The analytical and numerical results are presented in Section 3. Results and discussions are given in Section 4 and the validation has been performed in Section 5 while the study is summarized in Section 6.

    Consider the beam configuration attached to linear springs at both ends having length, as can be viewed through Figure 1. The material of the beam is made of stainless steel. The Young's modulus is 210 GPa and material density is 7850 Kg/m3. The length, width, and thickness of the beam are taken as 1 m, 0.02 m, and 0.003 m, respectively.

    Figure 1.  Beam-spring configuration.

    The equation of motion for the transverse deflection function in case of free beam's vibration is given by [22]

    EI4s(x,t)x4+ρA2s(x,t)t2=0. (2.1)

    Equation (2.1) together with the standard linear spring conditions of beam defined in [23] will be considered for the correct determination of eigenfrequencies and mode shapes. The objective is to provide a more optimal and accurate solution approach by considering the lumped and consistent mass matrices while providing the finite element solution in proceeding sections. The flexural boundary conditions at x=0 and x=L, representing the stiffness of the springs in the positive direction from left to right, are provided below:

    EI2s(0,t)x2=0, (2.2a)
    EI3s(0,t)x3=K1s(0,t), (2.2b)
    EI2s(L,t)x2=0, (2.2c)
    EI3s(L,t)x3=K2s(L,t). (2.2d)

    Equation (2.1) along with boundary conditions (2.2a–d) is solved by separation of variables to acquire the characteristics and mode shape equations where the roots of the characteristic equation are extracted through Mathematica-based code. These roots are termed as eigenfrequencies and will be useful in determining the eigenmodes in the subsequent section.

    As we seek to determine the natural frequency and mode shape of the vibrating beam as given by previously defined boundary conditions, we use an analytical and numerical approach to obtain the desired results. It is appropriate to note that a numerical approach is used to consider CM and LM matrices, which provides a justification for solving more complex problems with a similar scheme.

    We apply separation of variables to the governing problem given in Section 2 by letting

    s(x,t)=S(x)T(t). (3.1)

    Therefore, Eq (2.1) can be written as [24]

    c21S(x)4S(x)x4=1T(t)2T(t)t2=ω2, (3.2)

    where

    c=EIρA,

    and Eq (3.2) is further split as

    d4S(x)dx4η4S(x)=0 (3.3)

    and

    d2T(t)dt2+ω2T(t)=0, (3.4)

    where η is given by

    η=ωc. (3.5)

    Equation (3.3) yields the following equation

    S(x)=C1sin(ηx)+C2cos(ηx)+C3sinh(ηx)+C4cosh(ηx). (3.6)

    Using Eq (3.6) in Eq (2.2a–d), we obtain

    EIC2η2+EIC4η2=0, (3.7)
    EIη3C1+C2K1+EIη3C3+C4K1=0, (3.8)
    C1η2EIsin(ηL)C2η2EIcos(ηL)+C3η2EIsinh(ηL)+C4η2EIcosh(ηL)=0 (3.9)

    and

    C1[EIη3cos(ηL)K2sin(ηL)]+C2[EIη3sin(ηL)K2cos(ηL)]  +C3[EIcosh(ηL)K2sinh(ηL)]+C4[EIη3sinh(ηL)K2cosh(ηL)]=0. (3.10)

    By using Eq (3.7) in Eq (3.6), we obtain

    S(x)=C1[C2C1(cosh(ηx)+cos(ηx))+sin(ηx)+C3C1sinh(ηx)]. (3.11)

    The results obtained by using boundary conditions (2.2a–d) are given by Eqs (3.7)–(3.10) representing a system of four equations with four unknowns C1C4. In order to have a nontrivial solution, the determinant of the coefficient matrix must be zero, which leads to a frequency equation given as

    2η2EI[η3EIcosh(η)(EIη3cos(ηL)+sin(ηL)(K1+K2))   +(EIη3(K1+K2)cos(ηL)+2sin(ηL)K1K2)sinh(ηL)+E2I2η6]=0. (3.12)

    After finding the values of C3C1 and C2C1 from Eqs (3.8) and (3.10), respectively, and putting into Eq (3.11), the nth mode shape equation is determined. This equation is used to plot the mode shapes/eigenmodes subject to corresponding eigenfrequencies. The eigenfrequencies are computed by using Eq (3.5), after determining the eigenvalues (η) from Eq (3.12).

    The analytic solution for the beam attached to linear springs is achieved more conveniently. However, in case of more complex and challenging problems with added effects and flexural boundary conditions, and structural and material discontinuities [25], it becomes difficult to obtain analytical solutions. Therefore, we propose the finite element scheme with the consideration of LM and CM matrices to obtain numerical solutions for such problems in subsequent section. Albeit the scheme to be adopted herein will provide a reference model for the comparative analysis. In case of substantial problems related to beams, FEM provides a fast and easy method to address these problems.

    This section aims to provide the workings of FEM by CM and LM matrices to calculate the eigenmodes and eigenfrequencies.

    The first step in FEM is to discretize the beam into a finite number of elements. There are two end nodes, each with two degrees of freedom, for each beam element. As demonstrated in Figure 2, these degrees of freedom include both translational displacements (Vi, where i = 1, 2) and rotational displacements (θj, where j = 1, 2).

    Figure 2.  Beam element.

    The node's movement across the axis of the beam is indicated by the translational degrees of freedom, and its rotation about the corresponding axes is indicated by the rotational degrees of freedom. As the beam under examination is uniform, it is assumed that all elements, utilized to mesh the total beam, are indistinguishable. The subsequent step is to acquire the weak form of the differential equation. For this purpose, the weight functions are multiplied with the residual value of an approximate solution and are then integrated over the domain yielding zero value. While using the process of discretization and weak formation in Eq (2.1), it is found that

    L0Q(x)(EId4sdx4+ρAd2sdt2)dx=Q(x)EId3sdx3|L0EId2sdx2dQdx|L0+L0EId2sdx2d2Qdx2dx+L0Q(x)(ρAd2sdt2)dx=0. (3.13)

    It is noted that the highest order of derivative in Eq (3.13) is three; therefore, an approximate function of thrice differentiable is selected. A cubic interpolation polynomial fulfills this requirement [26] normally. Applying the Galerkin FEM, a weight function is equated with the approximate function

    Qi=Ni,

    where these cubic interpolation functions are called cubic spline functions, given as

    N1=13(xL)2+2(xL)3,N2=x(xL1)2,N3=(xL)2(32xL),N4=x2L(xL1). (3.14)

    While putting these shape functions, given by Figure 3, into Eq (3.13) and assuming

    s=4j=1sjNj,
    Figure 3.  Shape function for EBB.

    we get

    L0Q(x)(EId4sdx4+ρAd2sdt2)dx=EIsjNiNj,xxx]L0EIsjNj,xxNi,x]L0+L0EINi,xxNj,xxsjdx+ρAL0sj,ttNiNjdx=0. (3.15)

    Equation (3.15) can be written as

    [kϵ]sj+[mϵ]sj,tt=0, (3.16)

    where sj can be written as

    sj={¯sj}eiωt. (3.17)

    Putting Eq (3.17) in Eq (3.16), we obtain

    [kϵ]ω2[mϵ]=0, (3.18)

    where [kϵ] and [mϵ] are given as

    [kϵ]=EIL3[126L126L6L4L26L2L2126L126L6L2L26L4L2], (3.19)
    [mϵ]=ρAL420[15622L5413L22L4L2L3L3L25413L15622L13L3L222L4L2], (3.20)

    and the LM matrix [21] would be

    ρAL2[1000000000100000]. (3.21)

    Here, we clarify the computational procedures using MATLAB codes based on FEM with LM and CM matrices to calculate the eigenfrequencies and mode shapes of the beam subject to linear springs. Below is a detailed description of our approach:

    (I) Development of global matrices

    (1) The MATLAB code generates the global stiffness, and LM and CM matrices for the highest number of elements.

    (2) Boundary conditions are incorporated after forming the global stiffness and mass matrices.

    (II) Incorporating boundary conditions

    (1) The stiffness of the linear spring attached to the left end is added to the first entry of the first row of the global stiffness matrix.

    (2) Similarly, the stiffness of the spring at the right end is added to the second to last entry of the final row of the global stiffness matrix.

    (III) Calculation of eigenvalues and eigenfrequencies

    (1) Eigenvalues are calculated from the global stiffness and mass matrices according to Eq (3.18).

    (2) Eigenfrequencies are then obtained by taking the square root of these eigenvalues.

    This procedure follows the methods outlined in several studies, such as references [27,28,29,30,31].

    This section aims to discuss the eigenmodes/mode shapes of the structural dynamics of the beam, which actually describes the deformation that the beam component would exhibit if it vibrates at the eigenfrequency. The deformation usually takes place through an excitation, which leads to the overall vibration of a component of the beam that includes the individual shapes of vibration. Therefore, eigenfrequencies and mode shapes indicate how the beam structure behaves under certain boundary conditions. It is noteworthy that the eigenmode characteristic is suitable for the qualitative evaluation of the dynamics of the beam component. The tabular and graphical analysis are presented to discuss beam dynamics at length.

    Table 1 lists the comparison of eigenfrequencies determined by analytic method (AM), FEM by CM and LM matrices, receptively, with

    K1,K2=10000 N/m.
    Table 1.  Comparison of eigenfrequencies of first four modes for K1=K2=10000 N/m.
    Modes AM FEM by CM FEM by LM Percentage error
    For CM For LM
    N=5
    1st 6.90724849 6.90796199 6.91434953 0.01032973 0.10280563
    2nd 26.09674381 26.13470717 26.42771700 0.14547164 1.26825473
    3rd 52.84510702 53.16067633 54.27922231 0.59715900 2.71380904
    4th 81.45836541 82.55087709 74.63946703 1.34119028 8.37102285
    N=10
    1st 6.90724849 6.90729342 6.90918512 0.00065048 0.02803765
    2nd 26.09674381 26.09915748 26.18963094 0.00924893 0.35593375
    3rd 52.84510702 52.86511702 53.29345172 0.03786538 0.84841289
    4th 81.45836541 81.53039633 80.56105229 0.08842667 1.10156043
    N=50
    1st 6.90724849 6.90724856 6.90732785 0.000000101 0.00114894
    2nd 26.09674382 26.09674771 26.10054215 0.00001487 0.01455480
    3rd 52.84510702 52.84513933 52.86350476 0.00006114 0.03481446
    4th 81.45836541 81.45848351 81.43321124 0.00014498 0.03087979
    N=100
    1st 6.90724849 6.90724851 6.90726831 0.00000029 0.00028694
    2nd 26.09674382 26.09674412 26.09769399 0.00000115 0.00028694
    3rd 52.84510702 52.84510912 52.84970909 0.00000397 0.00870860
    4th 81.45836541 81.45837285 81.45215478 0.00000913 0.00762930

     | Show Table
    DownLoad: CSV

    The accuracy of eigenfrequencies are increased by increasing the number of elements of the beam. The percentage errors (with N = 5) for the CM matrix are 0.01032973, 0.14547164, 0.59715900, and 1.34119028, respectively.

    While 0.10280563, 1.26825474, 2.71380904, and 8.37102285 are percentage errors for the LM matrix, this indicates that the CM matrix gives better accuracy for nonclassical boundary conditions as compared to the LM matrix even for the lower number of elements. Table 1 clearly demonstrates that CM gives excellent agreement at

    N=50,

    but does not get the same accuracy for LM even by using

    N=100.

    It is noteworthy that

    N=10

    yields excellent accuracy; however,

    N=50

    and

    N=100

    are intended merely to demonstrate that increasing the number of elements improves accuracy.

    Table 2 is a comparison of natural frequencies calculated by analytical and numerical methods, respectively, by decreasing the spring stiffness of the right end. The table shows that natural frequencies and stiffness of the right end spring are in a direct relationship, where natural frequencies decrease with decreasing spring stiffness. In addition, the numerical results are in close agreement with the analytical results when comparing the numerical solution for the CM matrices instead of the LM matrices. Rather, FEM by the consideration of CM matrices gives excellent accuracy not only for lower modes but also for higher modes.

    Table 2.  Comparison of eigenfrequencies of first four modes for K1=10000 N/m, K2=1000 N/m.
    Modes AM FEM by CM FEM by LM Percentage error
    For CM For LM
    N=5
    1st 6.35701924 6.35757819 6.38367463 0.00879264 0.41930642
    2nd 18.88435515 18.89870611 18.60125288 0.07599391 1.49913654
    3rd 37.23312923 37.33388399 34.14209661 0.27060513 8.30183410
    4th 66.98883703 67.54369195 61.04497484 0.82827967 8.87291440
    N=10
    1st 6.35701924 6.35705430 6.36418013 0.00055151 0.11264540
    2nd 18.88435515 18.88526930 18.82742317 0.00484077 0.30147696
    3rd 37.23312923 37.23999014 36.48814214 0.01842689 2.00087155
    4th 66.98883703 66.92997846 65.34527890 0.08786325 2.45348061
    N=50
    1st 6.35701924 6.35701929 6.35730648 0.00000078 0.00451846
    2nd 18.88435515 18.88435660 37.20428112 0.00000767 0.01115087
    3rd 37.23312923 37.23314046 37.20428112 0.00003016 0.07747968
    4th 66.98883703 66.98890259 66.92288123 0.00009786 0.09845789
    N=100
    1st 6.35701924 6.35701923 6.35709101 0.00000015 0.00112898
    2nd 18.88435515 18.88435528 18.88383000 0.00000068 0.00278087
    3rd 37.23312923 37.23312992 37.22592579 0.00000185 0.01934685
    4th 66.98883703 66.98884110 66.97235365 0.00000607 0.02460615

     | Show Table
    DownLoad: CSV

    Table 3 indicates the comparison of eigenfrequencies by increasing the stiffness of the left end. From tables, it is observed easily by increasing the value of K1 that eigenfrequencies are also increased. A study has demonstrated that the CM matrices can give accurate eigenfrequencies for higher modes as well as lower ones, but LM cannot give the best accuracy for lower modes as well as for higher modes.

    Table 3.  Comparison of eigenfrequencies of first four modes for K1=1000000, K2=10000 N/m.
    Modes AM FEM by CM FEM by LM Percentage error
    For CM For LM
    N=5
    1st 6.96985529 6.97058784 6.97318699 0.01051026 0.04780157
    2nd 27.03932973 27.08117845 27.20251475 0.15476981 0.60350986
    3rd 57.08313134 57.47001406 57.52811675 0.67775315 0.77953924
    4th 92.56612927 94.07004264 80.41180184 1.62469078 13.13042635
    N=10
    1st 6.96985529 6.96990142 6.97085023 0.00066185 0.01427757
    2nd 27.03932973 27.04200859 27.09268028 0.00990727 0.19730722
    3rd 57.08313134 57.10818680 57.34259598 0.04389293 0.45453827
    4th 92.56612927 92.67041635 90.82713601 0.11238909 1.87864964
    N=50
    1st 6.96985529 6.96985536 6.96989697 0.00000100 0.00059800
    2nd 27.03932973 27.03933403 27.04157469 0.00001590 0.00830257
    3rd 57.08313134 57.08317194 57.09446492 0.00007112 0.01985452
    4th 92.56612927 92.56630243 92.51496052 0.00018707 0.05527805
    N=100
    1st 6.96985529 6.96985538 6.96986568 0.00000129 0.00014907
    2nd 27.03932973 27.03932996 27.03989174 0.00000085 0.00207849
    3rd 57.08313134 57.08313385 57.08597067 0.00000440 0.00497403
    4th 92.56612927 92.56614015 92.55346771 0.00001175 0.01367839

     | Show Table
    DownLoad: CSV

    Table 4 mentions the comparison of numerical and analytical results by increasing the spring stiffness of both ends and shows a similar behavior for natural frequencies and solution accuracy as in case of Tables 13.

    Table 4.  Comparison of eigenfrequencies of first four modes for K1=K2=50000 N/m.
    Modes AM FEM by CM FEM by LM Percentage error
    For CM For LM
    N=5
    1st 7.00984547 7.01058818 7.01069925 0.01059524 0.01217972
    2nd 27.72533453 27.77009228 27.75837598 0.01614326 0.11917421
    3rd 61.16521834 61.63086692 61.09088070 0.76129635 0.12153580
    4th 105.4525306 107.82633768 101.05828707 2.25106688 4.16703475
    N=10
    1st 7.00984547 7.00989023 7.01218744 0.00063853 0.03340972
    2nd 27.72533453 27.72813738 27.74771968 0.01010934 0.08273897
    3rd 61.16521834 61.19584117 61.39079599 0.05006575 0.08073897
    4th 105.4525306 105.60832597 106.45143402 0.14773974 0.94725408
    N=50
    1st 7.00984547 7.00984337 7.00986020 0.00002995 0.00021013
    2nd 27.72533453 27.72529534 27.72631753 0.00014135 0.00354549
    3rd 61.16521834 61.16521665 61.17571474 0.00002763 0.01716073
    4th 105.4525306 105.45258028 105.49998929 0.00004703 0.04500470
    N=100
    1st 7.00984547 7.00984334 7.00984750 0.00003038 0.00002895
    2nd 27.72533453 27.72529099 27.72554833 0.00015704 0.00077113
    3rd 61.16521834 61.16516979 61.16781319 0.00007937 0.00424236
    4th 105.4525306 105.45234022 105.46427897 0.00018055 0.03111408

     | Show Table
    DownLoad: CSV

    Figures 47 represent the first four natural modes for varying the spring stiffness of the underlying beam configuration. It is evident that the number of node points increases with each successive mode. The occurrence of node points indicates negligible deflection at certain locations of the beam. For higher natural modes, deflection becomes zero at multiple locations on the beam. Furthermore, the mode shapes clearly provide alternative symmetric and antisymmetric modes in case of even and odd modes, respectively. It is worth noting that the greatest amount of deflection occurs at the end of the beam because the spring beam system generally shows an initial deflection to obtain the equilibrium position. However, the spring no longer deflects after having static equilibrium.

    Figure 4.  Mode shapes of (mode 1): 1st eigenmode; (mode 2): 2nd eigenmode; (mode 3): 3rd eigenmode; (mode 4): 4th eigenmode; for K1 = K2=10000 N/m.
    Figure 5.  Mode shapes of (mode 1): 1st eigenmode; (mode 2): 2nd eigenmode; (mode 3): 3rd eigenmode; (mode 4): 4th eigenmode; for K1=10000 N/m,K2=1000 N/m.
    Figure 6.  Mode shapes of (mode 1): 1st eigenmode; (mode 2): 2nd eigenmode; (mode 3): 3rd eigenmode; (mode 4): 4th eigenmode; for K1=1000000 N/m, K2=10000 N/m.
    Figure 7.  Mode shapes of (mode 1): 1st eigenmode; (mode 2): 2nd eigenmode; (mode 3): 3rd eigenmode; (mode 4): 4th eigenmode; for K1=K2=50000N/m.

    The intent of this section is to demonstrate the validity of the results mentioned above. The underlying results for a few specific cases that have already been documented in the literature are rendered for this purpose. The simply supported EBB eigenfrequencies have been determined to be precisely comparable to those reported by Leissa [23] when

    K1=K2=1012

    has been taken into account, as shown in Table 5. Furthermore, letting the stiffness parameters be

    K1=K2=0
    Table 5.  Comparison of eigenfrequencies of first four modes for N = 10.
    Modes [23] FEM by CM FEM by LM
    K1=K2=1012
    1st 9.8696 9.8696 9.8695
    2nd 39.478 39.4782 39.4737
    3rd 88.876 88.8769 88.7669
    4th 57.914 157.7529 157.5231
    K1=K2=0
    1st 0.0000 0, 0000 0.0000
    2nd 0.000 0.0002 0.0041
    3rd 22.273 22.3740 21.7055
    4th 61.373 61.6881 58.6391

     | Show Table
    DownLoad: CSV

    verifies the findings of the free free EBB [23].

    In this article, the modal analysis of the Euler-Bernoulli beam subject to attached linear springs has been made. Eigenfrequencies and mode shapes were evaluated by comparing the analytic method and FEM using CM and LM matrices by varying the stiffness of the springs. It has been noted that more numbers of beam elements resulted in better accuracy of eigenfrequencies whereas consideration of the CM matrix showed excellent accuracy as compared to the LM matrix for N = 5 or N = 100. This has justified the preference of considering the CM matrix over the LM matrix even for a small number of beam elements for the greater advantage and better accuracy of the solution. Also, it has been observed that a large amount of deflection occurred at the end points of the beam which justified the dynamics of the spring beam system.

    While dealing with more complex and practical problems of beam dynamics, obtaining analytical solutions often becomes challenging and sometimes impossible. It has been noticed that the use of FEM with the consideration of the LM matrix is rather convenient in addressing more challenging and practical problems related to beam dynamics. The underlying study concluded with the aim to provide an effective way to treat such problems numerically in a more efficient way. Furthermore, it is important to note that the suggested method can be easily extended to specific two-dimensional structures such as plates and shells with common boundary conditions, albeit with numerical concerns. However, it is critical to recognize that adding plates and shell-like structures may create complexities that are beyond the capability of typical numerical methods. In such circumstances, mode-matching algorithms have emerged as viable options for addressing complex structures; see, for reference, [32,33]. While mode-matching methods may neglect some effects, such as break-out, they provide useful insights into vibrating processes and serve as benchmark answers for fully numerical systems. Traditionally, numerical and analytical methods are used to describe finite-length plate or shell-like structures. Numerical methods, such as the finite element method or the boundary element method, allow for the analysis of structures of various shapes and sizes. However, as the excitation frequency and structure dimensions increase, so does the number of degrees of freedom, making these approaches unsuitable even for relatively small structures. As a result, in situations where typical numerical approaches are limited, mode-matching solutions provide a desirable and precisely computed option for handling the complexity inherent in modeling finite-length plate or shell-like structures.

    M. Alkinidri: conceptualization, methodology, writing (review and editing), investigation, analysis, visualization, validation, resources; R. Nawaz: conceptualization, methodology, writing (original draft), investigation, analysis, visualization, validation, supervision; H. Alahmadi: methodology, writing (review and editing), investigation, analysis, visualization, validation, resources.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors express their gratitude to Dr. Gulnaz Kanwal for her valuable suggestions, particularly regarding the computational aspects of the study, which significantly improved the quality of the article.

    The authors declare no conflicts of interest.



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