Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation

: This article presents a modal analysis of an elastically constrained Rayleigh beam that is placed on an elastic Winkler foundation. The study of beams plays a crucial role in building construction, providing essential support and stability to the structure. The objective of this investigation is to examine how the vibrational frequencies of the Rayleigh beam are affected by the elastic foundation parameter and the rotational inertia. The results obtained from analytical and numerical methods are presented and compared with the conﬁguration of the Euler–Bernoulli beam. The analytic approach employs the technique of separation of variable and root ﬁnding, while the numerical approach involves using the Galerkin ﬁnite element method to calculate the eigenfrequencies and mode functions. The study explains the dispersive behavior of natural frequencies and mode shapes for the initial modes of frequency. The article provides an accurate and efﬁcient numerical scheme for both Rayleigh and Euler–Bernoulli beams, which demonstrate excellent agreement with analytical results. It is important to note that this scheme has the highest accuracy for eigenfrequencies and eigenmodes compared to other existing tools for these types of problems. The study reveals that Rayleigh beam eigenvalues depend on geometry, rotational inertia minimally affects the fundamental frequency mode, and linear spring stiffness has a more signiﬁcant impact on vibration frequencies and mode shapes than rotary spring stiffness. Further, the ﬁnite element scheme used provides the most accurate results for obtaining mode shapes of beam structures. The numerical scheme developed is suitable for calculating optimal solutions for complex beam structures with multi-parameter foundations.


Introduction
Structural elements such as beams are widely used in geotechnical, civil, and mechanical engineering because they can simulate the behavior of various structures. These structures are frequently used and modeled on elastic foundations for isolation purposes, to study the dynamics of buildings on the ground or in railway applications. To optimally design these structures, it is always necessary to know their dynamic characteristics. Given this, vibration analysis of beams on elastic foundations is a valuable study that can be used in various structural engineering applications.
To start with, the dynamic response of the beams without elastic foundations has been extensively studied by a number of researchers. Chun [1] presented the free vibration of the beam attached to the rotational spring at one end and by letting the other end free. Lee [2] derived the characteristic equation of a beam having a rotational spring at one end and the other free end with an attached mass. Lai et al. [3] used the Adomian decomposition method to solve the beam vibration problem. A sinc-Galerkin method was elastically constrained boundary conditions and material nonlinearity. Effectively addressing the complex dynamics associated with beam vibration under diverse conditions poses a formidable challenge in ensuring the reliability and accuracy of solutions. The accuracy of the obtained results is primarily contingent upon the assumptions made during modeling, the prescribed boundary conditions, and the chosen material properties within the study. In order to enhance the precision of predictions, researchers can strive to refine their models by employing more accurate assumptions or by leveraging advanced numerical techniques. Consequently, there remains a need for further research to bridge these gaps and develop more precise and efficient models for the analysis of beam vibrations on elastic foundations under various boundary conditions. The objective of the present study is aligned with this pursuit, aiming to address these research gaps and contribute to the development of improved methodologies for analyzing beam vibrations on elastic foundations.
This study is primarily focused on conducting an extensive modal analysis of a Rayleigh beam subjected to elastic constraints and positioned on an elastic Winkler foundation. The main aim is to investigate how the inclusion of rotational inertia influences the modal behavior and dynamic response of the beam. By carefully considering the combined effects of elastic constraints and the foundation's elasticity, the study aims to offer significant aid to the modal characteristics and overall behavior of the beam system. The solution to the underlying problem is derived with the best accuracy by using a finite element scheme for initial modes of the vibrating frequency with and without considering the Winkler elastic foundation. The frequency curve, natural frequencies, and corresponding mode shapes are sketched for various situations depicting the deflection behavior of the beam. The primary objective and novelty of this study is to establish a numerical scheme with the best accuracy so that the more complex nature of beam structures can be dealt with in such a scheme appropriately. The research is significant as it provides a prototype for obtaining optimal solutions for structural problems containing rotary and shear deformation effects simultaneously with dynamical boundary conditions. The findings of this study can be useful in determining optimal values of eigenfrequencies and eigenmodes for forced vibration problems of beams resting on multiparametric foundations. The study has applications in various fields such as civil engineering, mechanical engineering, and aerospace engineering. Understanding the vibrational behavior of beams on elastic foundations is crucial in the design of various structures, including buildings, bridges, and aircraft. The findings from this study can be applied to improve the accuracy of modeling and simulation tools used in structural analysis, which can lead to more efficient and cost-effective designs.
The rest of the article is organized as follows. Section 2 contains the governing problem. Section 3 states a working procedure for calculating eigenfrequencies, eigenvalues, and eigenmodes. The results are presented and discussed in Section 4, whereas the conclusion is provided in Section 5.

Statement of the Problem
Consider a Rayleigh beam (RB) that is attached to linear and rotational springs and resting on a Winkler elastic foundation as shown in Figure 1. Considering the Rayleigh beam theory, the equation of motion for a uniform Rayleigh beam [31] containing the homogeneous material properties is given by where ρ, I, A, ν, x, t, K, and E are mass density, second moment of inertia, the crosssection area of the beam, displacement of Rayleigh beam, space coordinate, time, stiffness of the Winkler elastic foundation per unit length, and Young's modulus, respectively. Rotational and linear springs are used to elastically restrained the beam resting on an elastic foundation. Accordingly, the boundary conditions (BC) are given as [31]: where L is the beam's length, τ 1 and τ 2 are linear spring constants and δ 1 and δ 2 are rotational spring constants. It is pertinent to mention here that the equation for the Euler-Bernoulli beam can be obtained by ignoring the rotary inertia effect in Equation (1). Additionally, the boundary conditions stated in Equations (2)-(5) render the classical boundary conditions as a special case by setting the spring constants accordingly. Therefore, the aim of this paper is to determine and analyze the frequency pattern and mode shape of the vibrating beam subject to the boundary condition (2)- (5). Note that the results for classical cases and the EBB are reduced as a special case. The next section explains the analytical and numerical procedure for determining eigenfrequencies and eigenmodes.

Determination of Natural Frequencies and Eigenmodes
In this section, we lay out the procedure for the determination of eigenfrequencies and eigenmodes. Numerous researchers used assorted techniques to handle similar problems with certain limitations and compromises on the accuracy of the approximate solutions. Separating the variables is suggested as a way to find frequency relations and eigenfunctions analytically. The root finding technique is then employed to determine eigenvalues and eigenfrequencies for determining respective eigenmodes. The finite element scheme is also used to determine a numerical solution whose validity is to be confirmed through validation.

Analytic Solution
The method of separation of variables is invoked herein to solve Equation (1). Accordingly, the displacement function is to be separated into two parts as Consequently, Equation (1) with the aid of Equation (6) ca be written as Furthermore, Equation (7) can be written as [32] −EIX (4) where ω is known as the natural frequency. Equation (8) can be further simplified to render and The solution to Equations (9) and (10) is given by where A, B, C, D, E, and F are constant coefficients to be determined. The parameters α and β above are defined by.
Equations (11) and (12), with the help of Equation (6) and the boundary conditions (2)-(5), lead to the system of equations, A system of four equations with four unknowns (A, B, C, and D) is represented by the Equations (15)- (18). The determinant of the coefficient matrix should be zero in order to find the non-trivial solution. It results in the characteristic equation, It is essential to note that the characteristic equation is used to calculate the eigenvalues α and β. Only when α is stated as a function of β or otherwise can the explicit values of α or β be found. The process outlined below is used to specifically determine the eigenvalues. Equations (13) and (14) define the dispersive relations, which can be expressed as where By using Equation (20), we have the expressions which together with (21) furnish Expressions in Equation (23) are made simpler, and the result is where the slenderness ratio u is described as Therefore, the eigenvalues of the RB are expressed in the form of a slenderness ratio, which indicates buckling (either pin-jointed or pivoted connections) failure in the beam structure beyond a certain limit. This implies that the eigenvalues in the case of the RB are dependent on the geometry contrary to the EBB,which only depends on the choice of boundary conditions. Hence, the eigenvalue expression (24) together with the slenderness ratio (25) can be written as where h is the inverse of the slenderness ratio. Given the above procedure, the characteristic Equation (19), together with Equation (26), yields the eigenvalues α and then β using a root finding procedure. This further helps in determining the eigenfrequencies using Equation (23). Thus, Equation (11), thanks to Equations (15)- (19), yields the mode function, The eigenvectors for determining A/D, B/D, and C/D are found by returning to the matrix equation rendered by Equations (15)- (18), and substituting the eigenvalues α and β into either of the three equations. Thus, the mode shapes are sketched and analyzed with the help of Equation (27).
As a special case, it is important to observe that by ignoring the stiffness of the elastic foundation and inverse of the slenderness ratio, i.e., h = K = 0, the problem is reduced to the EBB consideration where eigenvalues are explicitly determined and are independent of the beam geometry since both α and β become identical.

Formulation of Finite Element Method
A GFEM (Galerkin finite element method) is utilized to discretize the domain (length of the beam), which is divided into a set of finite line elements. In each beam element, there are two end nodes with two degrees of freedom each. A node can have nodal (vector) displacements or degrees of freedom, including translations (ν i ; i = 1, 2) and rotations (Ψj); j = 1, 2) as shown in Figure 2. Additionally, to obtain the differential Equation (1) in its weak form, multiply the residual by a weight function G(x) and integrate by parts to evenly distribute the differentiation orders G and ν. As a result, the equation is expressed as follows: After determining the weak form, approximate functions are selected for each element. In the weak form, ν(x, t) has the highest third order derivative. Therefore, thrice differential approximating functions are chosen. An interpolation polynomial would meet this requirement [33]. By using GFEM, the weight function can be equated with approximate functions G i = N i , and that cubic interpolation (see in Figure 3) function can be called a cubic spline (Hermite cubic interpolation function), given as On substituting Equation (29) into Equation (28) and ν := ∑ 4 j=1 ν j Nj, we obtain We can express Equation (30) as where k ij and m ij are the stiffness and mass matrices defined as. where and Therefore, if we consider harmonic time dependent ν j , i.e., by substituting Equation (34) into Equation (31), we obtain With the aid of a MATLAB code based on the GFEM, we calculate the natural frequencies and eigenmodes of beams subjected to elastic constraints. For a large number of elements, global stiffness matrices are straightforwardly produced. The Equation (35) can be utilized to determine the eigenvalues based on the stiffness and mass matrices, which further yield the eigen frequencies [31,34,35].

Results and Discussion
In this section, the proposed methods are used to determine the eigenfrequencies and eigenmodes of the elastically constrained RB and EBB with and without elastic foundation. Additionally, the frequency results of the proposed formulations are compared with the same results for the beams with classical boundary conditions available in the existing literature in order to verify its accuracy. Provided that spring parameters are given appropriate values, restrained boundary conditions degenerate into classical ones.

Graphical and Tabular Representations
This section presents the analysis of RB and EBB with and without Winkler elastic foundations having elastically constrained ends. The beams are made up of steel having L = 1 m, A = 0.0075 m 2 , E = 207 × 10 9 Pa, I = 14.063 × 10 −6 m 4 , and ρ = 76.5 × 10 3 kg/m 3 . Figures 4-7 depict the zeros (eigenvalues) of the dispersion relations for the RB and EBB having elastic constrained with and without elastic foundation, respectively. These eigenvalues are used to determine eigenfrequencies, which further help in determining the corresponding eigenmodes. Table 1 provides a comparison of four initial modes of natural frequencies of the RB and EBB for analytical and numerical results from higher to lower values of the stiffness parameters.  It is observed that the increase in the stiffness parameter yields an increase in the natural frequency and vice versa. The highest values of the stiffness parameters provide results for clamped-clamped edges while the lowest values of stiffness parameters give results for free-free edges of the beam. The RB and EBB results show that for smaller values of the stiffness parameters, one obtains rigid body modes, i.e., the translation or rotation of the beam takes place without undergoing any significant internal deformation. The comparison of the RB and EBB show that the presence of rotatory inertia yields lesser natural frequencies for the initial four modes that are less than 1%, 3%, 5%, and 7%, respectively, than that of the EBB. Hence, the rotatory inertia impacts the higher modes of the frequencies more than the lower modes. The comparison of the analytic and numerical results in percentage error (PE) is also made. Figure 8 shows the comparison of results for the RB (by ignoring rotatory inertia) with that of the EBB for the initial four modes while keeping the stiffness parameters identical. It is observed that the results of the RB are reduced to the EBB quite accurately. Figure 9 provides the comparison of the analytical and finite element results for the RB. The graphs show excellent agreement between analytical and numerical results in the absence of an elastic foundation. Table 2 presents the results of the initial four modes of the natural frequencies of the RB placed over an elastic foundation. A comparison is made between the analytic and numerical results in PE. Additionally, the results of EBB are stated for comparison purposes. According to the results, the increase in the stiffness of the elastic foundation increases the natural frequency. Moreover, this increase is relatively visible in the fundamental mode of the frequency.
Contrary to the beam that is not placed on an elastic foundation, no rigid modes are observed in this consideration. Figures 10 and 11 delineate the mode shapes of the RB that is placed over an elastic foundation for different values of the stiffness parameters of attached linear and rotational springs and elastic foundation. The analytical and finite element results are compared in Figure 10, indicating a good agreement, whereas Figure 11 shows the mode shapes of the RB obtained via FEM. In contrast to an independent beam, it is noted that the RB requires a higher value of frequency to vibrate when it is placed over an elastic foundation.   Tables 3-5 furnish the comparison of analytical and finite element results for varying the stiffness of rotational spring, elastic foundation, and linear spring, respectively. A decrease is observed in the natural frequency in each of the modes when one of the stiffness parameters is decreased and the other(s) is/are fixed. However, the fundamental frequency is considerably reduced as compared to higher modes frequencies.    Based on the analysis conducted, it can be deduced that manipulating the elastic foundation parameter allows for the adjustment of the vibrating frequency, thereby minimizing the duration for potential collateral damage to the vibrating structure. Consequently, placing the beam on an elastic foundation serves as a means to regulate its vibration and mitigate the risk of structural damage.

Validation of the Results
This subsection aims to provide the validity of the results obtained above. For this purpose, the underlying results are rendered for some special cases already reported in the literature. Rao [31] has outlined the natural frequencies of supported-supported RB and EBB by considering L = 1 m, A = 0.0075 m 2 , E = 207 × 10 9 Pa, I = 14.063 × 10 −6 m 4 , and ρ = 76.5 × 10 3 kg/m 3 , respectively. If the stiffness parameters of the linear and rotational springs are taken as τ 1 = τ 2 = 10 12 and δ 1 = δ 2 = 0, respectively, then the results obtained by Rao [31] are verified for simply supported edges. Additionally, the results of the EBB [36] and the RB for clamped-clamped edges are verified by letting the stiffness parameters as τ 1 = τ 2 = δ 1 = δ 2 = 10 12 . It is further observed that by equating the stiffness parameters of linear and rotational springs to zero, the obtained results are verified with that of free-free Euler-Bernoulli beam [36].

Conclusions
The frequency analysis of a beam resting on an elastic foundation and subject to rotary inertia effects has been studied. Analytical and finite element schemes have been used to determine the natural frequencies and corresponding mode shapes of the vibrating beam. The results have been obtained for the Rayleigh beam subjected to rotational and linear springs while the results for Euler-Bernoulli have been reduced as special cases. The key findings of the analysis are given as: • The eigenvalues obtained in terms of the slenderness ratio for the RB depend on the geometry, unlike the EBB where eigenvalues do not depend on the slenderness ratio. • The behavior of a beam under different conditions, such as the presence of rotatory inertia or placement on an elastic foundation, impacts its natural frequencies. • For smaller stiffness parameters, the beam undergoes rigid body modes without significant internal deformation.
• The inclusion of rotational inertia had a minimal effect on the fundamental mode frequency, but it had a significant impact on the higher frequency modes. • Placing the Winkler elastic foundation under the beam caused an increase in stiffness, leading to higher frequencies as the elastic foundation stiffness increased. • A detailed tabular and graphical analysis proved that the vibration frequencies and mode shapes are more affected by the linear spring stiffness compared to rotary spring stiffness. • Unlike independent beams, beams on an elastic foundation require higher frequencies to vibrate. Thus, by controlling the elastic foundation parameter, one can adjust the vibrating frequency to minimize collateral damage to the vibrating structure. • While comparing results with the existing ones in the literature, it has been observed that the finite element scheme provided the best accuracy for obtaining the mode shapes of the beam structure.
Therefore, it is concluded that the more complex nature of the beam structures can be treated with the numerical scheme established here. Optimal solutions for beams resting on multi-parameter foundations containing simultaneous shear deformation and rotational effects can be calculated considering forced vibration and dynamical boundary conditions. The strength of this study lies in the fact that its implications may lead to the development of more accurate and efficient numerical methods for analyzing beam structures, which can be used in building construction to provide essential support and stability to the structure. The findings may also be useful in designing beams that can minimize collateral damage to the vibrating structure by controlling the elastic foundation parameter. The research article may also pave the way for further research into the behavior of beams under different conditions, such as the presence of rotatory inertia or placement on an elastic foundation, and the impact on their natural frequencies. Contrarily, while the article provides detailed analysis and numerical methods for the vibrational frequencies of the Rayleigh beam, it is limited in terms of practical applications as it does not provide any experimental validation of the results.