Transverse and Longitudinal Damped Vibrations of Hydraulic Cylinder in a Mining Prop

Transcription

1 Vibraions in Physical Sysems Vol. 7 (06) Transverse and Longiudinal Damped Vibraions of Hydraulic Cylinder in a Mining Prop Wojciech SOCHACK nsiue of Mechanics and Fundamenals of Machinery Design Universiy of Technology, Czesochoa, sochacki@imipkm.pcz.pl Mara BOLD nsiue of Mechanics and Fundamenals of Machinery Design Universiy of Technology, Czesochoa, bold@imipkm.pcz.pl Absrac This sudy presens he influence of differen kinds of damping on ransverse and longiudinal vibraions of hydraulic cylinder in a mining prop. The dissipaion of vibraion energy in he model is caused by simulaneous inernal damping of viscoelasic maerial of beams ha model he sysem, exernal viscous damping and consrucional damping. Consrucional damping (modelled by he roaional viscous dampers) occurs as a resul of movemen resisance in he cylinder suppors. The eigenvalues of he sysem ih respec o changes in sysem geomery ih o values of load and for a seleced and variable damping coefficien values ere calculaed. Keyords: damped vibraion, hydraulic cylinder, ransverse vibraion, longiudinal vibraion. nroducion A hydraulic cylinder as an objec of research sudies on dynamics of mechanical sysems has been exensively invesigaed in he number of sudies. Mos of he published sudies focused on he ineracions beeen he cylinder ube and pison rod. Resuls of he invesigaions of he dynamic response of he model of a cylinder o axial impulse ere presened in paper []. The ork [] presens an analysis of he effec of iniial inaccuracy of connecion beeen he pison and cylinder ube on criical loading force in he cylinder. Many auhors analysed he effec of sealing or he medium on he cylinder's dynamics and dynamic sabiliy of cylinder. n sudy [3] calculaions of free vibraion frequencies ere exended ih he invesigaions of he dynamic sabiliy of he cylinder by means of deerminaion of geomerical parameers and load a he ime of losing he sabiliy ere presened. n paper [4] he problem of he sabiliy and free vibraions of a slender sysem in he form of a hydraulic cylinder subjeced o Euler's load as carried ou. The sudies [5] and [6] presen he effec of inernal damping on vibraions of a suppor beam ih a mass aached o a free end of he beam and on sabiliy of a suppor colu loaded ih a folloer force, respecively. The influence of small inernal and exernal damping on sabiliy of non-conservaive beam sysems is described in paper [7]. Equally ineresing publicaion concerning he effec of exernal damping on vibraion of beams ih sepped cross-secion is he sudy [8]. The effec of srucural damping of fixaions on free vibraion of he linear Bernoulli-Euler beam as presened in he sudy [9].

2 38 n sudy [0] dissipaion of vibraion energy in he model of hydraulic cylinder boom crane sysem occurs as a resul of simulaneous inernal damping of he viscoelasic maerial of he beam used in he model and he consrucional damping in he suppors of he cylinder and crane boom. The consrucional damping of suppors as modelled using roaional viscous dampers. The problem o be considered in he sudy [] is he naural vibraion of he sysem consising of o clamped-free rods carrying ip masses o hich several double spring-mass sysems are aached across he span. The sudy is concerned ih longiudinal vibraions of his mechanical sysem and he major conribuion of his sudy is o derive a general formulaion for he exac soluion of he sysem described by using he Green's funcion mehod. This sudy analyses he simulaneous effec of he consrucional damping, inernal damping, exernal damping and he influence of changes in sysem geomery on he ransverse and longiudinal vibraions of hydraulic cylinder in a mining prop. The resuls obained in he sudy ere presened in D figures and spaial presenaions.. Mahemaical Model A scheme of he considered sysem is presened in Fig.. The model of a hydraulic cylinder is composed of four beams. To of hem model a cylinder ube (l, l ) and o - pison rod (l, l ) in he cylinder. The liquid in he cylinder as adoped as he medium of load ransfer beeen he pison and he cylinder along he lengh filled ih liquid. The liquid rigidiy in he cylinder as modelled by he ranslaional spring. Siffness coefficien of spring as denoed by k S. n adoped model dissipaion of vibraion energy as caused by simulaneous inernal damping, exernal damping and consrucional damping. nernal damping of he viscoelasic maerial for individual pars of hydraulic cylinder as characerized by Young's modulus E and viscosiy coefficiens E. Exernal damping of medium surrounding he sysem ere denoed by coefficien c e. Consrucional damping occurs as a resul of movemen resisance in he pison and he cylinder suppors and i as modelled by he roaional viscous dampers. Damping coefficiens of roaional viscous dampers ere denoed by c R. The boundary problem conneced o he free vibraions of he considered nonconservaive (due o damping) sysem as formulaed on he basis of Hamilon s principle in he folloing form: d ò ( T-V ) d+ òdw d= 0 () here: T kineic energy, V poenial energy, δw N virual ork of nonconservaive forces originaing from damping. N

3 Vibraions in Physical Sysems Vol. 7 (06) 39 Figure. Diagram and beam model of a hydraulic cylinder ih damping The vibraion equaions for individual beams are knon and have he folloing form: here: - A J æ ç E è æ ç E è + r A 4 ö W ø W ö U ø 4 x x + c W + P x e W + r A = 0 U + = 0 here: m,n =, (c e = 0 for m = and n = ) W (x ransverse displacemen of beams ha model cylinder and pison rod U (x longiudinal displacemen of beams ha model cylinder and pison rod E Young's modulus for individual beams, E maerial viscosiy coefficien, J momen of ineria in beam cross-secions, A cross-secional areas of he beams, () (3)

4 30 r beam maerial densiy, c e viscous damping coefficien, P cylinder loading force (a he lengh l of he cylinder ube coverage ih he pison rod in he cylinder P=0) x spaial coordinaes, ime Soluions of equaions () and (3) are in he form: W U = ) e (4) = u ) e (5) here: ω he complex eigenvalue of he sysem, Subsiuion of (4) and (5) ino () and (3) leads o, respecively: V ) b )-g ) = 0 (6) here: a g Boundary condiions: u - (0) = u E (0) = J i= - + ) + u ) = 0 u a (7) r A æ ç ) è ce - r A ö ø =, J r =, b ) ) = - ) = ) =-c ) = 0, + P u ) = ) = (0) = )- R u (0) = (0) = (0) =-k u ) + (0) = = (0) = 0, u S E ), ) = ) = 0, J (0)- P u ) = u P + ie (0) = ) = (0) + (0) = 0, ) = (0) = c R u ), ( l u ) + ) = u ) ), u ) + u ) = P, (8) (9)

5 Vibraions in Physical Sysems Vol. 7 (06) 3 The soluion of equaions (6) and (7) are expressed in he form of funcions: here: lx -lx ilx -ilx = C e + Ce + C3e + C4e ) (0) u id x -d i x ) = D e + De () 4 4 b b, l = + + g b b l = g, () 4 4 The boundary problem is solved numerically for he eigenvalues ω. Depending on he soluion adoped, he roos ω are complex numbers (ha represen he damped vibraion frequencies Re(ω ) and damping m(ω ) in he considered sysem) and hey may accep posiive or negaive value. n his paper, presenaion of he resuls as based on posiive values of he real and imaginary pars of soluions. 3. Numerical Calculaion Resuls d = a Calculaions ere carried ou for a cylinder used in a mining prop. Compuaions ere carried ou for he daa conained in Table. Dimensionless damping parameers: η for inernal damping, µ for consrucional damping, and ν for exernal damping ere placed belo he able. Table. Geomerical and maerial daa adoped in he sudy Quaniy Symbol Uni Value Cylinder ube - exernal diameer D = D mm 90 Cylinder ube - inernal diameer d = d mm 50 Pison rod - exernal diameer D = D mm 60 Pison rod - inernal diameer d = d mm 0 Cylinder ube and pison rod densiy r kg/m e3 Young's modulus Damping parameers: h E ce LC C h =, n =, R P m =, p=, he d d PC = L E å 4 m, n= C å r E A J m, n= 3, d = LC år A EJ, m,n= Pa e here: P C he criical load of he cylinder exended o L C =4m and L C = l+ l+ l. ()

6 3 The resuls of he calculaions are presened in Figures o 5. The sysem as loaded ih he longiudinal force P (p=0 and p=0.3). The dependency of he eigenvalues (real pars Re(ω ) and imaginary pars m(ω )) on coefficiens of consrucional damping µ, exernal damping ν, inernal damping η and oal lengh of cylinder ha ranged from L C=.6m o L C=4m as also deermined. The relaionships beeen he firs eigenvalue of cylinder and changes is oal lengh L C and coefficien of consrucional damping µ a p=0.3 ihou inernal and exernal damping in he sysem are presened in he form of spaial diagrams in Figure. Figure. The dependency of he firs eigenvalue (Re(ω ) and m(ω )) for he cylinder on oal lengh L C and consrucional damping µ a η=0, ν =0 and p=0.3 As can be seen in he figure above, he higher value of m(ω n ) hen he more he ampliudes of a paricular (n) mode of vibraion are damped. Figure 3 presens he maximum values of m(ω max) for he firs mode of vibraion in he examined sysem depending on he hydraulic cylinder lengh L C for o values of loading. Figure 3. The relaionships beeen he maximum values of m(ω max) for he firs mode of vibraion in he cylinder and he exension oal lengh L C (for η=0 and ν=0) Nex invesigaions focused on consideraion of effec of differen kind of damping on cylinder vibraion. The dependency of real and imaginary pars of he firs eigenvalue of he hydraulic cylinder on exension oal lengh L C for seleced values of damping (η=0.0, ν=0.5, µ=0.5) and for o values of loading are presened in Figure 4.

7 Vibraions in Physical Sysems Vol. 7 (06) 33 Figure 4. The dependency of he firs eigenvalue (Re(ω ) and m(ω )) for he cylinder on exension oal lengh L C The nex figure (Figure 5) presens he change in he firs eigenvalue of he hydraulic cylinder depending on he exernal damping ν and inernal damping η ihou loading and loaded ih he force p=0.3 for seleced lengh of cylinder L C=3m. The invesigaions ere carried ou for opimal consrucional damping value µ=0.5. Figure 5. The dependency of he firs eigenvalue (Re(ω ) and m(ω )) for he cylinder on inernal damping η and exernal damping ν a µ=0.5 and L C =3 4. Conclusions This sudy presens a beam model of a hydraulic cylinder based on he sysem used in mining props. The compuaions for he model of ransverse and longiudinal vibraions in a hydraulic cylinder ih damping ere carried ou. The model of damping ook ino consideraion he inernal damping of he beams ha modelled a cylinder ube and a pison rod, exernal damping and consrucional damping ha modelled moion resisance in he suppors. Subsanial changes can be observed in he damped frequencies Re(ω ) and in degree of ampliude decay m(ω ) in he case of changes he lengh of hydraulic cylinder L C and coefficien of consrucional damping µ (Figure ). An increase in consrucional damping causes he increase in he values of degree of ampliude decay m(ω ) o maximum values, folloed by m(ω ) 0 here μ. These subsanial changes in boh Re(ω) and m(ω) are caused by considerable inervenion in he condiions of sysem fixaion (in exreme cases, he fixaion poins are changed from join mounings ino rigid mounings). The lengh of hydraulic cylinder

8 34 exension for hich he degree of vibraion ampliude decay is he highes allos for deerminaion of opimum lenghs of he hydraulic cylinder ih respec o minimum vibraion ampliudes in he sysem (Figure 3). can be concluded based on he calculaions ha inroducion of he inernal and exernal damping causes only insignifican changes in he firs eigenvalue (Figure 5). The resuls presened in he sudy help deermine he geomeric parameers and values of he coefficiens ha characerize damping of he sysem for hich he maximum degree of ampliude decay is mainained. Acknoledgmens This research as suppored by he Minisry of Science and Higher Educaion in 06, Warsa, Poland. References. L. Tomski, S. Kukla, Dynamical response of bar-fluid-shell sysem simulaing hydraulic cylinder subjeced o arbirary axial exciaion, Journal of Sound and Vibraion, 9() (984) P. J. Gamez-Monero, E. Salazar, R. Casilla, J. Freire, M. Khamasha, E. Codina, Misalignmen effecs on he load capaciy of a hydraulic cylinder, nernaional Journal of Mechanical Sciences, 5 (009) W. Sochacki, L. Tomski, Free vibraion and dynamic sabiliy of hydraulic cylinder se, Machine Dynamics Problems, 3(4) (999) L. Tomski, S. Uzny, A hydraulic cylinder subjeced o Euler's load in aspec of he sabiliy and free vibraions aking ino accoun discree elasic elemens, Archives of Civil and Mechanical Engineering, (3) (0) M. Gürgöze, A. N. Doğruoğlu, S. Aeren, On he eigencharacerisics of a canilevered visco-elasic beam carrying a ip mass and is represenaion by a spring-damper-mass sysem, Journal of Sound and Vibraion, 30 (007) S. U. Ryu, Y. Sugiyama, Compuaional dynamics approach o he effec of damping on sabiliy of canilevered colu subjeced o a folloer force, Compuers & Srucures, 8(4) (003) O. N. Kirrllov, A. O. Seyranin, The effec of small inernal and exernal damping on he sabiliy of disribued non-conservaive sysems, Journal of Applied Mahemaics and Mechanics, 69 (005) M.. Frisell, A. W. Lees, The modes of non-homogeneous damped beams, Journal of Sound and Vibraion, 4() (00) G. Oliveo, A. Sanini, E. Tripodi, Complex modal analysis of flexural vibraing beam ih viscous end condiions, Journal of Sound and Vibraion, 00 (997) W. Sochacki, M. Bold, Damped vibraion of he sysem of changing he crane boom radius, Journal of Applied Mahemaics and Compuaional Mechanics, 4() (05).. S. nceoǧlu, M. Gürgöze, Longiudinal vibraions of rods coupled by several springmass sysem, Journal of Sound and Vibraion, 34(5) (000)