Abstract
The aim is to derive an expression to calculate the natural frequencies and plot the mode shapes of a simplysupported beam with an overhang with an end overhang point mass by using the EulerBernoulli theory in the case of free transverse vibrations. The results are validated by finite element analysis. The importance of the system presented is that it can represent machine tool spindles or even machining tools like boring bars. The results are in good agreement with the results from the finite element analyses. The derived expression can be used in optimizing the value of the point mass and optimizing the support location for better performance of the system without the need to perform complex analysis to obtain the values of the natural frequencies and to plot the mode shapes.
1 Introduction
Beams with different configurations can be essential elements in representing engineering systems, starting from truck axels [1] to sandwich beams structures [2] to machinetool systems, mainly when it comes to machinetool spindle systems. Being able to calculate the natural frequencies of beam systems help optimize them in terms of weight, performance and cost. Double span beam systems attract a lot of researchers’ attention since it is the case, which is present in many engineering applications in addition to the traditional beam configurations. EulerBernoulli beam theory is widely used in the analysis of the mechanical vibrations of beams. Beams are analyzed using many methods and different techniques. The researcher He’s Variational Iterational Method (VIM) was used by Alima and Desmond [3] to analyze a EulerBernoulli beam on an elastic support in the case of free mechanical vibrations. Also, Lai et al. used the Adomian Decomposition Method (ADM) as an innovative eigenvalue solver for free vibration of a EulerBernoulli beam under different supporting conditions [4]. A method called Differential Transforms Method (DTM) was used by Ozgumus and Kaya [5] to analyze flapwise bending vibrations of a double tapered rotating EulerBernoulli beam. The natural frequencies and mode shapes of EulerBernoulli beams were determined by Yieh [6] using the singular value decomposition method. An approximate solution to the transverse vibration of the uniform EulerBernoulli beam under linearly varying axial force was derived by Naguleswaran [7].
In this study, transverse vibration analysis of a Continuous PinnedPinnedFree Beam (CPPFB) with a mass attached at the free end is carried out using the EulerBernoulli beam theory. Then the results of the natural frequency and mode shapes obtained by the analysis will be compared to the results of the Finite Element Analysis (FEA) of the same case.
2 Free vibration of a simply supported beam with an overhanging
2.1 Setting the boundary conditions
2.2 Substituting in the general form solution
2.3 Matrix form and eigenvalues
3 Validation of the model
A beam model with specific properties is analyzed using the analysis method in the previous sections to obtain the values of the natural frequencies for the first four eigenvalues and the first four mode shapes, then the same beam is analyzed using the FEA in order to compare the results and calculate the error percentage between both methods. Table 1 lists the parameters of the validation model.
The parameters of the validation model
Parameter  Value 
L  1 m 

7,850 kg m^{−3} 

2.05∙e^{+11} N m^{−2} 

0.628 kg 
I  1.333∙e^{−8} m^{4} 
A  4∙e^{−4} m^{2} 

0.3 m 
Since Eq. (24) has an infinite number of zeros and that the interest of this research is the first four modes, the parameters in Table 1 are substituted to Eq. (24) and then it is plotted for a domain of 4
Then the zeros of the plot are extracted and tabulated in Table 2 as follows:
Extracted beta values

Value 
1  1.99 
2  5.52 
3  9.53 
4  11.85 
Substituting
Natural frequencies values


1  18.72 
2  143.25 
3  426.33 
4  659.5 
In order to obtain the
The values of the coefficients from
4 FEA and comparison
The same model introduced in the last section is analyzed using the FEA method to find the values of the first four natural frequencies and for plotting the mode shapes. Table 4 lists the values of the natural frequencies of the FEA method and the proposed model in addition to the error percentage.
Theory and FEA method natural frequencies values with the error percentage
Mode  Theory [Hz]  FEA [Hz]  Error 
1  18.72  18.79  0.37% 
2  143.25  143.77  0.36% 
3  426.33  426.20  0.03% 
4  659.50  658.13  0.21% 
Figure 5 shows the normalized mode shapes obtained by the FEA.
5 Conclusions
A pinnedpinnedfree beam with a mass attached to the free end is studied. EulerBernoulli beam theory is used to derive the transcendental equation, which can be applied to different secondpin location and different attached mass values. The eigenvalues for calculating the natural frequencies and the mode shapes were obtained for a validation model. FEA of the same validation model were carried out and the first four natural frequencies were compared to the proposed method. The comparison of the results shows a very good match in the natural frequency values. However, the mode shapes for both methods take the same form with some differences in the peak’s values. The difference can be attributed to the normalizing process of the mode shapes.
The developed analytical model can be helpful in the design of many engineering applications like crane arms, machinetool spindles or even machinetool boring bars. Future work will include investigating different attached mass to beammass ratio in addition to the effect of the second pin location on the accuracy of the developed model, as well as analyzing the model under the application of force.
References
 [1]↑
F. Hajdu , P. Szalai , P. Mika , and R. Kuti , “Parameter identification of a fire truck suspension for vibration analysis,” Pollack Period. , vol. 12, no. 1, pp. 51–62, 2019.
 [2]↑
A. AlFatlavi , K. Jármai , and G. Kovács , “Theoretical and numerical analysis of an aluminum foam sandwich structure,” Pollack Period. , vol. 15, no. 3, pp. 113–124, 2020.
 [3]↑
A. Tazabekova , D. Adair , A. Ibrayev , and J. Kim , “Free vibration calculations of an EulerBernoulli beam on an elastic foundation using He’s variational iteration method,” in Proceedings of ChinaEurope Conference on Geotechnical Engineering, W. Wu and H. S. Yu , Eds, Springer Series in Geomechanics and Geoengineering, vol. 2, pp. 1022–1026, 2018.
 [4]↑
H. Y. Lai , J. C. Hsu , and C. K. Chen , “An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method,” Comput. Mathematics Appl., vol. 56, no. 12, pp. 3204–3220, 2008.
 [5]↑
O. O. Ozdemir and M. O. Kaya , “Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method,” Meccanica, vol. 41, no. 6, pp. 661–670, 2006.
 [6]↑
W. Yeih , J. R. Chang , C. M. Chang , and J. T. Chen , “Applications of dual MRM for determining the natural frequencies and natural modes of a rod using the singular value decomposition method,” Adv. Eng. Softw., vol. 30, no. 7, pp. 459–468, 1999.
 [7]↑
S. Naguleswaran , “Transverse vibration of an uniform EulerBernoulli beam under linearly varying axial force,” J. Sound Vibration, vol. 275, no. 1–2, pp. 47–57, 2004.