Equimomental systems representations of point-masses of planar rigid-bodies

This paper presents a study on the equimomental systems of point-masses of planar rigid bodies. In this work, the equimomental systems of three point-masses of planar rigid bodies are investigated using the concept of pseudo-inertia matrix. It is found that given a planar rigid body, it is always possible to determine an equimomental system of three equal masses located at the vertices of an isosceles triangle. A procedure is presented to determine equimomental systems with different masses, guaranteeing that the masses are positive. It is shown that it is always possible to choose an equimomental system of three point-masses located at the vertices of an isosceles triangle with a prescribed position of one mass. The conditions for prescribing the position of two and three point-masses are also investigated. A first idealized example shows the step-by-step procedure for determining an equimomental system of three point-masses of a planar rigid body. The proposed model is applied to a symmetric connecting rod in a second example due to its wide use in combustion engines. A third example shows an equimomental system of an asymmetric bucket of an excavator.


Introduction
A three-dimensional rigid body can always be modeled by a dynamically equivalent system of four rigidly connected point-masses [1].However, in the case of planar rigid bodies, it is always possible to construct a dynamically equivalent system with a three point-masses [2].These systems, having the same dynamic behavior, are known as equimomental systems [3].Wenglarz et al. [4] and Haung [5] introduce the concept of equimomental systems.Equimomental systems help define the inertial properties of rigid bodies and determine shaking forces, shaking moments, and input torques of mechanical systems [6,7].Laus and Selig [8] show formal proof of the existence of four point-mass equimomental to a given three-dimensional rigid body.The authors developed analytical expressions to determine the possible locations and values of the masses using modern methods.They also derived the equation of motion based on the screw theory of a rigid body in space.Similarly, in Chica et al. [9], the same result is derived in a more current language.The authors state that this important property seems to be forgotten, and up to the publication date of this paper, they had not found an appropriate demonstration.
Chaudhary and collaborators have extensively studied the equimomental systems of planar rigid bodies [6,[10][11][12][13].In the proposed models, it cannot be assured that the masses are always positive.However, this is not an obstacle for the rigid body representation process as long as the total mass and the moment of inertia about the center of mass give positive values [14].Jan de Jong et al. [15] propose a method to determine planar equimomental mechanisms through an inertia decomposition method.This method avoids infeasible solutions using constraints on the model parameters.The main disadvantage of this procedure is that it cannot be applied to mechanisms with more than one loop or to the mechanism where there are bodies connected with prismatic pairs.Gössner [16] proposes an equimomental model of point-masses located at the vertices of a polygon.The author believes that this polygonal system approach is new to the best of his knowledge.Future work can focus on the negative masses that occur here and their avoidance.
Equimomental point-mass systems are especially advantageous in the dynamic balancing of mechanisms and machines [17,18] and in the minimization of constraint forces and actuation moments [19,20].Where the equations of the dynamic model are greatly simplified and also allow reduced use of design parameters.An interesting application of equimomental systems in the field of biophysics is shown in Fábián et al. [21], where a simulation of the molecular dynamics of cholesterol is performed.The authors claim that with an equimomental arrangement, the convergence of the binding constraint is accelerated while preserving the original force field and dynamics of cholesterol.
Another possible and interesting application of point-masses equimomental systems is dynamic synthesis, which consists of determining the shape of a body given inertial properties and some restrictions on the body's form and maximum allowable stresses [22].Unfortunately, nowadays, dynamics is studied from an analytical perspective without a synthetic branch in the same way as in statics.Therefore, in inherently dynamic problems, quasi-static behavior is assumed [23].
The discussion of equimomental systems of point-masses dates back to the 19th century [24].However, nowadays, few researchers have taken this approach despite its advantages in the dynamic analysis of mechanisms and machines.Laus and Selig [8] have made significant advances in the case of three-dimensional rigid bodies, although the models developed have not yet been adapted to planar rigid bodies.These are the main reasons that motivated the realization of the present work, besides attempting to simplify the dynamic analysis of mechanisms and machines.
This paper presents a study on the equimomental systems of point-masses of a planar rigid body.By using the formalism of the pseudo-inertia matrix, it was found that an equimomental system of three equal point-masses always forms an isosceles triangle.If the radii of gyration are equal, the triangle formed by the three masses is equilateral; these results are stated as a theorem and a corollary with their respective proof.A procedure is also provided to determine equimomental systems with three different masses, and always guarantee positive masses.Furthermore, a theorem on the possibility of prescribing the position of one of the point-masses is shown, and the conditions for prescribing the position of two and three masses are established.Through a numerical example, the step-by-step procedure for determining a three-mass system equimomental to a planar rigid body is shown.An example is also shown where the results found are applied to determine an equimomental system of a symmetric connecting rod.Finally, in a third example, an equimomental system of three point-masses of an asymmetric planar rigid body is determined.
This paper begins with the presentation of the formalism for representing the inertial properties of a rigid body Sect. 2. Thereafter, in Sect.3, equimomental systems of three equal masses are studied, and a theorem and a corollary are presented with their respective proofs.Then, in Sect.4, equations are derived to determine all possible equimomental systems with unequal point-masses, obtained by three-dimensional rotations to the points of the equilateral triangle used to define the equimomental system with equal masses.Section 5 presents a theorem on the possibility of prescribing the position of one mass, and the conditions for prescribing the position of two and three masses are given.In Sect.6, the step-by-step procedure for determining an equimomental system of a planar rigid body is shown through an idealized numerical example.In a second extension example, the obtaining of an equimomental system of a symmetric connecting rod is shown, and in a third example, an application for an asymmetric planar rigid body is shown.Finally, the conclusions are given in Sect.7.

Inertia matrix
Two rigid bodies are called equimomental if they have the same inertial properties.In other words, two equimomental bodies have the same mass, their centers of mass coincide, and their inertia matrices are equal with respect to a fixed coordinate system [3].One of the ways to combine the inertial properties into a single matrix is through the homogeneous plane-distance inertia matrix, commonly known in a rather loose way as the pseudo-inertia matrix.The pseudo-inertia matrix appears naturally when determining the mean-squared distance of a rigid body to a plane, as presented in Selig and Martins [23].
The pseudo-inertia matrix of a body B is defined as follows: where p = x y z 1 T is the homogeneous position vector.Given the following definitions where m is the mass of the body; I i j corresponds to the elements of the traditional inertia tensor I, and x C , y C , z C are the coordinates of the center of mass.Then, the pseudo-inertia matrix can be expanded to Laus and Selig [8] argue that this matrix is a unique property and that two bodies will be equimomental if and only if they have the same homogeneous matrix.In a planar rigid body, the pseudo-inertia matrix Eq. ( 2) loses the third row and third column as the points of the body are constrained in the x y plane i.e. z = 0.Then, the pseudo-inertia matrix becomes The pseudo-inertia matrix of the planar body represented by n point-masses can be written as where m i is the i-th point-mass, and p i is the homogeneous extended position vector of point p i and has the form p i = x i y i 1 T [3].The rigid transformation can be performed by standard homogeneous representation where G is a homogeneous transformation matrix of the form with R the 2 × 2 rotation matrix and t its translation vector [3].
The following section presents a procedure to determine a system of three equal point-masses by scaling an equilateral triangle.

Equimomental system with equal point-masses
It is a classical theorem that there exists a translation that locates the center of mass at the origin of coordinates and a rotation that aligns the coordinate axes with the principal directions of inertia [25].That is, there is a homogeneous transformation G that diagonalizes a given pseudo-inertia matrix .So the pseudo-inertia matrix can be written as where a, b, are the gyration radii of the original system.The conditions for three points of equal masses to form an equimomental system will be deduced next.
We look for convenient canonical points q i that satisfy These points q i can be scaled by the diagonal matrix D = diag(a, b, 1), then the points p i = D q i are an equimomental system since Consider three point-masses located at triangle's vertices q 1 , q 2 , and q 3 , see Fig. 1.Following Eq. ( 8), the extended vectors satisfy the following conditions As a consequence of the above relationships, it will be shown that 3 i=1 qi qT i = 3I 3 is satisfied.This will then be used to derive an equimomental system of three point-masses of a given planar rigid body.The first condition, Eq. ( 9), yields the following constraint equations and the second condition, Eq. ( 10), yields The three constraints in Eq. ( 12) represent circles with radius R i = R = √ 2 for i = 1, 2, 3.The three constraints in Eq. (11) guarantee that the triangle is equilateral, since q T i q T j + 1 = 0 with i = j that is , and l i = l for i = 1, 2, 3.The radius R and the length of the sides of the equilateral triangle l are related by the equation R = l/ √ 3, then l = √ 6.Let us locate the equilateral triangle in an arbitrary orientation, as illustrated in Fig. 2.Where the homogeneous extended position vector of the triangle's vertices can be written as this homogeneous extended position vector satisfy where I 3 is a 3 × 3 identity matrix.
Theorem 1 Any planar rigid body is equimomental to a system of three equal point-masses located at the vertices of an isosceles triangle.
Proof Using the transformation shown in Eq. ( 5), it is always possible to find a transformation G that converts the pseudo-inertia matrix into a diagonal matrix of the form = m diag a 2 , b 2 , 1 .The points in Eq. ( 13) can be moved using a non-rigid transformation D = diag(a, b, 1) so that the extended position vectors of the points become pi = D qi for i = 1, 2, 3. Placing three equal masses m/3 at these points produces a system with the required inertia matrix, Then, all planar rigid body is equimomental to three equal points-masses, located at the vertices of a triangle which are obtained by deforming the equilateral triangle by the transformation D = diag(a, b, 1).Furthermore, the triangle is isosceles in the particular case when the triangle is oriented in such a way that it is symmetrical with respect to the y-axis, since p 12 2 = 3(a 2 + 3b 2 )/2, p 13 2 = 3(a 2 + 3b 2 )/2, and p 23 2 = 6a 2 as shown in Fig. 3.

Corollary 1
If a planar rigid body with inertia matrix = m diag a 2 , b 2 , 1 in respect to a reference frame with origin at the center of mass and coordinate axes coincident with the principal axes, and if a = b, then the rigid body is equimomental to a system of three equal point-masses located at the vertices of an equilateral triangle.
Proof The system of three-point masses is equimomental to the planar rigid body with inertia matrix = m diag a 2 , b 2 , 1 , as shown in Eq. (15).Furthermore, if a = b, then p 12 2 = p 13 2 = p 23 2 = 6a 2 shows that the three point-masses form an equilateral triangle.
It is possible to find solutions with different masses from the equimomental system of three equal pointmasses.These solutions with different masses will be analyzed in the next section.

Equimomental system with unequal point-masses
This section shows that it is possible to determine equimomental systems of three different point-masses that are not necessarily located at the vertices of an isosceles triangle.
If in Eq. ( 15) we use pi = DU qi instead of pi = D qi , where U ∈ O( 3) is a 3 × 3 orthogonal matrix, then the same pseudo inertia matrix is obtained as in Eq. ( 15), since that UU T = I 3 .This property can be used to determine a family of point-mass solutions.For this, we will parameterize U = U(β), where the parameter β is an angle around the y-axis.Let us define a 3D rotation around the y-axis, and, performing the rotation U q i for i = 1, 2, 3, yields, The right side of the Eqs.(18)(19)(20) can be used to find the point-masses and their position since that, 2 sin β + cos β) 2 q 3 q T 3 , then the masses are: 2 sin β) 2 , and 2 sin β + cos β) 2 .The extended position vectors are then, by eliminating the parameter β in vector q 1 it is determined that the mass m 1 describes a hyperbola defined by equation y 2 2 − x 2 = 1, see Fig. 4. Similarly, it is concluded that the masses m 2 and m 3 describe the same hyperbola defined by equation y 2 1/5 − x 2 = 1, see Fig. 4. By varying the parameter β from 0 to 2π, the masses start by forming the equilateral triangle studied above.The points move on the corresponding hyperbola to infinity and then reappear on the other branch of the hyperbola.When the parameter reaches the value of π, an equilateral triangle is formed again, but with points reflected with respect to the x-axis in relation to the initial triangle.Once the parameter reaches the value of 2π, the points return to the original position.Figure 5 shows the variation of the normalized mass, and it can be observed that the sum is constant and equal to 1.
Figure 5 shows the possible values of the masses for a fixed orientation of the equilateral triangle shown in Fig. 2. In order to determine all possible values of the masses, it is necessary to use two parameters, namely θ and β, where θ is the angle of plane rotation of the triangle taking as reference a triangle symmetric to the x-axis, see Eq. ( 13).Now we will use the transformation pi (θ, β) = DU(β) qi (θ ), then the masses and the position vectors become a function of two parameters, that is, m i = m i (θ, β) and q i = q i (θ, β).The masses are transformed according to Fig. 4 The effect of a 3D rotation on a triangle of point-masses and Fig. 6 illustrates the normalized mass.The three point will move according to

Equimomental systems with prescribed point-masses
In this section, we study the possibility of prescribing the position of one mass while guaranteeing that all masses are positive.In addition, the conditions for prescribing the position of two and three point-masses are also presented.

One prescribed point
Lemma 1 An equimomental system of three point-masses can always be chosen for a planar rigid body if one of the points is prescribed on one of the principal axes.
Alternatively, Lemma 1 can be rewritten as follow: "For any inertial system, if you prescribe a first point along one of the principal axes, there is always an equimomental system of three point-masses by a proper selection of the positions of two extra points." Proof Consider a rigid body with a pseudo-inertia matrix = m diag a 2 , b 2 , 1 relative to a reference frame with origin at the center of mass and coordinate axes coinciding with the principal axes.Let us define an x y coordinate system coinciding with the principal axes.If the rotation angle θ = 0, then point q 1 is on the x-axis according to Eq. ( 27).Now, if we fix the point p 1 , it obtains the following relation then, the points q 1 and q 2 of the non-scaled triangle can be written as From Eqs. ( 24), (25), and (26) we have that the masses are and according to Eq. ( 16) we have that 3 i=1 pi pT i = m diag a 2 , b 2 , 1 then the three point-masses form a system equimomental to the given rigid body.
Note that points p 1 and p 2 are symmetric with respect to the x-coordinate axis.Therefore, the system of point-masses forms an isosceles triangle with two equal masses, m 2 = m 3 .
Theorem 2 For a planar rigid body, an equimomental system of three point-masses located at the vertices of an isosceles triangle can always be chosen with a mass at an arbitrary point.
Proof Suppose a point is prescribed, and its coordinates are written relative to a coordinate system fixed to the body.In that case, the point coordinates can also be expressed relative to a reference system with the origin at the center of mass and coordinate axes coincident with the principal axes.In this coordinate system, the point can be rotated by an angle θ around the center of mass to be on the x-axis.It is now possible to determine an equimomental system with three point-masses located at the vertices of an isosceles triangle, as shown in Lemma 1. Finally, the prescribed point in the current position is rotated at an angle −θ that locates it in the original position.

Symmetrical planar rigid bodies
In the case of a planar rigid body with at least one axis of symmetry, the determination of an equimomental system is substantially facilitated since it is sufficient to define a point on an axis of symmetry exempt from the center of mass, then we obtain an isosceles triangle with two equal masses symmetrical to the axis of symmetry.

Two prescribed point
According to Theorem 2, two points can be prescribed as long as they are symmetric to an axis passing through the center of mass of the body.If the body is symmetric, then the two points must be symmetric to the axis of symmetry.

Three prescribed point
Some models, such as the one presented in Chaudhary [6], allow prescribing three point-masses with some restrictions, but it cannot be guaranteed that the masses are positive.It is clear that it is impossible to define three general points arbitrarily since the position of the masses must obey Eqs. ( 27), (28), and (29).

Examples
In the following, we first show an idealized example to demonstrate the procedure for determining an equimomental system of a planar rigid body.In the two subsequent examples, we consider a symmetric and an asymmetric rigid body, respectively.

Six point-masses rigidly connected
This idealized example shows the procedure to determine an equimomental system of three-point masses of a given rigid body.
Next we present the step-by-step procedure for determining the equivalent three point-mass system given a pseudo-inertia matrix.The pseudo-inertia matrix of a set of six point-masses, summing m = 1, located in the positions: here we are using equal masses for simplicity but we have no loss of generality.To determine the transformation G that converts the matrix into a diagonal one, we must be first translated the body center of mass.Then the eigenvectors of the matrix I that form a rotation matrix R are determined [26].Therefore, the transformation matrix has the form where t C is the position of the center of mass.Following the above procedure, we have The pseudo-inertia matrix calculated with these points is equal to Eq. (34), confirming that the two systems are equimomental.Figure 7 shows in red the set of six points, the undeformed triangle in blue, in green the deforming triangle representing the equimomental system, and the black point is the position of the center of mass.
The reference frame aligned with the principal axes is chosen in such a way that it satisfies the right-hand rule.Although G is not unique, this is not a problem since the different choices of G represent rigid rotations of the points p i .Therefore, the property of being an equimomental system is preserved (see Eq. ( 16)).

Symmetric connecting rod
A connecting rod is a mechanical component that connects a reciprocating piston to a rotating crank.Connecting rods are commonly used in internal combustion engines and compressors to convert reciprocating motion into circular motion and vice versa.Generally, connecting rods are symmetrical, which allows defining an equimomental system of two point-masses positioned along the axis of symmetry, as shown hereafter.
If the connecting rod is considered a straight rod with negligible diameter or width and thickness, see Fig. 8b, then we have I xy = I x x = 0, and I yy = I zz .Furthermore, the pseudo-inertia matrix with respect to the center of mass becomes = diag (I zz , 0, m) according to Eq. ( 3), and the scaling matrix is, therefore, D = diag √ I zz /m, 0, 1 = diag (a, 0, 1).Now fixing the mass m 1 at the point A with coordinates (x 1 , 0) and comparing this point with Eq. ( 27) we have that θ = 0 and we obtain the same relation as in Eq. ( 30), where a = √ I zz /m.The scaling of the point in Eqs. ( 27), (28), and (29) by means of pi = D qi produces The masses m 2 and m 3 collapse into the same point on segment AB so this system of three more pointmasses can be seen as an equimomental system of two point-masses, see Fig. 8b.Where the first and second masses are respectively, and located in the position with coordinates x 1 , and x 2 = −I zz /(mx 1 ).This model, deduced with modern techniques, is equivalent to the one shown in the classical literature as in [27].

Asymmetric planar body
Often, the rigid bodies that conform to a machine have complex geometries and in some cases, are asymmetrical.Thus, in the following example, a system of three masses equimomental to the bucket of a Komatsu PC400 Fig. 9 Komatsu PC400 excavator: a side view, and b Solidworks CAD of the bucket [28] excavator is determined.Figure 9a shows a side view of the excavator, and Fig. 9b shows the isometric drawing of the bucket generated in Solidworks.The inertial properties were obtained using Solidworks software and considering a steel alloy with a density of 7850 kg/m 3 ; where the computed mass of the bucket is m = 1787.17469476kg, and the coordinates of the center of mass are x C = 0.70554017 m, and y C = 0.11310906 m.Furthermore, the numerical entries of the inertia matrix with respect to the origin of coordinates O (see Fig.  Here, the three masses are equal and have a value of one-third of the total mass.Figure 10 shows in blue the undeformed equilateral triangle, and in green the position of the point-masses on the vertices of an isosceles triangle after the deformation of the equilateral triangle.

Conclusion
This work presents novel results on the equimomental systems of three-point masses of planar rigid bodies.The ideas studied in this research can contribute to the dynamics of planar rigid bodies and the concept of inertial synthesis [22], a less explored topic.
In the developed model, it is considered that the rigid body does not have a constant density and thickness; therefore, the expressions obtained are general.First, it is determined that the formed triangle is isosceles for an equimomental system of equal masses.Then, all possible solutions with unequal masses are determined, and the values of the normalized masses are shown graphically.Furthermore, a procedure is developed to determine an equimomental system with the position of a prescribed mass.Conditions are given to prescribing the position of two and three point-masses.Finally, three examples of applications of the ideas explored in this paper are presented.

Fig. 3
Fig. 3 Three point-masses located at isosceles the triangle's vertices

Fig. 5
Fig. 5 Variation of normalized masses from 0 to 2π

Fig. 6
Fig. 6 Three point-masses located at the triangle's vertices the diagonal matrix is written as = m diag a 2 , b 2 , 1 , and the non-rigid transformation of the triangle is D = diag(a, b, 1); therefore, we have that p i = D qi for i = 1, 2, 3.It is computed the position of the vertices' triangle in the current reference frame,The above points are written in the original reference frame using the transformation pi = G −1 p i ,

10
procedure as in Sect.6.1, we find the matrix G that diagonalizes the pseudo-inertia matrix.diagonal pseudo-inertia matrix is = G G T = diag (602.84288196,159.24583313, 1787.17469476), then, the scaling matrix becomes D = diag(0.58078924,0.29850426, 1).Thus, the position of the pointmasses in relation to the reference system Ox y is calculated by means of the equation pi = G −1 D qi ,