Open Journal of Applied Sciences, 2013, 3, 6569 doi:10.4236/ojapps.2013.31B1013 Published Online April 2013 (http://www.scirp.org/journal/ojapps) A Geometrical Theorem about the Static Equilibrium of a Commonpointforce System and its Application Guoquan Zhou Department of Physics, Wuhan University, Wuhan, Hubei, China Email: zgq @whu.edu.cn Received 2013 ABSTRACT A geometrical theorem for the static equilibrium of a commonpointforce system has been proven by means of vir tualwork principle: The equilibrium point of a commonpoint force system has a minimal weighted distance summa tion to every fixed point arbitrarily given on each force line with a weighing factor proportional to corresponding force value. Especially the mechanical simulating technique for its inverse problem has been realized by means of pulley block. The conclusions for the inverse problem derived from mechanic method are in accordance with that given by the pure mathematical method, and the selfconsistence of the theorem and its inverse problem has been demonstrated. Some application examples in engineering, economy and mathematics have been discussed, especially the possible ap plication in the research of molecular structure, has also been predicted. Keywords: Spatial Commonpointforce System; Static Equilibrium; Minimum Theorem；Principle of Virtual Work; Depot Problem; Transportation 1. Arising of the Problem and its Mathematical Model The socalled depot problem [15], one of the edging subject in the field of transportation science and modern architectural economy, is related directly to the economic efficiency and the industrial and commercial interests. The suitably chosen “depot” can save much man power and working time, decrease unnecessary loss of material resources. Recent years, with the fast development of mathematics, computer technology and operational re search, some methods have been found to solve the multidepot vehicle routing problem, such as the Integer Programming Model combined with the Hybrid Genetic Algorithm, the Nearest Neighbor Heuristic Method, and the Tabu Search Method based on Adaptive Memory Principle[15], and so on. One of the basic problems is just like the following example. Provided 12 P,P,, P are N disperse working sites for vehicle transportation, the possible location of the main traffic station P has mean i times of toandfro transportation between P and () each month, then which location P can have a minimal weighted distance summation to every working site and minimize the total fuel consumption ? n Pi1, 2,,iN This question can be expressed as a mathematical question: to look for a point P which can minimize the double weighted distance summation: 1 1122 2 2() N ii i N SnPP nPPnPPnPP (1) From the viewpoint of fuzzy mathematics, number i’s the weighing factors appeared in above ques tions are average values with respect to time and needn’t to be integers, they can be any positive real numbers. n A solution to such a problem is found to be related to a mechanical model—the static equilibrium of a com monpointforce system (CPF system for brevity), which is based on the principle of minimal potential for the static equilibrium of a conservative mechanical system in the gravitational field. Another equivalent method of virtualwork principle under ideal constraint is an alter native solution to the inverse case of above problem. In the following section, we will throw light on such a fact that static equilibrium method of CPF system can solve the problem of least weighted distance summation for both two and three dimensional cases. It can also be used to deal with cases involved in not only the straight path but also the blocked or curved path. 2. A Geometrical Theorem for the Balanced CPF System Based on the principle of virtual work, a geometrical theorem for the static equilibrium of a CPF system can Copyright © 2013 SciRes. OJAppS
G. Q. ZHOU 66 be easily proved. Then the correlation between the theo rem and above problems will be immediately established after a suitable mechanic model has been constructed. Theorem The equilibrium point of a commonpoint force system has a minimal summation of weighted dis tance from every fixed point arbitrarily given on each force line, with a weighing factor proportional to corre sponding force value. The above theorem means, for N balanced forces i ) with common point P and given values i respec tively (, and f is a common factor with force’s dimension) and for N fixed points nf (1 1, 2,iN ,2, N i in each force line respectively, the equilibrium point P of a CPF system must minimize the weighted distance Pi summation , i.e. for any other point 1 N i i PSn PP i P, min 11 NN ii i ii SnPPS nP i P ) (2) where satisfy following relation: (1,2, i ni N 12 12 :: ::: :FF F N nn n (3) By means of virtualwork principle about stable equi librium of a multiforce system, the above theorem can be easily proven as follows. For an arbitrary virtual displacement r (from 0) of the equilibrium position P and PP r i ri of each fixed point , suppose , the vir (1,2, i Pi N) ) i PP tual work of the resultant force is zero under 1 F= F N i i ideal constraint, that is 1 0Fr Fr N i i (4a) or 1 0Fr N ii i (4b) which is duo to following fact (1,2, ,rr ii N (5) On the same time, forces (1,2,) i iN N are all centric forces that respectively direct at fixed point (), so we have i P 1, 2,i () r Fi ii i nf r (6) where F ii nf, f is the common factor with dimension of force and the dimensionless positive real constant; i n ri i r is the unit vector along with or the direction of i PP force i : according to the property of centric force[79], its work can be express as (Fr ) iiii nf r (7) Then the condition (2) can be written as: 1 () N ii i nf r 0 (8) 1 0 N ii i nr (9) This is just the necessary and in fact also the sufficient condition for weighted distance summation 11 NN iii i ii nrnPP to attain its minimum. Then the above theorem get veri fied. For the special case that (a sufficient 1, N i jji nj n but not necessary condition), the CPF system can’t attain equilibrium state by itself, then point P will move to cer tain a point i and attain equilibrium by means of forced constraint reaction at . Thus, point is just P i Pi P the wanted point P to minimize the sum 1 N ii i nPP. The theorem (2) is actually the mathematical criterion for the static equilibrium of the CPF system. On the other hand, this geometry theorem also clearly reflects the geometric property of the CPF system having attained equilibrium state, and what is more, reflects the spatial property of the concerted fields of forces. On the other hand, a stable static equilibrium of the CPF system corresponds to the case of the minimal sum, and an unstable static equilibrium corresponds to the case of the maximal sum. 3. A Mechanical Model and Solution to the Inverse Problem There had ever been some discussion in history about the socalled depot problem or factorylocation problem similar to question in section 1. It is easily found to be the inverse problem of the above theorem, and can be solved by mechanic technique. Reference [6] had given two mechanic models to solve such a geometrical prob lem in the twodimensional case, one is the model of Figure 1. The balanced commonpointforce system. Copyright © 2013 SciRes. OJAppS
G. Q. ZHOU 67 surhe face tension of liquid cake films, the other is t model of ropes with suspended weights passing through smooth holes on a frictionless table. But in threedimen sional case, both models are invalid. In order to find a feasible technique to solve the problem in the threedi mensional case, we can always proportionately minify the real distribution of N given point 12 ,, PP P to a model of miniature distribution 12 ,, PP bora tory, shown as Figure 2, then instnless fixed pulleys at each point (1,2,) i NPiN . Each rope with length of L is conneto a suspended weight i G of mass i n at one end and to the common point P ate other endand passes by the fixed pulley at i P respectively. Note that each fixed pulley has a defi position and adjustable free axis. Obviously the whole CPF system is a pure gravitationdoingwork system un der ideal constraint, i.e. a conservative system. When the system attains static equilibrium, the whole system will possess the minimal total gravitation potential energy with respect to any given horizontal plane P in la frictio y all N velcted respecti , th nite , which is chosen to be the zero plane of potential energ(shown as Figure 2). If each of the point of 12 ,, y PP P has the height of 12 ,, hh h respectively w to plane ith respect , then each particle i G has its gravitation potential ergy of () iiii ng hPGspectively (1, 2,in ). Then the total gravenergy ) en re itation potential 1 ( iiii i ng hPG N must get the minimal for the equilibrium state, and i 1 ii i ng PG N must attain maximum because 1 ii i ngh ( N is a constant. Duo t, then ) o ii i PGL PP 1 i nL PP st attain its N i mu i maxii ition is that the whole commonpointforce sy must be themal, and 1 N i nP i P minimum. The cond stem attains its static equilibrium. And the equilibrium position P is just the wanted point to make the weighted distance summation 1 i i nPP N get its minimum. i It should be pointed out once more that the weighted distance summation may attain its minimum at the boundary point such as (1,2, ) i Pi N when the equilibrium state can only beans of forced constraint reaction at i P, then the point i P is just the wanted point satisfyinge minimum conditin. Here, we give a sufficient but not necessary condition, that is: When N, point P is just the wanted attained by me tho 1, ij jj i nn es the summa i 1 ii i The condition is that the whole coorce system attains its static equilibrium. And the equilibrium position P is just the wanted point to make the weighted distance summation 1 N ii i nPP get its minimum. point that maktion the minimal. monpointf out oncghted distance summation may attain its minimum at the bo PP m N Sn It should be pointed e more that the wei undary point such as (1,2,) i Pi N when the equilibrium state can only be attained by means of forced constraint reaction at i P, t i is just the wanted point satisfying the minimum condition. Here, we give a sufficient but noecessary conditionthat is: When 1, hen the point P t n, N ij jji nn, point i P is just the wanted point that m ists had ever this mechanical analogue of the depot problem in the 2 Concrete Conclusion for the m, the abovdirectly lead to some inter method: aketion N SnPP the minimal. For many years econom misinterpreted s the summa 1 ii i dimensional case, and mistakenly thought the center of mass c P to be the wanted point to make the weighted distance summation minimal. Actually, c P, the center of mass, leads to the minimal weighted square distance summation and not the minimal weighted distance sum mation. 4. The ThreeForce System For the simple case such as 3CPF or 4CPF syste e mechanical model will esting and valuable conclusions. In field of geometry, we can prove following conclu sion by means of pure mathematical For any triangle 123 PPP with its three inner angles not bigger than 120, there must be a point P within the triangle to minimize the distance summation 123 PPPPPP when and only when 12 PPP23 3123 PPPPPP . Figure 2. The mechanical model. Copyright © 2013 SciRes. OJAppS
G. Q. ZHOU 68 The same conclusion can be drawn by means of static equilibrium of commonpoint system, and what is more, the conclusion can be extended further. For 12312 3, ::::FFFNn he sufficient and necessary co 3 nn, it is easy to deduce tndition for the equilibrium of 3CPF system, that is ①, the three forces are coplanar. ②, following equation must be satisfied: 3 12 233112 sin(, ) sin(,) sin(,) F FF FFFFFF (10) Expression [7]. Here (10) is just the Lami’s theorem (F, )F ij expresses the angle between two vectors i and F (not larger than ). According the theorem (2), when expression (10) is satisfied he wanted point to make the weighted distance summation 11223 nPP nPPnP , corresponding point P is just t 3 m P inimal. Then it means 3 12 233112 sin(,)sin(,)sin( ,)FFFF FF n nn (11) Considering 1223 31 (,)(, )(,) 2FFFF FF , we make formation 1 π( a trans 32 ,) 23 1 312 π(,) π(,) F F F (12) Then 123 , and 123 ,, can be viewed as the three inner angles of certain a triangle. According the cosine theorem, following identity holds to the sine theorem and 222222 sinsin sin cos 2sin sin2 (.. (11)) kijki jk nn n cf Here can be permutated in turn within 1,2,3. Then ijk n(13) n ,,ij k 222 j ( )πarccos 2 FF ki jk k (14) So when the CPF system attains static equilibrium, the open angle of point P with respect to ,,PPPPPP respectively ca nnn nn ， n be calculated according to (14). Specially when according to (14) 1223 31 123 nnnn 123 3 , then 122331 2π , )(,)(,)3 F FF FF (15) Meanw 123 () SnPPPPPP just coincident with that g method. This is not only ex ou that makes 1 the minimal. This conclusion is iven by pure g one of the special application ) and the me pointing eometrical amples of CPF system, but also a convincing geomet rical side proof for the above theorem (2 chanical technique. By the way, it is worthy of t, when n=3 and is the needed point 12 nnn, 2 31 P 3123 P n PPnPnP (here 10 P be the minimal PP ), or min212313 PP thod, we ca nP. n ea Sn e me P sily extend our conclusions for case of to the case 5. Discussion of a ser e principle of virtual wo ential energy are s of t littom above because of the different ex ost intere en in civil engineering and in du By means of the sam 3n of 3n. Extension of Application Fields and Some The theorem (2) and the mechanical model given by this paper can be used to deal with both the coplanar CPF case and spatial CPF case. If we replace the pulley block at points of 12 ,n PP P by n smooth and frictionless small hole on the surphe crust, an ideal con straint case, thrk and the method of minimal total pottill valid. As a re su face lt, many geometric questions on the surface of a sphere can be solved at once. Just as what has been mentioned before, theorem (2) reflects the spatial propertyhe gravitation fields near to surface of earth. On a large space in the gravitation field, the conclusion must have a le difference fr pression of gravitation potential energy, and there must be also some spatial distortion effects resulting from the gravitation of the earth. One of the msting applications of the theorem and the mechanical model may be research of molecular structure. A stable molecular structure of multiatom molecule must obey a geometrical constraint condition. Every chemical bond has definite bond strength and a bond length under static equilibrium state, each bond strength (in terms of force) equals to the gradient of the potential energy along with corresponding chemical bond. Then there must be a constraint condition among each chemical bond in view of theorem (2) under the stable equilibrium. Other possible application fields of this paper include most of the problem aris stry design such as the factory location; the city drain age system; the choice of the trifurcation point of traffic roads. In a word, the mechanical model of the static equi librium of CPF system can solve most of the preliminary design that concerned problem of the minimal weighted distance summation. Its application can spare much man power and material resource, decrease social operation cost and promote the economy efficiency. Meanwhile, it can work as a powerful tool to study some problems of mathematics, mechanics and even the gravitation field. ( F hile the corresponding point P will make Copyright © 2013 SciRes. OJAppS
G. Q. ZHOU Copyright © 2013 SciRes. OJAppS 69 ledgements 6. Acknow This work was supported by the National Natural Sci ence Foundation of China under Grant No.10775105. REFERENCES [1] H. William, T. S. H. George, J. Ping and C. W. L. Henry, “A Hybrid Genetic Algorithm for the Multidepot Vehicle Routing Problem,” Engineering Applications of Artificial intelligence, Vol. 21, No. 4, 2008, pp. 548557. doi:10.1016/j.engappai.2007.06.001 [2] Benoit Crevier, JeanFrançois Cordeau and Gilbert La porte, “The Multidepot Vehicle Routing Problem with Interdepot Routes,” European Journal of Operational Research, Vol. 176, No. 4, 2007, pp. 756773. doi:10.1016/j.ejor.2005.08.015 Vol. 153, No. 3, [3] Rubén Ruiz, Concepción Maroto and Javier Alcaraz, “A Decision Support System for a Real Vehicle Rout Problem,” European Journal of Operational Research, ing 2004, pp. 593606. doi:10.1016/S03772217(03)002650 [4] Bruce L.Golden, Gilbert Laporte and Éric D “An Adaptive Memory Heurist . Taillard, ic for a Class of Vehicle Routing Problems with Minmax Objective,” Computers & Operations Research, Vol. 24, No. 5, 1997, pp. 445452. doi:10.1016/S03050548(96)000652 [5] Emmanouil E. Zachariadis, Christo Chris T. Kiranoudis, “An Adaptive M s D. Tarantilis and emory Methodol ogy for the Vehicle Routing Problem with Simultaneous Pickups and Deliveries,” European Journal of Opera tional Research, Vol. 202, No. 2, 2002, pp. 401411. doi:10.1016/j.ejor.2009.05.015 [6] C. D. Collinson, “Introductory Mechanics,” London: ing: , 1997, pp. 5152. c., 1983. Edward Arnold Ltd., Vol. 23, 1980, pp. 4142. [7] Z. X. Zhu and Q. Z. Zhou, “Theoretical Mechanics,” Beijing: Beijing University Press, Vol. 292293, 1982, pp. 313314. [8] G. Y. Yu and G. Q. Zhou, “Electrodynamics,” Beij Higher Education Press, Vol. 89 [9] K. David, C. Field and Wave, “Electromagnetics,” New York: AddisonWesley Publishing Company, In pp. 8485.
