The effects of irregularities and misalignments of the guide strings on stress levels in selected load-bearing elements of a conveyance

In order that the ultimate state methods can be applied in dimensioning of the load-bearing elements in a conveyance it is required that their design loads should be determined during their normal duty cycle and under the emergency braking conditions. This study aims to verify the predicted loading conditions obtained by theoretical analyses through experiments on a real object. Prior to the actual experimental procedure, limited to measurements of stress acting in the loadbearing cables in the conveyance, the system parameters have been identified. Special attention was given to parameters of roller guides and random irregularities of the guiding strings, which give rise to horizontal displacements of the conveyance and generate the conveyanceshaft steelwork interaction forces.


Introduction
With the assumption made that horizontal displacements of lumped masses are induced by random irregularities and misalignments of the guide strings [1,2], the analysis of the skip's dynamic behaviour in normal operating conditions (at fixed hoisting velocity V 0 =const) would yield the spectral densities of interaction forces between modelled masses as well as variances of the force amplitudes.
For the adopted calculation model of the skip (a 2D framework diagram), the magnitudes of forces were established that act upon the cross-sections of the load-bearing cables. In the work [3] the bending moments were determined. Thus obtained values of forces acting upon the cable cross-sections allowed the maximal stress levels acting upon a single cable to be determined [3].
The adequacy of the theoretical analyses of system dynamics and of fatigue endurance aspects has been verified through experiments on a real object [4,5], involving stress and strain measurements in cables. Measurement results are compiled in [6]. The analysis of measurement results reveals a major discrepancy between predicted data and experimental results. The study is limited to comparing and contrasting experimental results with predicated data including: -conveyance-shaft steelwork interaction forces -stress levels in load-bearing cables in the conveyance.
Underlying the theoretical analyses are the following assumptions made: -physical parameters of the guiding elements (roller guides) are based on the manufacturer's specifications; -horizontal displacements of modelled masses are induced by irregularities and misalignments of the guide strings.
In order to find the cause of such major discrepancy of results, the Authors first identified the real parameters of roller guides and examined the actual irregularities of the guiding strings.

Roller guides are the components of the investigated hoist installation
The schematic diagram of the hoist installation considered in the experimental procedure is shown in    Elasticity factor and viscous damping coefficient of the roller guides were obtained in laboratory tests performed as a part of the research study [7]. These parameters are: Elasticity of the skip structure in the plane determined by the guides' front surface is derived from the formula [4]:

Irregularities of the guiding strings
Irregularities and misalignments of the guiding strings in the hoistway were investigated in the shaft in which the experiment was carried out, involving the measurement of conveyance-shaft steelwork interaction forces and of stress levels in loadbearing elements of a conveyance [4,6]. Guiding strings, shown schematically in Fig 1, exhibit irregularities and certain misalignment, which were the subject of analysis. These irregularities of two guiding strings in the hoistway are designated as fx 1 and fx 4 (for one guide string) and fx 2 and fx 3 (for the other guide string). Measured misalignments of the guide string are plotted in Fig 3 and 4.   Measurement data were used to determine the function that correlates the irregularities of the guiding strings for varied hoisting velocities, given by the formula [8]: where: f(t) -the original signal, ̅ ( − ) -a correlated signal delayed by τ, ( ) -correlation function Calculating the correlation functions of irregularities and misalignments of the guide strings, we proceed to obtain the spectral density functions of those irregularities (see Fig 5,6,7).
Spectral density functions capturing guide string irregularities are plotted separately for each of the guide rails, for three values of hoisting velocity. The maximal values range from 1,5mms 2 ÷4mms 2 . A thorough analysis of these plots reveals a certain dependence associated with the actual graph shape. By applying the least square method approximation, the graphs can be substituted by the function [9]: Parameters D x and α were adjusted using the MATLAB software and ω is used to denote frequency given in   ) are compiled in respective figures. Parameters D x and α were adjusted for each hoisting velocity such that the mean square error involved in the approximation procedure should be as small as possible. As the value of spectral density function S x (ω) tends to increase with increasing velocity, the function used in further analyses will be that expressed as:
The product n•D x present in Eq(4) determines the parameters of the applied random excitations whilst the expression (5) is governed by parameters characterising the investigated object.
Variability ranges of parameters n 1 , n 2 , n 3 and n are adjusted basing on the calculation procedure highlighted in chapters 2 and 3. Parameters used in the procedure were those of the conveyance being the object of experimental testing [4,7]:  The value of I 50 read out from Fig. 8 and obtained for the skip parameters listed above is found to be I 50 =18.
In accordance with (4), spectral density of stresses in load-bearing cables due to irregularities and misalignment of the guide string becomes:

Conclusions
Predicted values of standard deviations of maximal stresses acting in load-bearing cables in a skip are decidedly lower than those measured experimentally [4,6]. Such significant discrepancy (30-fold) can be attributable to the fact that theoretical considerations relied on spectral densities of irregularities and misalignments of the guide strings reported in [6], and derived from normative reference documents [9,10] having relevance to conveyances in hoisting installations. Predicted data given in this study, based on irregularities and misalignments readouts in real conditions [11], are decidedly higher, yet the discrepancy between the theoretical and experimental results is still unacceptable. Fig. 9. Diagram of the conveyance -shaft steelwork interaction force [7]. Comparing and contrasting the experimental and predicted data will have its merits as long as all assumptions and conditions underpinning the dynamic analysis are fully satisfied. Actually, results of an experiment performed on the same object yet limited to measurements of the conveyance-shaft steelwork interaction forces summarised in [7] indicate the occurrence of impact effects in the systems whilst in fact the conveyance will run smoothly on the guide rails along short hoistway sections only. This behaviour is revealed in plots of conveyance-shaft steelwork interactions shown in Fig. 9 [7], where the peaks represent forces arising due to impacts between the skip and shaft steelwork.
It appears that when hoist operators attempt to distance the guides from the guide shoes, they do so for their own reasons. In order that theoretical results could be reliably verified experimentally [12], the conveyance needs to remain in contact with the guides, which is a requirement stipulated in relevant mining regulations.