CALCULATING THE NATURAL FREQUENCY OF HOLLOW STEPPED CANTILEVER BEAM

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 01, January 2019, pp , Article ID: IJMET_10_01_093 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed CALCULATING THE NATURAL FREQUENCY OF HOLLOW STEPPED CANTILEVER BEAM Luay S. Alansari, Hayder Z. Zainy Department of Mechanical Engineering The Faculty of Engineering, University of Kufa, Najaf, Iraq Aya Adnan Yaseen Al Mansour University College, Baghdad, Iraq Mohanad Aljanabi Department of Mechanical Engineering, the Faculty of Engineering, University of Kufa, Najaf, Iraq ABSTRACT Stepped or non-prismatic beams are widely used in many engineering applications and the calculation of their natural frequency is one of the most important problem. Several methods were used to calculate the natural frequency of stepped beam. In this work, the Rayleigh methods (Classical and Modified) and finite element method using software were used for calculating the natural frequency of Hollow stepped cantilever beam with circular and square cross section area. The comparison between the results of natural frequency and frequency ratio due to the increasing the length of the small part for these types of beams and for these three methods were made. The agreement between the classical Rayleigh results was better than the agreement between the Modified Rayleigh results for the two types of cross section area. The maximum error of was greater than that of and the maximum error of circular C.S.A. Was greater than that of square C.S.A. At the same dimensions. The natural frequency of circular C.S.A was smaller than that of square C.S.A. for the same dimensions Keyword head: Classical Rayleigh method, Modified Rayleigh method, Finite element method, workbench, Hollow beam, Stepped beam, Frequency. Cite this Article: Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi, Calculating the Natural Frequency of Hollow Stepped Cantilever Beam, International Journal of Mechanical Engineering and Technology, 10(01), 2019, pp

2 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi 1. INTRODUCTION Beams with variable cross-section are widely used in many engineering fields like mechanical engineering, aeronautical engineering, and civil engineering. There are many examples of structures that can be modeled as beam-like elements, such as robot arms, crane booms rotor shafts, columns, and steel composite floor slabs in the single direction loading case. The free vibrations for stepped or tapered beams (i.e. Non-uniform or Non-Prismatic beam) has been conducted using various approaches, such as the finite element, transfer-matrix, Adomian decomposition and other approximate methods [1 13]. Zhou and Cheung [1, 3] and Lu et al. [12] used the Rayleigh Ritz method for solving the bending vibrations of tapered beams and multiple-stepped composite beams respectively. For stepped beam, the free vibration analysis was studied by Jang and Bert [14, 15], Naguleswaran [16, 17], Ju et al. [18] and Dong et al. [19] with different boundary conditions. These papers and additional papers reviewed by Luay AL-Ansari [20] and Xinwei Wang and Yongliang Wang [21] were dealt with solid stepped beam and/or solid tapered beam. As theoretical analysis for hollow-sectional beams is more complicated than a solid beam and these beams are widely used in mechanical engineering and civil engineering. Generally, few researches dealing with natural frequency of hollow-sectional beams were conducted. Some researchers (like Murigendrappa et al. [22], Zheng and Fan [23], Naniwadekar et al. [24], and Peng Liping and Liu Chusheng [25]) focused on vibration of hollow-sectional beams with crack. Therefore, this paper will focus on calculating the natural frequency hollow-sectional cantilever beam with internal steps and the circular and square cross section area are used. Three numerical methods were used and these methods were Classical Rayleigh method, Modified Rayleigh method and Finite Element Methods using -Workbench (17.2). 2. ROBLEM DESCRIPTION The hollow cantilever beam with internal steps is shown in Fig. (1). The equation of motion of beam (i.e. Euler-Bernoulli and Timoshenko equations) cannot be solved analytically in this case because of varying in dimensions (i.e. area and Second Moment of Inertia) along the length of beam. Several researches were done for deriving new equation of motion described the variation in dimensions and /or solving it analytically. For calculating the fundamental natural frequency of the type of beam (or tube), classical Rayleigh method (), modified Rayleigh method () and the finite element method ( software) are used in this work in order to avoid the complexity in governing equation and its solution [20,26,27]. Figure 1. Geometry of hollow beam with internal steps used in this work 2.1. Rayleigh method (RM) The fundamental natural frequency of the system. The general formula of Rayleigh method was derived according to equate the potential and kinetic energy of any system and the fundamental natural frequency of this system can be estimated by the following equation [20, 26, 27] editor@iaeme.com

3 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam = (()) ( ) = () Where: (ω) is frequency (rad/sec), (y) is Deflection (m), (M) mass (kg), (A) is Cross Section Area (m 2 ), (ρ) is Density (kg/m 3 ), (E) is Modulus of Elasticity (N/m 2 ) and (I) is Second Moment of Inertia (m 4 ). As mentioned previously, the main problem of the vibration of stepped beam is the varying of the dimensions along the beam which leads to change in cross section area and second moment of inertia. Therefore, the methods described in references [20], [26] and [27] are used in order to calculate the equivalent second moment of inertia and these methods are: 1. Classical Method: The equivalent second moment of inertia for stepped beam with two internal steps can be found using the following equation [20, 26, 27]:! "# = ($ %&' ) ( ) *+, -( / *+ + - ( 0*+, - ( 1.,. + Where (LTotal) is the length of the beam, (LS) is the length of the beam, when the hollow width or diameter is (WS), calculating from free end and (LL) is the length of the beam, when the hollow width or diameter is (WL), calculating from free end and in this case equals (X T or LTotal).Numerical procedure. Modified Method: According to the idea described in [20] and [27], the equivalent moment of inertia at any point in the stepped beam can be calculated by applying the following: I 45 (x)= (7 89:;< ) ( ) *= > (?)-( / *= = - ( 0*=> (?)- ( = 2.3. Programming Rayleigh methods The Rayleigh Methods (i.e. Classical Rayleigh Method () and Modified Rayleigh Method ()) were programing using MATLAB code [20,25,26]. The general steps are: 1-Input the material properties (i.e. density and modulus of elasticity) and beam dimensions (see Fig. (1)). 2-Input number of divisions (N) and in this work N=8400 (i.e. the DX=0.1 mm). 3-Calculate the equivalent second moment of inertia according to the method (i.e. or ). 4-Calculate the mass matrix [m] (N+1). 5-Calculate the delta matrix [δ] ((N+1)* (N+1)) using Table (1) 6-Calculate the deflection at each node using the following equation and apply the boundary conditions. [y] (N+1) = [δ] ((N+1)* (N+1)) [m] (N+1) (1) (2) (3) editor@iaeme.com

4 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi Table 1 Formula of the deflections of the cantilever beam [20,25,26] B CD = EFG (HI F) KLM B DD = EIH HLM N OP = QRG (HS R) KTU 3. FINITE ELEMENT METHOD (FEM) In order to build 3D finite element model that shown in Fig. (2), Workbench (17.2) was used. Cantilever hollow beams with circular and square cross section were used in this work (see Fig. (2)). generally the number of Tetrahedrons elements was about (40,000) and the size of element was (2 mm) (see Fig. (3)). a. Square hollow beam. b. Circular hollow beam. Figure 2. Samples of beam geometry built in workbench software Figure 3. Samples of mesh and result in Workbench software editor@iaeme.com

5 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam 4. RESULTS AND DISCUSSION Generally, the length of beam used in this work is (0.84) m and the outer width (or diameter) is (0.04) m. Seven values of width (or diameter) for large part (i.e. WL or DL) (WL=0.01, 0.015, 0.02, 0.025, 0.03 and m) were used and in the same time the hollow width (or diameter) for small part (i.e. WS or DS) changed from (WL or DL) to (0.04) m. The dimensions of hollow stepped beams with square and circular cross section area, used in this work, can be summarized in Table (2). NO Length Beam (m) Table 2 Cases studied in this Work Length of Large Part (m) Length of Small Part (m) Width (or Diameter) of Large Part (m) Width (or Diameter) of Small From (0.015) to (0.035) From (0.02) to (0.035) From (0.025) to (0.035) (0.03) and(0.035) editor@iaeme.com

6 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi Fig. (4) illustrates the comparisons among natural frequencies of Hollow stepped cantilever beam calculated by classical Rayleigh method (), modified Rayleigh method () and the finite element method ( software), in additional to the comparison between circular and square cross section area when the inner width ( or diameter) of larger part is (0.01 m). When the length of small part is zero, the dimensions of cross section area are Wout (or Dout) = 0.04 m and Win=WL (or Din=DL) and when the length of small part is (L), the dimensions of cross section area are Wout (or Dout) = 0.04 m and Win=WS (or Din=DS). These points are found in each curve and represented the start and end points. When the length of the small part ( i.e. small cross section area) increases, the natural frequency increases and the natural frequency reaches to its maximum value when (XS=0.36 m) and then the natural frequency decreases. In the other hand, the natural frequency increases when the width (or diameter) of the small part increases and width (or diameter) of the large part is (0.01m). Also, the natural frequencies of square C. S. A. is larger than that of circular C. S. A. These points can be explain by considering two important parameters (mass and second moment of Inertia). When the length of small part increases, the mass of beam decreases and the equivalent second moment of Inertia decreases too and the decreasing rate of equivalent second moment of Inertia is larger than that of mass ( the equivalent second moment of Inertia depends on (length of small part)**4) while the mass depends on (length of small part) only). In the comparisons among the three calculating methods, the results are considered as exact results. For circular C.S.A., the error of results comparing with results increases when the length of small part increases and the maximum error is found when(xs=0.36 m). Also, the maximum error increases when the diameter of the small part increases. In method, the error, also, increases when the length and diameter of small part increase. Generally the maximum error of is greater than that of and the maximum error of circular C.S.A. Is greater than that of square C.S.A. At the same dimensions (see Table (3)). Also, the maximum error increases when the width (or diameter) of small part increases. In Figures (5) - (8), the comparisons among natural frequencies of Hollow stepped cantilever beam calculated by three calculating methods, for circular and square cross section area when the inner width ( or diameter) of larger part is (0.015, 0.02, and 0.03) m respectively. Generally, the same behavior can be noted in these figures but with increasing the values of natural frequencies when the width (or diameter) of small and large parts increase. For circular and square C.S.A., the variation of frequency ratio (ω/ ωs) [ where ω is the frequency of beam with any dimensions and ωs is the frequency for beam with dimensions of small part] are drawn and the comparison among the frequency ratios calculating by, and are illustrated in Figures (9) - (12). From these Figures, the frequency ratio changes along the dimensionless length of small part with the same way and the maximum frequency ratio of circular C.S.A equals to that of square C.S.A. for the same calculating method. Also, the maximum frequency ratio increases when the width (or diameter) of the small part increases and when the width (or diameter) of the large part increases editor@iaeme.com

7 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam D S =0.015 m W S =0.015 m D S =0.02 m W S =0.02 m D S=0.025 m W S=0.025 m D S =0.03 m W S =0.03 m editor@iaeme.com

8 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi D S =0.035 m W S =0.035 m Figure 4. Comparison among natural frequencies of hollow beams with circular and square cross section area due to change in length of the small step (x s) for different calculating method and different values of small width (or diameter) (W S or D S) when the large width (or diameter) (W L or D L) is (0.01) m. D S =0.02 m W S =0.02 m D S=0.025 m W S=0.025 m editor@iaeme.com

9 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam D S =0.03 m W S =0.03 m D S =0.035 m W S =0.035 m Figure 5. Comparison among natural frequencies of hollow beams with circular and square cross section area due to change in length of the small step (X s) for different calculating method and different values of small width (or diameter) (W S or D S) when the large width (or diameter) (W L or D L) is (0.015) m Ds=0.025 m W S=0.025 m editor@iaeme.com

10 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi D S =0.03 m W S =0.03 m Ds =0.035 m Ws =0.035 m Figure 6. Comparison among natural frequencies of hollow beams with circular and square cross section area due to change in length of the small step (X S) for different calculating method and different values of small width (or diameter) (W S or D S) When the large width (or diameter) (W L or D L) is (0.02) m D S =0.03 m W S =0.03 m D S =0.035 m. W S =0.035 m editor@iaeme.com

11 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam Figure 7. Comparison among natural frequencies of hollow beams with circular and square cross section area due to change in length of the small step (X S) for different calculating method and different values of small width (or diameter) (W S or D S) when the large width (or diameter) (W L or D L) is (0.025) m DS =0.035 m WS =0.035 m Figure 8. Comparison among natural frequencies of hollow beams with circular and square cross section area due to change in length of the small step (Xs) for different calculating method and different values of small width (or diameter) (Ws or Ds) When the large width (or diameter) (W L or D L) is (0.03) m Table 3 The maximum error between results and classical and modified Rayleigh method Width (or Diameter) of Large Part (m) Width (or Diameter ) of Small Part (m) Circle C. S. A. Classical R.M Modified R.M Maximum error % Square C. S. A. Classical R.M Modified R.M Maximum error % editor@iaeme.com

12 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi Figure 9. Comparison among frequency ratio of hollow beams with circular and square cross section area due to change in dimensionless (X S) for different calculating method and different values of large width (or diameter) (W L or D L) when the small width (or diameter) (Ws or Ds)is (0.02) m editor@iaeme.com

13 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam Figure 10. Comparison among frequency ratio of hollow beams with circular and square cross section area due to change in dimensionless (Xs) for different calculating method and different values of large width (or diameter) (W L or D L) when the small width (or diameter) (Ws or Ds)is (0.025) m editor@iaeme.com

14 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi Figure 10. Comparison among frequency ratio of hollow beams with circular and square cross section area due to change in dimensionless (Xs) for different calculating method and different values of large width (or diameter) (W L or D L) when the small width (or diameter) (Ws or Ds)is (0.025) m editor@iaeme.com

15 Calculating the Natural Frequency of Hollow Stepped Cantilever Beam Figure 11. Comparison among frequency ratio of hollow beams with circular and square cross section area due to change in dimensionless (Xs) for different calculating method and different values of large width (or diameter) (W L or D L) when the small width (or diameter) (W S or D S) is (0.03) m editor@iaeme.com

16 Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi Figure 12. Comparison among frequency ratio of hollow beams with circular and square cross section area due to change in dimensionless (X S) for different calculating method and different values of large width (or diameter) (W L or D L) when the small width (or diameter) (W S or D S)is (0.035) m 4. CONCLUSION From the previous results, the following point can be concluded: Generally the maximum error of is greater than that of and the maximum error of circular C.S.A. Is greater than that of square C.S.A. At the same dimensions. When the length of the small part increases, the natural frequency increases and the natural frequency reaches to its maximum value when (XS=0.36 m) and then the natural frequency decreases. The natural frequency increases when the width (or diameter) of the small part increases for the same width (or diameter) of the large part. In hollow stepped cantilever beam, the is better than the for calculating the natural frequency. While the is better than in stepped cantilever beam (see Ref. [20]). The natural frequency of circular C.S.A is smaller than that of square C.S.A. for the same dimensions. Finally, the modified Rayleigh method can be used for calculating the natural frequency for hollow stepped cantilever beam (with number of step larger than two), non-prismatic beam and beam with different cross section area. REFERENCES [1] Zhou, D., and Cheung Y. K. Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions, J. Appl. Mech. ASME, 68, 2001, pp [2] Wu, J.S., and Chen D.W. Bending vibrations of wedge beams with any number of point masses, J. Sound Vib., 262, 2003, pp [3] Zhou, D., and Cheung Y.K. The free vibration of a type of tapered beams, compute methods appl. Mech. Eng, 188, 2000,pp [4] Hooshtari. A., and Khajavi, R. An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler Bernoulli and Timoshenko beam elements, Eur. J. Mech. A/Solid, 29, 2010, pp [5] Wang. G, and Wereley, N. M. Free vibration analysis of rotating blades with uniform tapers, AIAA J., 42, 2004, pp [6] Kumar, A., and Ganguli, R. Rotating beams and nonrotating beams with shared eigenpair, J. Appl. Mech. ASME, 76, 2009, pp [7] Spyrakos, C. C., and Chen, C. I. Power series expansions of dynamic stiffness matrices for tapered bars and shafts, Int. J. Numer. Methods Eng., 30, 1990, pp editor@iaeme.com

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