ON DYNAMICALLY EQUIVALENT FORCE SYSTEMS AND THEIR APPLICATION TO THE BALANCING OF A BROOM OR THE STABILITY OF A SHOE BOX

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1 Proeedings of DEC 04 ASME 004 Design Engineering ehnial Conferenes and Computers and Information in Engineering Conferene September 8-Otober, 004, Salt Lake City, Utah, USA DE C ON DYNAMICALLY EQUIVALEN FORCE SYSEMS AND HEIR APPLICAION O HE BALANCING OF A BROOM OR HE SABILIY OF A SHOE BOX Just L. Herder 1 and Arend L. Shwab Delft University of ehnology, Department of Design, Engineering and Prodution 1: HuMan-Mahine Systems Group (MMS); : Laboratory for Engineering Mehanis Mekelweg, 68 CD Delft, he Netherlands V: , F: , E: j.l.herder@wbmt.tudelft.nl ABSRAC he stability of a rigid body on whih two fores are in equilibrium an be assessed intuitively. In more omplex ases this is no longer true. his paper presents a general method to assess the stability of omplex fore systems, based on the notion of dynami equivalene. A resultant fore is onsidered dynamially equivalent to a given system of fores ating on a rigid body if the ontributions to the stability of the body of both fore systems are equal. It is shown that the dynamially equivalent resultant fore of two given onstant fores applies at the intersetion of its line of ation and the irle put up by the appliation points of the given fores and the intersetion of their lines of ation. he determination of the ombined enter of mass an be onsidered as a speial ase of this theorem. wo examples are provided that illustrate the signifiane of the proposed method. he first example onsiders the suspension of a body, by springs only, that is statially balaned for rotation about a virtual stationary point. he seond example treats the roll stability of a ship, where the metaentri height is determined in a natural way. INRODUCION he stability of a two-fore system in equilibrium an be assessed intuitively at a single glane. Stati equilibrium is ahieved when the two fores are of equal magnitude, opposite sense, and have the same line of ation. However, although all in equilibrium, the rigid bodies in Fig. 1 have different stability. Assuming onstant fores (both in terms of magnitude and diretion), it is readily seen (a more rigorous derivation will follow) that stable equilibrium results if the fores are pointing away from eah other (Fig. 1a), whereas unstable equilibrium results if the fores are pointing towards eah other (Fig. 1b). In the borderline ase, the two points of appliation oinide, rendering the system in neutral equilibrium (Fig. 1). Evidently, the point of appliation of the fores on their line of ation is of vital importane to the stability, even though it does not affet the stati equilibrium itself. In an n-fore system in equilibrium, the judgment of the stability is muh less obvious. Figure shows an example of a rigid body on whih three fores are in equilibrium. One (a) (b) () Figure 1. wo-fore system in equilibrium ating on a rigid body: (a) stable equilibrium, (b) unstable equilibrium, () neutral equilibrium. 1 Copyright 004 by ASME

2 motion of a rigid body. In matrix form, the equations of motion for a rigid body under the influene of n external fores read: F me 0 i Ar F 0 I i i r (1) Figure A three-fore system ating on a rigid body. Stati equilibrium determines the magnitude and line of ation of the third fore, but the stability is not readily assessed. strategy to assess the stability of suh systems would be to ompose fores two by two until a two-fore system is obtained, the stability of whih an then be assessed as above. Clearly, the onventional proedure of fore omposition is not suffiient, sine it does not yield the point of appliation of the resultant fore. Sine the onventional proedure is aimed at equilibrium (not at stability), it yields in fat the statially equivalent fore system: an equivalent fore for whih the point of appliation on the line of ation is not relevant. herefore, in order to find the equivalent stability, a proedure is required to ompose fores in a dynamially equivalent way, i.e. in suh a way that the stability ontribution of the resultant fore is equal to the stability ontribution of the two original fores. his implies that in addition to the magnitude and line of ation of the resultant fore, also the point of appliation is to be found. his paper proposes a proedure for the determination of the dynamially equivalent resultant fore system, for any given fore system. he treatise will be limited to the planar ase of rigid body motion. he study will not be limited to the judgment of stability of stati equilibrium. It will be shown that the proposed proedure for the determination of dynamially equivalent fores is valid for the ontribution of fores to any state of motion. SAIC EQUIVALENCE his setion will use the Newton-Euler equations of motion to investigate the ontribution of fores to the nominal state of where m is the mass of the rigid body, E is the x identity matrix, I is the mass moment of inertia about the enter of mass C, r is the aeleration of the enter of mass, is the rotational aeleration of the rigid body, F i are external fores ating on the body, while the summation runs from i=1 through n, where n is the number of fores. he vetor notation is as follows. he vetor r i is the position vetor of the point of appliation P i of the fore F i relative to the inertial referene frame, whereas r i denotes a position vetor relative to point C of the rigid body, both expressed in the global oordinate system. he subsript denotes transposition, and the matrix A, whih reads: 0 1 A () 1 0 is in fat the rotation matrix for / radians and is used to effet the planar form of the vetor multipliation ri Fi. It is noted that the summation onvention is not used, but individual variables are onsidered. Suppose that Fi inludes F 1 and F among other fores, and that ( Ar i ) Fi inludes their moment ontributions ( Ar1 ) F1 and ( Ar ) F, among other terms. hen a single fore F r has the same ontribution to the nominal state of motion, as represented by Eq. 1, under the following onditions: Fr F 1 F (3) Ar / F Ar1/ F1 Ar / F r r (4) hese equations form the basis for the well-known onditions for stati fore omposition. he resulting transformed fore systems, often alled "equivalent fore systems" (e.g. [1]), are in fat statially equivalent fore systems. Moreover, the ontribution of the fore F r thus found is equivalent to the ontribution of the fores F 1 and F together, not only to the state of stati equilibrium but also to any nominal state of motion. Furthermore, it an be shown that the onditions 3 and 4 are valid with respet to any point of the rigid body (i.e. point C need not be the enter of mass). DYNAMIC EQUIVALENCE he vetor equation Eq. 3 determines the magnitude and diretion of the resultant fore F r, whereas the line of ation of F r is determined by the salar equation Eq. 4. However, Eqs. 3 and 4 do not determine the point of appliation of the Copyright 004 by ASME

3 fore F r. his is not important for the ontribution to the nominal state, but for the stability of the nominal state, the point of appliation of a fore is essential. herefore, it is important that stati equivalene is well distinguished from dynami equivalene. In this paper, a resultant fore will be onsidered dynamially equivalent to a system of fores ating on a rigid body if the ontributions to the stability of the body of the resultant fore and the original system of fores are equal. Stability essentially is a dynami phenomenon. herefore, small variations about the nominal state of the body will be onsidered to investigate the stability of the nominal state of a rigid body in the planar ase. he equations of motion for any nominal state of motion were given in Eq. 1. Expansion of these (for first order variations), subtration of the nominal state, and rearranging of terms, yields the equations for the variations about the nominal state: Mx Kx 0 (5) where the mass matrix is E m 0 M (6) 0 I and where the tangent stiffness matrix is K F F i i (7) i, r i,, r, ( Ar ) F ( Ar ) F i and where x [ r ]. A subsript with omma is used to denote partial derivatives, e.g. F, F /. he terms in the matrix K show that the fores and their moment ontributions must be differentiated with respet to the position and orientation of the rigid body, r and, respetively, implying that the harater of these fores affets the result. In the following setions, two kinds of fores will be addressed: onstant fores and entral linear fores. Constant Fores For onstant fores, i.e. fores due to a homogenous fore field, hene with invariant magnitude and diretion, most of the elements in the tangent stiffness matrix K vanish. he only remaining term is ( ( Ar / ) i Fi ),, representing the hange in moments due to a small rotation. o elaborate this term, a loal oordinate system is fixed to the body at point C suh that ri r r i r R ri, where os sin R (8) sin os is the rotation matrix desribing the transformation from the body fixed oordinates r into the spae fixed oordinates r as in r Rr (9) he rotation matrix is an orthogonal matrix (e.g. []), whih means that RR E (10) A diret result from this orthonormality is that the transposed 1 of R equals the inverse, as in R R. Furthermore, if we differentiate the identities in Eq. 10 with respet to we find: R, R RR, 0 or R, R ( R, R ) 0 (11) Indeed, the matrix R, R is a skew symmetri matrix. Moreover, the previously presented matrix A is in fat defined by: A R R (1), Returning to the term ( ( Ar i ) Fi ), of the stiffness matrix, we an now elaborate this term as follows: Ar i Fi ARri F /, i, Fi AR, ri Fi AR, R ri Fi AAri Fi ri (13) where the equality AA E is used. Consequently, the stiffness matrix of a set of onstant fores F i ating on a rigid body redues to: K Fi r (14) i his expression shows that the ontribution to the stability of a onstant fore is haraterized by the salar produt of the fore vetor and the position vetor of its point of appliation. he stability of the two-onstant-fore systems in Fig. 1 an now be investigated more rigorously. Evaluation of the stiffness maxtrix yields for the Fi ri term: r Fd i ri F1 r1/ F r/ F1 1 r F (15) where F1 d r1 r, i.e. the distane between points P 1 and P. hus, we find values of Fd, Fd and zero for the systems in Figs. 1a, 1b, and 1, respetively. Sine these terms are in fat the powers of the exponential solution to the linear differential equation Eq. 5, it an be onluded that the system in Figs. 1a, 1b, and 1 are stable, unstable, and neutrally stable, respetively, for small rotational disturbanes. Note that all of these systems are neutrally stable for small displaements. More interestingly, the tangent stiffness matrix K of Eq. 14 an be used to find the fore F r that is dynamially equivalent to two onstant fores F 1 and F, by demanding that their F and 3 Copyright 004 by ASME

4 ontributions to the tangent stiffness matrix K must be equal. his notion leads to the following equation: Fr rr F1 r1/ F r (16) hus an equation of salar produts is found, whih, together with the equation of fore vetors (Eqs. 3) and the equation of vetor produts (Eq. 4), uniquely defines the appliation point of the resultant fore F r yielding the same stability ontribution, when onstant fores are assumed. he appliation point found in this manner will be alled the dynamially equivalent appliation point (DEP) of the resultant fore. Equation 16 will be alled the stability equation for the ase of onstant fores. As is true for the fore and moment equations (Eqs. 3 and 4, respetively), it an be shown that the stability equation is valid for any point C on the rigid body. DEP of two onstant parallel fores o investigate the impliations of the stability equation, the speial ase of two parallel onstant fores is investigated first. Consider for example two gravity fores F 1 [ 0 m 1g] and F [ 0 m g] ating on a rigid body, as in Fig. 3a. Substitution of these expressions into Eqs. 3, 4, and 16, yields after elaboration: m m (17) 1 r rx r1 x/ r x/ m1 m m1 m m m (18) 1 r ry r1 y r y m1 m m1 m hus, the appliation point Pr ( r rx, r ry ) of the dynamially equivalent fore is found to be loated on the line onneting P 1 and P, in suh a way that P1 Pr / Pr P m / m 1 as shown in Fig. 3b. his result orresponds to the well-known proedure of finding the ombined enter of mass of two partiles, and demonstrates that the above derivation indeed yields equivalent dynamis. In fat, this result an be identified as a partiular ase of the proposed proedure. DEP of two onstant fores he more general ase of two onstant fores of arbitrary diretion (Fig. 4a) is onsidered next. he line of ation of the statially equivalent fore is known, and Eq. 16 is used to find its point of appliation. Graphial inspetion of this equation reveals a remarkable phenomenon. he DEP is loated on the irle defined by the appliation points of the original fores and the intersetion of their lines of ation, as is shown in Fig. 4b. Assuming that this is true, it will be shown that Eq. 16 results. First, it is realized that if Fr F 1 F, then the projetion of F r on any straight line is equal to the summation Figure 3. wo parallel onstant fores ating on a rigid body: (a) given situation, (b) dynami equivalent fore. his example demonstrates the analogy with the determination of the ombined enter of mass of two partiles. of the projetions of the fores F 1 and F on the same line. If the diameter of the irle through point is seleted, the following expression results for the projetions of the fores on this line (Fig. 4): Fr Dosr F1 Dos1 F Dos (19) where Fi Fi, and where eah term is multiplied by the diameter D. When is seleted as point C, the terms Dos i are equal to r i / ri /. Furthermore, as the vetors r i / and F i are ollinear, the terms FiD os i evolve into F i ri /. Sine Fr F 1 F, the treatise is valid with respet to any point of the rigid body. Consequently, Eq. 19 is equivalent to (a) (b) 4 Copyright 004 by ASME

5 Eq. 16, whih onludes the proof. hus, the irle onstrution is a onvenient way of finding the DEP of two given onstant fores. Stability of a three-fore system in equilibrium In the speial ase of a three-fore system in equilibrium, the assessment of the stability now beomes straightforward and onvenient by using the irle onstrution of Fig. 4. Figure 5 shows three ases of a rigid body whih is in equilibrium under influene of the fores F 1, F and F r. he body in Fig. 5a is in stable equilibrium, sine replaing fores F 1 and F by their dynamially equivalent resultant F r (not shown) yields a system similar to Fig. 1a. Similarly, the system in Fig. 5b is in unstable equilibrium, whereas the system in Fig. 5 is in neutral equilibrium. Note that it an now be onluded that the body in Fig. is in stable equilibrium, simply by onstruting the irle. Central linear fores his setion deals with fores generated by a entral linear fore field. One speial type of entral linear fores onsists of fores generated by zero-free-length springs [3], whih are of great benefit in the design of mehanisms in neutral equilibrium [4]. Due to the harater of these fores, the tangent stiffness matrix K will ontain more non-zero entries than in the ase of onstant fores. his setion will derive the onditions for the dynamially equivalent fore of two entral linear fores, or, in partiular, two zero-free-length springs. he entral linear fore generated by a zero-free-length spring an be expressed as: a r k a r Rr Fi k i i i i i i (0) (a) (b) where k i is the spring stiffness; a i is the position vetor of the fixed end of the spring (the origin of the entral linear fore field), and r i is the position vetor of the moving end of the spring. he moment ontribution of suh a fore with respet to an arbitrary point C of the body is: Ari Fi F R, ri k i ai r Rri R, ri i (1) From Eqs. 0 and 1, the ontributions due to this fore to the elements of the stiffness matrix K (Eq. 5) an be derived: F i r k i (), E Fi k i R, ri ki ARri ki i, Ar (3) F k R Ari i, r i, ri / ARr k (4) k i i i Ari Figure 4 wo onstant fores of arbitrary diretion ating on a rigid body: (a) given situation and statially equivalent fore that may apply anywhere on its line of ation, (b) dynamially equivalent point of appliation, as determined by the proposed irle onstrution, () proof of the irle onstrution. () 5 Copyright 004 by ASME

6 Figure 5. Systems of three onstant fores in equilibrium ating on a rigid body: (a) stable equilibrium, (b) unstable equilibrium, () neutral equilibrium. Ar F R r R r F R i i, k i, i, i i, ri k i r i ri Fi r i k i a i ri (5) where the equalities R, AR, R, R, E, rí ri, and R, R, are used. hus, the tangent stiffness matrix K for entral linear fores evolves into: ki E ki Ari K (6) ki Ari k i a i ri As ompared to the onstant-fore stiffness matrix (Eq. 14), the following differenes are apparent. Additional terms k i Ar i and k i Ar i are present as the off-diagonal elements. Consequently, the system is no longer indifferent with respet to arbitrary displaements. Pure translation is assoiated with a stiffness k i due to the term k i E. Furthermore, it is remarkable that the lower right term is not replaed by a ompletely different term but is expanded with the term k ir i ri (ompare Eq. 13 with Eq. 5), resulting in k i ri ri Fi ri ki ai ri. If now two entral linear fores are to be replaed by a single equivalent one, the ontribution to the stiffness matrix K due to the equivalent entral linear fore must be equal to the ontribution due to the two original ones. Considering Eqs. through 5 respetively, this leads to the following onditions for equal stability (stability equations): k r k 1 k (7) (8) k r Arr k1ar1/ k Ar k r r r rr Fr rr k 1r1/ r1/ F1 r1/ kr/ r/ F r (9) hus, when replaing two entral linear fores by one, a total of seven equations are found: the vetor equations Eq. 3 and Eq. 8, and the salar equations Eq. 4, Eq. 7, and Eq. 9, whih are to be solved for five unknowns (one salar, k r, and two vetors, r r and a r ). Consequently, no solutions are found in general. his leads to the onlusion that two entral linear fores annot generally be substituted by a single one in a dynamially equivalent manner. However, by imposing onstraints on the system, solutions for at least two speial ases are possible. he following setion will give the first speial ase, the seond one will be treated in the Examples setion. Speial Case 1: Common Attahment A first speial ase is possible when the zero-free-length springs are attahed to the rigid body at the same point (Fig. 6). hen r1 r whih, substituted in Eqs. 3 and 4, leads to rr r1/ r. ogether with Eq. 7, this immediately satisfies Eq. 8, while Eq. 9 now beomes: k r r F k 1 r r r rr k 1rr rr F1 rr krr rr F rr (30) he r r rr terms anel out, whih gives after rearranging: k 1 k r k1a ka rr 0 a 1 (31) his should be valid for any r r, and therefore, introduing the unit vetor e a and using the relations a 1 a aea and a 1 a a (see Fig. 6), Eq. 5 evolves into: 6 Copyright 004 by ASME

7 k1 k ar a1 a (3) k1 k k1 k From this result it is seen that a r traes the line A 1 A as the stiffnesses k 1 and k vary. It is also seen that a ak1 /( k1 k ) and a1 ak /( k1 k ). hus, the relation k1a1 ka defines the loation of point A r on the line A 1 A. So, two zero-free-length springs, k 1 and k, eah attahed with one end to a first rigid body and with the other end to a seond rigid body, an be omposed into a single zero-freelength spring k r in a dynamially equivalent way for any relative movement of the rigid bodies, under the following onditions: Firstly, k r must equal k1 k (due to Eq. 7), seondly, the free ends of the springs must be attahed to the same point of appliation P r, so rr r1/ r (assumed earlier); and thirdly, the fixed end A r of the dynamially equivalent zero-freelength spring must be loated on the line onneting A 1 and A, so that k1a1 ka (resulting from Eq. 3). Inversely, these equations an be used to resolve a single spring into two springs, where it is noted that this does not give a unique solution. Potential Fores he treatise above an be generalized when the applied fores an be derived from a potential funtion, i.e. when they are onservative. his is espeially useful when the stability is to be assessed in ases where the original fores and their points of attahment are not easy to identify. Examples are distributed loads, suh as hydrostati pressure. In the ase of a rigid body, only the potential of the external fores is to be onerned, so the equations of motion an be written as follows: V r, V, me 0 where V V r, 0 r I (33) is the potential energy of the body. he variations about the nominal state of motion an be found by extending Eq. 33 and subtrating the nominal state, whih results in: me 0 0 r V, I V, r r r V V, r, r 0 (34) his result is ompletely equivalent with Eq. 5. Depending on the situation, either of these may be more onvenient. EXAMPLES his setion will provide two examples. he first example is in fat the seond speial solution to finding a dynami equivalent of two entral linear fores. he seond one demonstrates the Figure 6. wo entral linear fores, ating at the same point of a rigid body, an be omposed into a single dynamially equivalent one. onvenient use of the potential using the example of the stability of a floating vessel. he Balaned Broom In addition to the ommon attahment, a seond speial ase of a dynamial equivalent of two entral linear fores is found when the motion is restrited to rotation (about a fixed point) only, and the two entral linear fores are not replaed by a resultant entral linear fore but by a onstant fore. Under these distint onditions, a solution an be found as follows. Due to the restrition to rotation, Eqs. 8 do not apply; and due to the replaement of two entral linear fores with a onstant fore, dynami equivalene is haraterized by: F r r r k 1r1/ r1/ F1 r1/ kr/ r/ F r (35) where the left side orresponds to the expression for onstant fores. Now, together with the Eqs. 3 and 4, a total of four equations (one vetor equation and two salar equations) are found to solve for four unknowns (two vetors, F r and r r ). he physial interpretation of this solution is still an open question, however an example onfirming this phenomenon is present in the Balaned Broom (a.k.a. Floating Suspension [5]). his is a statially balaned mehanism onsisting of one link with a mass at its end, suspended by two zero-free-length springs in suh a way that a stationary pivot is obtained (Fig. 7). Stati balane an be proved by using the potential. he total potential is the summation of the potentials of the springs and of the mass (with respet to O, see Fig. 7): a r r a r 1 r V k / a r r a r 1 k r / 7 Copyright 004 by ASME

8 mg r rm ez (36) where r i runs from point C to the points where the moving spring ends attah to the link. he equilibrium position an be found from the equilibrium of fores: V a r k a r mge 0, r k1 1 1 z (37) pae r e k pae r e mge 0 k1 1 z 1 r z r z (38) where the vetor ai r ai is the vetor from point C to the fixed spring attahment point A i, and where r 1 and r are defined aording to r1 r1 er and r re r, respetively. his results in the following two onditions that are valid for any e r or, equivalently, for any φ: k (39) 1r1 kr pa k pa mg 0 k (40) 1 1 In ase of equal springs k 1 k k, it is found from Eq. 39 that r 1 r r, and from Eq. 40 that p 1 mg / ka or z a / pa mg / k. his expression is independent of the orientation of the link, whih implies that the link has a stationary enter of rotation, even though no physial joint is present. hus, the link is restrited to rotation about a virtual pivot at C. Next, the rotation is investigated by differentiating the potential with respet to : 1R, r1/ a1 r R 1 R, r / a r Rr / mgr, rm ez, k r (41) V / k his equation an be simplified beause ( R, ri / ) Rri ri R, Rri r i Ari 0. In addition, making use of Fig. 7, it is observed that a 1 r (1 p) aez and a r pae z. Furthermore, if e is used as unit vetor perpendiular to e r, then R, r 1/ r 1/ e and R r. hus, Eq. 41 redues to:, r / e k1r1/ pa kr pa mgrm ez V, 1 / (4) where r m rm. Using Eq. 39, k1 r1/ kr kr, it is seen that this expression equals zero for any orientation if: kra mgr 0 or r m / r ka / mg (43) m It is thus shown that the broom is statially balaned for rotation by two zero-free-length springs, while at point C a stationary enter of rotation with no support fore is obtained! A solution to Eq. 35 is now possible when it is assumed that F is parallel to the line onneting the fixed ends of the r e Figure 7. he Balaned Broom, demonstrating the fat that two entral linear fores an be omposed in a single dynamially equivalent onstant fore for a rotating body. he name for this mehanism is due to a onfirmation experiment performed by Prof. Andy Ruina during a visit to our laboratory. An available broom was used by way of beam, while medial instrument overs out of latex served as approximate zero-free-length springs, with amazing result. springs and its point of appliation is loated on the line onneting the moving ends of the springs. Hene, Fr Fr ea and rr rr er. Under these onditions, Eq. 35 beomes: F rrr ea er k1a1r1 ea er kar a e e (44) A solution is found for any e r and therefore for any angle φ when: F r r r k (45) 1a1r1 kar his result onfirms that one solution for the dynamially equivalent fore of the two ideal spring fores is a onstant fore of magnitude F r and direted along e a, ating on the lever at point P, loated along the extension of P 1 P at a distane r r from point C. In partiular, this treatise proofs the neutral stability for rotation of the spring-suspended link when e a is set vertial and a gravity fore Fr mg, equal and opposite to the resultant of the spring fores, is applied at point P, where. r r r m Roll stability of a ship he metaenter of a ship is an example of a dynamially equivalent appliation point of a resultant fore, i.e. the hydrostati or buoyany fore (e.g. [6]). he position of the metaenter with respet to the enter of gravity of the ship, the r 8 Copyright 004 by ASME

9 metaentri height, determines the stability of the roll motion of a ship. he motion is stable if the metaenter is above the enter of gravity, in whih ase the metaentri height is taken positive. If we draw the free body diagram of a ship of retangular ross setion in the equilibrium position (Fig. 8a), that is at zero roll angle, it is lear that the resulting hydrostati fore must at on the enter line of the ross setion of the ship. Yet, at a glane, it is not obvious were the point of appliation of this resultant is loated in order to be dynamially equivalent with the hydrostati fores. A first, inorret guess would be the entre of gravity of the displaed water volume, also known as the entre of buoyany. Determining the DEP for the hydrostati fores is not so easy sine, for a rotated ship, these fores hange both in diretion as well as in magnitude. herefore a diret analysis as presented in Eq. 5 is rather umbersome. A muh easier approah is making use of the potential funtion for this onservative fore field and subsequently take derivatives as proposed in Eqs. 33 and 34 to obtain the dynamially equivalent fore system. Consider the ship in a displaed position (Fig. 8b). he potential funtion for the hydrostati fores is minus the potential of the displaed water volume whih is equal to the first moment of mass distribution with respet to the water line times the gravitational onstant g. We an divide the immerged ross setional area of the ship into two parts, a parallelogram A 1 and a triangle A as given by b b A 1 os w sin (46) os b b A sin (47) os where b is the width of the ship, the distane of the enter of gravity with respet to the bottom of the ship, and w the position of the enter of gravity with respet to the waterline. With the distane of the enters of mass with respet to the waterline of the two parts, respetively, as given by 1 b w 1 os w sin (48) b w os w sin (49) 6 the total potential of the hydrostati fores now beomes 1 1 Aw V g A w (50) where is the length of the ship and the density of water. he metaentri height of a ship is defined in the equilibrium Figure 8. Ship with retangular ross-setion: (a) stability is not easily assessed based on the physial fore system, (b) inlined vessel, () resulting dynamially equivalent fore onfiguration reveals the state of stability at a glane. position. In this position the roll angle will be zero due to symmetry. he submerged height h of the ship follows from the total weight of the ship G being equal to the buoyany, as in G g h b (51) Note that the buoyany also an be found as the first partial derivative of the hydrostati potential V with respet to vertial displaement w. With this height h, the displaement of the enter of gravity w at equilibrium is now w h (5) (a) (b) () 9 Copyright 004 by ASME

10 he dynamially equivalent fore system for the hydrostati fores follows diretly from the seond order partial derivative of the hydrostati potential V with respet to the roll angle (Eq. 34 resulting in 1 b 1 V, g h b h h (53) 1 h If we ompare this to the result for a onstant fore, i.e. the term Fd in Eq. 15, we onlude that we an replae the resulting hydrostati fores by a onstant fore F g h b (54) ating at a distane d from the entre of gravity, where the distane d is in this ase (a) (b) d 1 1 b h 1 h h (55) his distane d is the so-alled metaentri height of the ship. If we ompare this to the result from literature (e.g. [6]) where the metaentri height GM is given by I GM oo BG (56) Vol where Ioo is the area moment of inertia of the horizontal waterline area about the longitudinal axis of the ship, Vol is the displaed water volume, and BG the distane from the entre of buoyany to the entre of gravity of the ship, we find 3 1 b h GM (57) 1 bh his is in omplete aordane with Eq. 49. Note that the DEP of the hydrostati fores is not loated at the enter of buoyany but at a distane b 1h right above this point (Fig. 8). Stability of a shoe box A lear example of the differene between the enter of bouyany B and the metaenter M is present in the ase of a floating shoe box (Fig. 9a). We know from experiene that this is a stable onfiguration. Yet, in most ases, the enter of gravity G will be above the enter of buoyany B whih gives the impression of an unstable system. Substituting the dimensions as presented in Fig. 9a into Eq. 55 we alulate GM 3 h and BM 3h (58) Indeed, a very stable onfiguration! In onlusion, Fig. 9b shows the physial fore system, while Fig. 9 shows the dynamially equivalent two-fore system. Figure 9. Shoe box: (a) given onfiguration of enter of buoyane and enter of gravity, (b) physial fore system, () dynamially equivalent fore system. CONCLUSION Unlike two-fore systems, more omplex fore systems do not allow the assessment of their stability by inspetion. his paper presented a general method to determine the stability of omplex fore systems, based on the notion of dynami equivalene, where a resultant fore is onsidered dynamially equivalent to a given fore system ating on a rigid body if the ontributions to the body s stability of the resultant fore and the original fore system are equal. his demand is stronger than the demand for stati equivalene. Stati equivalene yields a resultant fore vetor and its line of ation. he loation on this line remains undetermined, as it does not affet the nominal state of a rigid body. However, for the assessment of the stability of this nominal state, the appliation point on the line of ation is essential. Demanding dynami equivalene pinpoints the attahment point of the resultant fore on the body in a unique manner. his point was alled the dynamially equivalent point of attahment. () 10 Copyright 004 by ASME

11 It was shown that the dynamially equivalent resultant fore of two given onstant fores applies at the intersetion of its line of ation and the irle put up by the appliation points of the given fores and the intersetion of their lines of ation. his result yields a onvenient graphial method for finding the dynamially equivalent appliation point of the resultant fore. he determination of the ombined enter of mass an be onsidered as a speial ase of this theorem. wo examples were given that illustrate the versatility and the signifiane of the proposed treatise. he first example onsidered the suspension of a body by springs only. It was shown that the body was statially balaned for rotation, while the enter of rotation proved to be a virtual stationary point. he seond example treated the roll stability of a ship. Using the proposed methodology, the metaentri height was determined in a natural and onvenient way. Future researh will be direted towards the impliation of dynami equivalene of spatial fore systems. Another field of appliation is present in roboti end effetors. In order to determine the stability of a grasp [7], the nature of the grasp fores is predominant, i.e. sliding of stiking; with fixed or floating line of ation; zero or non-zero free length springs, et. Consequently, future work omprises the dynamially equivalent omposition of other than onstant and entral linear fores, and the appliation to the synthesis of stable grasps. ACKNOWLEDGMENS Speial thanks are extended to Jan C. Cool, who was the first to disover the irle onstrution, to J. F. Besseling for his enouragement to formalize the proposed proedure, to Andy Ruina for his enthousiasm in performing the onfirmation experiment, and to Jaap Meijaard for pointing out that the enter of mass is a speial and well-known ase of a DEP. NOMENCLAURE a distane between fixed spring attahment points a i position vetor of a fixed spring attahment point relative to a fixed referene frame a i position vetor of a fixed spring attahment point relative to a loal referene frame A ross setional area A matrix used to effet the planar form of vetor multipliation b ship width distane of enter of gravity above ship floor d distane between two points e unit vetor E i identity matrix of rank i F magnitude of fore F fore vetor g aeleration of gravity h submerged depth of ship i index, ounter I mass moment of inertia with respet to point C I oo area moment of inertia of horizontal waterline area k spring stiffness K tangent stiffness matrix length of ship m mass M mass matrix n number of external fores p fration r link length r i position vetor of a moving spring attahment point relative to a fixed referene frame r i / position vetor of a moving spring attahment point relative to a loal referene frame R rotation matrix V potential, potential energy Vol displaed water volume w depth of enter of gravity below waterline first order variation density variable angle REFERENCES [1] Beer F.P., Johnston E.R. jr (1997) Vetor Mehanis for Engineers: Statis and Dynamis, MGraw-Hill, ISBN [] Bottema O., Roth B. (1979), heoretial Kinematis, North- Holland, Amsterdam. [3] Carwardine G. (193) Improvements in elasti fore mehanisms, UK Patent , Speifiations of Inventions, Vol. 773, Patent Offie Sale Branh, London. [4] Herder J.L. (001) Energy-free Systems; heory, oneption and design of statially balaned spring mehanisms, Ph.D. hesis, Delft University of ehnology, ISBN [5] Herder J.L. (1998) Coneption of balaned spring mehanisms, ASME DEC 5th Biennial Mehanisms Conferene, Sept , Atlanta GA, paper number DEC98/MECH [6] Besant W.H., Ramsey A.S. (1919) A reatise on Hydrodynamis, Part 1, Hydrostatis, 8th Edition, Bell, London. [7] Nguyen V-D (1989) Construting stable grasps, Int. J. Robot. Res., 8(1) Copyright 004 by ASME