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Technical Document 2643May 1994
Mathematical Analysisof Actuator Forces in aScissor Lift
H. Spackman DTIC
<'-O 94-27990
Approved for public releame; diseVbulon is unlimted.
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Technical Document 2643May 1994
Mathematical Analysisof Actuator Forces
in a Scissor Lift
H. Spackman
4
NAVAL COMMAND, CONTROL ANDOCEAN SURVEILLANCE CENTER
RDT&E DIVISIONSan Diego, California 92152-5001
K. E. EVANS, CAPr, USN R. T. SHEARERCommanding Officer Executive Director
ADMINISTRATIVE INFORMATION
This work was performed for the U.S. Army, Army Missile Command, RedstoneArsenal, AL 35898, under program element 0604370A and accession numberDN308274, by the Systems Engineering Branch, Code 536, of the Naval Command,Control and Ocean Surveillance Center (NCCOSC), RDT&E Division, San Diego, CA92152-5001.
Released by Under authority ofJ. P. Bott, Head D. W Murphy, HeadSystems Engineering Branch Advanced Systems Division
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1.0 ABSTRACT
In 1985. NCCOSC began development of a tele-operated vehicle as part of theU.S. Marine Corps' 'nd-Air Tele-Robotics Systems Program. One of the requiredvehicle components i rigid, light-weight, and compact lift mechanism capable ofdeploying a surveilluice package 10 feet above the vehicle bed. The lift mechanism thatwas eventually built and implemented was a 3-level scissor lift. In order to analyze theforces throughout the lift structure, a set of mathematical equations was derived. Fromthese equations it was discovered that prudent placement of a lift's actuator can signifi-cantly reduce the forces required of the actuator and the stress levels in the adjacent scis-sor members. The purpose of this paper is to present the equations that were derived foranalyzing the actuator forces. Using these equations, a designer can quickly determine theoptimal locations for mounting an actuator and the resulting forces.
2.0 - DUCTION
In reference (1), equations are deri, ,d fo- determining the reaction forces through-out a scissor lift. To facilitate analysis, reference , _') divides the problem into two parts. Inthe first part, equations for a basic scissor structure - a scissor structure with no actuatorsand with all four bottom joints pinned to "ground" - are werived. In the second part. equa-tions for calculating the actuator forces are derived. A later section dis.asses how theseequations can be used to calculate the reaction forces throughout the scisso. structure.
The actuator force equations in reference (1) are derived assuming taj: one or bothof the actuator ends lie along the longitudinal axis of the scissor members. In practice, thisis seldom the case because increased mechanical advantage and improved fitment can beobtained by offsetting the actuator ends. An attempt was made to derive equations that arevalid for any actuator placement, but the solutions were not found until after publication ofreference (1). The purpose of this paper is to present the equations for the more generalcase.
3.0 DERIVATION OF EQUATIONS
As in reference (1), the actuator force equations will be derived assuming conser-vation of energy and quasistatic equilibrium. The first assumption implies that frictionalforces are negligible, and the second that acceleration is negligible. In the material thatfollows, an equation referred to as the fundamental equation is first derived. This equationexpresses actuator force as a function of the derivative of lift height to actuator length, i.e.,
F =f (dh
Following this. equations for dh/dl are derived for the two possible methods of mountingactuators. Substituting the equations for dh/dl into the fundamental equation gives thefinal result.
3.1 DERIVATION OF THE FUNDAMENTAL EQUATION
An n-level lift is shown in figure (1) along with the possible loads that can beapplied to the top of the lift and some dimensional nomenclature. Of the six possibleapplied loads (see figure (1)) only Hx and Hy result in work as the lift elevates, and Hxresults in work only if the platform translates in the x-direction as the lift elevates.
Because the lift itself is a significant part of the load, its weight must be includedin the derivation. Let B equal the total weight of the lift, B, and Bz the components of B inthe x, and z directions respectively, and By the component of B in the negative y direction.As with the applied loads, only B, and By result in work as the lift elevates. If the lift'smass is evenly distributed then the weight of the lift can be modeled by placing By /2 at thetop of the lift and B,,/2 at the translating side of the lift. The validity of this statement isillustrated as follows.
Consider the mass shown in figure (2). The total potential energy of the mass canbe approximated by
E = lim "gPuwAvy,Am -- 0
where g = acceleration due to gravityp = density.
Passing to the limit gives
h
E = Jgpuwvdv.
Now, if the mass of the object is evenly distributed over its height then
Wpguw= h
where W = total weighth = height.
2
Making the appropriate substitution results in
h
E = Wy4dy
0
W h2hy Y 10
wit2
Now, if the block height increases but the total weight stays the same then
AE = E2-Et
- (h2 - hi)2
This is the desired result.Figure (3) shows a four level lift with the loads that result in work as the lift ele-
vates. Remember that Hx contributes only if the platform is attached to the joints thattranslate in the x-direction as the lift elevates. In most designs this is not the case, and Hxwould be disregarded.
Now, the assumption of conservation of energy requires that work in equals workout. In other words, if F = actuator force and 1 = actuator length then
1I hI u1JFdl = , J +-!,)dA- (H, +du
Taking the derivative of both sides with respect to 1 results in
F _ + nx + ( .)x 1d (EQ 1)
But an expression for duldl in terms of dh/dl can be found as follows. From figure (1) it isclear that
h = n(D2 -u2) 1/2
dh I n (D2 - u2) 1 /2 (-2u)ddl 2 dl
-nu dii(D 2 _- it2) 1/2 dl
-n dutan 0 dl
Solving for du/dl gives
du tan0dhdl ndl
Substituting this last equation into equation I results in
F- + + n .0 (EQ 2)
This is the fundamental equation.Having derived the fundamental equation, the next step is to derive expressions for
dh/dl as a function of actuator placement and 0 (see figure 1). Two options exist formounting actuators. The first option is to attach both ends of the actuator to scissor mem-bers. In this case both ends of the actuator translate as the lift elevates. The second optionis to attach one end of the actuator to a scissor member and the other end to a fixed point.In the next two sections equations for determining dh/dl are derived for each option.
3.2 DERIVATION OF dh/dl FOR OPTION 1
Figure (4) shows an n-level Aft at several stages of deployment. The fully retractedand fully extended positions are, of course, impractical but are included to show usefulterminology. The x-y coordinates of the offset point on the positively sloping member oflevel i are given by
x = x0 cos0 + x90 sin0
y = y90 sin0+y 0cos0.
(Notice that x90 is negative in figure (4)). However, these formulas are also valid for offset
7
o LEVEL 1
0
X00
o LEVEL i
o 000
oo Y
yo v 0 LEVELn
Sy xx 90
FULLY RETRACTED PARTIALLY EXTENDED FULLY EXTENDED
Figure 4. n-level lift at vi us stages of deployment.
8
points associated with positively and negatively sloping members of any level. Now given
any two points A and B. the distance between the points is given by
12 = (xB x.4) 2 + (.B -..- ) 2
= [ (XBo-X.40 ) cosO + (xB90 - X49o) sinO] 2 + I (.BQO - '.49o) sin0 + (YBo -- YAo) cosO] 2
(acosO - bsinO) 2 + (csinO + dcoso) 2 (EQ 3)
where
Ui = XBO- Y = YB90--YAQ.90
b = xB90 - X4 9 0 d YBo-YAO •
Taking the derivative of / with respect to h gives
21d [2 (acosO + bsinO) (-asinO + bcosO) + 2 (csinO + dcosO) (ccosO - dsinO) I LOh dh'
and solving for Lhresults indI
2 1dh
dl 2 (acosO + bsinO) (-asinO ý- bcos0) + 2 (csinO + dcosO) (ccosO -dsinO)
Solving for 1 in equation (3) gives
I = [ (acosO + bsino) 2 + (csinO +dcoso) 2]11 2 .
From figure (1) it is clear that
h = nDsin0 (EQ 4)
dh = nDcosOdO
Making the appropriate substitutions results in
dh ; nscosO [ (acosO + bsino) 2 + (csinO+dcoSO) 2 ] 1/2 (EQ 5)(acos8 + bsinO) (- asinO + bcosO) + (csin0 + dcose) (ccosO - dsinO)
Equations (2) and (4) together give the final result.
3.3 DERIVATION OF dh/dl FOR OPTION 2
The equation for dh/dl for the 2nd option is derived using a slightly differentapproach. Figure 5 shows the incremental displacement of an offset point Q (presumablywhere one end of the actuator is attached). The coordinates of point Q are given by
XQ = XaQCos 0 + XQ90sin0 (EQ 6)
YOC = yo 0 Sin0flu +.Q0 (EQ 7)
Taking the derivative of both equations results in
dx( ) _Q0sin0 + XQo 0COS0
dO
"dy() =.YoS0-Yo0sin0 (EQ 8)dO QCOOYSfl
dy dYQ' dxQ'
Now, dytQ can be found by dividing -- C by -d-. Doing so results indxo d
dyQ ) YQ9O cos 0 - YQO sin 0
dx, -XQ0 sin 0 + XQgCos0
An equation for ý (see figure (5)), then, is
( YQ9ocosO -YQosinf0""a(xQ0 sin 0 - Xo9ocos0J"
10
PATH OF ACTUATOR END
1\1dll
dxQ
/ /\
LONGITUDINALAXIS OF ACTUATOR
Figure 5. Incremental displacement of actuator end.
11
From figure 5 it is clear that
dv- = siný, (EQ 9)ds
and that
ds = ___
di sin 7i-- (EQ 10)
cos (71 + 90)-1
cos( +)
Taking the derivative of EQ (4) with respect to yQ results in
dh = nDcosO dOdyc, dyQ
But, from equation 6 we have
d_.A =Idv C Y0 90cosO-yQOsinO
which when substituted into the previous equation results in
dh = nD (EQ 11)dýV( YQ9O - YQ0tan0
12
Now, we can combine equations 9, 10, and 11 to obtain
dh A dh d4v) dsC dl dyv ds dl
-nDsin
(YQ90 -.Vo0 tan 0) cos (4? +
If the other end of the actuator is attached to a fixed point, P, then
(Yo-YPIatanxo.-xp
In summary
Xo = Xo0Cos 0 + XQ90 sin0
YQ = YQ90sin0 +YQocOSO
4 = atan (YQ--YP)xo - Xp
(Y090 COS 0 - Y00 sin 0-S=atan(Y, • OYoSfl
XQ0 sin0 - XQ 90 cos0)
dh A -nDsin•dl (yQgo -yQ0tan 0 ) cos ( +4)"
4.0 CONCLUSION
A critical component of scissor lift design is the placement of the lift's actuator(s).Prudent placement can reduce the force required of the actuator(s) and reduce stress levelsin the scissor structure. By assuming conservation of energy and quasi-static equilibrium,it is possible to derive an equation that gives actuator force as a function of the derivativeof lift height to actuator length, i.e., F =.fidh/dl). This equation is referred to as the funda-mental equation in this paper, and the result is equation (2) of section 3. 1. Having foundthis equation, the next step is to derive an equation for dh/dl as a function of angular dis-placement of the scissor members. However, there are two possible options for mountingan actuator, and the derivation of dh/dl is different for each option. The first option is toattach both ends of the actuator to scissor members, and the second is to attach one end of
13
the actuator to a scissor member and the other end to a fixed point. The derivations of dl
dl for the two options are given in section 3.2 and 3.3 respectively. By combining the fun-
damental equation with the appropriate equation for dh/dl, the designer can quickly deter-
mine the optimal locations for mounting the actuator.
14
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1. AGENCY USE ONLY (Leam blank) 2 REPORT DATE 3 REPORT TYPE AND DATES COVERED
May 1994 Final
4 TITLE AND SUBTITLE 5, FUNDING NUMBERS
MATHEMATICAL ANALYSIS OF ACTUATOR FORCES IN A SCISSOR LIFT PR: CH58
6. AUTHOR(S) PE: 0604370AH. Spackman WU: DN308274
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) a. PERFORMING ORGANIZATION
Naval Command, Control and Ocean Surveillance Center (NCCOSC) REPORT NUMBER
RDT&E DivisionSan Diego, CA 92152-5001
9- SPONSORING/MONITORING AGENCY NAME(S) AND ADORPSS(ES) 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
U. S. ArmyArmy Missile Command TD2643Redstone Arsenal, AL 35898
11. SUPPLEMENTARY NOTES
12a DISTRIBUTION/AVAJLABiLITY STATEMENT 12b. DISTRIBUTION CODE
Approved for public release; distribution is unlimited.
13. ABSTRACT (MJrnsum 200 words)
In 1985, NCCOSC began development of a teleoperated vehicle as part of the U.S. Marine Corps' Ground-Air Tele-RoboticsSystems Program. One of the required vehicle components was a rigid, light-weight, and compact lift mechanism capable ofdeploying a surveillance package 10 feet above the vehicle bed. The lift mechanism that was eventually built and implementedwas a 3-level scissor lift. In order to analyze the forces throughout the lift structure, a set of mathematical equations was derived.From these equations, it was discovered that prudent placement of a lift's actuator can significantly reduce the forces required ofthe actuator and the stress levels in the adjacent scissor members. The purpose of this paper is to present the equations that werederived for analyzing the actuator forces. Using these equations, a designer can quickly determine the optimal locations formounting an actuator and the resulting forces.
14. SUBJECT TERMS 15. NUMBER OF PAGES
Surrogate Teleoperated Vehicle (STV) 21Teleoperated Vehicle (TOV) 16. PRICE CODE
telerobotics unmanned ground vehicle
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFlCATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT
UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED SAME AS REPORT
NSN 750-01-280-5500 Standard form 298 (FRONT)
21 a NAME OF RESPONSIBLE, NOIVIUAL 21 b. TELEPHONE (inck-e/ eA Co0d) 21c. OFFICE SYMBOL
H. Spackman (619) 553-6110 Code 536
NSN 754001 -210-810 Sta04=f Ionl 2O (BAC1)
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