The Behavior of the Synchronous and Asynchronous Natural Frequencies for Asymmetric Double Beams

One of the important industrial applications in aerospace engineering and construction is the double beams because it has distinctive engineering properties such as resistance to stresses and high impacts on external surfaces, with resistance to bending stresses and buckling due to the elastic conduction layer while having a very important property of lightweight. Which made the researchers make their best efforts to analyze them in terms of dynamic loads and resistance to vibrations, especially for asymmetric types. Under arbitrary boundary conditions, Kim et al. [1] examined the free vibration of an elastically linked double-beam structure linked by an elastic layer with a homogeneous elastic stiffness. The vibration of the structure is modeled using Timoshenko theory, which considers the effects of shear deformation as well as rotational inertia. The vibration of linked double-beam with generalized elastic boundary conditions was investigated using the Haar wavelet discretization method. Hao et al. [2] used a modified Fourier–Ritz technique to analyze the vibration of a linked double beam with random boundary conditions and arbitrary fundamental parameters of beams. The displacement components were stated as Fourier cosine series with auxiliary polynomial functions. Hammed et al. [3] examined the dynamical responses of a double Euler-Bernoulli beam system under the influence of a moving distributed force, which is elastically coupled by a two - parameter Pasternak constructional work. The fourth order partial differential equations describing the beam motion were transformed into second order ordinary differential equations using the Finite Fourier sine transformation. Using the differential transformation approach, the dynamic response of the beams was estimated. Yang et al. [4] explored analytically the double-beam system, which consists of two generic beams with an assortment of symmetric boundary conditions and found the double beam mode shapes are similar to those of a single at identical boundary conditions and the amplitude of its for a double-beam system is doubled that of a single beam. He and Feng [5] developed a formula for the dynamic response of an elastically coupled multiple beam system under a moving oscillator using the finite sine-Fourier inverse transform. Stojanovi'c et al. [6] studied a universal approach for determining the buckling loads and natural frequencies for a collection of beam systems subjected to a compressive axial stress. The dynamical behavior of multi-layered microbeam systems in the presence of a moving mass was studied by Khaniki and Hashemi [7]. An analytical solution has been discovered for double- and three-layered microbridge systems utilizing the Laplace transform. A state space technique has also been employed for higher-layered microbridge systems. Abu-Hilal [8] discussed the dynamic behavior of a double-beam system passes by a moving load. The two simply supported beams are parallel, identical to one another, and joined by a viscoelastic layer that runs the length of the beam. Both beams' dynamic deflections are expressed in analytical closed forms. Atiyah and Abdulsahib [9] investigated the effect of four geometric and material characteristics on the vibration of twin beams. The qualities of the intermediate layer are mass density, thickness, and modulus of elasticity of the two beams. The frequencies of the twin beams were computed using the Bernoulli-Euler beam. Mirzabeigy and Madoliat [10] examined the influence of a nonlinear Winkler inner layer on small-amplitude free vibration. The well-known frequency solutions for double-beam systems were used, and it was discovered that the elastic inner layer had the greatest influence on the fundamental frequency, using the first mode of vibration. De Rosa and Lippiello [11] used the differential quadrature method to investigate the vibration of double beams linked by a Winkler-type elastic layer. Vertical translation and rotation elastic restrictions were applied to the ends of the double-beam. Abdulsahib and Atiyah [12] studied the effect of non-linear elasticity on the frequency of sandwich beams under arbitrary boundary conditions. The impact of the inner layer's non-linearity stiffness on those frequencies was calculated using the energy balancing approach. Most of the previous studies focused on the investigation of vibrations of symmetric beams and did not pay much attention to the effect of properties of connecting layers between the beams on the vibration characteristics. The behavior of the higher and lower natural frequencies of the asymmetric doubled beams will be studied under different 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 the two beams. To validate the present results, a comparison is achieved with previous results. The influence of material properties of connecting layer on the vibration asymmetric double beam is examined.

In order to validate the accuracy of Eqns. (22) and (23), comparison tests were made between the results of the present work and other references. Those comparisons are shown in Tables 1 and 2. An excellent identification between the present work and the results of the references [2, 3] can be observed.

Table 1. The natural frequencies of double beam at PP and CP boundary conditions

$\rho_1 A_1=\frac{1}{2} \rho_2 A_2=300 \frac{\mathrm{kg}}{\mathrm{m}}, E_1 I_1=\frac{1}{2} E_2 I_2=6 \times 10^6, L=8 \mathrm{~m}, K_e=2.5 \times 10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$

Table 2. The natural frequencies of double beam at P-P, C-C and C-F boundary conditions

$\rho_1 A_1=\frac{1}{2} \rho_2 A_2=100 \frac{\mathrm{kg}}{\mathrm{m}}, E_1 I_1=\frac{1}{2} E_2 I_2=4 \times 10^6, L=10 \mathrm{~m}, K_e=1 \times 10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$

The change of the natural frequencies with the difference in the stiffness values of the connected layer between the two beams can be noted in Figure 2 and Table 3. When the stiffness values of the elastic layer are increased from 100 kN/m² to 2,100 kN/m², an increase in the values of high frequencies (asynchronous) is observed up to 125% for PP beams, 40% for CC beams, 280% for CF beams, and 170% for CP beams. The different hardness values of the connected layer have no effect on the values of the lower natural frequencies of double beams, as these frequencies maintain their values despite the increase and decrease in the stiffness values of the connected layer. From the above, it can be concluded that the greatest effect of the stiffness layer is on the values of the higher natural frequencies of CF beams and the least effect is on the PP beams. Generally, the increase in the stiffness of the elastic connected layer leads to an increase in the values of the high frequencies of those beams and does not affect the values of the lower natural frequencies of them.

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Figure 2. Higher natural frequencies vs. stiffness of connected layer

Table 3. Higher natural frequencies (Hz) vs. stiffness of connected layer (N/m²)

$E_1=E_2=10 \mathrm{Gpa}, h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}$,$L=10 \mathrm{~m}, \rho_1=\rho_2=3000 \mathrm{~kg} / \mathrm{m}^3$

Table 4. Higher natural frequencies (Hz) vs. thickness of upper beam (m)

$E_1=E_2=10 \mathrm{Gpa} ., h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}$, $L=10 \mathrm{~m}, \rho_1=\rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}$

Figure 3 and Table 4 manifest the relationship between the increase in the upper beam thickness and the change in the values of higher natural frequencies. When the thickness of the beam is increased by about 20 times, a decrease in higher frequencies is noted about 9% for PP beams, 9% for CC beams, 9% for FF beams, 27% for CF beams, and about 4% for CP beams. The values of the higher natural frequencies vary in the same proportions when the ratio of the thickness of the lower layer of the beam changes for all types of beams. It was also found that the values of lower natural frequencies are not affected by the change in the thickness of the upper or lower layer of the beams. As a result, it can be concluded that the greatest effect of changing the thickness of one of the upper or lower beams, or the thickness of one of them to the other is for CF beams and the least effect is for CP beams. In general, the influence of thickness is small on the frequencies compared to the rest of the factors studied in this research.

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Figure 3. Higher natural frequencies vs. thickness of upper beam

Figure 4 and Table 5 represent the behavior of low frequencies with the change in beam length. When the length increases about 50%, the lower frequencies will decrease approximately 55% for all types of beams. While Figure 5 evinces the relationship of higher frequencies with the variation in length of beam. In this figure, it is observed that when the length of the beam is increased by 50%, the higher frequencies decrease 48% for PP beams, 44% for CC beams, 25% for CF beams, and about 52% for CP beams. As a result, the length of the beam affects the higher and lower frequencies in high and close proportions for almost all types of beams, and the least effect is only on the higher frequencies of CF beams.

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Figure 4. Lower natural frequencies vs. length of beam

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Figure 5. Higher natural frequencies vs. length of beam

The effect of varying the modulus of elasticity of the upper beam on the values of the lower frequencies of double beams is depicted in Figure 6 and Table 6. When the values of the elastic modulus are increased from 10 GPa. to 30 GPa., the lower frequencies increase about 6% for PP beams, 1% for CC beams, 26% for CF beams, and about 1% For CP beams. Also, from Table 6 and Figure 7, when the modulus of elasticity increased from 10 GPa. to 30 GPa., the higher natural frequencies increase about 58% for PP beams, 70% for CC beams, 21% for CF beams, and about 66% for CP beams. As a result, the influence of the change of the modulus elasticity is relatively small on the lower natural frequencies of all types of beams except for CF beams, and its influence is relatively large on higher natural frequencies of the most types of beams and comparatively less on the CF beams. The same effect was seen for the variation of the modulus of elasticity of the lower or upper beam on the natural frequencies of the beams in the same ratios.

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Figure 6. Lower natural frequencies vs. modulus of elasticity of upper beam

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Figure 7. Higher natural frequencies vs. modulus of elasticity of upper beam

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Figure 8. Lower natural frequencies vs. mass density of upper beam

Figure 8 and Table 7 elucidate the relationship between the changes in the mass density of the upper layer of the beam with the lower natural frequencies of the double beams. When the mass density increases from 1,000 kg/m³ to 3,000 kg/m³, the lower frequencies decrease about 5% for PP beams, 1% for CC beams, 14%, and about 2% for CP beams. The behavior of higher natural frequencies is displayed in Figure 9 and Table 7. In addition, while the mass density increases from 1,000 kg/m³ to 3,000 kg/m³, the higher frequencies decrease about 40% for PP beams, 42% for CC beams, 33% for CF beams, and about 40% for CP beams. Therefore, the influence of changing the values of mass density is comparatively small on the low natural frequencies of all types of beams except for CF beams, and its comparatively is relatively large on the higher natural frequencies of all types of beams and relatively less on the CF beams. Furthermore, there is no difference between the changes in the mass density of the upper or lower layer; both have the same effect and percentage change of frequencies.

Table 6. Natural frequencies (Hz) vs. modulus elasticity of beam (m)

$\begin{aligned} E_2 & =10 \mathrm{Gpa} ., h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}, \\ L & =10 \mathrm{~m}, \quad \rho_1=\rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}\end{aligned}$

Table 7. Natural frequencies (Hz) vs. mass density of beam (kg/m³)

$\begin{aligned} E_1=E_2 & =10 \mathrm{Gpa} ., h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}, \\ L & =10 \mathrm{~m}, \quad \rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}\end{aligned}$

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Figure 9. Higher natural frequencies vs. mass density of upper beam