Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD Sebastan Uzny, Łukasz Kutrowsk Insttute of Mechancs and Machne Desgn Fundamentals, Czestochowa Unversty of Technology Czestochowa, Poland uzny@mpkm.pcz.pl, kutrowsk@mpkm.pcz.pl Receved: 9 June 7; Accepted: September 7 Abstract. The boundary problem of free vbratons of a hydraulc telescopc cylnder, subjected to Euler s load was presented n ths work. The task was formulated on the bass of the Hamlton s prncple. The computatonal model, formulated by Tomsk, was taken nto account durng analyss. Numercal calculatons concern free vbratons of mult-stage hydraulc telescopc cylnder. Results were presented n the form of characterstc curves on the plane: external load-natural vbraton frequency. Non-dmensonal parameters of the structure were defned n reference to the characterstc parameters of the pston rod. Durng numercal smulatons, an nfluence of a non-dmensonal stffness parameter between followng elements on free vbratons of a hydraulc telescopc cylnder was determned for a dfferent number of cylnder stages. MSC : 74H45, 65P99 Keywords: free vbratons, hydraulc telescopc cylnder, slender system. Introducton Hydraulc cylnders are certan knds of motors, whch convert the energy of compressed hydraulc flud to mechancal energy. A typcal example s a lnear sngle-actng cylnder, whch s under consderaton n ths thess. One-stage and mult-stage (telescopc) cylnders can be found dependng on the number of stages. Telescopc cylnders are used when a long output range and a very compact retracted length are requred. They are used especally n dump trucks (dump body) and hydraulc elevators. Two dfferent computatonal models concernng the stablty and vbratons of hydraulc cylnders were developed by Tomsk [, ]. The frst model refers to free transversal vbratons and statc stablty of cylnders. It s used for cylnders wth a hgh slenderness rato []. The second model refers to free longtudnal vbratons
8 S. Uzny, Ł. Kutrowsk and forced vbratons. Ths knd of model s used for a cylnder wth a low slenderness rato []. The lmtng values of parameters, when the computatonal model has to be changed can be assumed when free transversal and longtudnal vbratons are equal. Free transversal vbratons of cylnders, formulated on the bass of the frst computatonal model, were consdered n works [-6]. The results of numercal smulatons accordng to the one-stage hydraulc cylnders for dfferent boundary condtons and parameters of structure can be found n those papers. Parameters of cylnders are: degree of coverage, stffness of fxed, stffness of pston rod and cylnders, stffness of sealng and gudng elements. In paper [3] longtudnal nerta of the pston rod has been taken nto account due to analyss. The effect of pressure along the cylnder, whch has sgnfcant mpact on ts strength was presented n work [4]. In papers [5, 6] the results of expermental analyss of one-stage hydraulc cylnder were carred out what confrmed the proposed computatonal model. In ths paper, the effect of rgdty between the followng elements (sealng and gudng elements for pston rod and all of cylnders) on characterstc curves s analysed. In the numercal research the dfferent number of stages (one, three, fve stage hydraulc cylnders) were taken nto consderaton.. Boundary value problem The consdered n-stage telescopc hydraulc cylnder subjected to Euler s load s presented n Fgure. Structure conssts of n cylnders and the pston rod. The overall length of the confguraton was defned as l C. In ths work, the telescopc hydraulc cylnder s consdered as fully extended and smply supported on both ends. The torsonal rgdty of sealng and gudng elements was modelled by rotatonal sprngs of C R stffness. Stffness of rotatonal sprngs are as follows: C R = C R = C R3 = C R4 = C R5 = C R. Geometrcal dmensons of the cylnders (outer dameter d z and nner dameter d w ) were defned as: d = d+ ( n ) g + ( n ) g z t U R d = d+ ( n ) g + ( n ) g w t U R (a,b) where g U and g R stand for the thckness of the sealng element and cylnder. Each element of the hydraulc cylnder s characterzed by adequate flexural rgdty (EJ) (E - Young s modulus, J - geometrcal nerta moment) and mass per length unt (ρa) (ρ - densty, A - area of cross secton). The mass of the hydraulc flud, whch flls the cylnders s (ρa) c ((ρa) cn = ), masses of sealng and gudng elements m were taken nto consderaton. Elements of the structure marked as =,,,n- correspond to cylnders and the n-element correspond to the pston rod.
The effect of torsonal rgdty between elements on free vbratons of a telescopc hydraulc cylnder 9 The boundary problem has been formulated on the bass of Hamlton s prncple. Fg.. Scheme of n-stage telescopc hydraulc cylnder subjected to Euler s load Potental energy (V) and knetc energy (T) are as follows: l (, ) n (, ) n l Y x t Y x t = x = x V = EI dx P dx + x= l (, ) (, ) Y x t Y x t + C n + + R = x x+ x+ = n l n x= l Y ( x, t) Y ( x, t) ( ρ ) = t = t T = A dx m + + n ( ρ A) l c = Y ( x, t) dx t () (3) Takng nto account the potental and knetc energy n Hamlton prncple, after approprate transformatons, dfferental equatons of moton and natural boundary condtons are obtaned. Dfferental equatons of moton after separaton of varables can be wrtten as follows:
S. Uzny, Ł. Kutrowsk ( ρ ) ( x) d y x d y x d y x EI + P A + 4 4 ( ρ ) ω dx dx dx d y A ω = for =... n c dx (4) where ω s free vbratons frequency. Boundary condtons (geometrcal and natural) are expressed as follows: ( EI) y () = ; y ( l ) = ; y ( l ) = y ( l ) n n + + x= ( x ) d yn xn n d x d xn ( EI) ( EI) d y x= l n n = ; = x= l x= l d y x dy x dy x + + ( EI) C R = dx dx dx+ x+ = x= l d y+ x+ dy x dy+ x+ + C R + dx+ dx x dx + = + x+ = ( x ) = + + dx+ x+ = 3 x= l 3 x= l d y x d y+ x+ dy x EI ( EI) P dx 3 + 3 dx + x dx + = + + dy x= l P + mω y ( x ) = (5a-h) Solutons of dfferental equatons (4) can be wrtten as: y ( x ) = A cosh( α x ) + B snh( α x ) + C cos( β x ) + D sn( β x ) (6) where: α k k k k 4 4 = + + Ω ; β = + + Ω 4 4 k P = ; Ω = ( EI) ( ρ A) ( EI) ω (7a-d) After substtuton of the soluton (6) nto boundary condtons, the system of equatons s obtaned. The matrx determnant of coeffcents equated to zero leads to the transcendental equaton, from whch the natural vbraton frequency of the system can be determned.
The effect of torsonal rgdty between elements on free vbratons of a telescopc hydraulc cylnder 3. Results Results of numercal smulatons of free vbratons of the consdered telescopc hydraulc cylnder were presented n the non-dmensonal form, defned as: ( ρ A) g C l Pl ω l ζ GU = ζ = = λ = Ω = (8a-e) d d EI EI EI 4 U gr R C C * n C ; GR ; c ; ; t t n n n Fg. a-c. Characterstc curves on non-dmensonal plane for dfferent parameters of stffness between elements (d t =. m, ζ GU =.5, ζ GR =.) In Fgure characterstc curves (λ cr (Ω*)) n case of: one-stage cylnder (Fg. a), three-stage cylnder (Fg. b), fve-stage cylnder (Fg. c) are presented. Numercal calculatons were carred out for chosen cylnder parameters (d t =. m, ζ GU =.5, ζ GR =.) and for dfferent values of a non-dmensonal parameter of rotatonal sprng stffness c (c =.8; 4; 8; 6; 3; 64; 8; 56; 58; /c = ). An nfluence of the consdered stffness of rotatonal nodes n whch the ends of ndvdual hydraulc cylnder elements are joned, on natural vbraton frequency magntude depends on ths stffness as well as on the number of stages.
S. Uzny, Ł. Kutrowsk 4. Conclusons The boundary problem of free vbratons of a telescopc hydraulc cylnder subjected to Euler s load was consdered n ths paper. The results were plotted n the form of characterstc curves. An nfluence of the gudng and sealng elements stffness on dynamc behavour were analysed. On the bass of the obtaned results, t can be concluded that the stffness of the sealng and gudng elements have great nfluence on vbraton frequency and crtcal load (n the presented problem the crtcal load corresponds to the zero magntude of the vbraton frequency - knetc stablty crteron). The smaller the rotatonal node stffness the greater ts nfluence on vbraton frequency and crtcal load. On the bass of the proposed non-dmensonal parameters, the obtaned relaton load - vbraton frequency s lnear. The characterstc curves are parallel to each other at the consdered confguraton of the system. Reducton of stffness of gudng and sealng elements due to wear or damage can have serous consequences n further explotaton of the hydraulc cylnders because the reducton of loadng capacty as well as vbraton frequency can be observed. Acknowledgement The study has been carred out wthn the statutory funds of the Czestochowa Unversty of Technology (BS/PB---3//P). References [] Tomsk L., Elastc carryng capacty of a hydraulc prop, Engneerng Transactons 977, 5(), 47-63. [] Tomsk L., Dynamka stojaków hydraulcznych obudów górnczych, Praca habltacyjna, Nr 7, Częstochowa 979. [3] Sochack W., Tomsk L., Free vbraton and dynamc stablty of a hydraulc cylnder set, Machne Dynamcs Problems 999, 3(4), 9-4. [4] Tomsk L., Elastc stablty of hydraulc props of longwall supports, Archves of Mnnng Scence 978, 3(3), 7-3. [5] Tomsk L., Uzny S., A Hydraulc cylnder subjected to euler's load n aspect of the stablty and free vbratons takng nto account dscrete elastc elements, Archves of Cvl and Mechancal Engneerng,, 3, pp. 769-785. [6] Uzny S., Free vbratons and stablty of hydraulc cylnder fxed elastcally on both ends, Proc. Appl. Math. Mech. 9, 9, 33-34.