THE INFLUENCE FUNCTION IN ANALYSIS OF BENDING CURVE AND REACTIONS OF ELASTIC SUPPORTS OF BEAM WITH VARIABLE PARAMETERS

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52,, pp , Warsaw 204 THE INFLUENCE FUNCTION IN ANALYSIS OF BENDING CURVE AND REACTIONS OF ELASTIC SUPPORTS OF BEAM WITH VARIABLE PARAMETERS Jerzy Jaroszewicz, Krzysztof K. Żur, Łukasz Dragun Bialystok University of Technology, Faculty of Management, Kleosin, Poland In the paper, theoretical knowledge about base solution of common differential equations with variable parameters is presented. The solutions are applied to mechanical problems of discrete, distributed and discrete-distributed homogeneous elastic models of structures and structural components. In this paper, the influence function and its properties are presented. Theinfluencefunctionisappliedtoanalysisofthebendingcurveofabeamwithconstants and variable parameters. The presented method of the influence function is based on the mathematical similarity of differential equations describing free vibrations and deflection of beams, which are fourth-order equations with variable coefficients. In this paper, examples of calculation of support reactions as function of stiffness of the beam and elastic supports are presented. Key words: influence function, bending curve, reactions of elastic supports. Introduction In innovation technology, solution methods of static and dynamic problems have application to design of drive assemblies of machines which can be modeled as elastic systems with variable parameters. Solutions to problems such as an increase of rotor velocity and acceleration or loading of bearings is very important(lazopoulos, 200; Jaroszewicza nd Żur, 202a,b,c). In a previous paper(jaroszewiczet et al., 2008), the authors analyzed the simplest lower estimator to calculate the basic frequency of axi-symmetrical vibration of plates with variable thickness of a circular diaphragm type. The existence of the simplest estimator of the actual value of the parameter depending on the frequency rate of change characterized by a thick plate wasanalyzed.theaccuracyofthemethoddifferedfromthefem,andinordertoimprovethe accuracyoftheestimators,itwasdecidedtouseahigherorder,inthatcasedouble.usingthe bilateral estimator, the similar problem arose in calculation of the exact solution in a paper by Conway(958). The aim of this paper is to present theoretical knowledge about the base solution(influence function) to common differential equations with variable parameters. The advantage of the influence function is the possibility of omission of the boundary of conjugation in the solution to the boundary value problem. The solutions are applied to mechanical problems of discrete, distributed and discrete-distributed homogeneous elastic models of structures and structural components like beams, shafts and plates(timoshenko, 940; Solecki and Szymkiewicz, 964). 2. The solution by means influence function In many works(zoryj, 987c) an important property of the influence function whoose base was constructed by solutions to the boundary value problem of the bending curve, vibrations and stability was presented. It is known that the base solution to the equation

2 248 J. Jaroszewicz et al. L[y(x)]=δ(x ) (2.) is the influence function φ(x, ) in which the linear operator have analytical parameters L[y(x)]=p 0 (x)y(x) n +p (x)y(x) n +...+p n (x)y(x) (2.2) whicharefiniteinintervals(a,b),wherea<x<b,a<<b,xistheindependentcoordinate, parameter,δ(x) Dirac sdelta,p i (x)>0 variablecoefficients. The fundamental solution(influence function) to Eq.(2.) is defined in the following form φ(x,)=k(x,)θ(x ) (2.3) whereθ(x)istheheavisidestepfunction,k(x,) Cauchyfunctionwhichisthesolutionto homogeneous Euler s equation L[y(x)] = 0 which satisfies following conditions K(x,)=K (x,)=...=k n 2 (x,)=0 K n (x,)= p 0 (x) Inthecaseofequations (2.4) L[y(x)]=δ j (x ) j=,2,3... (2.5) the solutions to them are partial derivativeswith respect to the parameter of the influence function y j (x)=( ) j j φ(x,) j In a general equation with the variable right part (2.6) L[y(x)]=g(x) (2.7) we have the following solution y= n k=0 A k k K k+y (x,) (2.8) wherea k arearbitraryconstants,y (x,) particularsolutiontoeq.(2.7)wellknownbythe Cauchy formula y (x,)= x K(x, s)g(s) ds (2.9) We can concludde that the partial derivative of the Cauchy function relative parameter (arbitrary steps) satisfies the equation L[y(x)]. Below, in Table, we present six formulas for the Cauchy function corresponding to chosen differential operators which have practical applications to mechanical engineering problems. 3. Influence function in the analysis of the bending curve and reactions of elastic supports of a beam with constant and variable cross-sections 3.. Exampleofageneralequationofthebendingcurveofthebeam The differential equation for deflection of an elastic beam in the static case has form(jaroszewicz and Zoryj, 997) (fy ) =G(x) (3.)

3 The influence function in analysis of bending curve Table. Formulas of Cauchy function(zoryj, 987) L[y(x)] K(x, ) (fy ) x (fy ) y + x y y + p f(x) y, p=const (f(x)y ) +py p=const x f(s) ds (x s)(s ) f(s) ds ln x ( p) k u k (x,) U(x,) k=0 u 0 (x,)=x u k (x,)= x p [x U(x,)] x ( p) k u k (x,) U(x,) k=0 u 0 (x,)=x u k (x,)= x x s f(s) u (k )(s,)ds, k=,2,... x s f(s) U(s,)ds x s f(s) u (k )(s,)ds, k=,2,... wheref=ej(x)isthebendingrigidityofthebeam(fig.),g(x) transversalloadingtakin into account all possible spatialy discrete compositions n G(x)= q i (x)[θ(x x i ) θ(x x i l i )]+P i δ(x x i )+M i δ (x x i ) (3.2) i= whereq i (x)isthedistributedloadactingonthesegmentx i <x<x i +l i,p i,m i discrete forceandmomentlocatedinsectionx=x i and0<x <x 2 <...<x n <l,l lengthofthe beam. Fig.. Bending of elastic beam(zoryj, 987) Onthebaseof(2.3)and(2.6),theparticularsolution y equation(3.)hasthefollowing form n y (x)= [θ(x x i )F i (x,x i ) θ(x x i l i )F i (x,x i +l i )] i= n φ ] + [P i φ(x,x i ) M i =xi i= (3.3) where

4 250 J. Jaroszewicz et al. φ(x,)=θ(x ) F i (x,)= x x x a φ(x,τ)q i (τ)dτ f(s) (x s)(s )ds (3.4) Theparticularsolutiony (0)forx=0satisfiestheboundarycondition y (0)=y (0)=y (0)=y (0)=0 (3.5) These expressions allow one to describe general solution(3.) in the following form x y(x)=y(0)+y (0)x M(0) 0 x s f(s) ds+q(0)φ(x,0)+y (x) (3.6) Expression(3.6)isageneralequationofthebendingcurveofthebeamwithavariablecrosssection.In(3.6),theinitialparameters:y(0),y (0),M(0),Q(0),i.e.deflection,angleofrotation, bendingmomentandtransversalforceontheleftendofthebeam(x=0),respectively,may take arbitrary values. Iff=EJ=constwehave φ(x,)= 6f (x )3 φ(x,) = 2f (x )2 (3.7) Considering(3.7),wecanobtainfrom(3.6)thewellknownequationofanelasticbeamof constantrigidity.generalequation(3.6)givesanewwayofthesolutionofafewproblemsof static multi-supported beams with rigidity distributed by steps Example of application of the influence function in the analysis of bending curve of the beam Figure 2 gives a model of an elastically supported beam(jaroszewicz and Zoryj, 994) subjecttoatransverseloadg.therigidityofthesupportatx=l isc.theflexuralrigidity isf=ej(x),wheretheyoungmodulusofelasticitye=const,asthematerialisassumedto be homogeneous and the plane moment of inertia J(x) is variable. The function /f(x) should be continuous, positive definite and should have a finite value and integral in[0, l]. Fig. 2. Model of an elastically supported beam WhenweconsiderthebeamshowninFig.2,thedeflectionisdefinedasfollows L[y]= Rδ(x l )+Gδ(x l 2 ) (3.8) wherel[y]=(fy ),δisdirac sfunction.

5 The influence function in analysis of bending curve The boundary conditions have the following form y(0)=0 fy (0)=0 fy (l)=0 cy(l )=R (3.9) The following solution to equation(3.8) can be proposed(jaroszewicz and Zoryj, 2000) y=c 0 +C (x )+C 2 K x0 +C 3 K x0 Rφ x +Gφ x2 (3.0) wherek=k(x,)isthecauchyfunctionforl[y(x)]=0whichcanbedefined K(x,)= x C i arrearbitraryconstantsand (x s)u(s,)ds U(s,)=s (3.) f(s) K x0 =K(x,0) K x0 = K φ x =φ(x,x )=K(x,l )θ(x l ) =0 The fundamental solution to equation(3.8) has form(2.3). AftercalculationofconstantsofintegrationC i,wehaveaparticularsolutionforthedeflectionofthebeamattheendx=linthefollowingform where y(l)= G l [ (l )(D 0+K l )+(l l )K l (3.2) K l (l,l )= l l l f(s) (x s)(s l )ds K l (l,)= D 0 = [(l l )K l ] l c K =(l,) Whenweassumetheconstantcross-sectionf(x)=f 0 =const f(s) (x s)(s )ds (3.3) K(x,)= 6f 0 (x ) 3 (3.4) we have the expression for deflection as follows [ l 2 x(l)=g cl 2 + l(l l ) 2] =0 (3.5) 3f 0 Inthecaseoftherigidsupportofthisbeamc,ithasbeenobtained y(l) 3f 0 Gl(l l ) 2 (3.6) 3.3. Influence function in the determination ofe elastic support reactions of the multi-supported beam We consider an elastic beam with free ends supported on elastic supports. we examine the displacementinpointswithcoordinatesa j (j=,...,n),so0<a <a 2 <...<a n <l.we apply the principle of release from constrains, and considering Hook s law, we have y(a j )=c j R j (3.7)

6 252 J. Jaroszewicz et al. wherer j isthemodulusofj-thsupportreactions,c j j-thsupportrigidity,y(a j ) deflection of the beam in j-th support. Boundary conditions for the considered beam are defined by equalingtozerothebendingmomentsandforcesatthebeamend (fy ) =M(0)=0 x=0 (fy ) =M(l)=0 x=l (fy ) x=0 =Q(0)=0 (fy ) x=l =Q(l)=0 From(3.6),bytakingintoconsideration(3.8),wehave (3.8) y(x)=y(0)+y (0)x+ỹ (x) (3.9) Weputintoformula(3.3)thesupportreactionR j n ỹ (x)=y (x) R j ϕ(x,a j ) (3.20) j= Substituting(3.9)toconditions(3.8) 2,wehavereceivedthefollowingequations n τ [ R j (l a j )= P i (l x i )+ j= i= n τ ( R j = P i + j= i= x i +l i x i q i (τ)dτ x i +l i x i (l τ)q i (τ)dτ+m i ] (3.2) Equation(3.2) presentstheconditionofequilibriumofmomentsatthepointx=l(leftend of the beam), the second condition equilibrium of transverse. Substituting(3.9) to(3.7), after transformation, we receive the follows equations y 0 +y 0 a R c = y (a ) y 0 +y 0 a 2 R (a 2,a ) R 2 = y (a 2 ) c 2 n y 0 +y 0 a n R n (a n,a j ) R n = y (a n ) c n j= (3.22) Thiswaywedeterminetwoconstantsy 0 =y(0)andy 0 =y (0)andwelookfornreactionsR i of the elastic supports through the system n + 2 linear algebraic equations(3.2),(3.22). If anyarbitrarysupportofthemisrigid(e.g.first)thenin(3.22)c weobtainthesystem considered in this case. Nowweconsiderthedeterminants n gotfromthesystemofequationsn=2,3,4,andwe find 2 =(a 2 a ) 2 3 =ϕ 2 (a 3 a 2 ) 2 +ϕ 32 (a 2 a ) 2 4 =ϕ 2 ϕ 32 (a 4 a 3 ) 2 +ϕ 2 ϕ 43 (a 3 a 2 ) 2 +ϕ 32 ϕ 43 (a 2 a ) 2 (3.23) Next,weproovethat n >0forn=2,3,4,whichimpliesthattheconsiderproblemhasonly one solution in all cases(rigid, elastic supports).

7 The influence function in analysis of bending curve ExamplesofcalculationofthereactionR i (,c i )fromtheelasticsupports We take into consideration the beam presented in Fig. 3 which has three elastic supports and constant stiffness(ej = const). The systems of equations(zoryj, 987; Jaroszewicz and Zoryj,994)canbeexpressedasfollows(a 0,a 2 =l,a 3 =2l) 2R +R 2 =2 R +R 2 +R 3 =2 y 0 R c =0 y 0 +y 0 l R l 3 6EJ R 2 c 2 = 4 24EJ y 0 +y 0 2l R (2l) 3 6EJ R 2l 3 6EJ R 3 = q(2l)2 c 3 24EJ (3.24) Fig. 3. Beam with three elastic supports(zoryj, 987) Fromthefirstequation(3.24)itfollowsthatR =R 3.Next,reducingtheconstantsy 0,y 0, we obtain R (,c i ) R 2 (,c i ) = R 3(,c i ) =2 = c + c c + 4c 2 + c c 2 c + 4c 2 + c (3.25) where=l 3 (EJ).ExamplesofcalculationsarepresentedinTable2. Table 2. Examples of calculation of reaction forces in the elastic supports of constant stiffness beam Variants I II III IV c c c R (,c i ) = R 3(,c i ) R 2 (,c i ) Inparticularcases,wefindfrom(3.25)wellknownvalues.Ifallsupportsarerigid(c i ), we have the well known results(niezgodziński and Niezgodziński, 202) R =R 3 = 3 8 R 2= 5 (3.26) 4

8 254 J. Jaroszewicz et al. Fortheabsolutelyrigidbaronelasticsupports,inthecaseEJ,=0,wereceive R =R 3 4c 2 c +4c 2 +c 3 R 2 =2 c +c 3 c +4c 2 +c 3 (3.27) ForEJ andc 2,R =R 3 0,R 2 =2andforEJ,c =c 3,wehave R =R 3,R 2 0,whichcanbeconcludedonthebaseofequations(3.2)and(3.22). Wecanalsoformulateconditionsforapplyingsimplerequations,e.g.c =c 3 =,c 2 =c.we presentthecorrespondingvaluesfoundfrom(3.25) intable3. Table 3. Results of calculation cl 3 /(EJ) R /() R 2 /() Wecannoticethatfor 0<cl 3 (EJ) <0.theinfluenceofrigidityofthesupportson reactionsattheendsupportisverysmall,andforcl 3 (EJ) >0 3 wecanconsiderthebeam as absolutely rigid and calculate the reactions by means of formulas(3.27). 4. Conclusions In this work, properties of the influence function applied to solution of mechanical problems of discrete, distributed and discrete-distributed homogeneous components were presented. The advantages of the influence function is the possibility of omission of the boundary of conjugation in the solution to the boundary value problem. The presented method gives possibility of solving the problem of bending curve deflection of a multi-supported beam with a variable cross-section in a closed analytical form. The presented method enables calculation of the force reactionr i (,c i )forelasticsupportsofbeamswithconstantandvariablestiffness(x). References. Conway H.D., 958, Some special solutions for the flexural vibrations of discs of varying thickness, Ingenieur-Archiv, 26, 6, Jaroszewicz J., Misiukiewicz M., Puchalski W., 2008, Limitations in application of basic frequency simplest lower estimators in investigation of natural vibrations circular plates with variable thickness and clamped edges, Journal of Theoretical and Applied Mechanics, 46,, JaroszewiczJ.,ZoryjL.,994,Analysisofthebendingcurveandcriticalloadofavariable cross-section beam by means of influence method, Journal of Theoretical and Applied Mechanics, 32, 2, Jaroszewicz J., Zoryj L., 996, Critical Euler load for cantilever tapered beam, Journal of Theoretical and Applied Mechanics,, 34, 4, Jaroszewicz J., Zoryj L., 997, Metody analizy drgań i stateczności kontynualno-dyskretnych układów mechanicznych, Rozprawy Naukowe Politechniki Białostockiej, Jaroszewicz J., Zoryj L., 999, The effect of non-homogenous material properties on transversal vibrations of elastic cantilevers, International Applied Mechanics, 35, 6, Jaroszewicz J., Zoryj L., 2000, Investigation of the effect of axial loads on the transverse vibrations of a vertical cantilever with variables parameters, International Applied Mechanics, 36, 9,

9 The influence function in analysis of bending curve Jaroszewicz J., Żur K.K., 202a, Limitation of Cauchy function method in analysis of estimators of frequency and form of natural vibrations of circular plate with variable thickness and clamped edges, Acta Mechanica et Automatica, 6, 2, Jaroszewicz J., Żur K.K., 202b, Metoda funkcji Cauchy w analizie drgań kuli sprężystej, Biuletyn Wojskowej Akademii Technicznej, 4, Jaroszewicz J., Żur K.K., 202c, Wpływ pierścieniowej masy skupionej na drgania własne płyt kołowych z typowymi warunkami brzegowymi, Biuletyn Wojskowej Akademii Technicznej, 4, Lazopoulos K.A., 200, Bending and buckling of thin strain gradient elastic beams, European Journal of Mechanics A/Solids, 29, 5, Niezgodziński M.E., Niezgodziński T., 202, Wzory, wykresy i tablice wytrzymałościowe, Wydawnictwo Naukowo-Techniczne, Warszawa 3. Solecki R., Szymkiewicz J., 964, Układy prętowe i powierzchniowe, obliczenia dynamiczne, Wydawnictwo Arkady, Warszawa 4. Timoshenko S., 940, Theory of Plates and Shells, New York 5. Štok B., Halilovič M., 2009, Analytical solutions in elasto-plastic bending of beams with rectangular cross section, Applied Mathematical Modelling, 33, 3, Zoryj L.M., 978a, K primeneniju obobshchennykh funkcji v analiticheskikh metodakh issledovania slozhnykh uprugikh sistem, Doklady AN USSR, Ser. A.,, Zoryj L.M., 978b, K razvityu analyticheskikh metodov issledovaniya zadan dinamiki uprugikh i gidrouprugikh system, Mat. Metody i Fiz.-Mech. Pola, 5, Zoryj L.M., 978c, Pro odnu fundamentalnu vlastyvist funkcii vplyvu, Dop. AN URSR, Ser. A., 9, Zoryj L.M., 987, Raschety i ispytaniya na prochnost, Metodicheskiya rekomendacji MR 23-87, GOSTSTANDART SSSR, Moskva Manuscript received January 29, 203; accepted for print September 9, 203