Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions

In this study, free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions: Clamped-Clamped (C-C), Clamped-Free (C-F), Clamped-Simply Supported (C-SS) and Simply Supported-Simply Supported (SS-SS). Analytical solution is carried out using Euler-Bernoulli beam theory and Newton Raphson Method. First, the equations of motion are provided. Then, solutions including the effects of the geometric characteristics, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes. To confirm the reliability of the vibration analysis carried out in the present paper as well, all the analytical results are checked with the corresponding numerical results obtained from the finite-element-method (FEM) based software called ANSYS. Numerical and analytical results are found to be good agreement.


Introduction
A beam is a slender horizontal structural member that resists lateral loads by bending, and this important element of engineering structures appears in various forms and comprises various artifacts, such as supporting members in high-rise buildings, railways, long-span bridges, flexible satellites, gun barrels, robot arms, airplane wings, etc. [1,2].
In many engineering applications, beams are subjected to dynamic loads, which can excite beam structural vibrations and cause durability concerns or discomfort because of the resulting noise and vibration. In addition, if the vibration exceeds certain limits, there is the danger of beam breakage or failure [3][4][5][6].
In this study, the free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions. Analytical solution is carried out using Euler-Bernoulli beam theory, in which material is assumed to be linear-isotropic, and Newton Raphson Method. This method is based on the simple idea of linear approximation, and used for finding the roots of equations. It is particularly useful for transcendental equations, composed of mixed trigonometric and hyperbolic terms. Such equations occur in vibration analysis. An example is the calculation of natural frequencies of continuous structures [26]. Solutions including the effects of the geometric characteristics, i.e., length and cross sectional area, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes. Furthermore, to confirm the reliability of the vibration analysis carried out in the present paper as well, all the analytical results are checked with the corresponding numerical results obtained from Finite Element Method (FEM)-based software called ANSYS [27], where the method is established on the idea of building a complicated object with simple blocks, or, dividing a complicated object into smaller and manageable pieces [28].
Present analysis can be used as a comparative study or data for the different solution methods of future works in the related field.

Basic Equations
Consider an elastic beam of length L , Young's modulus E , and mass density ρ with uniform cross section A , as shown in Figure 1.
Using Euler-Bernoulli beam theory, one can obtain the equation of motion of a beam with homogeneous material properties and constant cross section as follows [5]: where I is the area moment of inertia of the beam cross section, w is the transverse displacement, and t is time. (1) can be rearranged as follows: where A κ ρ = is the linear mass density of the beam. The solution of the Eq. (2) is sought by separation of variables. Assume that the displacement can be separated into two parts: one is depending on the position and the other is depending on time, as follows: where Λ and Ψ are independent of time and position, respectively.
Substituting Eq. (3) into Eq. (2) and after some mathematical rearrangements, the following equation is obtained: As observed from Eq. (4), the left side depends on the variable x, and the right side depends on the variable t, as previously noted. Consequently, the variables have been separated, and each side of (4) must equal a constant, denoted 2 ω − to have simple harmonic motion in the system. 4 2 If the position variable is separated where 4 2 If the time variable is separated Eq. (6) is solved as follows: where 1 4 ,..., C C are constants, and sinh and cosh are the hyperbolic sin e and cos e functions, respectively.
Eq. (8) is solved as follows: 5 6 ( ) sin cos where 5 6 C and C are constants. Thus, if Eq. (9) is multiplied by Eq. (10) to obtain ( , ) w x t , it yields eight combined constants as: where the constants 1 2 3 4 , , , C C C C can be obtained from the boundary conditions, and 5 6 , C C can be obtained from the initial conditions Finally, using Eq. (7) the natural frequency ( ) n f Hz of the beam is found as follows:

Particular Solution for C-C Beam
The boundary conditions satisfied by a C-C beam are as follows: When Eqs. (13)- (14) are considered in Eq. (9), after some mathematical operations, the coefficient matrix is obtained as follows: The non-trivial solution of the determinant of the coefficient matrix is as follows: where the subscript is an integer index.
Because the first three roots of Eq. (16) are calculated using the Newton-Raphson method, the following eigenvalues are obtained: where, δL, is the natural frequency parameter of the beam.

Particular Solution for C-F Beam
The boundary conditions satisfied by a C-F beam are as follows: When Eqs. (18)- (19) are considered in Eq. (9), after some mathematical rearrangements, the following coefficient matrix is obtained: The non-trivial solution of the determinant of the coefficient matrix is as follows: When the first three roots of Eq. (21) are calculated using Newton-Raphson method, the following eigenvalues are obtained:

Particular Solution for C-SS Beam
The boundary conditions satisfied by a C-SS beam are as follows: When Eqs. (23)- (24) are considered in Eq. (9), after some mathematical rearrangements, the following coefficient matrix is obtained: The solution of the determinant of the coefficient matrix is as follows: tanh tan

Particular Solution for SS-SS Beam
The boundary conditions satisfied by a SS-SS beam are as follows: When Eqs. (28)-(29) are considered in Eq. (9), after some mathematical rearrangements, the following coefficient matrix is obtained: The non-trivial solution of the determinant of the coefficient matrix is as follows:

Comparative Study
In this subsection, a comparative study was performed to validate the present numerical results. The analytical results were compared with the results of the FEM -based software called ANSYS [27]. In finite-element modeling, the beam type element is applied and meshed with 50 elements.
For the first three modes 100 nAnaly nFEM As shown in Figure 2, the numerical results of both methods are consistent, which shows the accuracy of the present formulation.

Free Vibration Analyses of Beams
In this section, four studies were performed to investigate the free-vibration behaviors of square cross-sectioned aluminum beams with different geometric characteristics under four different boundary conditions. The natural frequencies were obtained and discussed for the first three modes (n=1, 2, 3), including the effects of the geometric characteristics, i.e., length and cross sectional area, and the boundary conditions.

Study 1:
In Figure (n=1,2,3), the C-C boundary conditions were compared with the other boundary conditions. From this comparison, the following results were obtained: for n = 1, 2, and 3, i) the differences between C-C and C-F boundary conditions are 84%, 64%, and 49%, ii) the differences between C-C and C-SS boundary conditions are 31%, 19%, and 14%, and iii) the differences between C-C and SS-SS boundary conditions are 56%, 36%, and 27%, respectively. The percentages were calculated as follows: Therefore, the effect of the type of the boundary condition decreases with the increase in mode number, n.

Conclusions
In this study, the free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions. Analytical solution is carried out using Euler-Bernoulli beam theory, in which material is assumed Solutions including the effects of the geometric characteristics, i.e., length and cross sectional area, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes.
The following results were obtained: (1) The natural frequencies increase with the increase in mode number.