Compressive forces in ﬂ uence on the vibrations of double beams

: The in ﬂ uence of compressive forces on the lower and upper natural frequencies of the double beams has been studied in this article. Euler – Bernoulli ’ s hypoth - eses have been used to derive the natural frequency equa - tions. Two asymmetric beams were assumed in this work, and four di ﬀ erent boundary conditions were applied in these equations: Pinned – Pinned,Clamped – Clamped, Clamped – Free, and Clamped – Pinned. When the axial compressive force is increased about 18 times, it is observed that the lower natural frequencies decreased by 19% for PP beam, 8% for CC beam, 81% for CF beam, and 12% for CP beam. However, the greatest e ﬀ ect of the axial force on the higher frequencies is by reducing it in the CC beam by a ratio that does not exceed 2%. A rise in the values of axial compres - sive force causes a reduction in the lower natural frequen - cies, mostly for the CF beam, while it has a little e ﬀ ect on the higher natural frequencies. Similarly, when the com - pressive forces on the upper and lower beams ﬂ uctuate simultaneously, their e ﬀ ect is doubled on the frequencies when the axial compressive force on one of the two beams changes only.


Introduction
The double beam is considered one of the relatively recent developments in the field of creating materials that have unique properties of high resistance to stress and shock through the two outer layers and high flex-ibility with the light weight of the inner layer connecting between these two layers, which gave it a high resistance to bear the stresses of buckling and bending and gave these materials a wide space for engineering applications in aerospace fields such as aircraft structures, marine applications, the automotive industry, and structural applications. The vibration of beam constructions is essential in mechanical, civil, and aeronautical engineering. A double beam is a type of composite beam construction that is linked together to form a single beam. For different purposes, the beam's thickness and material qualities may vary. Zhang et al. [1,2] evaluated the vibrations of a coupled S.S double-beam system under compressive load using Bernoulli-Euler beam theory. A Winkler elastic layer is supposed to connect the two beams of the system indefinitely. The system's dynamic reactions to arbitrarily disperse continuous loads were determined, for two situations with specific excitation loadings. Zhao et al. [3][4][5] investigated the closed-form solutions of a Timoshenko double-beam forced transverse vibration under compressive axial stress. The two beams are represented by the Timoshenko model. The steady-state of the linked double-beam system was derived from the Laplace transform. Mao and Wattanasakulpong [6,7] evaluated the free vibration of a cantilever doublebeam system that is constantly linked. The differential equations for the double beam were expressed as a recursive algebraic equation based on the AMDM. Fei et al. [8][9][10] investigated the vibration properties of inclined double-beam. Two elastic beams make up the doublebeam system of varying mass which are connected by elastic springs. A tensile axial force was used on the beam with the higher rigidity mass. The dynamic equations of the double beam were developed by concurrently taking into account the effects of stiffness, sag, and other parameters. The element and stiffness matrix was generated from the governing equations to get the dynamic balance equations of the system. Sari et al. [11] studied the free vibration and stability assessments of single and double composite beams. A constant axially compressive or tensile force was applied to the closed-section beams. A layer of rotational and translational springs is supposed to link the twin beams. The discretization process was utilized to find the partial differential equations. On the mode forms, the critical loads, natural frequencies, impacts of elastic layer characteristics, axial forces, and boundary conditions have been examined. Stojanović et al. [12,13] investigated the effects of rotating inertia on the vibration and buckling of a double beam. The starting value and boundary value difficulties have been solved. An elastically coupled double-beam complex system's natural frequencies and amplitude ratios were determined. The influence of physical characteristics describing the vibrating system on the natural frequency, critical buckling load, and amplitude ratios was examined in the theoretical analysis. Kozić et al. [14] prepossessed an analytical method for defining the properties of coupled parallel beams subjected to axial force. For a complicated system, the amplitude ratio, natural frequencies, and critical buckling stress were calculated. Abdulsahib and Abbas Atiyah [15,16] investigated the influence of nonlinear elasticity on the frequency of sandwich beams with arbitrary boundary conditions. The energy balancing technique was used to calculate the effect of the inner layer's non-linearity stiffness on those frequencies. The behavior of the higher and lower natural frequencies of the asymmetric doubled beams will be studied under various boundary conditions, with the influence of a number of properties, such as the difference in thickness of the two beams, their mass densities, their elasticity modulus, the properties of the connected layer between them, or the length of beams. Milenković et al. [17] studied the natural frequencies of a Rayleigh double-beam system with a Keer layer in-between and the influence of axial stress. It was considered that the system's two beams being continually linked by a Keer layer. The system's equations of motion were defined by a number of differential equations. The standard Bernoulli-Fourier approach was employed to resolve these equations, and the Rayleigh theory was applied to derive the natural frequency and amplitude ratio of the examined model. Abbas Atiyah and Abdulsahib [18,19] examined the effect of four geometric and material properties on the twin beam vibration. The intermediate layer's properties, as well as the mass density, thickness, and modulus of elasticity of the two beams, were investigated. The Bernoulli-Euler beam equation was used to calculate the frequencies of the twin beams. In this article, the influence of the axial compressive forces on the lower and higher natural frequencies of the double beams was studied. Euler-Bernoulli's hypotheses have been used to derive the natural frequency equations. Two asymmetric beams were assumed in this work, and four different boundary conditions were applied in these equations: Pinned-Pinned, Clamped-Clamped, Clamped-Free, and Clamped-Pinned.

Theoretical work
Assuming two asymmetric beams joined by an elastic layer. Compressive axial loads are applied to both ends of each beam (F 1 ) and (F 2 ), respectively. Each beam has a different thickness (h), mass density (ρ), and modulus of elasticity (E), as shown in Figure 1. The two beams have the same length (L) and width (b). The Bernoulli-Euler beam theory was used to find the equations of motion as follows: where, A A E I I , , , , 1 2 , W W , and 1 2 are the cross-sectional areas, modulus of elasticity, moments of area, and the deflection for the upper and lower beam, respectively.
The following Boundary conditions were used in this work: For Cantilever (Pinned-Free) beam: For Pinned-Pinned beam: For Simply supported-Simply supported beam: ,́́, 0. For Free-Free beam: For Pinned-simply Supported beam: In order to solve equations (1) and (2), the following functions are assumed: . here: When substituting equations (15)-(17) into equations (1) and (2), one gets: n n n n n n n 1 (20) The differential equations (19) and (20) can be expressed as follows: Equations (20) and (21) can be expressed in matrix form as follows: The non-trivial solution of equation (23) is as follows: From equation (24), the lower natural frequency is: And, the higher natural frequency is: 3 Results and discussions Table 1 shows the first lower and higher eight natural frequency values for a double beam under compressive axial load for PP, CC, CF, and CP boundary conditions. The influence of the axial compressive force on the natural frequencies can be seen in Figures 2 and 3, and Table 2. When the axial compressive force is increased from 2,000 to 38,000 N, it is noticed that the lower natural frequencies decreased by 19% for the PP beam, 8% for the CC beam, 81% for the CF beam, and by 12% for the CP beam. While the greatest effect of the compressive force on the higher frequencies is by reducing it in the CC beam by a ratio that does not exceed 2%. Therefore, it can be concluded that the axial compressive force has a clear effect by reducing the lower natural frequencies, especially for the CF beam, and its effect is almost marginal on the higher frequencies. The same effect and behavior of the frequencies were observed when the axial compressive force on the upper beam of the double beam changed with the stability of the force value on the lower beam or when the effect was reversed by changing the compressive force on the lower beam and fixing it on the upper beam. Table 3 depict the behavior of the changing in natural frequencies when changing the values of the axial compressive forces on the upper and lower beams with the same values.

Figures 4 and 5, and
When the values of the axial forces are increased from 2,000 to 20,000, a decrease in the values of low natural frequencies is observed by 18% for the PP beam, 8% for the CC beam, 86% for the CF beam, and 11% for the CP beam. The greatest effect of changing the values of the axial compressive force on the higher natural frequencies when its values change symmetrically from 2,000 to 20,000 N for the CC beam causes it to decrease by less than 2%. A rise in the axial compressive force causes a lessening in the lower natural frequencies, particularly for the CF beam, while it has so little influence on the higher natural frequencies. Likewise, when the compressive forces on the upper and lower beams fluctuate at the same time, their effect is doubled on the frequencies when the axial compressive force on one of the two beams changes only.  Compressive forces influence on the vibrations of double beams  5 The behavior of natural frequencies when the axial compressive force on the upper beam changes and no force acts on the lower beam are shown in Figures 6  and 7 and Table 4. When the axial force is increased from 2,000 to 38,000, the lower natural frequencies decreased by 18% for the PP beam, 9% for the CC beam, 86% for the CF beam, and by 11% for the CP beam. Also, the effect of changing the value of the axial force in this case is so little on the higher natural frequencies and is almost imperceptible. In general, the natural frequencies change in the same proportion when the axial compressive force on one of the two beams changes, whether it is the upper or lower, and the value of the axial force remains constant on the other beam. This change is doubled when the two axial forces applied to the two beams change with the same value.
From the foregoing, it can be concluded that the effect of the axial force on the natural frequencies is clear when one end of the beam is free, and this effect is negligible if the two ends are fixed, especially if both are    clamped. This may be due to the buckling effect of the compressive axial force, so the more serious effect of the axial force is expected on natural frequencies when both ends are free. It was also noticed that the change in compressive force affects the lower natural frequencies (synchronous), while its effect is almost non-existent on the higher natural frequencies (asynchronous).

Conclusions
When the axial compressive force is increased from 2,000 to 38,000 N, it is noted that the lower natural frequencies decreased by 19% for the PP beam, 8% for the CC beam, 81% for the CF beam, and by 12% for the CP beam. While it is noticed that the greatest effect of the axial force on   the higher natural frequencies is by reducing it in the CC beam by a ratio that does not exceed 2%. A rise in the axial force causes a lessening in the lower natural frequencies, especially for the CF beam, while it has a small influence on the higher natural frequencies. Similarly, when the compressive forces on the upper and lower beams fluctuate at the same time, their effect is doubled on the frequencies when the axial compressive force on one of the two beams changes only. The natural frequencies change in the same proportion when the axial compressive force on one of the two beams changes, whether it is the upper or lower, and the value of the axial force remains constant on the other beam.
In this study, when both the upper and lower layers of the double beam are fixed symmetrically, it was found that the axial force has a significant effect on the lower natural frequencies (synchronous) and its effect is minimal on the higher natural frequencies (asynchronous), especially when one or both ends of the beam are free. In the future, it is important to study the change in the boundary conditions for the connection of the upper and lower layers to be asymmetrical by stabilization and the effect of this on the natural frequencies.
Conflict of interest: Authors state no conflict of interest.