In this paper, an analytical modeling approach for the flexural vibration analysis of the nonuniform doublebeam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints. Energy formulation is employed for the dynamic description of the coupling system. With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series. Supplementary terms with the smoothed characteristics are introduced to the standard Fourier series for the construction of displacement admissible function for each beam. In conjunction with the Rayleigh–Ritz procedure, the transverse modal characteristics of nonuniform doublebeam system can be obtained by solving a standard eigenvalue problem. Instead of solving the certain value of nonideal boundary conditions, the continuous spring stiffnesses of the boundary conditions are considered, and the rotational restrains are introduced in the coupling beam interface. Numerical results are then presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. This work can provide an efficient modeling framework for the vibration characteristics study of the complex doublebeam system, especially with arbitrary varying thickness and coupling stiffness.
The multiplebeam system has been extensively studied due to its wide application in various branches, such as mechanical and aeronautical engineering. A good understanding on its dynamic characteristics will be of great importance for the efficient design as well as vibration control of such complex system. For this reason, a lot of research attention has been devoted to the vibration behavior of multiplebeam structure by many researchers in the past decades.
For these multiplebeam structures, mechanical coupling between each beam component is usually taken into account through the introduction of translational spring across the beam interface [
In many occasions, the nonuniform beam component will be encountered with the background of optimal design, in which a better or more suitable distribution of mass and strength than the uniform beam is of great desire. Thus, vibration analysis of the nonuniform beam has been studied continuously [
In the current studies on the vibration analysis of the multiplebeam system, most of them are devoted to the uniform ones, and just little exception can be found in the literature for the vibration studies of nonuniform multiplebeam structure. For example, Mabie and Rogers [
From the practical point of view, elastic boundary restraint and other types of mechanical coupling strength should be considered. There is a clear gap in the literature on this aspect. In analysis of the real system, the nearest ideal boundary conditions such as clamped, free, and simply support is selected for the modeling. However, small deviations from ideal conditions in real systems indeed occur. For example, a beam connected at the ends to rigid supports by pins is modeled using simply supported boundary conditions which require deflections and moments to be zero. But the hole and pin assembly may have small gaps and/or friction which is called the nonideal boundary. In this paper, the spring stiffnesses of the boundary conditions are different and continuous, so the elastic boundary is used here which includes the aforementioned nonideal boundary. Moreover, when the beam is under axial force or in axial moving, the minimal rotations are emerged, and the rotation restrains are important. For the double beam system, when the similar rotation movements happen, rotational coupling spring is introduced to restrain the relevant displacement. Claeys et al. [
In this work, motivated by the current limitation in literature, an efficient modeling approach for the vibration analysis of the nonuniform doublebeam system with general elastic boundary condition is proposed, in which both the translational and rotational restraining springs are taken into account to describe the dynamic interaction between each beam interface. Energy principle is formulated for the analysis of the system motion equation, with the nonuniform thickness variation expanded into the Fourier series in a unified pattern. The transverse displacement admissible function is constructed as the superposition of the standard Fourier series and the boundary smoothed auxillary terms. In conjunction with the Rayleigh–Ritz procedure, all the modal parameters can be derived by solving an eigenvalue matrix. Numerical examples are then given to demonstrate the reliability and effectiveness of the current model. Finally, some concluding remarks are made.
Consider an elastically connected nonuniform doublebeam system, as illustrated in Figure
Nonuniform doublebeam system with arbitrary boundary conditions, in which both the translational and rotational coupling effects are taken into account.
For the coupled beam structure as shown in Figure
For the
The total kinetic energy of the
The coupling potential energy between the interfaces can be written as
In this work, in order to treat the crosssection area variation in a most general unified pattern, variables, such as arbitrary inertia
For various kinds of cross sections, the difference between them is merely the Fourier coefficients and the items which can be obtained easily through the Fourier transformation. For the traditional uniform beam,
Once the system Lagrangian is obtained, the other thing is to construct the appropriate admissible function. Differential continuity of the constructed function has significant effect on the final convergence and accuracy. Here, an improved Fourier series method is employed for this purpose, in which the additional functions are introduced to the standard Fourier series to remove all the discontinuities associated with the spatial differentiation of the displacement field functions. For each beam member, its flexural vibrating displacement function is expanded as [
It can be easily proven that the current constructed trigonometric function can satisfy the displacement and its higherorder differentiation continuity requirement in the interval (0,
Substituting the admissible function Equation (
In this section, the aforementioned modeling framework is programmed in the MATLAB environment. Several numerical results of different kinds of cross sections with various boundary conditions will be presented to demonstrate the effectiveness and advantage of the proposed model. As no research has been published about the free vibration of doublebeam system with uniform cross sections, the results of the current method will be the first compared with those from other relevant literatures to validate the correctness. In current solution framework, the elastic boundary conditions are easily obtained by setting the restraining stiffness coefficient into various values accordingly. Similarly, variation of the arbitrary inertia
Here, by setting the relevant parameters to constant in Equations (
The first six nondimensional frequency
Boundary conditions  Nondimensional frequency 


Beam 1  Beam 2  1  2  3  4  5  6 
SS  SS  9.8683  21.7345  39.4577  43.9572  88.7208  90.8309 
9.8696^{a}  21.7350  39.4784  43.9721  88.8265  90.9128  
CC  CC  22.3011  29.5464  61.4070  64.4416  120.1731  121.9079 
22.3733^{a}  29.5900  61.6728  64.6416  120.9034  122.4444  
CF  CF  3.5113  19.6809  21.9986  29.3130  61.5645  64.5643 
3.5160^{a}  19.6815  22.0345  29.3346  61.6972  64.6649  
SS  CF  8.7062  18.5912  25.7651  42.4830  62.6190  90.1489 
10.3911^{b}  20.2322  28.4861  42.4398  63.6832  89.9532  
SS  CC  16.2054  26.5768  42.3472  62.4802  90.1361  120.5933 
15.1683^{b}  29.4314  42.1633  63.6958  89.9281  121.3566  
CC  CF  10.8006  22.9898  29.1021  61.7083  64.3423  120.5406 
10.8090^{a}  23.0286  29.1356  61.8652  64.4954  120.9652 
^{a}Results from Ref. [
The first six nondimensional frequency
Boundary conditions  Nondimensional frequency 


Beam 1  Beam 2  1  2  3  4  5  6 
SS  SS  9.8683  11.5923  39.4576  41.2919  88.7207  90.5930 
9.8649^{a}  12.1168  39.4240  41.8523  88.5772  91.0371  
CC  CC  22.3008  23.3205  61.4048  62.8443  120.1720  121.8760 
22.3421^{a}  23.6691  61.4904  63.3157  120.2885  122.2928  
CF  CF  3.5113  5.3243  21.9985  24.5854  61.5643  63.8939 
3.5148^{a}  5.7683  22.0091  25.3841  61.5438  64.5943  
SS  CF  4.2204  10.9595  22.9299  40.6538  62.3415  89.9755 
4.8060^{a}  10.8866  23.8391  40.5768  63.1366  89.7648  
SS  CC  11.0137  22.6417  40.6693  61.8348  89.9830  120.5744 
10.9830^{a}  23.0436  40.6083  62.4392  89.7773  121.3189  
CC  CF  4.2334  22.2624  23.6115  61.6101  63.1334  120.5412 
4.7805^{a}  22.3691  24.3769  61.6224  63.8560  120.4110 
^{a}Results from Ref. [
In order to validate the proposed method for the case of variable cross sections, the coupling translational spring
Table
Nondimensional natural frequencies for a tapered nonuniform beam with clampedfree (CF) boundary condition and different crosssection parameters.


 






 
0.2  Current  3.60566  20.6030  56.1232  3.85241  21.0389  56.5570 
Ref. [ 
3.60827  20.6210  56.1923  3.85511  21.0568  56.6303  
0.4  Current  3.73471  19.0993  50.2941  4.31621  20.0358  51.2654 
Ref. [ 
3.73708  19.1138  50.3537  4.31878  20.0500  51.3346  
0.6  Current  3.93214  17.4767  43.9735  5.00654  19.0544  45.6680 
Ref. [ 
3.93428  17.4878  44.0248  5.00904  19.0649  45.7384  
0.8  Current  4.29053  15.7350  36.8378  6.19373  18.3793  39.7350 
Ref. [ 
4.29249  15.7427  36.8846  6.19639  18.3855  39.8336 
Moreover, a single beam with threestep changes in cross section is analyzed. The geometrical dimensions and material properties are
The first four dimensionless frequencies
BC 





SSSS  3.097  6.184  9.343  12.605 
3.096^{a}  6.184  9.343  12.605  
SSF  4.313  7.331  10.240  13.297 
4.313^{a}  7.331  10.240  13.297  
CC  4.536  7.663  10.808  14.068 
4.541^{a}  7.660  10.809  14.064  
CF  2.287  5.133  8.083  10.978 
2.285^{a}  5.133  8.083  10.978 
^{a}Results from [
The nonuniform double beam system is considered with CF boundary condition for both beams in which the upper beam’s crosssection function is the same as defined in Equations (
Nondimensional natural frequencies for the nonuniform doublebeam system under different coupling spring stiffness with







1  3.5132  3.5972  3.6543  3.6545  3.6641 
2  3.8524  4.1375  8.6261  8.7880  21.6424 
3  21.0389  21.2602  21.4288  21.4751  59.5703 
4  22.0129  22.2475  22.9277  23.3122  116.1607 
5  56.5570  56.7903  56.8261  57.0392  191.3209 
6  61.6177  61.7810  61.8310  62.0116  244.6118 
7  109.6314  109.8539  109.7742  109.9922  267.2176 
8  120.6885  120.8388  120.7933  120.9473  285.3005 
9  180.0414  180.2551  180.1284  180.3406  313.1571 
10  199.3790  199.5227  199.4420  199.5869  379.2597 
The influence of different coupling springs
The influence of the coupling translational spring
Finally, the nonuniform doublebeam system is considered, in which both beams are tapered beams as defined in Section
Nondimensional fundamental and second mode frequencies of the nonuniform doublebeam system with
In order to analyze the coupling influence of elastic boundary and taper ratio of the nonuniform beam, the fundamental frequency of the same structure is shown in Table
Nondimensional fundamental frequencies for the nonuniform doublebeam system under elastic boundary with




0  0.2  0.4  0.6  0.8  
0  3.5133  3.8526  4.3165  5.0070  6.1947 
10  3.5146  3.8544  4.3191  5.0113  6.2036 
10^{2}  3.5276  3.8720  4.3448  5.0537  6.2906 
10^{3}  3.6548  4.0431  4.5915  5.4483  7.0016 
10^{4}  4.7161  5.4056  6.3962  7.8291  9.5231 
10^{5}  9.4720  10.5052  11.4273  12.0812  11.7939 
10^{6}  15.7713  15.8984  15.6945  14.5994  12.3139 
10^{7}  20.5987  19.2828  17.4287  15.1154  12.3731 
10^{8}  22.1393  19.9912  17.6770  15.1730  12.3791 
10^{9}  22.3299  20.0700  17.7029  15.1788  12.3797 
Nondimensional fundamental frequencies of the nonuniform doublebeam system with
In this paper, an efficient modeling approach for the vibration analysis of nonuniform doublebeam system with general boundary condition is established, in which the fullcoupling on the common interface are taken into account by introducing both the translational and rotational restraining springs across the beams. The elastic boundary condition of the beams and various rotational restrains in the beam’s interface can be studied easily in the current modeling. In order to treat the nonuniform beam profile in the most general pattern, the arbitrary thickness variation functions are all expanded into Fourier series. Energy formulation is employed for the description of doublebeam dynamics, with the admissible function constructed as the superposition of Fourier series and boundary smoothed auxiliary terms. In conjunction with the Rayleigh–Ritz procedure, all the modal parameters can be derived by solving a standard eigenvalue problem.
Numerical examples are then presented to demonstrate the correctness and reliability of the current model for predicting the modal frequencies of the uniform doublebeam and nonuniform beam structures. Based on the model established, the influence of boundary condition and coupling strength on the modal characteristics of the nonuniform beam is investigated and addressed. The results show that the rotational coupling stiffness can also play an important role in affecting the doublebeam system, which has received little research attention in the existing literature. It can be also found that the variation of the thickness profile can be utilized to adjust the system modal parameters when the change of boundary and/or coupling conditions may be difficult to perform. Although just simulation is implemented for the doublebeam system, this approach can be very easy for handling the multiplebeam structure of any number. This work can provide an efficient modeling approach for the dynamic study of the multiplebeam system with complex boundary, coupling, and thickness variation conditions.
All data generated or analyzed during this study are included in this published article.
The authors declare that they have no conflicts of interest.
This work was supported by the Fok Ying Tung Education Foundation (Grant no. 161049).