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  1. 1. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131 Dynamic Reanalysis of Beams Using Polynomial Regression Method * B. Rama Sanjeeva Sresta, **Dr. Y. V. Mohan Reddy * (Department of Mechanical Engineering, G Pulla Reddy Engineering College, Kurnool, A.P., India,) ** (Department of Mechanical Engineering, G Pulla Reddy Engineering College, Kurnool, A.P., India,)ABSTRACT: The paper focuses on dynamic reanalysis the complete set of modified simultaneous equations.of simple beam structures using a polynomial The solution procedures usually use the originalregression method. The method deals with the response of the structure.stiffness and mass matrices of structures and canbe used with a general finite element system. This SOLUTION APPROACH- FINITEmethod is applied to approximate dynamic ELEMENT METHODreanalysis of cantilever simple beam structure and Initially the beam is divided into smallerT-structure. Preliminary results for these example sections using successive levels of division. Analysisproblems indicate the high quality approximation of each section is performed separately. Using theof natural frequencies can be obtained. The final finite element technique, the dynamic analysis ofresults from regression method and Finite element beam structure is modeled.method are compared. [K-λM] [X]=0 ----------------- (1) Where k, m are the stiffness and mass matrixKeywords: mass matrix, stiffness matrix, respectively.natural frequency, dynamic reanalysis, The dynamic behavior of a damped structurepolynomial regression. [4] which is assumed to linear and discretized for n degrees of freedom can be described by the equationINTRODUCTION: of motion. Reanalysis methods are intended to analyze M +C +Kx=f-------------(2)efficiently new designs using information from Where M, C = αM+βK, and K are mass,previous ones. One of the many advantages of the damping and stiffness matrices, , and X aresubstructure technique is the possibility of repeating acceleration, velocity, displacement vectors of thethe analysis for one or more of the substructures structural points and “f” is force vector. Undampedmaking use of the work done on the others. This homogeneous equation M +Kx=0. Provides therepresents a significant saving of time when Eigen value problem (k-λm) = 0.modifications once are required. Modification is Solution of above equation yields the matrices Eigeninvariably required in iterative processes for optimumdesign never the less, in the case of large structures values λ and Eigen vectorsthe expenses are still too high.[1] Therefore, development of techniques which λ = , = [ 1, 2….. n]are themselves based on previous analysis, and whichobtained the condensed matrices of the substructures The eigenvector satisfy the orthonormalunder modification, with little extra calculation time, conditions M =I, K =λ, C = αI+βλ=ξ,can be very useful. “General Reanalysis Techniques” Using the transformation X = q in the equation ofare very useful in solving medium size problems and motion, and premultiplying by one obtains,are totally essential in the design of large structures. M + C + K q= f -------------(3)Some steps in a dynamic condensation process areparticularly characterized by their computational It is important note, that the matrices,effort, as for instance: = M , = C , K are not Stiffness matrix factorization usually diagonalised by the eigenvectors of the Resolution of certain systems of linear equation original structure [3] Given an initial geometry and Resolution of an eigen problem to obtain the assuming a change ΔY in the design variables, the normal vibration modes. modified design is given by Reanalysis methods [2] are intended to Y = +ΔY. ------------------ (4)analyze efficiently structures that are modified due to The geometric variables Y usually representchanges in the design. The object is to evaluate the coordinates of joints, but other choice for thesestructural response for such changes without solving 2126 | P a g e
  2. 2. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131variables is sometimes preferred. The displacement The parameter, α, is a constant, often calledanalysis equations for the initial design are the “intercept” while b is referred to as a regression r = R. coefficient that corresponds to the “slope” of the line.where = stiffness matrix corresponding to the The additional parameter ε accounts for the type of error that is due to random variation caused bydesign , R= load vector whose elements are usually experimental imprecision. The regression procedureassumed to be independent of the design variables assumes that the scatter of the data points about theand r= nodal displacements computed at . The best-fit straight line reflects the effects of the errorstiffness matrix and mass matrix of a typical plane term,[12-15] and it is also implicitly assumed that εtruss element are follows a Gaussian distribution with a mean of 0. Now, however, we will assume that the error isK= and Gaussian Figure 2 illustrates the output of the linear model with the inclusion of the error term. 3.1.3 Multiple Linear RegressionsM= The straight line equation is the simplest form of the linear regression given as Y=α+βX+ε Where α+βX represents the deterministic part and ε is„A‟ is the cross sectional area, ‟l‟ is member length, the stochastic component of the model.of the beam „ „ is density of the beam. The simple linear population model equationREANALYSIS OF REGRESSION indicating the deterministic component of the modelMETHOD that is precisely determined by the parameters α and The statistical determination of the β, and the stochastic component of the model, ε thatrelationship between two or more dependent represents the contribution of random error to eachvariables has been referred to as a correlation determined value of Y. It only includes oneanalysis, [6]whereas the determination of the independent variable. When the relationship ofrelationship between dependent and independent interest can be described in terms of more than onevariables has come to be known as a regression independent variable, the regression is then definedanalysis. as “multiple linear regression.” The general form of the linear regression model may thus be written as:1.1 Regression Y = +…….+ + ε. --------------- The actual term “regression” is derived from (6)the Latin word “regredi,” and means “to go back to” Where, Y is the dependent variable, and X1,or “to retreat.” Thus, the term has come to be X2 … Xi are the (multiple) independent variables.associated with those instances where one “retreats” Multiple linear regression models also encompassor “resorts” to approximating a response variable polynomial functions:with an estimated variable based on a functional Y= +……. + +, ------------- (7)relationship between the estimated variable and one The equation for a straight line is a first-orderor more input variables. In regression analysis, the polynomial. The quadratic equation,input (independent) variables can also be referred to Y= ------------------ (8)as “regressor” or “predictor” variables. is a second-order polynomial whereas the cubic equation,3.1.1 Linear Regression Linear regression involves specification of a Y= -------------- (9)linear relationship between the dependent variable(s) is a third-order polynomial.and certain properties of the system under Taking first derivatives with respect to each of theinvestigation. Linear regression deals with some parameters yields:curves as well as straight lines. = 1, = X, = ---------------- (10) The model is linear because the first derivatives do not include the parameters. As a3.1.2 Ordinary Linear Regression consequence, taking the second (or higher) order The simplest general model for a straight derivative of a linear function with respect to itsline includes a parameter that allows for inexact fits: parameters will always yield a value of zero. Thus, ifan “error parameter” which we will denote as . the independent variables and all but one parameter Thus we have the formula: are held constant, the relationship between the dependent variable and the remaining parameter willY = α +β X +  ----------------- (5) always be linear. It is important to note that linear 2127 | P a g e
  3. 3. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131regression does not actually test whether the data user. Figure (2) shows an example of creating a finitesampled from the population follow a linear element for a cantilever beam.relationship. It assumes linearity and attempts to findthe best-fit straight line relationship based on the datasample. The dashed line shown in the figure (1) is thedeterministic component, whereas the pointsrepresent the effect of random error. Figure 2: Descretized Element The polynomial equation for regression method, + + . These 3 values for both case studiesFigure 1: A linear model that incorporates a Young‟s modulus (E) 0.207× N/stochastic (random error) component. Density (ρ) 7806 Kg/ Cross section of area (A) 0.029×0.0293.1.4 Assumptions of Standard RegressionAnalyses 1.2 Case Study 1 The subjects are randomly selected from a larger The Cantilever Beam of 1m length, shown population. The same caveats apply here as with in figure () is divided into 4 elements equally element correlation analyses. The observations are Stiffness Matrix and Mass Matrix are extracted. independent. The variability of values around the Natural frequencies of the cantilever beam at each line is Gaussian. node are found from MATLAB program by X and Y are not interchangeable. Regression considering two situations- models used in the vast majority of cases attempt a) width alone is increased by 5% and to predict the dependent variable, Y, from the b) width and depth of the beam are increased by 5% independent variable, X and assume that the each. error in X is negligible. In special cases where Reanalysis of the beam is done by this is not the case, extensions of the standard Polynomial regression and the percentage errors are regression techniques have been developed to listed in the table. account for non negligible error in X. The relationship between X and Y is of the correct form, i.e., the expectation function (linear or nonlinear model) is appropriate to the data being fitted. There are enough data points to provide a good sampling of the random error associated with the Experimental observations. In general, the minimum number of independent points can be no less than the number of parameters being Figure 3: Cantilever beam with nodes and estimated, and should ideally be significantly elements higher. First natural frequencies of cantilever beamNUMERICAL EXAMPLES by increasing depth of the beam by 5%. The The polynomial regression method is polynomial regression equation is given byapplied to a simple beam structures. In finite element + +method, Discretization means dividing the body into Fitting target of lowest sum of squared absolute erroran equivalent system of finite elements with = 8.7272727289506574 ,associated nodes. The element must be made small -3.6333 , = -enough to view and give usable results and to be 1.053659large enough to reduce computational efforts. Smallelements are generally desirable where the results are = 7.76275 , = -changing rapidly such as where the changes in 3.05561 ,geometry occur. Large elements can be used where = 8.07983 C6 = 2.2512E+01the results are relatively constant. The discretizedbody or mesh is often created with mesh generation First natural frequencies of cantilever beamprogram or preprocessor programs available to the by increasing width and depth of the beam by 5%. 2128 | P a g e
  4. 4. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131Fitting target of lowest sum of squared absolute error= 1.8470358974358964 , -6.30489 , = 4.2 Case Study 24.028518 , The dimensions of a T-structure are given in figure (4). The Cantilever Beam is divided into 6 = 4.028518 , = - elements. Then Element Stiffness Matrix, mass1.0347691 , Matrix and natural frequencies are determined using = -1.0347691 , = - MATLAB. Polynomial regression method is applied1.0347691 , to this structure. The results from polynomial regression and FEM are compared for closeness.Table 1: Increasing the Depth of the Beam Natural NaturalWidt Frequenc Frequency %Erro Heighth y Regression(H r Fem(Hz) z)0.029 0.029 22.53 22.52 -0.044 0.03040.029 23.65 23.64 -0.042 50.029 0.0319 24.78 24.69 -0.363 0.03330.029 25.91 25.88 -0.115 50.029 0.0348 27.03 27.01 -0.074 Figure 4: T-Structure with nodes and elements 0.03620.029 28.16 28.13 -0.106 5 The results are as follows: First natural0.029 0.0377 29.29 29.27 -0.020 frequencies of cantilever beam by increasing depth of 0.0391 the beam by 5%.0.029 30.41 30.40 -0.033 5 Fitting target of lowest sum of squared absolute error0.029 0.0406 31.54 31.53 -0.032 equal to 1.0240640782836770 , 0.0420 -8.043993 , = -0.029 32.67 32.66 -0.03 5 1.35761 ,0.029 0.0435 33.79 33.76 -0.089 = 1.53493 , = - 6.65185Table 2: Increasing Width and Depth of the BeamWidth Height Natural Natural %Erro = -3.99731 , = Frequen Frequency r 4.446583 . cy Regression( Fem(Hz Hz) Natural frequencies of cantilever beam by ) increasing width and depth of the beam by 5%,0.029 0.029 22.47 22.47 0 Fitting target of lowest sum of squared absolute error0.0304 0.0304 23.60 23.61 0.0423 equal to 7.2417839254080818 ,5 5 70.0319 0.0319 24.72 24.75 0.1213 -3.483496 = 50.0333 0.0333 26.08 26.89 3.1058 7.71712 ,5 5 20.0348 0.0348 26.97 27.03 0.2224 = 7.71712 , =- 60.0362 0.0362 28.09 28.16 0.2491 1.84785 5 90.0377 0.0377 29.22 29.59 1.2662 5 = -1.8478 =-0.0391 0.0391 30.34 30.38 0.13185 5 3 1.84780.0406 0.0406 31.56 31.90 1.0773 10.0420 0.0420 32.59 32.70 0.33755 5 20.0435 0.0435 33.80 33.83 0.0887 5 2129 | P a g e
  5. 5. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131Table 3:Increasing Depth of the Beam CONCLUSION Natural Natural From this work the following conclusions Frequen Frequenc are drawn. The FEM method is applied for dynamicWidt Height cy y %Error analysis of cantilever beam and T-structure using theh Fem Regressi MAT lab. Natural frequencies are obtained for (Hz) on (Hz) cantilever and T-structure beams using FEM.the0.02 4.4975× polynomial regression method is used for obtaining 0.029 444.69 444.71 natural frequencies of cantilever beam and T-90.02 0.0304 4.2834× structure by varying width and depth for dynamic 466.91 466.93 reanalysis.9 5 2.0443× The results obtained from reanalysis using0.02 0.0319 489.16 489.17 regression method are close to results obtained using9 FEM. The minimum and maximum errors in0.02 0.0333 1.9550× regression method when compared with the results 511.49 511.509 5 obtained by FEM are0.02 3.7479× 0.0348 533.63 533.659 Minimum Maximum0.02 0.0362 1.7990× -0.36319 (with -0.02000 555.85 555.86 Cantilever increasing width) 3.105829 50.02 3.4600× beam 0 (with increasing 0.0377 578.02 578.04 width and depth)9 1.6657× 0(with increasing0.02 0.0391 4.4975× 600.33 600.34 width)9 5 T-structure 0.302235 0(with increasing0.02 width and depth) 0.0406 622.57 622.57 090.02 0.0420 3.1019× 644.76 644.78 REFERENCES:9 5 [1]. M.M.Segura and J.T.celigilete, “ a new0.02 2.0084× dynamic reanalysis technique based on 0.0435 667.02 667.049 model synthesis” “Centro de studios e investigaciones tecnicas”, deTable 4: Increasing Width and Depth of the guipuzcoa(CEIT) apartado 1555, 20009 sanBeam(T-Structure) Sebastian, spain (1994).Width Heigh Natural Natural %Error [2]. Z. Xie, W.S. Shepard Jr,” Development of a t Frequen Frequency single-layer finite element and a simplified cy Regression( finite element modelling approach for Fem(Hz Hz) constrained layer damped structures”, Finite ) Elements in Analysis and Design 45 (2009)0.029 0.029 444.36 444.36 0 pp530–537.0.030 0.030 465.79 466.43 0.1374 [3]. Uri kirsch , Michaelbogomolni , “analytical45 45 00 modification of structural natural0.031 0.031 487.97 488.81 0.1721 frequencies”Department of civil and9 9 41 environmental engineering ,Technion- Israel0.033 0.033 514.76 515.18 0.1787 institute of technology, Haifa 32000, Israel35 35 2 [2006].0.034 0.034 532.30 533.55 0.2347 [4]. Uri kirsch, G.Tolendano,”Finite Elements in8 8 41 Analysis and Design “Department of Civil0.036 0.036 554.51 555.93 0.2560 and Environmental engineering , Technion25 25 81 Israel institute of technology, Haifa 32000,0.037 0.037 576.72 578.30 0.2739 Israel.7 7 63 [5]. M. Nad‟a, “structural dynamic modification0.039 0.039 598.87 600.68 0.3022 of vibrating system”* a Faculty of Materials15 15 35 Science and Technology, STU in Bratislava,0.040 0.040 629.67 630.65 0.1556 Paulínska 16, 917 24 Trnava, Slovak6 6 35 Republic.0.042 0.042 643.26 645.43 0.1818 [6]. Tirupathi R.Chandrupatla , “Ashok05 05 86 D.Belegundu ,Rowan University Glassboro”, new jersey and the0.043 0.043 667.06 667.79 0.1094 Pennsylvania state university, university5 5 35 park, Pennsylvania. 2130 | P a g e
  6. 6. B. Rama Sanjeeva Sresta, Dr. Y. V. Mohan Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue4, July-August 2012, pp.2126-2131[7]. Arthur Christopoulos and Michael J. Lew [11]. Hill, A.V., “The combinations of Critical Reviews in Biochemistry and haemoglobin with oxygen and with carbon Molecular Biology, 35(5):359–391 (2000). monoxide.” I, Biochem. J., 7, 471– 80,1913.[8]. Hill, A.V., The combinations of [12]. Motulsky, H.J., “Intuitive Biostatistics”, haemoglobinwith oxygen and with carbon Oxford University Press, New York, 1995. monoxide. I, Biochem. J., 7, 471 80, 1913. [13]. Motulsky, H.J., “Analyzing Data with[9]. Wells, J.W., “Analysis and interpretation of GraphPad Prism”, GraphPad Software Inc., binding at equilibrium, in Receptor-Ligand San Diego,CA, 1999. Interactions”: A Practical Approach, E.C. [14]. Ludbrook, J., “Comparing methods of Hulme, Ed. Oxford University Press, measurements”, Clin. Exp. Pharmacol. Oxford, 1992, 289–395. Physiol.,24(2), 193–203, 1997.[10]. Hill, A.V., “The possible effects of the [15]. Bates, D.M. and Watts, D.G., “Nonlinear aggregation of the molecules of Regression Analysis and Its Applications”, haemoglobin on its dissociation curves”, J Wiley and Sons, New York, 1988. . Physiol. , 40, iv–vii,1910. 2131 | P a g e