三家子煤礦主井提升設備選型設計【單繩纏繞式提升機的設計】
三家子煤礦主井提升設備選型設計【單繩纏繞式提升機的設計】,單繩纏繞式提升機的設計,三家子,煤礦,提升,晉升,設備,裝備,選型,設計,纏繞
Internal damping characteristics of a mine hoist cable undergoing non-planar transverse vibration
by A.A. MANKOWSKI*
SYNOPSIS
The work described in this paper is an attempt to increase present-day knowledge of fatigue in mine hoisting cables, particularly the internal energy loss arising from interwire/strand friction in a cable undergoing periodic non-planar transverse vibration. Such frictional energy loss is known to be one of the major influences limiting the useful working life of hoisting cables in use today, and is responsible for the large capital outlay required to maintain the high safety factors prescribed by the mining industry.
The experimental method employed identifies two mechanical characteristics of cables that are independent of amplitude and frequency, and are primarily attributed to the type of cable construction. Interest is focused on
the time rate of change of curvature as the major parameter influencing the internal damping mechanism. Empirical results confirm that amplitude and mode number play an important role in quantifying the internal losses, and also reveal that a critical radius of curvature exists below which damage due to vibration fatigue rises exponentially to potentially high levels.
INTRODUCTION
The problem of damage due to vibration fatigue continues to impose limits on winding velocities, depths of wind, and payloads in modern, deep South African gold mines. Until the mechanism of internal energy loss inherent in the transverse vibration of cables is thoroughly understood, such damage will continue to have a marked effect on the running costs and efficiency of South
African mining operations.
Two major reasons impeding engineering breakthroughs in this area are the complex nature of the internal damping mechanism and the nonlinearity of the dynamic response of the cable to time-dependent boundary conditions. To date, a purely mathematical solution to the problem appears intractable, and it has become necessary to give increasingly more serious consideration to experimental results. Accordingly, the primary objective of the investigation described here was to determine experimentally (by laboratory simulation) the internal losses of a mine hoisting cable undergoing non-planar transverse vibration of large amplitude in the spectral neighbourhood of its fundamental and higher harmonic frequencies.
The scope of the investigation was limited to the mine geometries most likely to be encountered in practice in deep South African mining operations, namely the single-drum and the Blair multi-drum winding systems. The length of cable extending from the winding drum to the headsheave, commonly referred to as the catenary, suffers the most violent transverse vibration in practice, and hence served as the section of cable to be modelled in this investigation. The symbols used are defined at the end of the paper.
HISTORICAL NOTE
The groundwork on the fundamental and analytical aspects of the internal damping characteristics of structural cable and their influence on transverse vibration was conducted in the early 1950s by Yul. The cable used was a 7-wire specimen (0,4 kg. m-I) formed by 6 helical wires stranded round a single-core wire. All the constituent wires were zinc-coated and of similar chemical composition, the nominal diameter being approximately 9,5 mm, the overall length 2000 mm, and the lay length 127,0 mm. Yu's investigation concentrated on the determination of hysteretic damping characteristics of a family of these specimens undergoing planar vibration in a state of zero tension.
Although the specifications and experimental method employed were distinctly far removed from the geometry and dynamic conditions of present-day mine hoisting cable, the following observations from that early investigation are relevant and describe the basic nature of the internal damping of stranded cable undergoing free planar vibration.
(1) The solid internal friction of the wire material is small.
(2) For practical purposes, it can be assumed that only dry friction exists (interstrand friction).
(3) The damping capacity (dissipation of energy per cycle) associated with internal dry friction is a linear function of amplitude.
(4) A critical amplitude seems to exist, above which the curve of specific damping capacity begins to rise hyperbolically.
In the past three decades, it appears that little independent research has carried Yu's pioneering efforts further in an attempt to expand present knowledge on the damping characteristics of mine hoisting cable. A number of
investigations, however, have dealt with the static and dynamic response of massive guy cables. A detailed account of developments in this field is given by Davenporf, in which he points out that Yu's conclusions clearly establish an equivalent viscous damping to be of the order of 2 to 7 per cent of critical damping. While this may be true for dry cables of simple geometry, its application to massive guy cables and mine hoisting cables is questionable on the grounds that these cables are much more complex in their construction: concentric left- and right-handed helices containing inner cores that deform
in the plastic regions (polypropylene, sisal, and hemp impregnated with bitumen-based lubrication).
Simplified models of stranded cables employing viscous damping mechanism proportional to velocity are decidedly more popular in the literature mainly because of the relative ease of formulation and solution. However, when the analyses account for tension gradients along the length of a cable in addition to internal structural damping proportional to amplitude and frequency, a nonlinear response manifests itself in the form of drag-out an jump phenomena3. These phenomena primarily describe the response of the medium to forced vibration of varying frequency passing through resonant conditions.
Vanderveldr also cites the work ofYu1, and adds that no simple model taking into account the transverse damping behaviour can be assumed. Furthermore, he contends that at least both the usual structural and viscous types of damping must be included in any analysis attempting to predict the attenuation of transverse waves that are propagated in a stranded cable. Vanderveldr surmounted this mathematical difficulty by assuming a frequency-dependent coefficient of viscous damping. In this way, and providing the excitation is periodic, any other type of internal damping mechanism present is assumed to be contained in the damping coefficient. His
theoretical and experimental results show particularly good agreement and, where relevant, are seen to complement Yu's experimental results as follows.
(a) For a metallic core, the internal damping is affected by the tensile load. (Radial forces and inter-strand stress increase with increasing axial tension so that dry-friction damping also shows an increase.)
(b) For non-metallic cores, the damping capacity increases as the axial loads decrease.
It is worth while mentioning here that, although YUl commented on the dependence of damping on amplitude, neither Davenporf nor Vanderveldr explicitly considered the effect of curvature and its time rate of change as a parameter influencing the dissipation of energy. The mathematical form of this parameter is given by Kolsky: considering a plane distortional (bulk) wave that is propagated in the positive x direction with its particle motion in the y direction, the governing equation of motion can be shown to be
with general solution
where band C are both frequency-dependent, m is the mass density, u the shear modulus, and a the shear viscosity. Attention is drawn to the last term of Equation (1), which clearly associates the time rate of change of curvature with the shear viscosity.
PRELIMINARY DISCUSSION
In Fig. 1 the envelope of a length of cable undergoing free non-planar transverse vibration in the fundamental mode is shown over one complete cycle in increments of one-quarter periods. The length of the span is 2. Land
the amplitude at mid-span is S. To within first-order terms, the mathematical curve traced out by the cable during vibration at anyone instant can be approximated by a parabolic arc having its axis perpendicular to the chord joining the supports at the boundaries. This mathematical approximation of the curve, as pointed out by Dean6, introduces errors that are small when the chord is horizontal and the sag-to-span ratio is less than 0,02. When the chord is not horizontal, symmetry is lost, and the cable will hang in the mathematical trace of a truncated catenary in its equilibrium position. However, for relatively small sag-to-span ratios, the approximation to a shallow parabolic arc is sufficiently accurate and does not introduce significant errors in the analysis. Parabolic-for- catenary approximations are frequent in the literature, particularly for the dynamic analysis of massive guy cables having inclined spans.
Boundary Conditions
When the diameter of the cable is large enough compared with the span, and the radius of curvature of the vibrating cable is sufficiently small, a local gradient in flexure stress will be set up in the cable. Depending on the type of boundary conditions, two gradients in flexure stress are possible: (i) a constant gradient and (ii) a time-dependent gradient varying with the mode of vibration.
In the following analysis, both types of gradients are considered and are the result of ball-and-socket arrangements connecting the vibrating cables to the support, the types of rotational constraints imposed at the boundaries being the sole controlling influence on the gradients.
Fig. 1-Envelope of cable undergoing free non-planar
transverse vibration in the fundamental mode
Constant Gradient in Flexure Stress
In this example the ball-and-socket joints are constrained in a manner that allows the cable to rotate about its geometric centre and revolve round the span (Fig. 2). Thus, the boundary conditions employed here allow the ball joints 3 degrees of rotational freedom within the sockets. This is tantamount to a rigid length of cable whirling round the span defined by the supports. Fig. 2
shows the circular orbit of a plane section of this cable occurring at mid-span in the y-z plane; the span here is taken normal to the page. The letter A is assumed fixed to the transverse section of cable, where, it is noted, the letter A revolves about the span and is seen to rotate about its geometric centre relative to an inertial reference fixed to the supports. The gradient in flexure stress occurring at the apex of the letter A is also shown in Fig. 2 and is seen to be constant for all time t. The indicators (c - ) and (T + ) in Fig. 2 represent the relative compressive and tensile states respectively occurring on the surface of the sections indicated as it continues its cycle.
From basic beam theory the flexure stress here is tensile owing to the fact that the apex remains at the outermost fibres of the circular section during vibration. The nature of the constraints also prevents the neutral axis (NA) from moving relative to the fixed indicator A. The constant value of the flexure stress in this example is attributed to centrifugal effects of the whirling cable combined with the bending effects.
Time-dependent Flexure Stress
The boundary conditions in this example are similar to those above with the exception that rotation about the X axis is constrained. As a consequence, the reference letter A, as shown in Fig. 3, now becomes irrotational. This fact is borne out by the unchanging vertical orientation of the letter A over one complete cycle. Furthermore, the reference of the apex of letter A experiences
a flexure-stress cycle as it completes one revolution round the span. Noteworthy here is the time-dependent orientation of the neutral axis where it is seen to rotate relative to the fibres comprising the cable. The variation in flexure stress occurring at the apex of A is plotted against time over a period of two cycles in Fig. 3. Again, the constant component of tensile stress is attributed to the centrifugal effects arising from the increase in arc length as the cable balloons to a dynamically stable configuration.
Fig. 3-Flexure stress occurring at a fixed point on the cable,
irrotational motion
EXPERIMENTAL APPARATUS
The specifications of the mine cable used in this investigation were 43,5 mm (nominal diameter) with construction 6 x 32(14/12/6 tri)F and linear mass density 8,00 kg' m-I. In Fig. 4, a length of cable is shown suspended from supports of unequal height. The boundary conditions restricted the motion of the cable at the supports to pure rotation about the central longitudinal axis of the cable. Full thrust bearings were used for this purpose.
A predetermined tension and cable geometry were obtained by a hydraulic jack positioned at the lower end and locking devices fixed to the bearing casings at the lower and upper support ends. The movement of the jack and lower support were constrained to horizontal translation in the vertical plane defined by the suspended cable. The upper thrust bearing was hinged to accommodate any desired slope and, once the inclination of the upper bearing
had been adjusted to match the slope of the cable, the bearing casing was locked into position. In this way, both thrust bearings were subject to purely axial thrust (tension) through their axial centres.
The suspended cable was excited by rotating the lower end by an electric motor, gear-reduction transmission, a series of chain drives, and a Reynold coupling. The Reynold coupling was situated between the driven end of the cable and the driving unit, and had the advantage of isolating the dynamic response of the cable from the excitation. This is desirable since mechanical feedback in the form of reflected longitudinal and transverse waves could (given sufficient build-up time) modulate the frequency and amplitude of the excitation, especially in the neighbourhood of resonant conditions.
The speed of the electric motor was controlled by a 3,7 kW three-phase variable-frequency driving unit, and monitored by an electro-optical revolution counter. The horizontal component of tension was measured by an inline hydraulic transducer, which was situated behind the lower thrust bearing and rotated with the cable. The applied torque to the Reynold coupling was determined by a mechanical equivalent of a floating field dynamometer. The motor, transmission, and chain drives were housed in a single unit, which was mounted on trunnions and, under torque reaction, this unit could rotate about the trunnion bearings and be counterbalanced by movable masses. Thus, the relative movement of the balancing masses served as an indication of the applied torque.
To rid the central dynamometer carriage of its natural pendulum-type vibration, an extension arm fixed to the carriage was immersed in motor oil, the immersed section being a flat paddle placed normal to the direction of oscillation.
Fig. 4-lsometric sketch of the layout of the test apparatus
CONCLUSION
The use of an empirical approach in this investigation gave rise to a highly comprehensive and efficient technique for quantifying the complex damping mechanism occurring in mine hoisting cables undergoing non-planar transverse vibration in all the harmonic modes. In the author's experience to date, neither theoretical nor experimental evidence based on the (irrotational-rotational) mechanical equivalence has been encountered in the literature. Because the basis of the method lies in the testing of the actual cable whose internal frictional characteristics are sought, any cable having suitable laboratory dimensions can be subjected to the dynamic test described here.
Two flexural-type damping characteristics were identified by an experimental method. The precise form that the damping took was dictated mainly by convenience in computation. It is, however, consistent with the damping encountered in analogous systems, and conforms qualitively to a type of shear-viscosity damping, or one that is proportional to the time rate of change of curvature. In the absence of a large body of concrete information
on the exact nature of damping in this situation, the type of damping characteristics identified are justified: they have the virtue by being simple and ensure that a quantitatively correct assessment of the internal power loss is achieved.
It should be emphasized that the following conclusions are based on the testing of the dynamic response of a single mine hoisting cable of fixed construction. As a result, there may be some difficulties in the application of the following observations to other mine hoisting cables having different geometric construction. However, in spite of these potential differences, certain qualitative trends can be generalized and summarized as follows.
(1) The internal power loss of a mine hoisting cable can be characterized qualitatively and quantitatively by two experimentally determined parameters: a damping capacity coefficient, Cl, and a curvature characteristic, C2. A mathematical relationship was developed that makes it possible, given these two coefficients and the dynamic environment of the cable (amplitude, span, and frequency), to assess the amount of internal power loss.
(2) A critical radius of curvature exists above which the internal power loss increases linearly with increasing amplitude-to-span ratio. Experimental evidence also shows the losses in this linear region to increase as the square of the mode number of vibration.
(3) For radii of curvature below the critical value, the internal losses rise exponentially, and no attempt was made to investigate the losses occurring in that region.
(4) For typical mine installations, the higher modes of non-planar transverse vibration have a significant influence on the undesirable effects associated with damage from vibration fatigue. This observation is based on the application of the empirical results obtained here to the dynamic conditions of an existing gold mine.
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