Imad A. Abdulsahib* | Qasim A. Atiyah
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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Vibration of double beams with an elastic connected layer has been studied in this paper by assuming that the beam is a Bernoulli-Euler beam. The natural frequencies equations of the symmetric double beam have been computed at arbitrary boundary conditions. The behavior of those frequencies has been investigated with a change in the stiffness of connected layer, modulus of elasticity of beam, length of beam, mass density of beam, and thickness of beam. The high effect of the elastic connected layer on the higher natural frequencies of a cantilever double beam is less than that in the clamped and free double beams. The increase in the thickness of upper and lower beams made a high increase in the values of lower natural frequencies in all types of beams. The change in the modulus of elasticity values of double beam becomes high on the lower natural frequencies but without enlarging the influence on the higher frequencies, especially in the cantilever double beam. The similar effect of change in the mass density of the beam resulted in the same influence on the higher and lower natural frequencies in all types of beams. The length of the beam enlarges the influence on the higher natural frequencies of clamped and free.
double beam, natural frequencies of beam, vibration of beam
In many current engineering applications, double beam systems are commonly employed such as aircraft structures and civil buildings. As a result, researchers continue to be interested in the dynamic behavior of double beam structures. Li and Sun [1] developed a numerical approach for analyzing the mode shapes and the natural frequencies of a double beam structure with a general boundary and any beam mass, made up of double beams bonded by an elastic layer that is uniformly distributed between them. Hao et al. [2] enhanced an analytical method for investigating the vibration characteristics of a double beam under different boundaries. The current framework provides the impact of the connected layer stiffness on the vibration characteristics of double beam. Lai et al. [3] used a mix of finite sin–Fourier transforms and numerical Laplace transforms depending on Durbin transform to investigate the displacement response in the time domain of a double simply supported Euler–Bernoulli beam system with elastic connection. The Bernoulli–Euler beam theory was used by Zhang et al. [4] to study the characteristics of buckling and vibration of a double beam structure under a compression force. The results indicated that the system's critical buckling load is linked to the compression ratio of elastic connected layer and beams, and that the axial compressions have a significant impact on the parameters of the system's free transverse vibration. The vibrational properties of double beams under compressive stress were studied by Kozic et al. [5]. The system's two parallel beams are easily and regularly connected by a Kerr-type three-parameter. The impact of non-linear elasticity on the frequencies of sandwich beams under varied boundary conditions was demonstrated by Abdulsahib and Atiyah [6]. The energy balance technique based on Galerkin-Petrov (EGP) and the Homotopy Perturbation Method were used to study the influence of the inner layer's non-linearity stiffness on those frequencies (HPM). By distinguishing between the synchronous and asynchronous movements of beams, Mirzabeigy and Madoliat [7] explored the influence of nonlinearity in connected layer on the vibration of double beam studied and, concluded the high frequencies are more accurate if ignoring the effect of the nonlinearity of the elastic layer. Mao [8] analyzed the frequencies behavior of double beams using the AMDM technique, The suggested technique was applied to systems containing any number of beams to compute vibration characteristics with varied parameters. Oniszczuk [9] demonstrated the continuous vibration characteristics of double beams. A uniform set of dynamic equations was solved using analytical technique to characterize the system's motion. The analytical technique was used to determine the ultimate shape of the vibrations. The vibration properties of a double beam were examined by Rezaiee-Pajand and Hozhabrossadati [10]. This structure consists of two beams, one end is elastic and the other is free, as well as the two beams are connected by a mass-spring mechanism. The impact of four geometric and material parameters on the vibration of twin beams was examined by Atiyah and Abdulsahib [11]. Those parameters of two beams were mass density, thickness, modulus of elasticity, and the properties of the intermediate layer. The Bernoulli-Euler beam was used to compute the frequencies of the double beams.
In this paper, a number of variables of the elastic connecting layer that are believed to affect the vibration behavior of the double beams, which were not fully studied previously are investigated. Those parameters of two beams were mass density, thickness, modulus of elasticity, and the properties of the intermediate layer. The Bernoulli-Euler beam was used to compute the frequencies of the double beams. the equations of motion are derived to calculate symmetric and asymmetric frequencies at different boundary conditions, which are the most common in various engineering applications, with calculating the effect of a number of connecting layers variables on those frequencies.
Two beams are connected by an elastic layer with arbitrary boundary conditions. The two beams are symmetric and have the same length, as shown in Figure 1. The Bernoulli-Euler beam theory for free vibrations is used to describe the equations of motion [1]:
$\frac{\partial^2}{\partial x^2}\left(E_1 I_1 \frac{\partial^2 W_1}{\partial x^2}\right)+K\left(W_1-W_2\right)+\rho_1 A_1 \frac{\partial^2 W_1}{\partial t^2}=0$ (1)
$\frac{\partial^2}{\partial x^2}\left(E_2 I_2 \frac{\partial^2 W_2}{\partial x^2}\right)-K\left(W_1-W_2\right)+\rho_2 A_2 \frac{\partial^2 W_2}{\partial t^2}=0$ (2)
where, A1, A2, ρ1, ρ2, E1, E2, I1, and I2 are the cross-sectional area, mass density, modulus of elasticity, and moment of area for the upper and lower beam, respectively, k is the stiffness of elastic layer, and W1, W2 are the deflection of the upper and lower beam, respectively.
Figure 1. Double-beam with elastic connected layer
The boundary conditions in general form for clamped beams are assumed as follows: $W_i(0, t)=\acute{W_l}(0, t)=W_i(l, t)=W_l(l, t)=0, i=1,2$.
The boundary conditions for simply supported beams are: $W_i(0, t)=\acute{\acute{W_l}}(0, t)=W_i(l, t)=\acute{\acute{W_l}}(l, t)=0, i=1,2$.
In addition, the boundary conditions for free beams are: $\acute{\acute{W_l}}(0, t)=\acute{\acute{\acute{W_l}}}(0, t)=\acute{\acute{W_l}}(l, t)=\acute{\acute{\acute{W_l}}}(l, t)=0, i=1,2$.
And, the boundary conditions for cantilever beam are: $W_i(0, t)=\acute{W_l}(0, t)=\acute{\acute{W_l}}(l, t)=\acute{\acute{\acute{W_l}}}(l, t)=0, i=1,2$.
The natural frequencies of the system will be got by solving Eqns. (1) and Eq. (2). Assume the time-harmonic motion with the above boundary conditions and by the separation of variables, the solutions of Eqns. (1) and (2) can be written as follow:
$W_i(x, t)=\sum_{n=1}^{\infty} x_n(x) \cdot T_{n i}(t), i=1,2$ (3)
where,
$X_n(x)=\cosh \left(k_n x\right)-\cos \left(k_n x\right)-\sigma_n\left[\sinh \left(k_n x\right)-\right.\left.\sin \left(k_n x\right)\right], k_n=\frac{\pi(2 n+1)}{2 l}, n=1,2,3, \ldots \ldots \sigma_n \cong 1 \ldots \dots$ (4)
For clamped beams,
$X_n(x)=\sin \left(k_n x\right), \quad k_n=\frac{n \pi}{l}, \quad n=1,2,3, \ldots \ldots$ (5)
For simply-supported beams,
$X_n(x)=\cosh \left(k_n x\right)+\cos \left(k_n x\right)-\sigma_n\left[\sinh \left(k_n x\right)+\right.\left.\sin \left(k_n x\right)\right], k_n=\frac{\pi(2 n+1)}{2 l}, n=1,2,3, \ldots . \sigma_n \cong 1$ (6)
For free beams,
$X_n(x)=\cosh \left(k_n x\right)-\cos \left(k_n x\right)-\sigma_n\left[\sinh \left(k_n x\right)-\right.\left.\sin \left(k_n x\right)\right], k_n=\frac{\pi(2 n-1)}{2 l}, n=1,2,3, \ldots . \sigma_n 1 \ldots .$ (7)
For cantilever beam, the assumed general forms for time functions are:
$T_{n i}=C_i e^{j w_n t}, i=1,2$ (8)
Substituting the above expression into Eqns. (1) and (2) will get:
$\left(E_1 I_1 k_n^4+K-\rho_1 A_1 \omega_n^2\right) C_1-K C_2=0$ (9)
$\left(E_2 I_2 k_n^4+K-\rho_2 A_2 \omega_n^2\right) C_2-K C_1=0$ (10)
These equations can be solved when the two beams are symmetric; the lower and higher frequencies will be obtained as follows:
$\omega_{1 n}=\sqrt{\frac{E I k_n^4}{\rho A L^4}}$ (11)
$\omega_{2 n}=\sqrt{\frac{E I k_n^4+2 K L^4}{\rho A L^4}}$ (12)
A convergence test is utilized to compare the accuracy of Eqns. (11) & (12) with the results in reference [1]. The numerical values are used as in reference [11], such as EI=4×106 N.m2, L=10 m, ρA=1×102 kg.m-1, K=1~5×105 N.m-2, and ωn (Hz). Table 1 shows the comparison results between the present results and the Ref. [1] when the boundary conditions of double beams are simply supported. This table shows a good convergence of results between the present work and the literature by Li and Sun [1]. The maximum difference between the present and results of reference [1] is less than 1%. this convergence confirms the validity of the derived equations in this work.
The natural frequencies for the clamped double beam are same for the free double beam, because it has the same dimensionless natural frequency function but it has another mode shape. Therefore, the behavior is same for the both cases in the all figures of this paper. Figure 2 manifests the behavior of higher frequencies with the change in stiffness of the elastic connected layer (k), in all cases of boundary conditions. The change in the values of k has not affected the lower frequencies (synchronous), but it caused an increasing in the higher natural frequencies (asynchronous) in all modes when increasing the k values.
In Table 2, it is seen that when k is increased from 100000 to 1800000 N/m2, the higher natural frequencies (asynchronous) increased about 367% in the simply supported beam, about 265% in the clamped and free double beams, and about 415% in the cantilever double beam. As a result, there is a high effect of elastic connected layer on the cantilever double beam, and this effect is less in the clamped and free double beams.
Figure 3 and Table 3 elucidated that for the clamped and free double beams, the higher natural frequencies decrease when the ratio (h/b) increases from 1 to 12 times. However, if the ratio is more than 12, the higher natural frequencies start increasing. The same behavior for the simply supported beam can be seen for the ratio (h/b) between 1 to 22 times, but the higher frequencies decrease with the increase in this ratio for more than 22 times. In the cantilever double beam, the higher frequencies also decrease with the (h/b) ratio increase. The behavior of lower natural frequencies (synchronous) with the changes in the values of thickness in the upper and lower beams (h1 & h2) is depicted in Figure 4 and Table 3. When the thickness increases from 0.02 to 0.38 m (h1=h2), the lower natural frequencies increase about 3000% in the simply supported beam and increase in the same ratio approximately 3000% in the clamped, free, and the cantilever beams. Consequently, the effect of the change in thickness on the higher natural frequencies (asynchronous) is higher in the simply supported, and the clamped, free double beams, but it generally causes an increase in those frequencies with the thickness increase in a cantilever beam. The increase in thickness of the upper and lower beams made a great increase in the values of the lower natural frequencies in all types of beams.
The effects of changing the modulus of elasticity of the upper and lower beams (E1 & E2) on the higher natural frequencies are demonstrated in Figure 5 and Table 4. When the modulus of elasticity changes from 10 GPa to 140 GPa (E1=E2), the frequencies of simply supported beams increase about 28%, the frequencies of clamped and free beams increase about 93%, and the frequencies of cantilever beam increase just 4%. Figure 6 and Table 4 portrays the behavior of lower natural frequencies when the modulus of elasticity of upper and lower beams changes. When the elasticity modulus increases from 10 GPa to 140 GPa, the lower natural frequencies increase about 275% in all types of beams. The change in the values of the elasticity modulus of double beam has a great effect on the lower natural frequencies but not as much as that effect on the higher frequencies, especially in the cantilever double beam.
Figures 7-8 and Table 5 reveal the effect of changing the mass density of the upper and lower beams (ρ1 & ρ2) on the higher and lower natural frequencies, respectively. The natural frequencies (higher & lower) of the simply supported, clamped, free, and cantilever beams decrease about 45% when the mass density of beam (ρ1=ρ2) changes from 1500 kg/m3 to 5000 kg/m3. Accordingly, the same effect of the change in the mass density of the beam results in the same effect on the higher and lower natural frequencies in all types of beams.
Figure 9 and Table 6 exhibit the effect of changing the length of the upper and lower beams (L1 & L2) on the higher natural frequencies. When the length of beam (L1 =L2) changes from 5 m to 14 m, the higher frequencies of simply supported beam decrease about 58%, the frequencies of clamped and free beams decrease about 75%, and the higher frequencies of cantilever beam decrease about 23%. The behavior of lower natural frequencies with change in length of the beam is shown in Figure 10 and Table 6. If the length of beam changes from 5 m to 14 m, the lower natural frequencies for the all types of beams decrease about 83%. Consequently, the length of the beam enlarges the effect on the higher natural frequencies of the clamped and free beams and makes the same effect on the lower natural frequencies of the all types of double beams.
Figure 2. Higher natural frequencies versus stiffness of elastic layer
Figure 3. Higher natural frequencies versus thickness of double beam
Figure 4. Lower frequencies versus thickness of beam
Figure 5. Higher natural frequencies versus modulus of elasticity of double beam
Table 1. A comparison test for present work with reference [1]
|
No. of mode |
k=1×105 N/m2 |
k=2×105 N/m2 |
k=3×105 N/m2 |
k=4×105 N/m2 |
k=5×105 N/m2 |
|||||
|
Present |
Li & Sun [1] |
Present |
Li & Sun [1] |
Present |
Li & Sun [1] |
Present |
Li & Sun [1] |
Present |
Li & Sun [1] |
|
|
1 |
19.739 |
19.74 |
19.739 |
19.74 |
19.739 |
19.74 |
19.739 |
19.74 |
19.739 |
19.74 |
|
2 |
48.884 |
48.88 |
66.254 |
66.25 |
78.957 |
78.94 |
78.957 |
78.96 |
78.957 |
78.96 |
|
3 |
78.957 |
78.96 |
78.957 |
78.96 |
79.935 |
79.96 |
91.595 |
91.59 |
101.930 |
101.93 |
|
4 |
90.742 |
90.74 |
101.164 |
101.16 |
110.608 |
110.61 |
119.307 |
119.31 |
127.413 |
127.41 |
|
5 |
177.653 |
177.65 |
177.653 |
177.65 |
177.653 |
177.65 |
177.653 |
177.65 |
177.653 |
177.65 |
|
6 |
183.195 |
183.20 |
188.575 |
188.58 |
193.805 |
193.81 |
198.898 |
198.90 |
203.864 |
203.86 |
Table 2. Higher natural frequencies (Hz) versus stiffness of elastic connected layer $E=70 \times \frac{10^9 \mathrm{~N}}{\mathrm{~m}^2}, L=10 \mathrm{~m}, b=0.02 \mathrm{~m}, \rho=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, h=0.02 \mathrm{~m}$
|
K *105(N/m2) |
Simply supported |
Clamped, Free |
Cantilever |
K*105(N/m2) |
Simply supported |
Clamped, Free |
Cantilever |
|
1 |
106.601 |
154.617 |
93.369 |
10 |
293.877 |
314.493 |
289.340 |
|
2 |
140.346 |
179.554 |
130.580 |
11 |
307.729 |
327.474 |
303.399 |
|
3 |
167.423 |
201.427 |
159.325 |
12 |
320.983 |
339.960 |
316.835 |
|
4 |
190.693 |
221.147 |
183.624 |
13 |
333.712 |
352.003 |
329.723 |
|
5 |
211.417 |
239.248 |
205.063 |
14 |
345.972 |
363.647 |
342.127 |
|
6 |
230.283 |
256.072 |
224.465 |
15 |
357.813 |
374.930 |
354.096 |
|
7 |
247.717 |
271.857 |
242.317 |
16 |
369.274 |
385.883 |
365.674 |
|
8 |
264.002 |
286.774 |
258.942 |
17 |
380.390 |
396.534 |
376.896 |
|
9 |
279.339 |
300.953 |
274.562 |
18 |
391.191 |
406.906 |
387.794 |
Table 3. Natural frequencies (Hz) versus thickness of beam $E=70 \times 10^9 \mathrm{~N} / \mathrm{m}^2, L=10 \mathrm{~m}, b=0.02 \mathrm{~m}, \rho=3000 \mathrm{~kg} / \mathrm{m}^3, K=10^5$
|
Thickness of beam, h (m) |
Simply supported N.F. |
Clamped, Free N.F. |
Cantilever N.F. |
|||
|
Lower |
Higher |
Lower |
Higher |
Lower |
Higher |
|
|
0.020 |
2.753 |
408.258 |
6.240 |
408.296 |
0.980 |
408.249 |
|
0.060 |
8.258 |
235.847 |
18.719 |
236.444 |
2.941 |
235.721 |
|
0.100 |
13.763 |
183.092 |
31.198 |
185.221 |
4.902 |
182.640 |
|
0.140 |
19.268 |
155.502 |
43.677 |
160.366 |
6.863 |
154.456 |
|
0.180 |
24.773 |
138.319 |
56.156 |
147.214 |
8.824 |
136.369 |
|
0.220 |
30.278 |
126.761 |
68.636 |
140.934 |
10.785 |
123.563 |
|
0.260 |
35.783 |
118.747 |
81.115 |
139.284 |
12.746 |
113.943 |
|
0.300 |
41.288 |
113.207 |
93.594 |
140.964 |
14.707 |
106.430 |
|
0.340 |
46.793 |
109.515 |
106.073 |
145.105 |
16.668 |
100.408 |
|
0.380 |
52.298 |
107.271 |
118.553 |
151.085 |
18.629 |
95.493 |
|
0.420 |
57.803 |
106.196 |
131.032 |
158.448 |
20.590 |
91.435 |
|
0.460 |
63.308 |
106.086 |
143.511 |
166.859 |
22.551 |
88.062 |
|
0.500 |
68.813 |
106.779 |
155.990 |
176.067 |
24.512 |
85.250 |
|
0.540 |
74.318 |
108.148 |
168.469 |
185.889 |
26.472 |
82.907 |
|
0.580 |
79.823 |
110.085 |
180.949 |
196.187 |
28.433 |
80.967 |
|
0.620 |
85.328 |
112.504 |
193.428 |
206.859 |
30.394 |
79.374 |
|
0.660 |
90.833 |
115.330 |
205.907 |
217.826 |
32.355 |
78.086 |
|
0.700 |
96.338 |
118.502 |
218.386 |
229.029 |
34.316 |
77.068 |
|
0.740 |
101.843 |
121.969 |
230.865 |
240.423 |
36.277 |
76.292 |
Table 4. Natural frequencies (Hz) versus modulus of elasticity of double beam $L=10 \mathrm{~m}, b=0.02 \mathrm{~m}, h=0.4 \mathrm{~m}, \rho=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K=10^5 \mathrm{~N} / \mathrm{m}^2$
|
Modulus of elasticity E N/m2 |
Simply supported N.F. |
Clamped, Free |
Cantilever |
|||
|
Lower |
Higher |
Lower |
Higher |
Lower |
Higher |
|
|
1E+10 |
20.806 |
93.628 |
47.166 |
102.752 |
7.411 |
91.587 |
|
2E+10 |
29.425 |
95.912 |
66.704 |
113.061 |
10.481 |
91.886 |
|
3E+10 |
36.038 |
98.143 |
81.695 |
122.505 |
12.837 |
92.185 |
|
4E+10 |
41.613 |
100.324 |
94.333 |
131.271 |
14.823 |
92.482 |
|
5E+10 |
46.525 |
102.459 |
105.468 |
139.488 |
16.572 |
92.779 |
|
6E+10 |
50.966 |
104.551 |
115.535 |
147.247 |
18.154 |
93.074 |
|
7E+10 |
55.050 |
106.601 |
124.792 |
154.617 |
19.609 |
93.369 |
|
8E+10 |
58.850 |
108.612 |
133.408 |
161.651 |
20.963 |
93.663 |
|
9E+10 |
62.420 |
110.588 |
141.501 |
168.392 |
22.234 |
93.955 |
|
1E+11 |
65.797 |
112.528 |
149.155 |
174.873 |
23.437 |
94.247 |
|
1.1E+11 |
69.008 |
114.435 |
156.435 |
181.122 |
24.581 |
94.538 |
|
1.2E+11 |
72.077 |
116.312 |
163.391 |
187.163 |
25.674 |
94.828 |
|
1.3E+11 |
75.020 |
118.158 |
170.063 |
193.014 |
26.722 |
95.118 |
|
1.4E+11 |
77.852 |
119.976 |
176.482 |
198.694 |
27.731 |
95.406 |
|
1.5E+11 |
80.585 |
121.7673 |
182.677 |
204.2161 |
28.70496 |
95.69382 |
|
1.6E+11 |
83.22783 |
123.5322 |
188.668 |
209.5923 |
29.64635 |
95.98041 |
|
1.7E+11 |
85.78928 |
125.2722 |
194.4745 |
214.834 |
30.55876 |
96.26615 |
|
1.8E+11 |
88.27644 |
126.9884 |
200.1126 |
219.9509 |
31.44471 |
96.55104 |
|
1.9E+11 |
90.69543 |
128.6818 |
205.5962 |
224.9514 |
32.30636 |
96.83509 |
Table 5. Natural frequencies (Hz) versus mass density of double beam $E=70 \times 10^9 \frac{\mathrm{N}}{\mathrm{m}^2}, L=10 \mathrm{~m}, b=0.02 \mathrm{~m}, h=0.02 \mathrm{~m}, K=10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$
|
Mass density of beam (ρ) kg/m3 |
Simply supported N.F. |
Clamped, Free N.F. |
Cantilever N.F. |
|||
|
Lower |
Higher |
Lower |
Higher |
Lower |
Higher |
|
|
1500 |
77.853 |
150.757 |
176.483 |
218.661 |
27.732 |
132.044 |
|
1750 |
72.077 |
139.574 |
163.391 |
202.441 |
25.674 |
122.249 |
|
2000 |
67.422 |
130.559 |
152.839 |
189.366 |
24.016 |
114.354 |
|
2250 |
63.566 |
123.093 |
144.098 |
178.536 |
22.643 |
107.814 |
|
2500 |
60.304 |
116.776 |
136.703 |
169.374 |
21.481 |
102.281 |
|
2750 |
57.498 |
111.341 |
130.341 |
161.492 |
20.481 |
97.521 |
|
3000 |
55.050 |
106.601 |
124.792 |
154.617 |
19.609 |
93.369 |
|
3250 |
52.890 |
102.419 |
119.896 |
148.551 |
18.840 |
89.706 |
|
3500 |
50.966 |
98.694 |
115.535 |
143.148 |
18.155 |
86.443 |
|
3750 |
49.238 |
95.347 |
111.617 |
138.294 |
17.539 |
83.512 |
|
4000 |
47.675 |
92.319 |
108.073 |
133.902 |
16.982 |
80.860 |
|
4250 |
46.251 |
89.563 |
104.846 |
129.904 |
16.475 |
78.446 |
|
4500 |
44.948 |
87.040 |
101.892 |
126.244 |
16.011 |
76.236 |
|
4750 |
43.749 |
84.718 |
99.175 |
122.877 |
15.584 |
74.203 |
|
5000 |
42.642 |
82.573 |
96.664 |
119.766 |
15.189 |
72.324 |
|
5250 |
41.614 |
80.583 |
94.334 |
116.879 |
14.823 |
70.581 |
|
5500 |
40.657 |
78.730 |
92.165 |
114.192 |
14.482 |
68.958 |
|
5750 |
39.763 |
77.000 |
90.139 |
111.682 |
14.164 |
67.442 |
|
6000 |
38.926 |
75.379 |
88.241 |
109.331 |
13.866 |
66.022 |
Table 6. Natural frequencies (Hz) versus length of double beam$E=70 \times 10^9 \frac{\mathrm{N}}{\mathrm{m}^2}, h=0.02 \mathrm{~m}, b=0.02 \mathrm{~m}, \rho=3000 \frac{\mathrm{kg}}{\mathrm{m}^3}, K=10^5 \frac{\mathrm{N}}{\mathrm{m}^2}$
|
Length of beam, L (m) |
Simply supported N.F. |
Clamped, Free N.F. |
Cantilever N.F. |
|||
|
Lower |
Higher |
Lower |
Higher |
Lower |
Higher |
|
|
5 |
220.200 |
238.372 |
499.169 |
507.447 |
78.437 |
120.356 |
|
5.5 |
181.984 |
203.596 |
412.536 |
422.515 |
64.824 |
111.962 |
|
6 |
152.917 |
178.092 |
346.645 |
358.463 |
54.470 |
106.303 |
|
6.5 |
130.296 |
159.092 |
295.366 |
309.151 |
46.412 |
102.408 |
|
7 |
112.347 |
144.759 |
254.678 |
270.544 |
40.019 |
99.674 |
|
7.5 |
97.867 |
133.833 |
221.853 |
239.900 |
34.861 |
97.717 |
|
8 |
86.016 |
125.427 |
194.988 |
215.299 |
30.639 |
96.292 |
|
8.5 |
76.194 |
118.907 |
172.723 |
195.362 |
27.141 |
95.236 |
|
9 |
67.963 |
113.808 |
154.064 |
179.079 |
24.209 |
94.443 |
|
9.5 |
60.997 |
109.791 |
138.274 |
165.689 |
21.728 |
93.837 |
|
10 |
55.050 |
106.601 |
124.792 |
154.617 |
19.609 |
93.369 |
|
10.5 |
49.932 |
104.051 |
113.190 |
145.414 |
17.786 |
93.004 |
|
11 |
45.496 |
101.996 |
103.134 |
137.731 |
16.206 |
92.714 |
|
11.5 |
41.626 |
100.330 |
94.361 |
131.291 |
14.827 |
92.483 |
|
12 |
38.229 |
98.969 |
86.661 |
125.871 |
13.618 |
92.297 |
|
12.5 |
35.232 |
97.850 |
79.867 |
121.293 |
12.550 |
92.146 |
|
13 |
32.574 |
96.925 |
73.841 |
117.413 |
11.603 |
92.022 |
|
13.5 |
30.206 |
96.155 |
68.473 |
114.114 |
10.760 |
91.919 |
|
14 |
28.087 |
95.510 |
63.669 |
111.297 |
10.005 |
91.834 |
Figure 6. Lower natural frequencies versus modulus of elasticity of double beam
Figure 7. Higher natural frequencies versus mass density of double beam
Figure 8. Lower natural frequencies versus mass density of double beam
Figure 9. Higher natural frequencies versus length of double beam
Figure 10. Lower natural frequencies versus length of double beam
In this paper, a good convergence in the results between the present work and the reference Li and Sun [1] is evinced. The great effect of elastic connected layer on the cantilever double beam, and this effect is less than that in the clamped and free double beams. The effect of the change in thickness on the higher natural frequencies (asynchronous) is higher in the simply supported, and the clamped, free double beams, but it generally causes an increase in those frequencies with the thickness increase in a cantilever beam. The increase in the thickness of upper and lower beams made a great increase in the values of lower natural frequencies in all types of beams. The change in the values of the elasticity modulus of double beam has a great effect on the lower natural frequencies but not as much as that effect on the higher frequencies, especially in the cantilever double beam. The higher and lower frequencies of the all types of beams decrease when the mass density of beam increases. The lower frequencies decrease with the increase in the beam length. The length of the beam enlarges the effect on the higher natural frequencies of clamped and free beams.
In all cases, the same behavior in all modes is recognized; therefore, only the first mode has been studied in this work, because there is no large difference in behavior occurring in 2nd, 3rd or nth mode.
In the future, it is possible to study the effect of the properties of the elastic connecting layer of asymmetric double beams.
|
B |
dimensionless heat source length |
|
CP |
specific heat, J. kg-1. K-1 |
|
g k |
gravitational acceleration, m.s-2 thermal conductivity, W.m-1. K-1 |
|
Nu |
local Nusselt number along the heat source |
|
Greek symbols |
|
|
$\alpha$ |
thermal diffusivity, m2. s-1 |
|
$\beta$ |
thermal expansion coefficient, K-1 |
|
$\phi$ |
solid volume fraction |
|
Ɵ |
dimensionless temperature |
|
µ |
dynamic viscosity, kg. m-1.s-1 |
|
Subscripts |
|
|
p |
nanoparticle |
|
f |
fluid (pure water) |
|
nf |
nanofluid |
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