International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN 2249-6890 Vol. 3, Issue 4, Oct 2013, 107-118 TJPRC Pvt. Ltd. CALCULATING STATIC DEFLECTION AND NATURAL FREQUENCY OF STEPPED CANTILEVER BEAM USING MODIFIED RAYLEIGH METHOD LUAY S. AL-ANSARI Faculty of Mechanical Engineering, University of Kufa, An Anjaf, Iraq ABSTRACT Rayleigh method is one of classical methods used for calculating the natural frequency of the beam but it is not accurate when the beam is a stepped beam. Rayleigh method was modified using a new method for calculating the equivalent moment of inertia of stepped beam. In order to verify the new method, the static deflection and natural frequency of four types of beam were calculated using classical Rayleigh method, modified Rayleigh method and Finite Element Method (FEM) using ANSYS. The four types of beams were circular beam, square beam, rectangular beam with stepping in width only and rectangular beam with stepping in height only. The comparison between the results of static deflection and natural frequency for these four types of beams and for these three methods were made. A good agreement was found between the results of static deflection calculated by ANSYS and modified Rayleigh methods for each type of beam except the square beam specially when the length of larger step is more than half of the length of beam. Also, a good agreement was found between the results of natural frequency calculated by ANSYS and modified Rayleigh methods for each type of beam. That means the new method, used for calculating the equivalent moment of inertia, is a good method for considering the effect of change in moment of inertia when the natural frequency is calculated. KEYWORDS: Natural Frequency, Static Deflection, Stepping Cantilever Beam, Rayleigh Method, Modified Rayleigh Method, Finite Elements Method, ANSYS, Stiffness of Beam, Equivalent Moment of Inertia, Point Equivalent Moment of Inertia, Circular Beam, Square Beam, Rectangular Beam INTRODUCTION Variable cross-sectionbeams with and/or material properties are frequently used in aeronautical engineering (e.g., rotor shafts and functionally graded beams), mechanical engineering (e.g., robot arms and crane booms), and civil engineering (e.g., beams, columns, and steel composite floor slabs in the single direction loading case).over the years, a lot of researches have been done with regard to the vibration of beam structures in many different configurations and complexities. Free vibrations of a uniform and non-uniform beam according to the Timoshenko theory are the subject of research of many authors, for example the papers [1-6] are devoted to these vibration problems. The exact and numerical solutions for fundamental natural frequencies of stepped beams for various boundary conditions were presented Jang and Bert [7, 8]. Wang [9] analyzed the vibration of stepped beams on elastic foundations. Lee and Bergman [10] studied the vibration of stepped beams and rectangular plates based on an elemental dynamic flexibility method. They divided the structure with discontinues into elemental substructures and obtained the displacement field for each in terms of its dynamic Green s function. Based on the Rayleigh Ritz method, Lee and Ng [11] computed the fundamental frequencies and critical buckling loads of simply supported stepped beams by using two algorithms. Rosa et al. [12] performed the free vibration analysis of stepped beams with intermediate elastic supports. Naguleswaran [13] analyzed the vibration and stability of an Euler Bernoulli stepped beam with an axial force.an exact analytical solution for a cantilever beam of nonuniform cross-section and carrying a mass at the free end has been obtained by Rossi et al. [14].
108 Luay S. Al-Ansari Banerjee et al. [15]usedthe dynamic stiffness method to investigate the free bending vibration of rotating beams with linearly changed cross-section. Ece et al. [16] solved the problem of the vibration of simply supported and clampedbeams with varying cross-section width for three different types of boundary conditions associated, and free ends. Exact displacement interpolation functions tolinearly changed cross-section beams were solved and then used to derive the accurate stiffness matrix [17, 18]. They usedthe exact displacement interpolation functions to solve varied cross-section beam problems is a straightforward way; however, they focused on the beam with linearly and continuously changed crosssection. Solutions to the free vibration problem of stepped beams were presented by using the properties of Green s function[19, 20]. Jaworski and Dowell [21] conducted an experiment of free vibration analysis of a stepped cantilevered beam and compared the experiment results with ANSYS software and the classical Rayleigh-Ritz method, component modal analysis, the local boundary conditions and non-uniform beam effects were discussed. Lu et al.[22] used the composite element method to analyze free and forced vibrations of stepped beams and comparedthe theoretical results with experimental results. Mao and Pietrzko [23] used the Adomian decomposition method to investigate the free vibrations of a two-stepped beam, considering different boundary conditions, step locations, and step ratios. Zheng and Ji [24] presented an equivalent representation of a stepped uniform beam to simplify the calculation of static deformations and frequencies. In this work, Rayleigh method are modified by calculating the equivalent moment of inertia at each point along the beam. The equivalent moment of inertia at each point is calculated by a new method that we called " the Point Equivalent Moment of Inertia". For verifying the new method, the static deflection of stepping cantilever beam with four different cross sections are calculated using Rayleigh method, Finite Element Method (FEM) using (ANSYS) software version 14.0 and Modified Rayleigh method. The comparison between these results are done in order to understand the effect of changing in moment of inertia. The natural frequency of stepping cantilever beam with four different cross sections are also calculated using Rayleigh method, FEM using ANSYS software and Modified Rayleigh method. The comparison between these results are done in order to understand the effect of changing in moment of inertia on the natural frequency. RAYLEIGH METHODS Rayleigh method is a simpler method for finding the natural frequencies for uniform beam. It includes calculating the potentialenergy and kineticenergy. Thepotential energy by integrating the stiffness through the length of the beamand the kinetic energy can be calculated by integrating the mass through length of the beam. So one can get [21, 25, 26, 27]: 2 l 2 d y( x) EI dx dx n1 2 0 i1 l n1 2 0 2 A( y( x)) dx g i1 m y i i m y i 2 i (1) Where: ( ) is frequency, (E) is Modulus of Elasticity, (I) is Moment of Inertia, (ρ)is Density, (A) is Cross Section Area, (m) mass, and (y) is Deflection. For stepped cantilever beam, the beam must be divided into several parts. In this work, the beam is divided into (100) parts in other word, there is (101) mass. The magnitude of the first mass is ((ρ*a i *ΔX)/2) when (i) refers to the node or mass number and (i=1). While the magnitude of the last mass is ((ρ*a i *ΔX)/2)when(i=100). When the node is the node between the two steps the magnitude mass is (((ρ*a i *ΔX)/2)+((ρ*A i+1 *ΔX)/2)) (see Figure (1)).
Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified Rayleigh Method 109 Figure 1: The Dividing Scheme of the Stepping Cantilever Beam By calculating the deflection of the beam(y(x)) using the following steps [21, 25, 26, 27]: Dividing the length of the beam into (n) parts (i.e. (n+1) nodes). Calculate the primary deflection (i.e. delta (δ)) at each point and then the delta matrix [δ] ((n+1)* (n+1)) using Table (1). Calculate the mass matrix [m] ((n+1)) using the distribution technique using in [21, 25, 26, 27]. Calculate the deflection at each node by multiplying delta matrix and mass matrix ([y] (n+1) = [δ ]((n+1)* (n+1)) [m] ((n+1)) after applying the boundary conditions. Table 1: Formulae of the Deflections of the Cantilever Beams[21, 25, 26, 27],, THE POINT EQUIVALENT MOMENT OF INERTIA 27]: In the classical Method, the equivalent moment of inertia can be calculated by the following equation [21, 25, 26, (2) and for Nth stepped beam equation (2) can be written as: (3) In the point equivalent moment of inertia, the parameters, affect the value of equivalent moment of inertia, are the length of steps and the dimensions of cross section area of the steps. The length of small part can be expressed as a function of distance along thebeam. Now if the beam is only large part then the equivalent moment of inertia will be (I 2 ). But if the beam consists of two steps, the equivalent moments of inertia at the points that lie on the large step are (I 2 ).
110 Luay S. Al-Ansari While the equivalent moment of inertia at the points that lie on the small step are changed and depend on the position of step change. The equivalent moment of inertia at any point lies on the small step can be written as [27]: (4) NUMERICAL APPROCH (FINITE ELEMENT METHOD) In this method, the finite elements method was applied by using the ANSYS Software version 14.0. The three dimensional model were built and the element (Solid Tet 10 node 187) were used [21, 28]. In this work, Four different types of cross section area of beam are used (see Figure (2)): The first one is the circular cross section area. The diameter of larger area is (0.01) m and the diameter of smaller area is (0.005) m and the length of the beam is (1 m) (see Figure (2-a)). The second is the square cross section area. The width of larger area is (0.01) m and the width of smaller area is (0.005) m(see Figure (2-b)). The third is rectangular cross section area. The dimensions of larger area are(the width =0.02 m) and (the height =0.02 m) and the dimensions of smaller area are (the width =0.01 m) and (the height =0.02 m)(see Figure (2-c)). The fourth is rectangular cross section area. The dimensions of larger area are(the width =0.02 m) and (the height =0.02 m) and the dimensions of smaller area are (the width =0.02 m) and (the height =0.01 m)(see Figure (2-d)). (a) Circular Stepped Beam (b) Square Stepped Beam (c) Rectangular Stepped Beam (3rd Type) (d) Rectangular Stepped Beam (4th Type) Figure 2: Types of the Beams Using in this Work RESULTS AND DISCUSSIONS As mentioned previously, four types of beam were used in this work. In each one of these beams, the length of the larger step changes from (0.1) m to (0.9) m with step (0.1) m. At the beginning, the equivalent moment of inertia for each type of beam and for each step of the length of the larger step must be calculated using the new method. Then, the static
Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified Rayleigh Method 111 deflection, at the position of (100 N) applied force must be calculated using the classical Rayleigh Method, the modified Rayleigh Method and the Finite Element Method (ANSYS). Finally, the natural frequency for each type of beams and for each step of the length of the larger step must be calculated using the classical Rayleigh Method, the modified Rayleigh Method and the Finite Element Method (ANSYS).Therefore, the results can be divided into: Equivalent Moment of Inertia Figures (3-6) show the variation of the equivalent moment of inertia, calculated by new method, along the length of the beam for the four types of beam. From Figure (3), the value of the equivalent moment of inertia change depends on the length of the larger step in addition to the distance (x). If the length of the larger step increases the final value of the equivalent moment of inertia is closer to the value of moment of inertia of the larger step. Also the slope of the curve decreases when the length of the larger step increases. The same behavior can be seen in Figure (4) but with the different value of the maximum moment of inertia (i.e. moment of inertia of the larger area). In Figure (5) and Figure (6), the slope of the curves change with the rate differ than that in Figure (3) and Figure (4). Also, the rate of change in slope in Figure (5) differ than that in Figure (6) because the stepping happens in width only in Figure (5) and in height only in Figure (6). Figures (7-10) show the comparison between the equivalent moment of inertia of the four types of beams, calculated by new method, along the length of the beam when the length of the larger step is (0.2, 0.4, 0.6 and 0.8) respectively. The rectangular beam with stepping in width only has a smaller rate of change in the value of equivalent moment of inertia than that of the rectangular beam with stepping in height only, square beam and circular beam. In order to understand that the equations used for calculating the moment of inertia must be remember. Also, the value of the equivalent moment of inertia of square, rectangular beam with stepping in width only and rectangular beam with stepping in height only will be close together when the length of the larger step increases. Static Deflection Figures (11-14) show the comparison between the value of static deflection, calculated by ANSYS, classical Rayleigh method and modified Rayleigh method, of circular beam along the position of the applied force when the length of the larger step is (0.2, 0.4, 0.6 and 0.8) respectively. The values of static deflection calculated by modified Rayleigh method are much closer to ANSYS results than that calculated by classical Rayleigh method. The agreement between the results of static deflection calculated by modified Rayleigh method and ANSYS increases when the length of the larger step decreases. But, The agreement between the results of static deflection calculated by classical Rayleigh method and ANSYS increases when the length of the larger step increases. Figures (15-18) show the comparison between the value of static deflection, calculated by ANSYS, classical Rayleigh method and modified Rayleigh method, of square beam along the position of the applied force when the length of the larger step is (0.2, 0.4, 0.6 and 0.8) respectively. The agreement between the results of static deflection calculated by modified Rayleigh method and ANSYS decreases when the length of the larger step increases and the same behavior happens for the agreement betweenclassical Rayleigh results and ANSYS results. The same behavior, that appears in square beam, can be seen in static deflection of rectangular beam (3rd type and 4th type) (see Figures (19-26)). This lead us to say that "the modified Rayleigh method is better than the classical Rayleigh method for calculating the static deflection of stepped cantilever beam". Natural Frequency Figures (27-30) show the comparison between the value of natural frequency calculated by ANSYS, classical
112 Luay S. Al-Ansari Rayleigh method and modified Rayleigh method, of circular, square and rectangular beam (3rd type and 4th type) beam along the variation of the length of the large step. For the natural frequency, the results of modified Rayleigh method are much closer to ANSYS results than the classical Rayleigh results.this lead us to say that "the modified Rayleigh method is better than the classical Rayleigh method for calculating the natural frequency of stepped cantilever beam". CONCLUSIONS AND RECOMMENDATION From the previous results, the following point can be concluded: The point equivalent moment of inertia is a good method for calculating the moment of inertia of stepped cantilever beam. This method has a good sensitivity for the change in the length of the larger step. The slop of the equivalent moment of inertia depends on the length of the larger step, shape of stepping (i.e. two dimensions stepping or one dimension stepping) and the shape of cross section area (i.e. the equation of moment of inertia). The modified Rayleigh method is better than the classical Rayleigh method for calculating the static deflection of stepped cantilever beam. The modified Rayleigh method is better than the classical Rayleigh method for calculating the natural frequency of stepped cantilever beam. Finally, the modified Rayleigh method can be used for calculating the static deflection and natural frequency for stepped beam (with number of step larger than two) and non-prismatic beam. Figure 3: The Equivalent Moment of Inertia along the Length of the Circular Beam for Different Length of Larger Step Figure 4: The Equivalent Moment of Inertia along the Length of the Square Beam for Different Length of Larger Step Figure 5: The Equivalent Moment of Inertia along the Length of the Rectangular Beam(3 rd Type) for Different Length of Larger Step Figure 6: The Equivalent Moment of Inertia along the Length of the Rectangular Beam(4 th Type) for Different Length of Larger Step
Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified Rayleigh Method 113 Figure 7: The Comparison between the Equivalent Moment of Inertia along the Length of the Four Types of Beam When the Length of Larger Step is (0.2) m Figure 8: The Comparison between the Equivalent Moment of Inertia along the Length of the Four Types of Beam When the Length of Larger Step is (0.4) m Figure 9: The Comparison between the Equivalent Moment of Inertia along the Length of the Four Types of Beam When the Length of Larger Step is (0.6) m Figure 10: The Comparison between the Equivalent Moment of Inertia along the Length of the Four Types of Beam When the Length of Larger Step is (0.8) m Figure 11: The Static Deflection of the Circular Beam Calculated by Different (0.2) m Figure 12: The Static Deflection of the Circular Beam Calculated by Different (0.4) m
114 Luay S. Al-Ansari Figure 13: The Static Deflection of the Circular Beam Calculated by Different (0.6) m Figure 14: The Static Deflection of the Circular Beam Calculated by Different (0.8) m Figure 15: The Static Deflection of the Square Beam Calculated by Different (0.2) m Figure 16: The Static Deflection of the Square Beam Calculated by Different (0.4) m Figure 17: The Static Deflection of the Square Beam Calculated by Different (0.6) m Figure 18: The Static Deflection of the Square Beam Calculated by Different (0.8) m
Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified Rayleigh Method 115 Figure 19: The Static Deflection of the Rectangular Beam (3 rd Type) Calculated by Larger Step is (0.2) m Figure 20: The Static Deflection of the Rectangular Beam (3 rd Type) Calculated by Larger Step is (0.4) m Figure 21: The Static Deflection of the Rectangular Beam (3 rd Type) Calculated by Larger Step is (0.6) m Figure 22: The Static Deflection of the Rectangular Beam (3 rd Type) Calculated by Larger Step is (0.8) m Figure 23: The Static Deflection of the Rectangular Beam (4 th Type) Calculated by Larger Step is (0.2) m Figure 24: The Static Deflection of the Rectangular Beam (4 th Type) Calculated by Larger Step is (0.4) m
116 Luay S. Al-Ansari Figure 25: The Static Deflection of the Rectangular Beam (4 th Type) Calculated by Larger Step is (0.6) m Figure 26: The Static Deflection of the Rectangular Beam (4 th Type) Calculated by Larger Step is (0.8) m Figure 27: The Variation of Natural Frequency of the Circular Beam Calculated by Different Methods with the Length of Larger Step Figure 28: The Variation of Natural Frequency of the Square Beam Calculated by Different Methods with the Length of Larger Step Figure 29: The Variation of Natural Frequency of the Rectangular Beam (3 rd Type)Calculated by Different Methods with the Length of Larger Step Figure 30: The Variation of Natural Frequency of the Rectangular Beam (4 th Type) Calculated by Different Methods with the Length of Larger Step
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