tecnical note 3 general articulation angle stability model for non-slewing articulated mobile cranes on slopes * J Wu, L uzzomi and M Hodkiewicz Scool of Mecanical and Cemical Engineering, University of Western ustralia, ert, Western ustralia BSTRCT: In recent years tere ave been reports of a number of non-slewing articulated mobile (NSM) crane tipping accidents in ustralia. Recent work by te autors explored te stability of NSM cranes at two specific articulation configurations. Tis sort note develops a general static model for te stability of NSM cranes on slopes wit various orientations. KEYWORDS: Static stability; rollover; safety. REFERENCE: Wu, J., uzzomi,. L. & Hodkiewicz, M. 04, general articulation angle stability model for non-slewing articulated mobile cranes on slopes, ustralian Journal of Mecanical Engineering, Vol., No., pp. 3-38, February, ttp://dx.doi.org/0.758/ M-09.04... NOTTION FL FR RL RR b C D m b f r tot contact point between ground and left front tyre contact point between ground and rigt front tyre contact point between ground and left back tyre contact point between ground and rigt back tyre alf track widt [m] centre of gravity (CO) of te total crane including te load distance from te tot vector to te overturn line [m] weigt force of te load [N] weigt force of te boom [N] weigt force of te anterior body and te front drive axle [N] weigt force of te posterior body and te rear drive axle [N] total weigt force of te crane including te mass [N] * aper M-09 submitted 6//; accepted for publication after review and revision 6/08/3. Corresponding autor ndrew uzzomi can be contacted at andrew.guzzomi@uwa.edu.au. b f r l boom lifting point eigt [m] boom CO eigt [m] anterior body CO eigt [m] posterior body CO eigt [m] longitudinal distance from te front drive axle to te centre of articulation joints [m] l longitudinal distance from te rear drive axle to te centre of articulation joints [m] l 3 longitudinal distance from te anterior body CO to te front drive axle [m] l 4 longitudinal distance from te posterior body CO to te rear drive axle [m] l 5 longitudinal distance from te load CO to te front drive axle [m] l 6 longitudinal distance from te boom CO to te front drive axle [m] l 7 distance between te load CO to te boom lifting point [m] O projection middle point of front drive axle O projection middle point of rear drive axle W b weelbase [m] slope gradient [ ] articulation angle [ ] angle between te front drive axle and slope gradient line [ ] Institution of Engineers ustralia, 04 ustralian Journal of Mecanical Engineering, Vol No
3 general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz INTRODUCTION Non-slewing articulated mobile (NSM) cranes are common in te ustralian construction, manufacturing and mining industries. Tey are used for general purpose pick-and-carry of eavy components because of teir ig-manoeuvrability. typical NSM crane, sown in figure, is comprised of: an extendable and luffing boom (luffing permits te boom to be raised and lowered) an anterior (front) body a front drive axle wit tyres a posterior (rear) body a rear drive axle wit tyres. Te anterior and posterior bodies are connected by two co-linear articulation joints. Steering is acieved troug a cange of te articulation (yaw) angle (, normally between 40 (left turn) and 40 (rigt turn)) of te two bodies of te veicle using two symmetric ydraulic actuators. Te front and rear drive axles are connected to te bodies wit a pair of semi-elliptic leaf springs. Wen in te lifting and carrying mode, te front axle is normally locked rigidly to te anterior cassis. NSM cranes are mainly designed to be operated on firm, flat and level ground (to witin % gradient). However, tey are allowed to be operated on side slopes of up to 5 (8.75% gradient) wit reduced rated capacities. But construction, manufacturing and mining sites seldom present suc perfect conditions. More importantly, no control device for te stability of NSM cranes as been developed. number of accidents involving rollover of suc cranes ave been reported in recent years. Tis led te autors to begin a teoretical investigation into te stability of te NSM crane (Wu et al, 04). Tat work considered two specific orientations of across te slope wit = 90 or 70 at te maximum articulation angle = ±40. Te model was owever limited. ltoug it sufficed in suggesting tat locating te articulation joints at te centre of te weelbase is not ideally suited to te roll-over stability of te NSM cranes, te model cannot be used to investigate stability of a general orientation. Tus tis note presents a general static model wic allows te stability of te NSM cranes wit different orientation ( between 0 and 360 ) and different articulation angles ( between 40 and 40 ) on a slope ( variable) to be investigated. METHOD Tis paper considers te stability of a NSM crane on an inclined surface (slope) under static conditions. Te approac taken to model te stability of te NSM cranes focuses on determining te geometric relationsip between te combined CO (C, denoted as ) of te crane including te load and te overturn lines (OLs) wit different orientation angles () from 0~360. Under tese limited conditions, it is assumed tat instability occurs wen crosses te nearest OL downslope, given by D in figure. 3 MODELLIN In developing te teoretical model, te following assumptions are made (Wu et al, 04): Te articulation joints are at te centre of te weelbase and only permit te yaw degree of freedom. Figure : Scematic representation of a NSM crane wit component labels. ustralian Journal of Mecanical Engineering Vol No
general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz 33 Figure : Scematic of a NSM crane on slope wit orientation angle () from 0~360. (a) (b) Figure 3: Scematic of a NSM crane across a side slope wit individual CO locations, geometry and coordinated systems indicated. Te crane bodies are assumed symmetric and rigid and can be geometrically represented as lines and concentrated point masses. Tus te COs of te boom, te anterior and posterior bodies are located on te mid plane of te straigt crane resting on orizontal ground. Bot te front and rear axles are assumed to be rigidly connected to te veicle s bodies respectively. It is expected tat accounting for tyre and/or suspension deflection may increase instability on an inclined flat slope and tus te model likely overstates te true stability. Te slope is assumed to be firm and flat, but not necessarily orizontal. Te instability is caused statically by te combined CO moving across an OL. Te scematic for a non-straigt NSM crane across a slope is sown in figure 3. ustralian Journal of Mecanical Engineering Vol No
34 general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz From figure 3(a), it is possible to define te individual CO positions of te crane elements in coordinate system O Y Z. Tus, for te supported load: m ( ll5)cos l7sin m 7 cos Y l ( ll5)sin m Z Te boom: b ( l l6)cos b Y b ( ll6)sin b Z Te anterior body: f ( 3)cos l l f Y f ( ll3)sin f Z Finally, for te posterior body: r ( l l4)cos( ) r Y r ( l l4)sin( ) r Z Since c = ( i i /), Y c = ( i Y i /) and Z c = ( i Z i /) (Kleppner & Kolenkow, 973), te coordinate of te combined CO (C) in coordinate system O Y Z can be defined as in equation (5), below. Te coordinates of eac tyre contact point in coordinate system O Y Z are as follows. For te front left tyre: FL l cos bsin FL 0 Y ( l sin bcos ) FL Z For te front rigt tyre: FR l cos bsin FR 0 Y ( l sin bcos ) FR Z () () (3) (4) (6) (7) For te rear left tyre: RL ( l cos( ) bsin( )) RL 0 Y l sin( ) bcos( ) RL Z For te rear rigt tyre: RR ( l cos( ) bsin( )) RR 0 Y l sin( ) bcos( ) RR Z s te governing OL canges depending on te different orientation angle () and te articulation angle (), it is important to define te limit angles wic define te zones corresponding to eac OL. To acieve tis, te orientation angle () wic results in te CO being in line wit eac down slope tyre is determined. Tis defines a region of applicability for eac OL. Hence, FL, FR, RL and RR can be defined as in equations (0) to (3), next page. Terefore, te OL wit different orientation angles can be represented as: OL OL OL OL OL OL 3 4 (0, RL) ( RL, FL) ( FL, FR) ( FR, RR) (,360 ) RR (8) (9) (4) ccording to figure 3(b), it is possible to transfer te coordinates of te combined CO (C) and OLs from coordinate system O Y Z to coordinate system O Y Z and project on plane O Z as: C cos Ysin C C C C Z Z (5) Ten project OL, OL, OL 3 and OL 4 from coordinate system O Y Z onto O Y Z. From te projected points, eac OL can be determined as: OL K M (6) OL K M (7) OL K M (8) 3 3 3 3 m(( ll5)cos l7sin ) b( ll6)cos f( ll3)cos r( l l4)cos( ) tot C m( l7 cos ) bb ff rr C Y tot C Z m( ll5)sin b( l l6)sin f( l l3)sin r( l l4)sin( ) tot (5) ustralian Journal of Mecanical Engineering Vol No
general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz 35 ( l l )sin b r 4 tot FL arctan m( ll5) b( ll6) f( ll3) r( l l4)cos totl ( l l )sin b r 4 tot FR arctan m( ll5) b( ll6) f( ll3) r( l l4)cos totl ( l l )sin ( bcos l sin ) r 4 tot RL arctan m( ll5) b( ll6) f( ll3) r( l l4)cos tot( lcos bsin ) r 4 tot RR arctan m( ll5) b( ll6) f( ll3) r( l l4)cos tot( bsin lcos ) ( l l )sin ( bcos l sin ) (0) () () (3) OL K M (9) 4 4 4 4 were: K M K cos sin cos l sin cos( ) sin( )cos Table : Example NSM crane values as used by Wu et al (04). m = 440 kg l = 950 mm = 6000 mm b = 000 kg l = 950 mm b = 400 mm f = 4000 kg l 3 = 500 mm f = 800 mm r = 0,000 kg l 4 = 500 mm r = 800 mm tot = 0,40 kg l 5 = 3600 mm = [ 360, 360 ] b = 0 mm l 6 = 500 mm = 5 l 7 = 000 mm = 0, ±0, ±40 M l sin( ) ( lcos bsin l)sin ( bcos lsin b)cos K3 [( l sin bcos b)sin ( l cos bsin l )cos ]cos ll sin ( ll) bcos b sin M3 ( l cos bsin l )cos ( l sin bcos b)sin ( lcos bsin l)sin ( blsin bcos )cos K4 [( l sin bcos b)sin ( l cos bsin l )cos ]cos ll sin ( ll) bcos b sin M4 ( l cos bsin l )cos ( l sin bcos b)sin In terms of static instability, te distance downslope from C to te individual OL is of interest. Tis distance for different orientation angles can be represented as: Figure 4: Scematic representation of configurations according to table. For typical NSM cranes [ 40, 40 ] and [0, 5 ]. M CZ C (0, ) RL K M3 CZ C ( RL, FL) K3 M CZ D C ( FL, FR) K M4 CZ C ( FR, RR) K4 M CZ C ( RR,360 ) K (0) 4 DISCUSSION Equation (0) is te general equation to determine te distance from te combined CO to te critical OL for different orientation angles. For bot a qualitative and quantitative understanding of tis equation, te example parameters from te previous work (Wu et al, 04) are used and reported in table. Tese values were cosen as tey seemed representative of tese type of cranes (Wu et al, 04). Figure 4 accompanies table. Using tese input values and logging te cases in MTLB, te relationsips ustralian Journal of Mecanical Engineering Vol No
36 general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz between te distance (D) from combined CO to te nearest downslope OL and te orientation angle () were calculated and are reported in figure 5. Figure 5(a) sows ow te stability of te NSM crane is related to te individual OLs. Te cosen case is for = ±40 and = 5. s expected te grap is symmetric about te = 0 (360 ) orientation. Under te conditions wen te NSM crane is on te slope wit te: anterior body iger tan te posterior body at 40 articulation angle ( = 90 wit = 40 or = 70 wit = 40 ), te distance from te combined CO to te OL (D) is 843. mm. Tis is te same as te result from figure (c) in Wu et al (04) wen l =.95 m (l = l/). posterior body iger tan te anterior body at 40 articulation angle ( = 90 wit = 40 or = 70 wit = 40 ), te distance from te combined CO to te OL (D) is 94 mm. Tis is te same as te result from figure (c) in Wu et al (04) wen l =.95 m (l = l/). Hence, te general metod developed ere replicates te specific cases of Wu et al (04). Figure 5(b) sows te effect of articulation angle on te stability of te NSM crane on a 5 gradient slope. In particular te: stability of te NSM crane decreases sligtly wen te articulation angle () canges from 0 to 0. Te stability decreases markedly wen te articulation angle () canges from ±0 to ±40. orientation angle () of 80 is te critical angle for te tip-over stability. Te critical angle for te roll-over stability is dependent of te articulation angle (). Terefore, te orientation condition wen te crane is across te slope wit te articulation angle ( 0 ) cannot be regarded as (a) (b) (c) Figure 5: Distance from CO to OL as a function of orientation angle for (a) = ±40 on = 5 slope sowing critical OL zones; (b) for different on an = 5 slope; and (c) = 40 and canging slope gradient. ustralian Journal of Mecanical Engineering Vol No
general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz 37 te critical condition to ensure tat crane will not roll over during te operation. Figure 5(c) explains te relationsip between te slope inclination angle on te stability for a NSM crane articulated to ±40. s te slope gradient increases te stability reduces, eventually reacing te limit of stability at 9. It is noted tat te model does not take into account dynamic effects and tyre and/or suspension flexibility. Tese effects would be present in actual operation and would likely reduce te stability furter and ence result in a lower. Wen interpreting figure 5, te apparent stability of te crane at orientation angles near zero degrees would likely be overstated. rollover event is still possible in suc situations sould a sufficient lateral force be applied. Suc a force would arise, for example, from dynamic sifting of load and or cornering forces. It sould also be noted wen interpreting te graps tat te sape of te graps provides an indication of te sensitivity of te stability around te point of interest. One sould terefore not focus just on te minimum value of te plot. 5 SUMMRY ND FUTURE WORK In tis note a general model for te static stability analysis of NSM cranes on inclined slopes was developed. Equation (0) is te general equation to determine te distance from te combined CO to te OL measured downslope. Te results of tis work suggests tat te orientation of 90 (70 ) corresponding to a straigt NSM crane being across te slope, cannot always be regarded as te critical condition leading to rollover during operation. Instead, te rollover stability is a function of te articulation angle (). It sould owever be noted tat te metods developed in tis work are limited to static cases. Future work sould focus on te development of a general dynamic model wic permits stability analysis on flat and inclined slopes and allows te CO to cross any of te four OLs, not just te downslope one. Suc an approac would facilitate te development of stability index based on te minimum value of te perpendicular distance from te CO to eac OL. Tis work sould be informed by te stability index approaces used in te general field of veicle dynamics. REFERENCES Kleppner, D. & Kolenkow, R. 973, n Introduction to Mecanics, nd edition, Mcraw-Hill. Wu, J., uzzomi,. L. & Hodkiewicz, M. 04, Static stability analysis of non-slewing articulated mobile cranes, ustralian Journal of Mecanical Engineering, Vol., No., pp. a-b, ttp://dx.doi.org/0.758/ M-08.04... ustralian Journal of Mecanical Engineering Vol No
38 general articulation angle stability model for non-slewing... Wu, uzzomi & Hodkiewicz JIN WU Jing Wu is a mecanical engineer at Westrac ty Ltd. His researc interests are in mining equipment, and articulated mobile macinery design and modification. He completed a B(Hons) from Tongji University in Mecanical Engineering and ten a MEngSc in Mecanical Engineering at te University of Western ustralia. NDREW UZZOMI ndrew uzzomi is an assistant professor in te Scool of Mecanical and Cemical Engineering at te University of Western ustralia (UW). He obtained is BE (Hons) and D in powertrain dynamics from UW. He as completed a UW ostgraduate Teacing Internsip and been a UW Witfeld Fellow. From 007 to 00 e was a postdoctoral researcer at te University of Bologna (Italy). Tere e worked in te departments of mecanical engineering and agricultural engineering. t te end of 00 e was invited back to UW as an Honorary Researc Fellow. His researc interests include kinematics, dynamics, vibration and mecanical design, and, in particular, teir application to agricultural engineering. MELIND HODKIEWICZ Melinda Hodkiewicz is a professor in te Scool of Mecanical and Cemical Engineering at te University of Western ustralia (UW). Her researc interests are in asset management, maintenance and prognostics. Se completed a B(Hons) from Oxford University in Metallurgy and Science of Materials, followed by 0 years as an engineer in te mining industry and ten a D in Mecanical Engineering at UW. Se eads te Engineering sset Management rogram at te UW and sits on te ISO committee for sset Management Standards. ustralian Journal of Mecanical Engineering Vol No