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The effect of vibrations on asymmetric double beams is a common engineering problem in various engineering applications. In this paper, the synchronous (lower) and asynchronous (higher) natural frequencies of the asymmetric double beams are calculated using the BernoulliEuler method. Where the traditional methods are used to find the frequency equations at different boundary conditions, such as Pinned beam, clampedClamped beam, ClampedFree beam, and ClampedPinned beam. The increase in the stiffness of the elastic connected layer leads to an increase in the values of the high frequencies of double beams. The greatest effect of changing the thickness of one of the upper or lower beams is for CF beams and the least effect is for CP beams. 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. The influence of the modulus elasticity change is relatively small on the lower natural frequencies of all types of beams except for CF beams, and its effect is relatively large on the higher natural frequencies of the most types of beams and comparatively less on the CF beams. The effect of varying the values of mass density is relatively small on the low natural frequencies of all types of beams except for CF beams, and its effect is comparatively large on the higher natural frequencies of all types of beams and relatively less on the CF beams.
double beam, vibration of asymmetric beam, synchronous and asynchronous mode
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 doublebeam 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 doublebeam 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 EulerBernoulli 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 doublebeam 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 doublebeam 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 sineFourier 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 multilayered 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 threelayered microbridge systems utilizing the Laplace transform. A state space technique has also been employed for higherlayered microbridge systems. AbuHilal [8] discussed the dynamic behavior of a doublebeam 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 BernoulliEuler beam. Mirzabeigy and Madoliat [10] examined the influence of a nonlinear Winkler inner layer on smallamplitude free vibration. The wellknown frequency solutions for doublebeam 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 Winklertype elastic layer. Vertical translation and rotation elastic restrictions were applied to the ends of the doublebeam. Abdulsahib and Atiyah [12] studied the effect of nonlinear elasticity on the frequency of sandwich beams under arbitrary boundary conditions. The impact of the inner layer's nonlinearity 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.
Figure 1 shows the asymmetric double beam of different properties (ρ, E, b, and h). An elastic layer between them having the elastic stiffness (K_{e}) connects these beams. The BernoulliEuler beam theory for vibrations is utilized to relate the equations of motion [12, 13]:
$\frac{\partial^2}{\partial x^2}\left(E_1 I_1 \frac{\partial^2 Y_1}{\partial x^2}\right)+K_e\left(Y_1Y_2\right)+\rho_1 A_1 \frac{\partial^2 Y_1}{\partial t^2}=0$ (1)
$\frac{\partial^2}{\partial x^2}\left(E_2 I_2 \frac{\partial^2 Y_2}{\partial x^2}\right)K_e\left(Y_1Y_2\right)+\rho_2 A_2 \frac{\partial^2 Y}{\partial t^2}=0$ (2)
where, A_{1}, A_{2}, ρ_{1}, ρ_{2}, E_{1}, E_{2}, I_{1}, I_{2}, Y_{1} and Y_{2} are the crosssectional area, mass density, modulus of elasticity, moment of area and the deflection for first and second beam, respectively.
Figure 1. Asymmetric double beam
Assuming the timeharmonic motion as follow [14]:
$W_i(x, t)=\sum_{n=1}^{\infty} x_n(x) \cdot T_{n i}(t), i=1,2$ (3)
where, [15],
$\zeta=\frac{x}{L}$ (4)
$y_n(\zeta)=\cosh \left(\Omega_n \zeta\right)\cos \left(\Omega_n \zeta\right)$$\sigma_n\left[\sinh \left(\Omega_n \zeta\right)\sin \left(\Omega_n \zeta\right)\right]$,
$\Omega_n=\frac{\pi(2 n+1)}{2}$, (5)
$n=1,2,3, \ldots \ldots, \sigma_n \cong 1$ For Clamped beams [15]
$y_n(\zeta)=\sin \left(\Omega_n \zeta\right), \quad \Omega_n=n \pi, \quad n=1,2,3, \ldots ..$ (6)
For Pinned beams [15]
$y_n(\zeta)=\cosh \left(\Omega_n \zeta\right)+\cos \left(\Omega_n \zeta\right)$$\sigma_n\left[\sinh \left(\Omega_n \zeta\right)+\sin \left(\Omega_n \zeta\right)\right]$,
$\Omega_n=\frac{\pi(2 n+1)}{2}$, (7)
$n=1,2,3, \ldots \ldots, \sigma_n \cong 1$ For Free beams [15]
$\begin{aligned} & y_n(\zeta)= \cosh \left(\Omega_n \zeta\right)\cos \left(\Omega_n \zeta\right) \\ &\sigma_n\left[\sinh \left(\Omega_n \zeta\right)\sin \left(\Omega_n \zeta\right)\right], \\ & \Omega_n= \frac{\pi(2 n1)}{2}, \\ & n=1,2,3, \ldots \ldots, \sigma_n \cong 1 \text { For Cantilever beam }[15]\end{aligned}$ (8)
$y_n(\zeta)=\cosh \left(\Omega_n \zeta\right)\cos \left(\Omega_n \zeta\right)$$\sigma_n\left[\sinh \left(\Omega_n \zeta\right)\sin \left(\Omega_n \zeta\right)\right]$,$\Omega_n=\frac{\pi(4 n+1)}{4}$,
$n=1,2,3, \ldots \ldots, \quad \sigma_n \cong 1$ (9)
For ClampedPinned beams [15]
The time functions are assumed as follow [14]:
$T_{n i}=D_i e^{j w_n t}, i=1,2$ (10)
The double beams have the following Boundary conditions:
Substituting the above expressions in to Eqns. (1) and (2) will get:
$\left(E_1 I_1 \Omega_n^4+K_e\rho_1 A_1 \omega_n^2\right) D_1K_e D_2=0$ (15)
$\left(E_2 I_2 \Omega_n^4+K_e\rho_2 A_2 \omega_n^2\right) D_2K_e D_1=0$ (16)
For simplifying the solution of Eqns. (15) and (16), the following parameters are assumed:
$\Omega_{n 1}=K_e+\Omega_n^4 E_1 I_1$ (17)
$\Omega_{n 2}=K_e+\Omega_n^4 E_2 I_2$ (18)
$A=\left(\rho_2 A_2\right) \cdot \Omega_{n 1}+\left(\rho_1 A_1\right) \cdot \Omega_{n 2}$ (19)
$B=\left(\rho_2 A_2\right)^2 \cdot \Omega_{n 1}^2+\left(\rho_1 A_1\right)^2 \cdot \Omega_{n 2}^2$ (20)
$C=2\left(\rho_1 A_1\right) \cdot\left(\rho_2 A_2\right) \cdot\left[2 K_e{ }^2\Omega_{n 1} \cdot \Omega_{n 2}\right]$ (21)
The lower and higher (synchronous and asynchronous) natural frequencies for asymmetric double beams at arbitrary boundary conditions can get as follow:
$\omega_{1 n}=\sqrt{\frac{A\sqrt{B+C}}{2\left(\rho_1 A_1\right) \cdot\left(\rho_2 A_2\right)}}$ (22)
$\omega_{2 n}=\sqrt{\frac{A+\sqrt{B+C}}{2\left(\rho_1 A_1\right) \cdot\left(\rho_2 A_2\right)}}$ (23)
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
$\rho_1 A_1=\frac{1}{2} \rho_2 A_2=300 \frac{\mathrm{kg}}{\mathrm{m}}, E_1 I_1=\frac{1}{2} E_2 I_2=6 \times 10^6, L=8 \mathrm{~m}, K_e=2.5 \times 10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$
No. of Mode 
PinnedPinned 
ClampedPinned 

Present 
Ref. [2] 
Present 
Ref. [2] 

1 
21.8090 
21.8090 
34.0697 
34.0697 
2 
41.5407 
41.5407 
49.0993 
49.0994 
3 
87.2358 
87.2358 
110.4076 
110.4080 
4 
94.1280 
94.1281 
115.9303 
115.9300 
5 
196.2806 
196.2810 
230.3563 
230.3560 
6 
199.4394 
199.4390 
233.0537 
233.0540 
7 
348.9432 
348.9430 
393.9221 
393.9220 
8 
350.7298 
350.7300 
395.5055 
395.5060 
Table 2. The natural frequencies of double beam at PP, CC and CF boundary conditions
$\rho_1 A_1=\frac{1}{2} \rho_2 A_2=100 \frac{\mathrm{kg}}{\mathrm{m}}, E_1 I_1=\frac{1}{2} E_2 I_2=4 \times 10^6, L=10 \mathrm{~m}, K_e=1 \times 10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$
No. of mode 
PinnedPinned 
ClampedClamped 
ClampedFree 

Present 
Ref. [3] 
Present 
Ref. [3] 
Present 
Ref. [3] 

1 
19.7392 
19.7392 
44.7466 
44.7451 
7.0320 
7.0320 
2 
43.4699 
43.4699 
59.1799 
59.1790 
39.3630 
39.3630 
3 
78.9568 
78.9564 
123.3457 
123.3403 
44.0690 
44.0690 
4 
87.9442 
87.9439 
129.2832 
129.2791 
58.6692 
58.6688 
5 
177.6529 
177.6508 
241.8068 
241.7925 
123.3943 
123.3918 
6 
181.8256 
181.8239 
244.8888 
244.8888 
129.3296 
129.3297 
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/m^{2} to 2,100 kN/m^{2}, 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.
Figure 2. Higher natural frequencies vs. stiffness of connected layer
Table 3. Higher natural frequencies (Hz) vs. stiffness of connected layer (N/m^{2})
$E_1=E_2=10 \mathrm{Gpa}, h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}$,$L=10 \mathrm{~m}, \rho_1=\rho_2=3000 \mathrm{~kg} / \mathrm{m}^3$
K_{e} 
PP 
CC, FF 
CF 
CP 
100000 
201.998 
418.555 
111.598 
295.928 
200000 
221.667 
428.394 
144.179 
309.688 
300000 
239.728 
438.012 
170.648 
322.863 
400000 
256.521 
447.424 
193.531 
335.520 
500000 
272.280 
456.641 
213.980 
347.717 
600000 
287.175 
465.677 
232.639 
359.500 
700000 
301.335 
474.540 
249.908 
370.909 
800000 
314.859 
483.240 
266.059 
381.978 
900000 
327.826 
491.787 
281.284 
392.734 
1000000 
340.298 
500.188 
295.726 
403.204 
1100000 
352.330 
508.450 
309.495 
413.409 
1200000 
363.964 
516.580 
322.678 
423.368 
1300000 
375.237 
524.584 
335.342 
433.098 
1400000 
386.182 
532.467 
347.545 
442.614 
1500000 
396.825 
540.236 
359.334 
451.929 
1600000 
407.189 
547.894 
370.748 
461.057 
1700000 
417.297 
555.447 
381.821 
470.007 
1800000 
427.165 
562.898 
392.582 
478.790 
1900000 
436.810 
570.253 
403.056 
487.415 
2000000 
446.247 
577.513 
413.264 
495.890 
2100000 
455.488 
584.683 
423.227 
504.222 
Table 4. Higher natural frequencies (Hz) vs. thickness of upper beam (m)
$E_1=E_2=10 \mathrm{Gpa} ., h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}$, $L=10 \mathrm{~m}, \rho_1=\rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}$
h_{1} 
PP 
CC 
CF 
CP 
0.010 
212.061 
423.503 
128.922 
302.887 
0.015 
205.407 
420.211 
117.656 
298.266 
0.020 
201.998 
418.555 
111.598 
295.928 
0.025 
199.924 
417.558 
107.800 
294.517 
0.030 
198.530 
416.892 
105.191 
293.572 
0.035 
197.528 
416.416 
103.288 
292.896 
0.040 
196.773 
416.058 
101.837 
292.387 
0.045 
196.184 
415.780 
100.694 
291.991 
0.050 
195.712 
415.557 
99.770 
291.674 
0.055 
195.324 
415.375 
99.008 
291.414 
0.060 
195.001 
415.223 
98.368 
291.197 
0.065 
194.727 
415.094 
97.824 
291.014 
0.070 
194.491 
414.984 
97.355 
290.856 
0.075 
194.287 
414.888 
96.946 
290.720 
0.080 
194.108 
414.805 
96.587 
290.600 
0.085 
193.950 
414.731 
96.270 
290.495 
0.090 
193.810 
414.665 
95.986 
290.401 
0.095 
193.684 
414.606 
95.732 
290.317 
0.100 
193.571 
414.554 
95.503 
290.242 
0.105 
193.468 
414.506 
95.295 
290.173 
0.110 
193.375 
414.462 
95.105 
290.111 
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.
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.
Figure 4. Lower natural frequencies vs. length of beam
Figure 5. Higher natural frequencies vs. length of beam
Table 5. Natural frequencies (Hz) vs. length of beam (m)
$E_1=E_2=10$ Gpa. $_{.}, h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}$, $\rho_1=\rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}$
L (m) 
Lower Frequency 
Higher Frequency 

PP 
CC 
CF 
CP 
PP 
CC 
CF 
CP 

8.0 
281.552 
638.248 
100.302 
439.838 
295.982 
644.743 
135.624 
449.212 
8.2 
267.986 
607.493 
95.469 
418.644 
283.107 
614.314 
132.090 
428.482 
8.4 
255.376 
578.909 
90.977 
398.946 
271.202 
586.063 
128.880 
409.257 
8.6 
243.636 
552.296 
86.795 
380.606 
260.177 
559.790 
125.963 
391.401 
8.8 
232.688 
527.477 
82.894 
363.503 
249.954 
535.318 
123.308 
374.790 
9.0 
222.461 
504.294 
79.251 
347.526 
240.463 
512.490 
120.889 
359.316 
9.2 
212.894 
482.607 
75.843 
332.581 
231.640 
491.165 
118.682 
344.882 
9.4 
203.931 
462.289 
72.650 
318.579 
223.431 
471.216 
116.668 
331.400 
9.6 
195.523 
443.228 
69.654 
305.443 
215.783 
452.531 
114.826 
318.793 
9.8 
187.623 
425.321 
66.840 
293.103 
208.653 
435.007 
113.141 
306.990 
10.0 
180.194 
408.478 
64.193 
281.496 
201.998 
418.555 
111.598 
295.928 
10.2 
173.196 
392.617 
61.701 
270.566 
195.781 
403.090 
110.183 
285.550 
10.4 
166.599 
377.661 
59.350 
260.259 
189.970 
388.538 
108.884 
275.805 
10.6 
160.372 
363.544 
57.132 
250.531 
184.533 
374.830 
107.691 
266.644 
10.8 
154.487 
350.204 
55.035 
241.338 
179.442 
361.907 
106.594 
258.026 
11.0 
148.920 
337.586 
53.052 
232.642 
174.673 
349.710 
105.584 
249.911 
11.2 
143.649 
325.637 
51.175 
224.407 
170.201 
338.190 
104.653 
242.264 
11.4 
138.653 
314.311 
49.395 
216.602 
166.006 
327.299 
103.794 
235.053 
11.6 
133.913 
303.566 
47.706 
209.198 
162.068 
316.995 
103.001 
228.248 
11.8 
129.412 
293.363 
46.103 
202.166 
158.369 
307.238 
102.268 
221.821 
12.0 
125.134 
283.666 
44.579 
195.484 
154.893 
297.992 
101.590 
215.748 
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.
Figure 6. Lower natural frequencies vs. modulus of elasticity of upper beam
Figure 7. Higher natural frequencies vs. modulus of elasticity of upper beam
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/m^{3} to 3,000 kg/m^{3}, 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/m^{3} to 3,000 kg/m^{3}, 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)
$\begin{aligned} E_2 & =10 \mathrm{Gpa} ., h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}, \\ L & =10 \mathrm{~m}, \quad \rho_1=\rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}\end{aligned}$
E_{1} (Gpa.) 
Lower Frequency 
Higher Frequency 

PP 
CC 
CF 
CP 
PP 
CC 
CF 
CP 

1.0E+10 
180.194 
408.478 
64.193 
281.496 
201.998 
418.555 
111.598 
295.928 
1.1E+10 
183.815 
412.358 
65.740 
285.691 
206.716 
434.384 
112.540 
305.153 
1.2E+10 
186.013 
412.927 
67.175 
287.016 
212.522 
452.664 
113.519 
316.676 
1.3E+10 
187.343 
413.131 
68.508 
287.572 
218.898 
470.555 
114.533 
328.463 
1.4E+10 
188.188 
413.234 
69.744 
287.869 
225.491 
487.875 
115.581 
340.061 
1.5E+10 
188.758 
413.296 
70.891 
288.051 
232.117 
504.635 
116.661 
351.369 
1.6E+10 
189.163 
413.338 
71.955 
288.174 
238.688 
520.872 
117.770 
362.372 
1.7E+10 
189.464 
413.368 
72.942 
288.263 
245.163 
536.627 
118.907 
373.077 
1.8E+10 
189.696 
413.390 
73.857 
288.330 
251.524 
551.938 
120.069 
383.499 
1.9E+10 
189.879 
413.408 
74.707 
288.382 
257.763 
566.839 
121.254 
393.657 
2.0E+10 
190.027 
413.422 
75.496 
288.424 
263.879 
581.360 
122.459 
403.566 
2.1E+10 
190.149 
413.433 
76.228 
288.458 
269.875 
595.530 
123.681 
413.243 
2.2E+10 
190.252 
413.442 
76.908 
288.487 
275.755 
609.371 
124.920 
422.703 
2.3E+10 
190.339 
413.451 
77.541 
288.511 
281.522 
622.906 
126.172 
431.958 
2.4E+10 
190.414 
413.457 
78.130 
288.532 
287.181 
636.154 
127.435 
441.021 
2.5E+10 
190.480 
413.463 
78.678 
288.550 
292.737 
649.132 
128.708 
449.904 
2.6E+10 
190.537 
413.469 
79.190 
288.566 
298.195 
661.856 
129.989 
458.615 
2.7E+10 
190.588 
413.473 
79.667 
288.580 
303.558 
674.341 
131.276 
467.166 
2.8E+10 
190.633 
413.477 
80.113 
288.592 
308.832 
686.599 
132.568 
475.564 
2.9E+10 
190.673 
413.481 
80.530 
288.603 
314.021 
698.642 
133.863 
483.817 
3.0E+10 
190.710 
413.484 
80.920 
288.613 
319.127 
710.481 
135.161 
491.932 
Table 7. Natural frequencies (Hz) vs. mass density of beam (kg/m^{3})
$\begin{aligned} E_1=E_2 & =10 \mathrm{Gpa} ., h_1=h_2=20 \mathrm{~mm}, b_1=b_2=40 \mathrm{~cm}, \\ L & =10 \mathrm{~m}, \quad \rho_2=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K_e=100 \frac{\mathrm{kN}}{\mathrm{m}^2}\end{aligned}$
ρ_{1} (kg/m^{3}) 
Lower Frequency 
Higher Frequency 

PP 
CC 
CF 
CP 
PP 
CC 
CF 
CP 

1000.0 
189.558 
413.363 
74.737 
288.263 
332.586 
716.391 
166.025 
500.532 
1100.0 
189.464 
413.354 
74.271 
288.234 
317.265 
683.068 
159.290 
477.285 
1200.0 
189.361 
413.343 
73.795 
288.203 
303.924 
654.005 
153.494 
457.016 
1300.0 
189.246 
413.331 
73.307 
288.168 
292.177 
628.366 
148.453 
439.140 
1400.0 
189.119 
413.317 
72.809 
288.129 
281.739 
605.528 
144.032 
423.223 
1500.0 
188.976 
413.302 
72.302 
288.084 
272.392 
585.017 
140.124 
408.935 
1600.0 
188.815 
413.285 
71.786 
288.034 
263.967 
566.465 
136.650 
396.019 
1700.0 
188.633 
413.264 
71.262 
287.976 
256.332 
549.578 
133.544 
384.273 
1800.0 
188.426 
413.241 
70.731 
287.909 
249.385 
534.124 
130.755 
373.533 
1900.0 
188.187 
413.213 
70.195 
287.829 
243.041 
519.913 
128.240 
363.671 
2000.0 
187.911 
413.180 
69.654 
287.735 
237.235 
506.789 
125.964 
354.578 
2100.0 
187.590 
413.140 
69.109 
287.621 
231.914 
494.624 
123.898 
346.170 
2200.0 
187.213 
413.090 
68.561 
287.481 
227.039 
483.311 
122.016 
338.376 
2300.0 
186.766 
413.025 
68.012 
287.305 
222.579 
472.761 
120.298 
331.141 
2400.0 
186.235 
412.940 
67.462 
287.078 
218.514 
462.902 
118.725 
324.426 
2500.0 
185.600 
412.823 
66.912 
286.775 
214.832 
453.679 
117.282 
318.207 
2600.0 
184.839 
412.651 
66.363 
286.356 
211.527 
445.054 
115.956 
312.483 
2700.0 
183.932 
412.378 
65.816 
285.755 
208.596 
437.024 
114.734 
307.287 
2800.0 
182.860 
411.888 
65.272 
284.858 
206.038 
429.658 
113.607 
302.700 
2900.0 
181.613 
410.862 
64.730 
283.496 
203.845 
423.241 
112.564 
298.865 
3000.0 
180.194 
408.478 
64.193 
281.496 
201.998 
418.555 
111.598 
295.928 
Figure 9. Higher natural frequencies vs. mass density of upper beam
An excellent agreement was found between the numerical results obtained from the current proposed mathematical model and the results of a number of previous literatures. 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 their lower natural frequencies. The greatest influence 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 effect of thickness is small on the frequencies compared to the rest of the factors studied in this research. 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 influence is only on the higher frequencies of CF beams. The effect 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 effect is comparatively large on the higher natural frequencies of the most types of beams and relatively less on the CF beams. The same influence was observed 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.
The effect 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 influence is relatively large on the higher natural frequencies of all types of beams and comparatively less on the CF beams. In addition, 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.
A_{1} 
Crosssectional area of upper beam 
A_{2} 
Crosssectional area of lower beam 
E_{1} 
Modulus of elasticity of upper beam 
E_{2} 
Modulus of elasticity of lower beam 
I_{1} 
Second moment of area of upper beam 
I_{2} 
Second moment of area of upper beam 
K 
Modulus of elasticity of elastic layer 
L 
Length of the beams 
$W 1$ 
Transverse deflection of upper beam 
$W 1$ 
Transverse deflection of lower beam 
Greek symbols 

$\rho_1$ 
Mass density of upper beam 
$\rho_2$ 
Mass density of lower beam 
$\omega_{1 n}$ 
Lower (synchronous) natural frequency for i^{th} mode of the two beams 
$\omega_{2 n}$ 
Higher (asynchronous) natural frequency for i^{th} mode of the two beams 
$\Omega 1$ 
Dimensionless natural frequency for i^{th }mode 
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