Calculation of the Track Width of Ropeways. G. Oplatka and M. Volmer 1

Transcription

1 alclation of the Trac Width of Ropeways G. Oplata and M. Voler Introdction The trac width of a ropeway is defined as the distance between the tracs that carry the cabins. In soe installations the tracs ay not be parallel to each other. This is often so in jig-bac traways. In this case the trac width will not be constant over the trac length. Figre is a setch of the featres of a jig-bac traway and illstrates the definition of the trac width and other ters. pper terinal trac wide trac rope hal rope traway axis tower contercable carrier (c abin) lower ter inal The athors Prof.dr. dr. h.c. Gabor Oplata and M. Voler are ebers of the Institte of Lightweight Strctres and Ropeways of the Swiss Federal Institte of Technology (ETHZ) Zürich ETH Zentr LEO / ILS Leonhardstrasse 7 H 809 Zürich Phone: Fax: Eail: voler@ils.avt.ethz.ch

2 Figre. Definition of the trac width of a jig-bac traway The trac width of other ropeways, sch as continosly circlating ones is defined analogosly. The pward and downward travelling cabins pass by in doble trac jig-bac ropeways and in continosly circlating ones. abins experiencing a side-wind will be displaced in the direction of the wind. This ay case the distance between the cabins travelling in opposite directions to decrease, even to the extent that a collision ay occr. The probability of sch an accident st be ept within reasonable liits by eeping the tracs far enogh apart or by other eans. Ths the cabins ay be strealined, gides at the towers ay be provided and other provisions ay be ade at the design stage. With increased trac width the length of the cross girders on the towers and the size of the terinals increase as does the cost of the whole installation. One tries therefore to eep the trac width as sall as possible. For safety on the other hand the trac width st be large enogh to avoid the danger of collision and that of deropeent at the towers.. Definition of the proble Up to now the trac widths of jig-bac traways and continosly circlating ropeways were calclated according to heristic forlas prescribed in Switzerland by the Swiss Federal Traffic Athority (BAV Bndesat für Verehr), based on the wor of Prof.Bittner of Vienna, Astria. The application of these forlas to large trac widths and new types of ropeways (sch as Fnitel, 3S) has been qestioned. Also the physical odelling of these forlas is insfficient. To jdge the characteristics of a rope span pair and to estiate the probability of two ropes toching or even cabins carried on the ropes colliding, two areas have to be considered. These are: the behavior of the rope span nder dynaic wind load. Both the spacial distribtion of the wind velocity and its behavior in tie are iportant. the effect of the dynaic pressre on the cabins. The sallest effective width of the trac will depend on the axi expected displaceent of the rope span (possibly with a shift of phase) and the clearance gage of the cabins. Rope displaceents are cased by gsts of side-wind. The pressre distribtion on rope spans cased by sch winds is still nnown. This is especially tre for long spans. An additional difficlty is that the reslting odes of oscillation have not been stdied sfficiently..3 Ai of this stdy

3 In this stdy we ai to define a ethod to deterine trac width of a ropeway as a fnction of the type of ropeway, the eteorological conditions (especially wind), the different loading conditions and the longitdinal profile. These vales st be fond at a confidence level that allows to qantify the probability of collision of two cabins travelling in opposite directions. The wind odel To find the syste behavior of a ropeway a sitable odelling of the wind forces is iportant. The Reynold nbers that occr in the air flow pattern arond ropeways indicate clearly that the flow is trblent. The flow pattern can be defined in the direction of its height soewhat ore exactly by defining a bondary layer region and a gradient region. Figre. Wind profiles for edi wind velocity blowing over grond with different degrees of roghness [] The velocity characteristic on the left ay be taen to apply in the typically rogh grond of the ontains. The thicness of the bondary layer is therefore abot 500 eters, that is, ropeways experience an air crrent in the bondary layer region. It shold be noted that the flow in the bondary layer is ostly stochastic. Methods of probability theory were therefor sed to describe the flow velocity []. This has the advantage that stochastic processes can be described as sch. They have the disadvantage on the other hand that they are prely descriptive and they do not help to explain the physical process leading to trblence. The wind velocity at a specific point in the air flow pattern is generally characterized (for instance in [3]) by its power spectr. Noralizing this spectr with the variance of 3

4 the wind velocity and its freqency gives a relative spectr of the wind velocity that is a diensionless qantity. To a good approxiation this qantity is the sae at all points of the flow pattern. The ost faos exaple of this is the Davenport spectr, as follows: f S ( f ) x = () ( + x ) where x L x = f (0) () represents a diensionless freqency. Besides () other spectra are sed, sch those called after Harris, Hino, Sii and Kaial. They each give ore or less differing vales that, differ also fro easred ones. These differences can affect the accracy of calclating trac width. Figre 3 gives a coparison of ost of the spectra sed today ETHZ 0.3 f*s(f)/siga^ [-] Hino Si i Davenport Ha rris Ka i a l [Hz] Figre 3. oparison of wind power spectra sed today Eqation () defines the excitation of the ropeway as a syste so to say copletely. Yet it is clearly the air velocity differs at different points along the rope span. In other words we are looing for a fnction that describes the effect of the wind on the rope as a fnction of strctral diensions. In other words this fnction describes the air strea 4

5 at its varios points. The air strea itself is defined ainly by the strength of the wind gst. We can iagine of the trblent air flow consisting of both: the wind and the gsts. The latter have a velocity patterns arond the edian, while they all are in the range of edi wind velocities. Figre 4 shows this odel. gst of wind (shown withot scale) Figre 4. Trblence strctre The ellipses of Figre 4 represent the size of a gst of wind. This is defined by L p 0 = ρ ( x) dx (3) Here ρ (x) is the relative spatial atocorrelation fnction of the air flow pattern. Using this definition of the size of the gst and considering typical diensions of ropeway strctres it is possible to calclate the so called aerodynaic transfer fnction of a large size rope with sag, as follows: A( f ) 4 M ( c f ) + = (4) M is the oent applied to the rope span by the edian wind velocity, c is the rope chord of the rope span, is the edian wind velocity, is a factor that depends on the size of the wind gst and shold be easred in the field. 5

6 It shold be ephasized that eqation (4) is valid for a large size rope with sag. A ropeway installation incldes also eleents with a cboid geoetry, that is, cabins. Another aerodynaic transfer fnction, called after Vicery [4] is sed for these ones. It is as follows: 4 F A ( f ) = (5) 4 3 f A + Here A is the srface area of the side of the cabin, F is the force on the side of the cabin cased by the edian wind velocity. There are other aerodynaic transfer fnctions in se, aong others those of Davenport and Rscheweyh [5]. They describe the forces on cbic strctres and also high rectanglar ones, sch as towers and chineys. Figre 5 gives a coparison aong aerodynaic transfer fnctions Vicery appa [-] Davenport Rscheweyh ETHZ f [Hz] Figre 5 A coparison of aerodynaic transfer fnctions 3 Modelling the ropeway 6

7 The behavior of the ropeway has still to be described. A rope span can oscillate in two tally perpendiclar directions. hanges in the rope travel velocity de to the drive of the ropeway, sch as eergency braing case longitdinal oscillations and ones in the vertical plane (transverse oscillations). Only transverse oscillations in the direction of the y-axis of Figre 6 affect the diensioning of the trac width. horizontal oscillation (y- axis) longitdinal osc illation (x- axis) z x y vertial oscillation (z- axis) Figre 6. Directions of oscillation of the rope span A ore profond atheatical analysis of the oscillations in a rope span with sag shows that the swinging of the rope span arond the rope chord (c) is echanically decopled both fro the transverse waves along the z-axis and fro the longitdinal waves along the x-axis. The wind cannot case higher freqency oscillations (higher odes) in rope spans as shown in Figre 6 becase the spectr of the exciting forces (see Section 4) is too narrow. Therefore the rope span ay be regarded as a swinging oscillator with several degrees of freedo. The first of these is that of the rope span itself, others ay be allocated to cabins placed in it. Figre 7 shows a possible configration with three cabins in span. Syste (rope span) Pendl f 3 The rope as pendl f f l l l l l 7 l

8 Figre 7. Model of a rope span with three cabins as a for ass pendl Note that Figre 7 shows a standing still ropeway, that is, with zero travelling speed. This siplification is perissible becase we consider only low freqency oscillations in the rope span. The general eqation of oveent of the ropeway as a echanical syste is as follows: M & + D & + K = (t) (6) Here M is the atrix of the asses, D represents the daping, K is the atrix for stiffness of the echanical syste, represents the pendl vector of the angle of the displaced syste shown in Figre 8 and (t) is the excitation by the side wind as a fnction of tie. q ds l 4 3 l rope span cabin l Figre 8. Pendl syste of Figre 7 shown anglarly displaced It is difficlt to apply eqation (6) becase the daping factor D is sally nnown, while daping has a sbstantial inflence on the displaceent of the syste. The reason for this is that according eqation (6) a force with a continos freqency spectr excites the syste. Excitation will tae place at resonance freqencies of (6) therefore, and the syste response will be highly dependent on the daping D. The vale of this, or at least of soe of its coponents, is estiated in easreents carried ot on an existing Fnitel installation (refer to Section 6). 8

9 Perforing a Forier transforation on eqation (6) reslts in the freqency dependent agnification fnction G (ƒ) of the ropeway as a echanical syste. It describes copletely the freqency dependent transfer behavior of the installation as follows: [ 4 f M + πif D + G( f ) = π K ] (7) 4 Aplitde spectr Up to now we dealt separately with the definition of the proble of wind spectra and with the transfer fnctions wind - ropeway on the one hand and the echanical behavior of the ropeway on the other. In this section we shall cobine these sing the tools of syste theory. The exiting by side wind is a stochastic process and can therefore be described conveniently as a range of freqency. Becase all partial probles as the power spectr of the wind, and the aerodynaic and echanical transfer fnctions of the ropeway are already nown, we can therefore cobine these as follows, sing syste theory: S ( f ) = G( f ) A( f ) S ( f ) A * ( f ) G * ( f ) (9) Here S ( f ) is the aplitde spectr of the syste response, and G ( f ), A ( f ) and S ( f ) are the fnctions described in section and 3. Eqation (9) is displayed cotatively as follows: 9

10 tie fnctions wind velocity (t) M(t) loading (t) response a(t) g(t) M t t t freqency fnction freqency fnction p() wind spectr (Davenport) x aerodynaic transfer fnction p(m) = load spectr x echanical transfer fnction M p( ) = spectr of the response M Probability densities Figre 9. Scheatic illstration of the calclation of the response spectr To calclate the trac width we shall se the aplitde of the oscillations. Its variance, fro eqation (9), is as follows: * = G( f ) A( f ) S ( f ) A ( f ) G ( f ) df (0) Note that is a atrix whose eleents are the variances and covariances of the response process. It has the following internal strctre: * = i n i i n n () 0

11 5 Trac width

12 The characterization of the process in Section 4 allows s to calclate the trac width of the ropeway. To do this we have to propose the trac width as defined spacing. There are soe ways to do this. Here we shall regard trac width as a state variable as illstrated in Figre 0. bondary line loaded span epty span eqilibri position withot Wind eqilibri position with Wind Wind direction U V S= tra c wid th Figre 0. Definition of trac width S in free span The trac width is the s of the two partial lengths U and V. These are the probability rates of the axi displaceent of the pward and downward traveling cabins. The trac width S is therefore defined as S = U + V () Each qantity S, U, and V consists of its edian S, U, and V and a tie dependent fnction S', U' and V'.

13 Becase sally the pward travelling cabins are not eqally loaded as the downward travelling ones, the sags of the corresponding span are different too. Prespposing sfficient difference in the loads of the cabins by strong, gsty wind one cold therefore iagine a sitation shown in Figre. Here cabins travelling in the two directions pass nder (or over) each other withot colliding. downward travelling pward travelling possibility of a collision Figre. abins passing nder each other withot colliding In the operational sitation of Figre probles ay arise on a change in speed of the installation as occrs on an eergency stop for instance. Then the rope spans oscillate with coparatively large aplitdes in the vertical plane that ay case a collision of the cabins. Becase it st be possible to stop a ropeway in any operational sitation, cross-over of the two tracs is therefore not perissible. This is indicated in Figre 0. We regard the qantities U, V, S defined in Figre 0 as rando variables characterized by probability density. The wind velocity, itself a rando variable, has an aplitde distribtion that is noral (Gassian) as shown by easreents and by the liit vale theore [], [7]. Figre shows this distribtion. 3

14 Gassian distribtion t p() U Figre Aplitde distribtion density of wind velocity It can be shown that the type of distribtion density of the inpt fnction is conserved in linear transfer systes, as considered in Section 4. This eans that the distribtion density of the aplitde of oscillations in the rope span considered st be Gassian. The Gassian distribtion is particlarly convenient to se in calclations as it is flly characterized by its edian (expectation) and the variance of the process it represents. Both the edian and the variance of the process we considered have been calclated. The variance () has been constrained to the degrees of freedo that are of interest here. We need to consider only the inflence of those cabins at idspan of continosly circlating ropeways and those at the passing point in jig-bac traways. The inflence of cabins otside these positions ay be neglected. The ltivariable probability densities indicated in Figre 0 ay be redced therefore to the eleents shown within the frae in (). Note that the nbering of the atrix eleents ay not correspond to the degrees of freedo of the echanical syste. = n n i i n i (3) 4

15 This allows s to describe copletely the copond distribtion except of the angle of dislocation, whose edian is considered separately. T P ( ) = e (4) n (π ) det This does not yet define the trac width S or its coponents U and V (Figre 0) becase (4) is dealing only with distribtion of displaceent angle. The distribtion densities for U and V are calclated with the help of (4) fro the distribtions for the pward and downward travelling sections of carrying rope. The behavior of (4) obtained for the ropes travelling in the two directions will, of corse, norally differ as the trac loadings are different. The expression (4) is sitable to transfor the rando variables U and V. Next step will be sed to transfor distribtion densities of U and V to transfor the into distribtion density of trac width S, becase it is a rando variable itself. The atheatical procedre is illstrated scheatically in Figre 3. P(, ) P(, ) S K S K K loaded span S K S Linear transforation P() P(v) S = + v V P(s) Figre 3 Scheatic illstration of the calclation of the distribtion density of S The derivation is long and we present only its reslts. The distribtion densities for U and V are: S 5

16 and p ( ) e = (5) π p v v ( v) e = (6) π v with the variances = l + D D f l + f (7) and v = l + U U f l + f (8) Here the raised indices indicate the pward (U) and the downward (D) travelling sections of the carrying rope, ƒ i stands for the sag in idspan. The variance of the trac width can now be expressed fro (7) and (8) as follows: = + s v (9) The distribtion density of the trac width is p ( s s ) s ( s) e = (0) π s The qestion now is how to calclate fro eqation (0) the trac width that is to be bilt. Eqation (0) defines only the (sall) probability that the bondary line in the configration shown in Figre 0 will be violated, given a side wind with a edian velocity. The probability of this bondary line violation can be defined (Figre 4) forally as follows: [ S asgeführt ] = W S p( s) ds () S asgef. This probability of bondary line violation shold be sitably defined. 6

17 p(s) Probability of bondary line violation Figre 4 Distribtion density of the trac width. The trac width can be written fro Figres 0 and 4 as s Trac width as bilt S S = S + S + D B () where B is the cabin width, S D is the dynaic coponent of the trac width. The stochastic processes acting on the ropeway described in Section are responsible for. S D The vale of the probability of bondary line violation S D is norally set based on practical experience. Epirical vales sed in the steel constrction indstry, in particlar for high rise chineys are based on S D = γ (3) S where the vale of γ is set at: γ = 3.5 ` (4) The probability of bondary violation will then be, fro (): W [ S γ ] = Φ( γ ) = (5) S In other words the probability of violating the bondary line shown in Figre 0 is, expressed as a percentage, 0.03%. Using eqations () and (3) we can write the trac width as 7

18 S = S + γ S + B (6) The edian vale S ay be calclated fro the edian wind velocity. The edian trac width S being the difference of the two statical displaceents of both tracs, its vale for eqal loading of the two tracs (pward and downward) is zero ( S = 0 ). With neqal loads the vale of S is sally sall copared to that of S D. For the sae of copleteness, it st be entioned that for the ropeways, where the vehicles passing towers possess transverse swinging freedo, their trac with st also be proved. The ethod forerly described can be applied in principle. Ths there are no additional difficlties for the calclation of the trac width on the tower. 6 Monitoring of oscillations of the rope span on the Fnitel in Montana Several vales have to be assed to calclate properly the trac width of a ropeway sing the calclations described above. An exaple is the daping in the syste, a qantity whose vale is not available at the design stage. Another qantity that is only ncertainly to estiate is the correlation factor that is iportant in the calclation of the aerodynaic transfer fnction (eqation (4)). In the following we shall describe the procedre of carrying ot easreents on an existing installation and estiating the issing vales of paraeters so that they can be sed in the calclation of new installations of a siilar design. The installation chosen was a Fnitel one in Montana (Switzerland). This was eqipped with instrents to easre the wind velocity and the sideways displaceent of the rope span. The Fnitel in Montana has two fields with extree length of span of abot. each. It is therefore particlarly sited for onitoring the pendl oscillation of these rope spans. The span nearer the lower terinal was chosen for the easreents, becase both the spply of power to the easring instrent and the signal transission fro it is easier fro there. The following easring concept was eployed: The rotating cp aneoeter (aneograph) installed peranently on no. tower are sed for recording the wind velocity on the rope span. A video caera is installed on no. tower to observe the rope span. When the wind velocity is high enogh the caera switches on in single frae ode to provide three iages per second of the span. Both the data fro the aneoeter and the video iages are transitted throgh a glass fiber lin to the lower terinal and stored in a personal copter. This copter is connected throgh an ISDN lin with another personal copter at the Swiss Federal Institte of 8

19 Technology (ETHZ) in Zürich. Here the stored data and iages can be called p for evalation. The easring syste is shown scheatically in Figre 5 no. tower aneograph video c aera no. tower glass fiber lin for data transfer lower terinal Violettes copter (P) for data captre ISDN - lin Swiss Federal Institte of Technology (ETHZ) in Zrich, Switzerland opter (P) for data evalation Figre 5. Measring set-p at Fnitel in Montana The iages fro the video caera are annotated in the copter in Montana for data captre with the crrent tie and date and the oentary vale of wind velocity. Each iage can be identified with the tie of day and wind velocity. The inforation gives the vales for the paraeters in the eqations of Section describing rope oveent. The sideways displaceent of the tracs can be read fro the video iages. Applying a Forier transforation to the tie fnction of the wind velocity and to the rope 9

20 displaceent gives according to eqation (7), a way of describing the whole syste free of paraeters. S ( f ) = S ( f ) A( f ) G( f ) (7) fro the aneoeter estiate of paraeter Estiate of syste daping Fro iage of the video caera The easreents are crrently rnning. Therefore there are no definite reslts yet. Figre 6 is a typical iage fro the video caera. Figre 6 Iage fro the video caera onted on no. tower of the Fnitel in Montana The syste paraeters of the installation fro this test set-p ay be sed in the design of an installation of the sae type. For the design of other types of installation 0

21 sch as jig-bac traways and continosly circlating onocables the easreents shold be repeated on a corresponding type of installation. It shold be possible to design the trac width of "Fnitel" type installations in the ftre on a scientific basis, sing the procedre described in this paper. 7. Bibliography [] Rscheweyh H. Dynaische Windwirng an Baweren ( Bände) Baverlag GbH Wiesbaden nd Berlin 98 [] Prof. H. Bühlann Wahrscheinlicheitsrechnng nd Statisti AMIV Verlag 98 [3] Zrsansi J.A. Windeinflüsse af Baonstrtionen Verlagsgesellschaft Rdolf Müller Köln Bransfeld 98 [4] Vicery B.J., Kao K.H. Drag or Longwind Response of Slender Strtres Jorn. Of the Strct. Div. Proc ASE Vol. 93 [5] Galeann T. Rscheweyh H. Unterschng winderregter Schwingngen an Stahlschornsteinen Forschngsbericht 63 Stdiengesellschaft Stahlanwendng e.v. Düsseldorf 99 [6] Schlitt H. Systetheorie für regelose Vorgänge Springer Verlag 960 [7] Rotta J.. Trblente Ströngen B.G. Tebner Stttgart