Simple models for rope substructure mechanics: application to electro-mechanical lifts

Size: px

Start display at page:

Download "Simple models for rope substructure mechanics: application to electro-mechanical lifts"

Brendan Charles
6 years ago
Views:

1 Journa of Physics: Conference Series PAPER OPEN ACCESS Simpe modes for rope substructure mechanics: appication to eectro-mechanica ifts To cite this artice: I Herrera and S Kaczmarczyk 016 J. Phys.: Conf. Ser View the artice onine for updates and enhancements. Reated content - SUBSTRUCTURE WITHIN CLUSTERS OF GALAXIES.. J. Geer and T. C. Beers - SUBSTRUCTURE, DYNAICS AND EVOLUTION IN CLUSTERS OF GALAXIES Christina. Bird - Fundamenta Consideration of Piezoeectric Ceramic uti-orph Actuators Yoshiro Tomikawa, Koichi asamura, Sumio Sugawara et a. This content was downoaded from IP address on 6/04/018 at 14:5

2 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 Simpe modes for rope substructure mechanics: Appication to eectro-mechanica ifts I Herrera 1a, S Kaczmarczyk b a Escuea de Ingenierías Industriaes, Universidad de Extremadura, Avda. de Evas s/n, Badajoz, Spain b Schoo of Science and Technoogy, The University of Northampton, St. George s Avenue, Northampton, NN 6JD, UK 1 iherrera@unex.es, stefan.kaczmarczyk@northampton.ac.uk Abstract. echanica systems modeed as rigid mass eements connected by tensioned sender structura members such as ropes and cabes represent quite common substructures used in ift engineering and hoisting appications. Specia interest is devoted by engineers and researchers to the vibratory response of such systems for optimum performance and durabiity. This paper presents simpified modes that can be empoyed to determine the natura frequencies of systems having substructures of two rigid masses constrained by tensioned rope/cabe eements. The exact soution for free un-damped ongitudina dispacement response is discussed in the context of simpe two-degree-of-freedom modes. The resuts are compared and the infuence of characteristics parameters such as the ratio of the average mass of the two rigid masses with respect to the rope mass and the deviation ratio of the two rigid masses with respect to the average mass is anayzed. This anaysis gives criteria for the appication of such simpified modes in compex eevator and hoisting system configurations. 1. Introduction We focuses our attention on the mechanica substructure shown in figure 1 in which two umped mass components 1 and are connected by an eastic tensioned member/rope-piece of ength which can be the equivaent mechanica mode of a number of common substructures in ift engineering and hoisting equipment. We are particuary interested in traction ift suspension ropes substructures. It shoud be mention that the system in figure 1 is a semi-definite system. Thus, there is a rigid body mode with zero frequency. The prediction of dynamic response of the suspension ropes is essentia to understand and mode the eevator system. In a typica ift instaation the car/counterweight suspension system is formed by a number (n R ) of equay tensioned stee wire ropes acting in parae, attached to the car frame at one end, passing over a traction sheave and diverter pueys to the counterweight at the other end. The contact between the rope and traction sheave and diverter pueys is speciay critica, not ony for the neccesity to produce the desired motion of the car but for its contro, safety, and maintenance of reevant traction condition according to EN If we assume that no sipping exists between the ropes and the traction sheave both sides of the rope can be represented by the simpified substructure shown in figure 1. Thus: 1 woud correspond to the traction sheave equivaent mass (rotary or inear) Content from this work may be used under the terms of the Creative Commons Attribution 3.0 icence. Any further distribution of this work must maintain attribution to the author(s) and the tite of the work, journa citation and DOI. Pubished under icence by Ltd 1

5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) 01008 doi:10.

3 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 for both sides and woud correspond to the car mass for the frame side substructure or the counterweight mass for the counterweight side substructure. Another mode from the opposite side to that free ( 1 )-free ( ) mode shown in figure 1 is the fixed ( 1 )-free ( ) mode which was extensivey studied by Leung []. This mode coud be the best aproximation for static stopped eevator sings(brakes pressed) but its response woudn t be affected at a by the umped mass ( 1 ). 1 x dm dx EΩ ρ Figure 1. Substructure of two rigid masses tensioned by a rope. Figure. Eevator system schem. The dynamic behavior of the mass suspension system can be anayzed by treating the ropes as inear distributed-parameter eastic members. To give criteria about the reative infuence of the mechanica characteristics of the rope-pieces with respect to the joined up rigid masses properties and the method of anaysis, severa substructuring techniques were studied [-3] and the system shown in figure 1 wi be anayzed. The dynamics of suspended ropes has been intensivey studied. Severa contributions from Zhu et a. [4-5] and Kaczmarczyk [6-7] give a wide and compete soution to the rope modeed as inear eastic continua. The assumptions for the car and counterweight as assembies to be treated as rigid bodies are based on the ow ongitudina eastic moduus of the rope whist perfect fexura fexibiity and high eastic moduus of the metaic components. Typica anaysis capabe of generaization and moduarity of compex systems is the appication of the finite eement method [8-9]. The continuous description of the rope is substituted by a finite number of sections/ eements, with their deformations described by equations in terms of unknowns dispacements. In the simpified modes it is proposed to discretize the suspended rope- substructure using ony one eement. This has the advantage of maintaining sma number of unknowns and equations and faciitates any further synthesizing strategy. In such an approach these modes describe the eevator response we without significant oss of accuracy. Particuary, specia interest is devoted to justify the appication to suspended rope-substructures which were discretized to adopt a 6 degrees of freedom discrete mode of the eevator system [1]. In a typica ift or eevator system the car and the counterweight are mounted in a frame/sing (see figure ). Both are generay made of stee sections and fixed to each counterpart by bots and nuts or by weding techniques [1]. Specia attention is devoted to tighten the pieces to avoid noise during trave. Each of these components can be treated as an idea rigid body constrained by the ift guiding system. In this arrangement the six degrees of freedom of a rigid body moving in space are reduced to ony one (vertica transation) assuming idea siding joints between car-frame or counterweight and the guide rais.

4 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/ Boundary-vaue probem soutions Two rigid components of masses 1 and are connected by an eastic tensioned rope-piece of ength (figure 1). The rope-piece is characterized by its density ρ, net cross-section area Ω and apparent eastic moduus E and then its tota mass m (m=ρω). Let us denote by the average mass of 1 and. Then, two characteristics ratios of the system are intended to be anayzed: the ratio of the average rigid masses with respect to the rope mass m, /m and the ratio of the semi-difference mass ( 1 - )/ with respect to the average mass, d: 1 d (1 d ); (1 d ) 1 (1) The ongitudina vibrations (u) of the rope-piece are characterized by the differentia equation [10]: u 0 () b u where the dot denotes the differentiation with respect to the time t and the prime denotes derivative with respect to x, and b E. The soution of (1) for stationary vibrations can be determined if u has the form u X( x ) T(t ) where X is a function of x ony and T is a function of t ony. Substituting this in () gives b X T X T (3) By separating the variabes, T Acos( t ); X C sin x Dcos x (4) b b where A, B and C are integration constants. Combining the constants yieds: u C sin x Dcos xcos( t ) b b (5) where C AD and D AD. The constants C and D are determined by the boundary conditions of the rope-piece and by the initia conditions of the vibration. The boundary conditions are: 1 at x 0 : E u 0,t u 0,t at x : E u,t u,t (6) which eads to: E C cos t D cos 1 t b E C cos D sin cos t C sin Dcos cos t b b b b b (7) 3

5 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 This set of equations is compatibe and undetermined if: 1 m m tg 1 1 m m (8) where α=ω/b. The soution of equation (8) gives n eigenvaues α i (i=1,,n). Every α i defines the frequency of the system ω i in terms of the parameters 1 /m, /m, and b, so that the compete soution of () is of the form: b b (9) i i i u i i C sin x D cos x cos( t ) i i i1 Equation (8), which can be re-written in terms of parameters and d as Equation (10), gives the eigenvae α 1 corresponding to the fundamenta frequency ω 1 (ω 1 =α 1 b/). m tg m (10) d d 1 1d 1 m m m The eigenvaue α 1 which corresponds to the owest root of (10) is isted for a number of mass /m and d ratios in the tabe 1. It is evident that the first frequency of the substructure wi be infuenced by both the mass ratio and the deviation ratio. The imitation of this exact soution is the difficuty to be extended to damped and non steady state soutions. An aternative to the exact soution based on the frequency equation (10) capabe of increasing compexity and moduarity is that based on the finite eement method (FE). Tabe 1. First frequency factor α 1 for a number of /m and d ratios. /m d=0 d=0.3 d=0.5 In the FE approach the rope-piece is subdivided into a number of finite axiay oaded bar eements of ength e and inear interpoation for the spatia part X(x) of the unknown dispacement u is proposed which gives the oportunity to incude the infuence of damping, essentia for modeing rea eevator 4

6 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 systems. This drives to a finite consistent eement with two degrees of freedom characterized dynamicay [1] by the matrices [m e ], [k e ] and [c e ] as foows: 3 6 e e E e c [ m] m ;[ k ] ;[ c ] e e 6 3 (11) where c is the viscous damping coefficient in the rope that is assigned depending of the suspended overa mass and wi be defined further on. If we discretize the rope-piece shown in Figure 1 by ony one eement foowing the Finite Eement method, assembe together to the concentrated masses 1 and and express the governing equations using the matrices of the system [], [K] and [C], force vector [F] and unknown dispacements [u] we get: where: [] u [C] u [K] u F (1) m m E c [] ;[K] ;[C] m m 6 3 (13) Another coarse simpified mode for the dynamics shown in Figure 1 is based on the discrete approach in which the tota mass of the rope is redistributed into two concentrated masses on its ends and these two masses are connected with and idea equivaent spring/damper. Then we get the matrices as foows: m 0 1 E c [] ;[K] ;[C] m 0 (14) Finay, if we consider the rope massess, the matrices of the governing equations are: 0 E c 0 1 [] ;[K] ;[C] (15) We can compare these simpified modes of the substructure shown in Figure 1 with the exact soution soving: -ω []+[K] = 0 (16) which is the condition for getting the natura frequencies of the undamped system. The eigenvaue probem represented by Equation (16) can be easiy soved using estabished numerica techniques 5

7 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 (impemented in software toos such as the eig ATLAB function). The anaytica soution for the massess rope mode gives the foowing fundamenta frequency: 1 d 1 E (17) which is independent of the mass ratio /m. ω* d= Exact Soution -- + Consistent Approximations -- assess rope approximation -- o Discrete approximation d=0.65 d=0.35 d=0 /m Figure 3. The first frequency dimensioness factor ω* of the substructure shown in figure 1 against the mass ratio /m for a number of deviation ratios d and modes. It can be shown that there is a simpe cosed-form formua for the modes (14) and (15) k (18) 1 m eff 6

8 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 where k E and mode (14). m eff mm 1 m m 1 where m i = i, i=1, for mode (15) and m i = i +m/, i=1, for 3. Resuts and Concusions Figure 3 shows the dimensioness parameter ω* defined which is reated to the first frequency of the substructure ω 1 (ω 1 =α 1 b/), as a function of the mass ratio /m for a number of deviation ratios. E E m m m m *,d 1 (19) It is observed how the consistent simpification gives the best approximation in practicay every condition of mass and deviation ratios with respect to the other simpification modes. Considering the rope massess -equations (15)- or using the discrete aproximation equations (14)- gives a considerabe error in the range of mass ratio which appies to eevator appications (/m= and d=0: α 1 =0.960 and ω*=0.960*()^0.5= and by (19) ω* aprox = ()^0.5, that is, an 3.98% error), though it can be acceptabe for coarse approximations. We have studied the consistent approximation for a number of typica configurations of residentia eevators (number of passengers n P ranging from 4 to 16) in the car-frame sing which is worse case than that of the counterweight sing. We have considered both the rotary and the transationa ift drive-car frame substructures (See tabe ). The mass of the sing m has been computed according to the number of ropes n R and roping system (n:1). It is observed that the mass deviation d is aways beow 75.1% and the maximum mass ratio is above 1.46 (/m>1.46) giving an error in the fundamenta frequency in the range 0.3%-1.4% after computing equation (10) and soving equation (16). The question that must be discussed now is whether the same resuts woud arise by anayzing higher natura frequencies of the substructure. Tabe. Appication to car-frame sing of residentia eevators n P Car-frame (empty car) 1 Lift Drive +bedpate 1 =0.75 I/R Rotary ift drive n R (n:1) ρ [kg/m] m =ρhnn R (h=30m) d /m x7 : x110: x87: x x10: x x7: x x10: x130: (1:1) 4(:1) 4(1:1) 5(.1) 6(1:1) 6(:1) 6(1:1) 8(:1) 6(1:1) 9(:1) This was studied by Leung in the case of ongitudina vibrations in a fixed-free bar []. It is obvious that p natura frequencies can ony be predicted by a p degree-of-freedom mathematica mode. It is evident that the accuracy of the predicted frequencies decreases with the increase of the number of degrees of freedom. The consistent simpification is the best one eement-substructure. The appication of simpe modes to suspended rope-substructures of ony 6 degrees of freedom discrete mode of an eevator system seems to be adequate and coherent. In future work, the effect of increasing the number of 7

9 5th Symposium on the echanics of Sender Structures (oss015) Journa of Physics: Conference Series 71 (016) doi: / /71/1/01008 eements of the rope-piece using the FE method for increasing knowedge in the dynamic response of the eevator system, speciay to higher frequencies and retardation estimations, must be anayzed. Acknowedgements Authors woud ike to thank r. José Luis Yus from P Ascensores for his hep with the compiation of weight distribution data of commercia eevators. References [1] Siciia J R, Aonso F J and Herrera I 015 echanica Discrete Simuator of the Eectromechanica Lift with n:1 roping The 5 th Symposium on echanics of Sender Structures (oss015) 1- September 015 Northampton, UK. [] Leung A Y T 1993 Dynamic Stiffness and Substructures Springer-Verag London [3] Craig RR 1995 Substructure ethods in Vibration Transactions of the ASE 117 [4] WD Zhu, H Ren An Accurate Spatia Discretization and Substructure ethod With Appication to oving Eevator Cabe-Car Systems Part I: ethodoogy Journa of Vibration and Acoustics october [5] Zhu WD and Ren H 013 An Accurate Spatia Discretization and Substructure ethod With Appication to oving Eevator Cabe-Car Systems Part II: Appication. Journa of Vibration and Acoustics october [6] Kaczmarczyk S and irhadizadeh S 014 odeing, Simuation and Experimenta Vaidation of Noninear Dynamic Interactions in an Aramid Rope System Appied echanics and aterias Specia Issue: Dynamics and Contro of Technica Systems [7] Kaczmarczyk S and Picton P D 013 The Prediction of Noninear Responses and Active Stiffness Contro of oving Sender Continua Subjected to Dynamic Loadings in Vertica Host Structures The Internationa Journa of Acoustics and Vibration 18(1) [8] Stokey W F 00 Vibration of systems having distributed mass and easticity Shock and Vibration Handbook Chapter 7. Fifth Edition. Cyri. Harris. c Graw Hi. [9] Aarcón E 1990 Cácuo Dinámico Chapter 4 Cácuo atricia de Estructuras Editoria Reverté. [10] Beuzi O 1971 Ciencia de a Construcción IV Editoria Aguiar 8

UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE

UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641,