Vibration Analysis of Radial Drilling Machine Structure Using Finite Element Method

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1 Advanced Materials Research Online: ISSN: , Vols , pp doi: / Trans Tech Publications, Switzerland Vibration Analysis of Radial Drilling Machine Structure Using Finite Element Method Rajiv Kumar 1, a, M.P.Garg 2, b, Dr. R.C.Sharma 3, c 1 Department of Mechanical Engineering,MVEC,Yamuna nagar,haryana,india 2 Department of Mechanical Engineering,MMU,Mullana,Haryana,India 3 Department of Mechanical Engineering,MMU,Mullana,Haryana,India a rajivpansra@gmail.com, b mpgarg@yahoo.com, c er_rcs2000@yahoo.co.in Keywords: Vibration analysis; Drilling machine; Finite element method. -shaped structure; Two-noded frame element; Abstract Manufacturing industries now a days have stringent expectation from the machine tools in terms of productivity as well as quality of products.vibration plays an important role in determining the quality of product.if the pattern of vibration prevailing in the machine tool during cutting is known,then machine tool structure can be designed in such a way so that natural frequency of machine tool structure can be isolated from the forced frequency.so, this study is focussed on finding the natural frequency and mode shapes of radial drilling machine structure.finite element analysis has been done to find out the natural frequencies and mode shapes of radial drilling machine structure.assembled mass and stiffness matrices are obtained for each element and solved by using inverse iteration technique. Introduction The increasing demand for accuracy and output has caused vibration problems to gain importance. Both Forced and Sustained Vibrations are known to be prejudicial to accuracy and output. All machine tools give rise to vibrations. Deterioration in machine condition always produces a corresponding increase in vibration levels. Vibration signals have proved to be one of the most reliable parameter used in machine health monitoring to check machine condition. The purpose of vibration analysis of machine tool structure is to sustain the useful oscillations and destroy the unwanted ones.the motive behind this study is that if natural frequency of a machine tool structure is known before hand,then forced frequencies obtained during cutting can be isolated from the natural frequency to avoid resonance. A no. of researchers has adressed sililar types of problems.bellman et al. [1] purposed Differential quadrature (DQ) method for vibration analysis of plates. This method was later used by Leiw et al. [2] for vibration analysis of rectangular plates.wei [3] used discrete singular convolution (DSC) method for analysis of mechanical behavior of plates. R.S. Bais et al. [4] developed updated FE models of a drilling machine using analytical and experimental results. Analytical FE modeling had been done using I-DEAS software. Nanda et al.[5] found the natural frequencies and mode shapes for vibration analysis of upright drilling machine. A method for analyzing lateral vibrations in a milling machine spindle was presented including finite-element modeling (FEM), by M. Rantatalo et al. [6].Dynamic characteristics of XK717 CNC milling machine were analyzed using the finite element method by Weiwei et al. [7]. H. Nguyen-Xuan et al. [8] studied an edge-based smoothed finite element method (ES-FEM) with the discrete shear gap (DSG) method for static, free vibration and buckling analyses of Reissner Mindlin plates using 3-node triangular elements.e.s. Ventsel et al. [9] described the method of determining the normal modes of vibrations and natural frequencies of elastic shells of revolution with an arbitrary meridian, partially filled with a fluid. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-12/05/16,14:03:06)

2 2718 Advanced Manufacturing Technology, ICMSE 2012 Problem Formulation of Radial Drlling Machine Structure The basic matrix equation for dynamics of a system is given by [M]{ẍ}+[C]{ẋ}+[K]{x}=[q] (1) Assuming un-damped free vibration, harmonic motion,equation (1) becomes to where [K]=stiffness matrix,[m]=mass matrix [K]-λ[M] {X} = 0 (2) Equation (2), eigenvalue problem can be solved using inverse iteration scheme for finding the natural frequencies and corresponding mode shapes. When the length of the element is parallel to x-axis stiffness matrix for a frame element is given by [k] e = A 0 l [B] T [D] [B] dx (3) Where [K] e = Elemental stiffness matrix, [B] = [L] [N], [L] = [-d 2 /dx 2 ], D= E*I Where [L]= Shape function of frame element, I= mass moment of inertia and E= Youngs modulus of elasticity. When the length of the element is parallel to x-axis mass matrix for a frame element is given by [m] e = A l 0 ρ [N] T [N] dx (4) When the length of the element is parallel to y-axis, then rows and columns of elemental mass and stiffness matrices, corresponding to x and y directions are interchanged. The simplified structure of radial drilling machine ( ), which is considered for FE analysis is shown in figure 1 with dimensions. All dimensions of the structure are in mm. Figure1 Structure for FE analysis Figure 2 The - Structure with node numbers Eigen values and eigen vectors are found by using a program in C language.. ANALYSIS PROCEDURE The various steps in the finite element vibration analysis of a stature as follows: 1. Structure is divided into seventeen two-noded frame element and eighteen nodes. Each node has two displacements and a rotational degree of freedom. Therefore each element has six degree of freedom and total structure is having 54 degrees of freedom. The discretized structure is shown in figure Elemental connectivity table is prepared for the structure, which shows that which element is connected to which pair of nodes. 3. Elemental mass matrices and stiffness matrices are computed using equations 3 and Using conventional rules of assembly in FEM,system mass and stiffness matrices are obtained. 5. System mass and stiffness matrices are modified according to boundary condition, which for the present work is that, the displacements at the fixed node are zero. 6. The eigen value thus obtained can be solved using inverse iteration technique.

3 Advanced Materials Research Vols Vibration Analysis of Radial Drilling Machine Structure Following assumptions are made in the vibration analysis of radial drilling machine structure: The drilling machine structure is considered two dimensional. The joints are assumed to be rigid and the structure is considered to be one piece. Damping is neglected. Switches,broader base of the machine, motor driving the drill are not considered, since these in no way contribute towards the rigidity of the drill machine structure. Displacements at the grounded mode are assumed to be zero. Figure 3 shown a radial drilling machine. It is modeled as a Structure as shown in figure 2. Framed structure(figure2) is discretized into 17 elements and 18 nodes. Element numbers 8,9,10,11,12,13 of the framed structure(figure 2) represent the column of drilling machine structure. The column of the drilling machine is tapered more width near to the pillar and less width at end. So, total length of arm is divided into six frame elements having same length and different width as shown in figure 4 Figure 3 Radial Drilling Machine and figure 5. The FE model of the radial drilling machine structure is obtained as discussed below: Figure 4 Column of drilling machine structure Figure 5 Frame elements representing the column of drilling machine These six elements will approximate the shape of column of drilling machine structure. Element numbers 14, 15, 16 of the framed structure (fig. 2) represent the drilling head structure of radial drilling machine. Frame type elements having rectangular cross section are taken. Element numbers 1,2,3,4,5,6,7 of the framed structure represent pillar of drilling machine structure. Pillar of drilling machine is having circular cross-section. Base for drilling machine is represented by element no. 0 of the framed structure. The element is having circular cross-section. RESUTLS AND DISCUSSION From figure 6, it is seen that in the fundamental mode shape (corresponding to fundamental natural frequency Hz) node no. 1,2,3,4,5,6,7 & 8 (refer fig 2 & all correspond to pillar of radial drilling machine structure) shows displacements along positive x-axis from the corresponding positions on the un-deformed machine structure. Node no. 9, 10,11,12,13 and 14 (which corresponds to column of drilling machine) shows displacements below the un-deformed structure.node no. 15,16,12 and 17(drilling head of radial drilling machine structure) shows displacements to the right side of the corresponding positions on the un-deformed machine structure.

4 2720 Advanced Manufacturing Technology, ICMSE 2012 Table1First three natural frequencies obtained from C program Mode No. Frequency (Hz) Figure 6 First three mode shapes of the structure In the 2 nd mode shape (around 2 nd natural frequency Hz), node no. 9, 10,11,12,13 shows displacements above the un-deformed machine structure, where as node no. 14(all correspond to column of radial drilling machine) is overlapping with the un-deformed machine structure. Nodes corresponding to pillar of drilling machine (i.e. node no. 2,3,4,5,6,7,8) shows displacement along negative x-axis from the un-deformed structure and node no.1 is overlapping with the un-deformed structure.drilling head(i.e. element no.14,15,16 fig. 2) vibrates left side of the un-deformed structure. In the 3 rd mode shape (corresponding to third natural frequency Hz), column of radial drilling machine (i.e. node no. 9,10,11,12,13,14) vibrates above the un-deformed drilling structure. Nodes corresponding to pillar of drilling machine (i.e. node no. 1,2,3,4,5) shows displacement along negative x-axis from the un-deformed structure, where as node no.6,7 are overlapping with un-deformed structure and node no. 8 shows displacement in positive x-axis from the un-deformed machine structure. Element no. 14, 15 vibrates right side, where as element no.16 (all correspond to drilling head of machine structure) vibrates left side of the un-deformed machine structure.so, from fig.6, by studying the first three mode shape, the vibration pattern of drilling machine structure can be visualized. CONCLUSION Based upon the results and discussions presented in this paper, following conclusions may be drawn. a) It is observed that at fundamental frequency of the machine structure at various positions, pillar of radial drilling machine structure vibrates most of times along positive x-axis with maximum displacement, column of radial drilling machine structure vibrates below and drilling head of radial drilling machine structure vibrates right side of the corresponding positions on the un-deformed structure. b) At higher natural frequencies i.e. from 3 rd natural frequency and onwards, pillar of radial drilling machine structure vibrates very close to un-deformed positions along positive or negative x-axis, column of radial drilling machine structure vibrates below the corresponding positions on the un-deformed structure and drill head of radial drilling machine structure vibrates right side with some elements and at the same time left side with rest of the elements from the corresponding positions on the un-deformed structure. c) In the 2 nd mode shape (around 2 nd natural frequency), it is observed that pillar of radial drilling machine structure vibrates along negative x-axis, column of radial drilling machine structure vibrates below and above and drilling head of radial drilling machine structure vibrates left side of the corresponding positions on the un-deformed structure.

5 Advanced Materials Research Vols References [1] R.E.Bellmon,J.Casti: Differential quadrature and long term integration, Journal of Mathematical Analysis and Applications, Vol. 34 (1971), pp [2] M.K. Liew, T.M Teo: three dimensional vibration analysis of rectangular plates using differnetial quadrature method, Journal of Sound and vibration (1997) pp [3] G.W.Wei: Vibration analysis by singular convolution, Journal of Sound and Vibration, Sec. 32(1998)pp [4] R.S. Bais,A.K. Gupta,B.C.Nakra,T.K.Kundra: Studies in dynamic design of drilling machine using updated finite element models,mechanism and Machine Theory39(2004) [5] J.S. Saini, M.P. Garg,T. Nanda: Finding the Natural Frequencies and Mode Shapes for Vibration Analysis of Upright Drilling Machine, IE (I) Journal, MC, vol. 86( 2005)pp [6] Matti Rantataloa, Jan-Olov Aidanpa, Bo Go ranssonc, Peter Normand: Milling machine spindle analysis using FEM and non-contact spindle excitation and response measurement, International Journal of Machine Tools & Manufacture 47 (2007) Elsevier Ltd [7] Wang Weiwei:Finite Element Analysis of Dynamic Characteristic for the XK717 CNC Milling Machine, International Conference on Measuring Technology and Mechatronics Automation (2009), IEEE. [8] H. Nguyen-Xuan, G.R. Liu, C. Thai-Hoang, T. Nguyen-Thoi:An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner Mindlin plates, Comput. Methods Appl. Mech. Engrg. 199 (2010) [9] E.S. Ventsel, V.Naumenko, E.Strelnikova, E.Yeseleva: Free vibrations of shells of revolution filled with a fluid, Engineering Analysis with Boundary Elements 34 (2010) Elsevier Ltd

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Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian

Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method: