Mathematical Analysis Of Scissor Lifts
K ILCOPY04NCN4Technical Document 1550June 1989Mathematical Analysis ofScissor LiftsH. M. SpackmanApproved for pubic release; distribution Is unlimited.k04IL
NAVAL OCEAN SYSTEMS CENTERSan Diego, California 92152-5000R. M. HILLYERTechnical DirectorE. G. SCHWEIZER, CAPT, USNCommanderADMINISTRATIVE INFORMATIONThis work was performed for the U.S. Marine Corps, Development/EducationCommand, Quantico, VA 22134, under program element number 0603635M and agencyaccession number DN308 274. The work was performed by the Advanced Technology Development Branch, Code 612, Naval Ocean Systems Center, San Diego, CA 92152-5000.Released byR. E. Glass, HeadAdvanced TechnologyDevelopment BranchUnder authority ofD. W. Murphy, HeadAdvanced SystemsDivisionFS
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGEREPORT DOCUMENTATION PAGEIa. REPORT SECURITY CLASSIFICATIONlb. RESTRICTIVE MARKINGSUNCLASSIFIED3. DISTRIBUTION/AVALABLITY OF REPORT2a. SECURITY CLASSIFICATION AUTHORITY2b. DECLASSIFICATION/DOWNGRADING SCHEDULEApproved for public release; distribution is unlimited.5. MONITORING ORGANIZATION REPORT NUMBER(S)4. PERFORMING ORGANIZATION REPORT NUMBER(S)NOSC TD 15506b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATIONLCode 612Sa. NAME OF PERFORMING ORGANIZATIONNaval Ocean Systems Center7b. ADDRESS IV4, %.aWZC)Se. ADDRESS ft. SAhaWZCv)San Diego, California 92152-50008a. NAME OF FUNIN/SPONSORNG ORGANIZATION 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERU.S. MarineCorpsI10. SOURCE OF FUNDING NUMBERSPROGRAM ELEMENT NO. PROJECT NO.BC, ADDRESS ft &i%6aW2dCod.)DevelopmentlEducation CommandQuantico, VA 2213411. TITLE (S0603635MTASK NO.AGENCYACCESSION NO.CH58DN308 274)Mathematical Analysis of Scissor Lifts12. PERSONAL AUTHOR(S)H. M. Spackman13a. TYPE OF REPORT13b. TIME COVEREDFinal16. SUPPLEMENTARY NOTATIONFROM17. COSATI CODESFIELD15. PAGE COUNT6318. SUBJECT TERMS (Culmmulnmmsfld. A/'hbtathv)GROUPSUB-GROUP. .-.h .19. ABSTRACT (14. DATE OF REPORT (Var, UwA D')June 1989TO.Sactuatois l- .umuij'm/,it-ew'.C",' ,l/',.P.[i:,)This document presents mathematical techniques for analyzing reaction forces in scissor lifts. It also presents several design issuesincluding actuator placement, member cross-section, and rigidity.,20. OISTRIBUTION/AVALABUTY OF ABSTRACT[]UNCLA88IFEDUNUMOTED21. ABSTRACT SECL4ITY CLASSIFICATIONE] SAME AS RPT22&. NAME OF RESPONSIBLE INOIDUALH. M. SpackmanDD FORM 1473, 84 JANE]DTIC USERSUNCLASSIFIED22b. TELEPHONE (iwkfAw0**)(619) 553-411083 APR EDITION MAY E USED UNTILEEXHAUSTEDALL OTHER EDITIONS ARE OBSOLETE22c. OFFICE SYMBOLCode 612UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE
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CONTENTS1.0IN T R O D U CT IO N .2.0N O M E N C LA T U R E .I3.0GENERAL SCISSOR LIFT EQUATIONS .3.1. Equations for the Basic Scissor Structure .3.1.1 Load 1: Centered Load in the Negative y Direction .43.2.1553.1.23.1.3Load 2: Moment About the z Axis .Load 3: Centered Horizontal Load in the Positive x Direction .9123.1.4Load 4: Centered Load in the Positive z Direction .193.1.5Load 5: Moment About the x Axis .243.1.6Load 6: Moment About the y Axis .26Equations for the A ctuator . .3.2.1 Derivation of dh/df Assuming Both Actuator Ends arePinned to a Scissor M em ber .30353.2.2 Derivation of dh/df Assuming One Actuator End isP inned to G round .394.0CALCULATION OF REACTION FORCES .455.0ACTUATOR PLACEMENT .476.0STRENGTH AND RIGIDITY .537.0CO N C L U S IO N .568.0BIB L IO G R A P H Y .56FIGURES1.2.3.4.5.6.7.8.9.Nomenclature: applied loads and scissor lift dimensions .Nomenclature: reaction forces .Scissor lift m odels .Basic scissor structure with load applied in the negative y direction .Freebody diagrams for load applied in the negative y direction .Basic scissor structure with a moment about the z axis .Freebody diagrams for applied moment about the z axis .Basic scissor structure with load applied in the positive x direct;ci .Freebody diagrams for load applied in x direction withthe back joints pinned . . .i234770101313
FIGURES (continued)10.II.12.13.14.15.16.17.1819 .20.21.22.23.24.25.26.27.Freebody diagrams for load applied in x direction withthe front joints pinned . . .Basic scissor structure with load applied in the positive z direction .Deflections and reaction forces for load applied in the positive z direction .Freebody diagrams for load applied in the positive z direction .Basic scissor structure with applied moment about the x axis .Freebody diagrams for an applied moment about the x axis .Basic scissor structure with applied moment about the y axis .Freebody diagrams for an applied moment about the y axis .Four level lift loaded in the x and y directions .U n ifo rm m ass .n-level lift with actuators attached between the bottom joints .C oordinates of a point on the lift .Path traveled by a point on a positively sloping scissor member .Length of an actuator attached between two points on a lift .Actuator position and displacement nomenclature .A ctuato r m od els .D etailed n-level scissor lift .P ossible actuator locations .dh28.29.30.31.32 .172020222525272731313335363840434647vsq-. 49Weightless n-level lift with actuator between left joints of level i .N orm al and tangential forces . .Sum m ary of reaction forces . .C ro ssb ra cing .49525455TABLEI.Comparison of dh/df for several actuator locations .Acc:6; 9 nn ForA.lr,: p:, ;-'-Codes48
1.0 INTRODUCTIONThe purpose of this document is to present mathematical equations for analyzingreaction forces in scissor lifts and to discuss several design issues including actuator placement, and strength and rigidity. In section 2.0 the nomenclature is presented. In section 3.1equations are derived for the scissor members whose reaction forces are not affected by theactuators. In section 3.2, equations for calulating the actuator forces directly are given. Insection 4.0 the equations from sections 3.1 and 3.2 are combined into a single method ofdetermining the reaction forces throughout the lift. Forces obtained from this analysis canbe used in selecting the appropriate material and cross-section of the scissor members, andto select suitable actuators. In the remaining sections the design issues listed above arediscussed.2.0 NOMENCLATUREFigure 1 shows an n-level scissor lift with the six possible applied loads. The letterH is used instead of F for the linear forces in order to avoid confusion with later terminology. Notice that H. and H, act in the positive x and z directions, respectively, whileHy acts in the negative y direction. The direction of H . was selected to correspond withloads normally encountered. At each joint there are also six possible reaction forces andmoments. In order to distinguish between joints, the nomenclature shown in figure 2 willbe used. The subscripts L, R, F, B, and M are used to denote left, right, front, back andmiddle, respectively. This system of subscripting was selected because the reaction forcesare usually symmetric allowing some of the subscripts to be dropped. One slight inconsistency in this nomenclature is the M subscript. All other subscripts come in pairs, ane ,obe totally consistent one more subscript could have been included to denote bottom. Thereasons for omitting this subscript are that symmetry never exists between the bottom andmiddle joints, and the bottom joints already have an excessive number of subscripts. Thereare cases in the paper where the L, R, F, and B subscripts are all dropped from the reactionforces at the bottom of the scissor resulting in X i , Yi, or Z i . The reader may mistakenlyassume that the equations for X i , Yi, and Z i also apply to the middle joints. This is nottrue. X i, Yi, and Z i apply only to the bottom joints. The M subscript is never dropped.Notice that the reaction forces at the top of the i and i-I scissors are not shown infigure 2. These forces are omitted because they are equal and opposite the forces at thebottom of the previous scissor, and therefore are not unique. Also notice that none of thepossible reaction moments at the joints are shown. The moment about the z axis is omittedbecause the joints are assumed to be frictionless. The reason this assumption is possible isbecause the radius through which the friction forces act is small and, if the joints are lubricated, the coefficient of friction is small. The reaction moments about the x and y axis' arezero for loads applied in the x-y plane. These loads include H. H', and M. It is believedthat they are also negligible for the out-of-plane loads Mr, M, and H. especially if crossbracing is used.
yIHyMYpinszslides/rollersx-level 1level 2hLeftlevel ndBack-oFront* wRight I-uFigure 1. Nomenclature: applied loads and scissor lift dimensions.
Detail AB-See detailA for loadsi-iXBU-1MRi-1YF-ZU-'ML'-M-IFi1ZU1X -i-XB i-1XF i-1ZMRi-1ZBR1ZFRi-1See detailB for loadsYMUMRiYFYFIYZBRi%L-FZF-XMiXFUZMRIZFRiDetail BZBUXBUReactionX-ForceY-ForceZ-ForceMUiiForcein pos. x dirin pos. y dirin pos. z e 2. Nomenclature: reaction forces.3Level1 thru nXMLI
3.0 GENERAL SCISSOR LIFT EQUATIONSAn n-level scissor lift with a single actuator in the i and i l levels is shown in figure3a. One possible way of calculating the reaction forces throughout the lift is to begin atthe top of the lift where the applied loads are known, and, using equations of static equilibrium, solve for the reaction forces in the first scissor (level 1). The forces at the top of thesecond scissor are now known since they are equal and opposite the forces at the bottom oflevel 1, and the forces in the second level can now be calculated. This process continues tolevel i. At this level there are more unknowns than equations because the actuator adds anunknown variable. The analysis now shifts to the bottom of the lift. The reaction forces atthe bottom of the lift can be found by doing a freebody analysis of the entire lift. Once theforces at the bottom of the lift are known, the reaction forces from level n to the bottom oflevel i l can be calculated. Now that the forces are known at the top of level i and at thebottom of level i 1, the remaining reaction forces can be determined using equations ofstatic equilibrium.level121n2n-Xa.modelleveloriginallift levellevelb.Figure 3. Scissor lift models.4i 21-i 11Xc.
This method of solution is almost impossible to do by hand because any error willbe carried on to later calculations. Also the number of calculations is overwhelming. It istherefore desirable to find a simpler method of analysis. Before developing this alternatemethod, two critical observations about the lift are necessary. First, notice that the reactionforces in levels i through i-I and in levels i 2 through n are completely unaffected by theplacement of the actuator as long as it is confined within the i and i l levels. In fact, thescissor members in levels 1 through i-I can be modeled as a scissor structure that has noactuators and is pinned to "ground" at all four bottom joints as shown in figure 3b. Thisstructure will be referred to as "the basic scissor structure" throughout this paper. Thisthought process can be continued by considering levels i 2 through n. Notice that thesemembers meet the criteria of a basic scissor structure except that the structure is upsidedown. This portion of the lift can therefore be modeled as the basic scissor structure shownin figure 3c. In this model the lift will have negative weight. The loads at the top of thismodel are easily determined by doing a freebody analysis on the entire lift. The second critical observation is that if frictional losses are assumed to be negligible, then the principle ofconservation of energy applies. This allows the actuator forces to be calculated directly.In section 3.1 general equations that give the reaction forces in the basic scissorstructure are derived. In section 3.2 equations for calculating the actuator forces arederived. A discussion about the proper application of these formulas is given in section 4.0.3.1 EQUATIONS FOR THE BASIC SCISSOR STRUCTUREThe possible loads on a scissor lift were illustrated in figure 1. In this section reaction force equations for the basic scissor structure are derived separately for each load.3.1.1 Load 1: Centered Load in the Negative y DirectionThe first load that is considered is H ,. This load can be made more general byincluding the distributed weight of the scissor lift. Let B equal the weight of the lift. If thelift is on an inclined surface then the weight of the lift will have components in the x, y,and z directions which will be denoted by BV, BY, and B., respectively. Positive B is inthe negative y direction whereas positive B and B are in the positive x and z directions,respectively. Let HY0 equal the applied load at the top of the lift, and let H,.i equal H .0 plusthe weight of the lift in the negative y direction up to and including level i. If all the levelshave the same weight and this weight is denoted by b. then,H0(V)H o ibxwhere,b.,B,,In 5
This loading condition is shown in figure 4. A platform is shown at the top of the structurebut the type of connection between the platform and the top joints is not shown. It isassumed that either the front or back joints are attached via slides/rollers and the othervia pins. The joints that are attached via slides/rollers cannot support any load in thex direction. This assumption is used for all loads except for load 6.Because the load is centered and vertical, the reaction forces on the left side of thelift are identical to the right side. Therefore, only one side of the lift needs to be analyzed,and the L and R subscripts can be dropped. Also, if the scissor lift is rotated 180 degreesabout the y axis, the problem is exactly the same. This implies that the reaction forces aresymmetric about the y-z plane. In mathematical terms this means thatYBi YFiXF, - "i,XB,YMi Z 0.Figure 5a shows a freebody diagram of the right side of the lift. The equation for Yi isderived using principles of static equilibrium as follows.From figure 5a,IFY 0,21- ib22 H-i4Yi -H)- 0 iby44The reactio,' forces acting on level i are shown in figures 5b and 5c. Referring to figure 5c,E 0,d-XRi -sin 0 -42XXB ,I--- cos2 Hi44Id-2cos 0 XB, I -d2sin 0 0iIHvi" 4 tan 6This equation only gives the change of XB with each succeeding level. In order to find thegeneral equation for XRi letH Hi I4 tan 06
HYOlevel12Figure 4. Basic scissor structure withload applied in the negative y direction.HyoHyo44 levelxBi-12BiHyiIHyi 1Hyi-1444II.Ii'XB\\P"byYIYIa.4d%b.c.Figure 5. Freebody diagrams for load applied in the negative y direction.
If the equation is written for all the levels up to i, the following set of equations is obtained,Xsi - XBi- I' f(i) ,XBi I-XBi2:f(i- 1),XBI - XB0 f(l).Notice that if the above equations are added together, all of the terms on the left of theequal sign cancel except for XBi and XB0. Performing the addition results iniXBi-XBo I f(k),k 1Hyk Hvk I4 tan 0k IAnd from equation IXBl- X 0 - vH. O kb. H 1.0 (k - l) bl.4 tan 0k1l 2aa(Hv,k ! 0(2k - ) by4tan0tan0(- ,oib -iHy2 tan 0" k-2 tan60iHb2tan04 tan 0 b4tan 0,/i(i 1)2i2 biiH,2tan0Hk-4 tan 0k IH, 4tan 0bl2 ()I'(2 12 tan 08
Since one pair of joints at the top of the lift is connected to rollers, XB0 equals zero andix (Hyo ib )2 (2 tan 0Solving now for XMi from figure 5cIFx 0,XM,-XBi - XBi-1gMiXB, XBi-,X0I H2(2 tan 0 ' 4 tan(2i - 1)HO (2i 22 tan 0In summary(2 02.Y1 -14 tan 0 )- 2i I) b,4 tan 0(, i-XFi XBiYi HvO(HrO 2tan6 ib ,,4YMi - )H2 tan 0(2i2 -2i 1)b,4 tan 00 .Z i Z i 0It should be noted that the reaction loads are completely independent of the lengthof the scissor members and only depend on the applied load (including the distributedweight of the lift) and the angle of the scissor members from horizontal.3.1.2 Load 2: Moment About the z AxisThis loading condition is shown in figure 6a. In order to analyze this case, a criticalobservation needs to be made. Typically, the front or back joints at the top of the lift arepinned to the ,platform, and the other pair is attached via slides/ rollers. Because of this,only the pinned pair can support loads in the x direction. Since the applied load is amoment and has no net linear force, and because of symmetry, the resulting forces in thex direction at the pinned joints are zero. A moment can be represented by a force couplewhere the two forces are equal and opposite but not colinear. Because the forces on the top9
joints of the lift can have no x component, the couple used to represent the moment mustbe composed of forces acting in the y direction as shown in figure 6b. The equation relatingFto M, isMz Fdcos 0,F-Mz(2)d cos 0Since the load acts in the x-y plane the reaction forces on the right side of the platform areidentical to the left side, and the L and R subscripts can be dropped. The loads on one sideof level I are shown in figure 7.FMzFlevelIIIIII1Ma.b.Figure 6. Basic scissor structure with a moment about the z axis.F/2F/2F/2 M1R t-YBXFlXFlF1YF1a.b.Figure 7. Freebody diagrams for applied moment about the z axis.10
Referring to figure 7,XMR 0F-2d cos O YFI d cos 0YFIY-0F2- 0 YBIYsl F2 YF1F20YFIF2I M, 0dFd--0CCOSYFI-os dXFI-sin00XFI 0.IF, 0XB1 XFXB 0 0 .The above equations show that the forces at the top of level two are identical to theloads at the top of level one. This implies that reaction forces at all levels are the same aslevel one, and only the forces at level I need to be determined. The remaining equations arederived as follows.IFY0F- YMI YF 0FYMI--YFI F.II
XF, 0XMI XFI 0XMI -XFI 0.Since the reaction forces are the same for all levels, the general equat
The analysis now shifts to the bottom of the lift. The reaction forces at the bottom of the lift can be found by doing a freebody analysis of the entire lift. Once the forces at the bottom of the lift are
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