Mathematical Modeling of the Effect of Temperature on the Dynamic Characteristics of a Cantilever Beam with Flexible Root

This study presents the development of an analytical solution for the dynamic response of a cantilever beam with a flexible root taking into account the influence of temperature. The investigated cantilever beam has a uniform rectangular cross-section with finite lengths. The dynamic response of the cantilever was investigated under three conditions, namely, rigid root, resilient root, and resilient root accompanied by different surrounding temperatures. The selected lengths for the beam were 0.3175, 0.1588, 0.1058, 0.0794, 0.0635, 0.0529, 0.0454, 0.0397, 0.0353, and 0.03175 m. The chosen linear spring coefficients were 0.01, 0.1, 100, and ∞ N/m while rotational spring coefficients were 0.01, 0.1, 100, and ∞ N·m/rad. The surrounding temperatures for the third condition were −100, 25, 100, and 200°C. A MATLAB code was developed to calculate the fundamental natural frequency under different surrounding temperatures and spring coefficients. The proposed mathematical solution was validated with real experimental data and the verification findings revealed a good match between them. For the rigid condition, the finding revealed good matching between the analytical model and experimental results, particularly at the length range of 0.3175−0.1058 m. For the resilient condition, the fundamental natural frequencies were found to be highly affected by decreasing beam length and increased at 100 N/m and 100 N·m/rad and higher coefficients. Finally, there was a reduction in the calculated natural frequencies with increasing temperature.


Introduction
Mathematical modeling of dynamic behavior for the structural part is an important topic. It is quite necessary to predict the natural frequency during the design stage by deriving a reliable analytical solution. Historically, distinguished eforts have been made in the feld of estimation of natural frequencies as well as mode shapes of fexible root cantilever beams [1][2][3][4][5][6].
Qiao et al. [7] presented an exact solution for solving single and two degrees of freedom for the free fexural vibration of a nonuniform Euler Bernoulli beam. Te presented model was efcient in terms of computation with a signifcant reduction in determinant order compared with other methods. Te vibration behavior of a functionally graded beam was investigated by Hein and Feklistova [8] using both the Euler-Bernoulli theory and the Haar technique. Some classic wavelets were applied to simplify and transform the governing equation of the beam system. Te beam was investigated under various cross-sections, mass density, rigidity, and diferent coefcients of translational and rotational stifness. Te Haar wavelet approved its capability through the achieved results where the applied approach was accurate, simple, and soft. Ahmed [9] conducted an experimental and numerical study to analyze the free vibration behavior of a cantilever beam with notched and unnotched geometrical conditions. Te beam was made of Kevlar-reinforced epoxy. Good agreement was obtained when the results were compared with published works. Te author found a reduction in the computed natural frequencies with increasing notch depth. Also, fber orientation has some infuence on the convergence and divergence between numerical and experimental fndings. Majeed et al. [10] modeled a fexible smart cantilever beam based on Euler-Bernoulli and piezoelectric theories using state space and fnite element techniques. Te aim of the modeled smart structure was to reduce beam vibration and settling time where the achieved results showed efective performance of the proposed technique. Also, a sliding mode observer was designed for vibration suppression of the fexible cantilever beam by Al-Samarraie et al. [11].
Zhang et al. [12] derived an analytical solution for the free vibration analysis of a nonuniform fexible Timoshenko beam with multiple discontinuities. Te model results were accurate when verifed with the fndings of fnite element simulation and the literature. A rare case of free vibration for the Timoshenko beam under elastic restraints as stated by Shi et al. [13] was investigated. An exact solution was achieved by applying the Fourier series where simultaneous satisfaction was obtained from the governing equation and boundary condition for any defned level of accuracy. New results for the elastic retrained beam were presented to be used as a benchmark solution for future work. An improved Fourier-Ritz method was adopted by Wang et al. [14] for the free vibration analysis of an axially loaded cantilever beam made of the laminated composite under diferent boundary conditions. Tey reported that the model derived using this approach was accurate and reliable and had fast convergence.
Pham and Nguyen [15] employed Euler-Bernoulli's theory for a 3D cantilever beam having a moving hub and fexibility. Tey applied Hamilton's principles to derive the equation of motion and also used the Galerkin approach to reduce the model order. Both simulation and experimental data validated the derived dynamical model where good matching was achieved. Te free vibration problem of the double beam under restrained and coupling conditions was solved by Chen et al. [16] by using an improved Fourier approach. Te reliability of the proposed analytical solution was compared with a numerical model. An improved Fourier-Ritz method was applied by Hao et al. [17] to analyze the characteristics of free vibration for the double beam under restrained stifness and elastic layer conditions. Te displacement discontinuities and its derivative were removed using the Fourier series and polynomial function to speed up the model convergence rate. Also, numerical simulation was performed for the investigated beam using diferent shapes and material properties. Te mathematical model was compared with the published works to confrm its reliability and accuracy. Chen and Du [18] took into account the efect of restrained stifness and rotation speed when deriving an analytical solution based on the Fourier series for a rotational beam. Te efectiveness of the analytical solution was approved by comparison with numerical results.
Zhao [19] applied the shape function method to solve the free vibration problem of the axially loaded beams under arbitrary elastic support with concentrated masses and nonconventional boundary conditions. Te free vibration governing equation was reformulated by using Dirac's delta function to solve the shape function approach. Good agreement was obtained when the results of the derived mathematical model were with the fndings of the published works. Also, Zhao [20] studied the special geometry of double beams having concentrated masses. Te author obtained the mode shapes of free and forced vibration for the investigated beam by applying the shape function method. A new orthogonality form condition was derived and the reliability of the model was achieved via comparison with a numerical example.
Te free vibration characteristics of the functionally graded material beam were investigated by Kim et al. [21] using the Haar wavelet method. Te governing equation was constructed by applying Hamilton's principles to make a generalization of the boundary conditions for the four locations along the beam. Validation of the derived model with published results and numerical fndings approves its accuracy and efectiveness.
Paridie et al. [22] developed an artifcial neural network (ANN) model to predict the natural frequency of a cantilever beam with various cross-sections and under the efect of magnitude and load location. Te ANN model was trained and tested with fnite element (FE) simulation data for the beam under investigation. Good matching was achieved between ANN and FE models.
Analytical analysis methods showed and proved their capability by solving various engineering problems. Particularly, the analytical solutions to fnd the dynamic behavior for vital engineering parts such as cantilever beams have found a wide range of engineering applications (e.g., rotating, blade, cantilever bridges, balconies, cranes, overhanging roofs such as stadium roofs, and shelters). In general, cantilever beams are used in diferent environments; therefore, providing an analytical solution to fnd the dynamic response under diferent environments is a crucial task. A comprehensive search in the available literature was achieved, and it was found that the compound efects of thermal and root stifness on the dynamic response of the cantilever beam have not been addressed before by any researchers. Terefore, this study investigates the combined efect of temperature and fexible root for a cantilever beam to estimate the dynamic response represented by the fundamental natural frequency. Te investigation is conducted based on developing an analytical solution whose results are verifed with real fndings.

Development of the Analytical Solution for Free Vibration Cantilever Beam under Surrounding Temperatures and Resilient Conditions
In this section, an analytical model will be developed to identify the fundamental natural frequencies for the cantilever beam, taking into account the efect of a fexible root at one end of the beam and the infuence of temperature. Te fexible root in this study is taken as linear and rotational springs.
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Derivation of the Analytical Solution of the Free Vibration
Cantilever Beam. Te diferential equation that governs the sinusoidal vibrational motion of the cantilever beam is as follows [2]: where E represents the beam modulus of elasticity, I xx is the area moment of inertia, and (f n ) is the natural frequency.
Te prime symbols at Y refer to the diferentiability per nondimensional parameters; let us call it Z, where Z � z/L, where L is the total length of the cantilever beam.
Calculating the frequency variable is possible by using the following equation: where ρ is the beam density, A is the beam cross-sectional area, ω is the angular natural frequency, and T is the temperature in°C. As stated at the beginning of this section that the cantilever beam under investigation has a resilient root represented by linear and rotational springs at the left end, in this analytical solution, it was taken into consideration the efect of the surrounding temperature, where any change in the surrounding temperature will signifcantly afect the material properties of the beam (modulus of elasticity). Terefore, it was assumed that the modulus of elasticity of the beam is a function of surrounding temperatures E(T). Tis surrounding temperature leads to a change in the magnitude of stifeness of the structure/ beam, and eventually, the values of natural frequency will be changed too. Terefore, this study aims to show how the dynamic response of a cantilever beam is infuenced by the change in the surrounding temperature assuming the resilient root conditions (the linear and rotational springs at the root of the beam). Tese springs have stifness of K1 and K2. If boundary conditions are applied to such beam depicted in Figure 1, the following can be attained: where where S T and S R represent the coefcients of linear and rotational springs, respectively.
Te mode shape given in equation (5) satisfes the boundary conditions: Here, When we substitute equation (5) into equation (2) while minimizing the resulting error using Galerkin integral, we get the following equation to calculate the fundamental natural frequency: where If the stifness of linear spring approaches to infnity (S T ⟶ ∞), then S * T � 0 and equation (7) will become Similarly, if the stifness of the rotational spring approaches to infnity (S R ⟶ ∞), then S * R � 0 and equation (7) will become Finally, if the rigid conditions exist (i.e., both S * R and S * T equal zero), the fundamental natural frequency will be f n � 1.878854. Te Scientifc World Journal

Development of MATLAB Code.
To program the differential equation so as to calculate the fundamental natural frequencies of the cantilever beam under the given conditions, a MATLAB code was written to perform this task. Te fow chart of the developed code is depicted in Figure 2. It consists of several steps as follows: (1) Enter the dimensions of the cantilever beam and its density (2) Enter the modulus of elasticity for the beam based on the selected surrounding temperatures (3) Enter the linear and rotational stifness coefcients (4) Determine the moment of inertia and other geometrical parameters (5) Calculate the fundamental natural frequencies using equations (7)- (14) based on the boundary conditions (6) Present the natural frequency under a given surrounding temperature (7) Select another temperature within the specifed range and follow steps 2-6 (8) Plot the fundamental natural frequencies as a function of linear and torsional stifness coefcients and temperatures In fact, the time spent understanding and analyzing the mathematical model of the problem and then formulating the equations of motion, taking into account the thermal and root stifness (linear and rotational) efects, is considerable (approximately four months). At this stage, it was verifed that the accuracy of the results of the analytical solution was verifed by comparing it with the results of other researchers who used other approaches. Te last stage is to build the code based on the analytical solution to reduce the computational time to minimum. Tus, the execution time (computational cost) was relatively short (approximately 30 min and maybe more according to the case study and the range of the variables) to run and collect the data of the developed MATLAB code.

Results and Discussion
Tis section provides and analyzes the achieved fndings of the developed analytical solution that was developed essentially to determine the fundamental natural frequency of the cantilever beam. Te dynamic response of the cantilever beam was investigated under the following conditions: (1) Cantilever beam with the rigid root (2) Cantilever beam with the resilient root (linear and rotational springs) (3) Cantilever beam with rigid and resilient roots under the efect of temperature.
Te results of the abovementioned conditions have been verifed with real experimental data as will be seen in the following subsections.

Efect of the Rigid Root on the Dynamic Response of the Cantilever Beam.
Tis subsection presents the results of the calculated fundamental natural frequency for the cantilever beam under free vibration, ambient temperature, and rigid root conditions. Te f n was calculated at diferent beam lengths, and the results were compared with experimental and theoretical fndings [2] Meanwhile, the error % was calculated to show how the results of the derived analytical solution were close or near to the published fndings. Te three results are tabulated and given in Table 1. On one side, it can be seen that there is a very close matching between the analytical model and the theoretical model. On the other side, there is good matching with the experimental work, particularly at 0.3175−0.1058 m. After that range, some divergence is noticed. Te experimental work is normally associated with some noise factor that shifts its results from the theoretical and analytical solutions. It includes human error, environmental conditions, and the accuracy of the utilized apparatuses.

Efect of the Resilient Root on the Dynamic Response of the Cantilever Beam.
When the fexible root is attached to one end of the cantilever beam, the dynamic response changes according to the conditions of the impeded resilient root. Te efect of both linear and rotational coefcients on the natural frequency is included in the analytical model. Te fundamental natural frequencies are plotted against linear and rotational spring coefcients at diferent beam lengths as depicted in Figures 3-12.
Investigation of Figures 3-12 reveals the following points: (1) Te fundamental natural frequency is highly afected by decreasing the cantilever beam length (2) When increasing the linear spring coefcient coactively with the rotational coefcient (CRS), the fundamental natural frequency is highly increased at 100 N/m and 100 N·m/rad and higher coefcients. To sum up, there are no high diferences in fundamental natural frequencies at a length range of 0.3175 m, regardless of linear and rotational coefcients. Based on the mentioned mathematical formula, it is well known that there is an inverse proportion between frequency and the taken period (time). Terefore, a long cantilever beam takes more time to come back to the original coordinates. In other words, a low natural frequency is produced. Terefore, the length of the cantilever beam must be taken into consideration. Also, to resist the dynamic bending of the beam, a precaution has to be considered in the material selection for the cantilever beam. A more stif material is preferred to maintain the stability of the beam. On the other hand, at low linear or rotational coefcients (0.01 N/m or 0.01 N·m/ rad), the values of natural frequency are low and comparable, and vice versa, for higher linear and rotational coefcients. Te higher values of linear and rotational spring coefcients mean that these springs are stifer than those with low coefcients. Terefore, a cantilever beam with a stifer spring means that it returns faster to the relaxed position. Quick pull-back motion produces an overshoot, which generates a high amount of oscillation and accordingly a higher natural frequency. In contrast, slowly pulling back to the original position is done by the cantilever beam attached to the springs having low linear and rotational coefcients. Terefore, low resistance to the oscillational motion is obtained because the cantilever beam makes a slow response and hence low natural frequency is produced.  Two alloys were taken as a case study to illustrate the infuence of temperature on the fundamental natural frequency, namely, N-based alloy and AA5054 aluminum alloy. Te material properties represented by the Young modulus of elasticity were taken as a function of temperature. Te selected temperatures were −100, 25, 100, and 200°C, respectively. Tey include subzero, normal, and high temperatures to show how the dynamic response of the cantilever beam will be impacted under these conditions (i.e., variable temperatures and fexible root). Table 2 presents the modulus of elasticity (E) of    those two materials as a function of the selected temperature range.
When E-values are replaced in the derived fundamental natural frequency of equation (7), Tables 3 and 4 are obtained. Tese tables reveal a reduction in fundamental natural frequency with increasing temperature for both materials. As indicated in equation (2), the material properties of the cantilever beam represented by Young's modulus of elasticity were taken as a function of temperature. Consequently, the fundamental natural frequency will be changed accordingly.  Te Scientifc World Journal

Conclusions and Remarks
In this study, a mathematical model was developed to investigate the efects of linear and rotational spring coefcients conjugated with temperature infuence. Based on the presented results and discussion, the following conclusions can be drawn: (1) Te analytical model was successfully derived.     (6) Te natural frequency was afected by increasing temperature as the beam modulus of elasticity was taken as a function of temperature.
Te current study can be extended in future work to fnd the dynamic response of the rotating cantilever beam (turbomachine blades) working in diferent environmental conditions with diferent root stifness. Also, the infuence of the defect in the cantilever beam on the dynamic response under diferent working conditions can be investigated. Furthermore, the current analytical solution can be enhanced to study the vibration characteristics of a microcantilever working under high-temperature conditions.

Data Availability
Te data that support the fndings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.