Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.


Introduction
The common method to obtain the equations of motion for a finite element belonging to an elastic element of a multibody system is represented by Lagrange equations. In this way, the most important step in the analysis of the multi-body systems with elastic elements is solved, the rest of the procedures being the classical ones currently used in the finite element method FEA and verified by practice. So, it is possible to determine the dynamic response of a single finite element. Using the known assembly methods and introducing loads and proper boundary conditions, the set of differential equations describing the behavior of the entire elastic system is obtained.
In this process, the type of finite element chosen will determine the shape functions that will be used and which will give the final form of the coefficients of the matrix appearing in the equations of motion. In such an analysis, it is considered that the deformations are small enough to not have influence on the general, rigid motion of the multibody system. Thus, a large class of systems has been studied, having in their structure elastic elements modeled with one-dimensional, two-dimensional or three-dimensional finite elements. One-dimensional finite elements were among the first studied. We can mention here [1][2][3][4][5][6][7]. The method was then extended naturally to bi-and three-dimensional elements [8][9][10].
The problems regarding the accuracy of the results and the accuracy of the models used in the case of the mechanisms, which have significant displacements and deformations, were not developed in the paper. These very interesting and current study domains themselves represent a distinct field of research. In this paper, we only dealt with the presentation of an alternative method, which presents advantages of calculation, of determining the equations of motion for a single element. The other operations used then in the method of finite elements were considered the ones currently used in practice.

Gibbs-Appell Formalism
In the case of scleronomic liaisons, the acceleration of a point i of a system of material points having n degree of freedom has, generally, the expression: where r i is the position vector of the point and q j , j = 1, n represent the independent coordinates. In the following, the dot placed above a scalar, vector or matrix size means the derivative of the respective size with respect to time. The notion energy of accelerations for a system of material points is defined as [27]: The form of this expression is similar to the expression of kinetic energy but the mechanical significance is different. In the case of a solid body, the definition can be extended to the whole domain of the body as: where the acceleration of a certain point of the solid is similar to Equation (1): Introducing Equation (3a) in Equation (3) we get: where: contains terms that have no accelerations, represents the terms in which the accelerations intervene linearly and: represents the quadratic terms in accelerations. GA equations are given by relations: where Q j it represents the generalized force corresponding to the generalized coordinate q j .

Basic Hypothesis
Now consider a finite element modeling a domain of an elastic element from the structure of a multibody system. The aim of the paper is to determine the motion equations of this element. The generalized coordinates will be the displacements of the nodes. The displacements of an arbitrary point of the continuous domain modeled by the finite element shall be expressed using a finite number of nodal coordinates. The finite element shall be related to the mobile reference frame, which participates in the general rigid motion (   Z o ), the acceleration of the origin of the local reference frame relative to the global reference frame OXYZ. We shall note with ω(ω x , ω y , ω z ) the angular velocity and with ε(ε x , ε y , ε z ) the angular acceleration. Generally, a multibody system consists of several solids. Every solid is characterized by these two vectors, and the number of pairs of such vectors is equal to the number of distinct bodies. The relations between the components of a vector in the local coordinate system {t} L and the components of the same vector in a global coordinate system {t} G are obtained using a matrix of rotation [R]: Symmetry 2020, 12, 321 5 of 13 Symmetry 2020, 12, x FOR PEER REVIEW 5 of 13 where   where the matrix elements   ) , , ( z y x N , the shape functions, will depend on the type of the finite element chosen and   L  are the independent nodal coordinates. The velocity of point M' will be: (12) and the acceleration will be: These expressions will be used to determine the energy of accelerations. In the future, the index G mark the vector with the components expresses in the global reference system, the index L mark the vector with the components expresses in the local reference frame and the non-indexed vectors are considered to be written in the local coordinate system. The derivatives of the positional matrix   R occur in the previous relations. These derivatives will define angular velocities and angular accelerations. From the orthogonality conditions written for the positional matrix we have: where   E is the unit matrix, the following relation can be obtained: If an arbitrary point M of the finite element has a displacement f L changing into M', we may write: where {r M } G is the position vector of point M' with components expressed in the global coordinate system. The continuous displacement field noted f L is approximated, using the finite element method, by relation: where the matrix elements [N(x, y, z)], the shape functions, will depend on the type of the finite element chosen and {δ} L are the independent nodal coordinates. The velocity of point M' will be: and the acceleration will be: These expressions will be used to determine the energy of accelerations. In the future, the index G mark the vector with the components expresses in the global reference system, the index L mark the vector with the components expresses in the local reference frame and the non-indexed vectors are considered to be written in the local coordinate system. The derivatives of the positional matrix [R] occur in the previous relations. These derivatives will define angular velocities and angular accelerations. From the orthogonality conditions written for the positional matrix we have: where [E] is the unit matrix, the following relation can be obtained: which is a skew symmetric matrix angular velocity, with components in global coordinate system, corresponding to the angular velocity: Obviously, we have, too: The angular acceleration operator is defined by: . and: It will result after some elementary calculus: .. whereas: We have, too: These relations will be useful in the following considerations.

Motion Equations
In the following, the equations of motion for a finite element will be established using the GA formalism. For this, it is necessary to first determine the energy of the accelerations of a finite element. If we use Expression (13) determined previously for the acceleration of a point, the energy of the accelerations for the considered finite element is given by the expression [30]: .. The quadratic terms can be easily identified in the accelerations of the independent coordinates .. δ L , noted with E a2 : ..
the terms in which these accelerations appear linear E a1 : . δ L dV (25) and the terms that do not contain at all accelerations and which do not have importance in GA equations, E a0 : The potential energy (internal work) has a classical form: where {σ} is the stress tensor and {ε} the strains tensor: Writing the Hooke law as follows: (29) and of the differential relations between strains and finite deformations: where [a] represents the differentiation operator [31] and, using Equations (13), (15) and (16), the internal work becomes [31]: where [k] is the classical stiffness matrix: If noted with p = p(x, y, z) , the vector of body forces per unit volume, then the external work of these is: The concentrated forces q L in the nodes of the element will give an external work: The equations of motion are obtained by applying the GA equations previously presented: By ∂E ∂δ e , we understand: In our case: and: After a series of elementary calculations and rearranging of terms, we get the equations of motion for the finite element considered: where: therefore: Symmetry 2020, 12, 321 where it is denoted: The term E a0 does not contain .. δ L and, consequently: Finally, the motion equations can be written in the condensed form: [m] ..
These equations are related to the local system of coordinates. Similar formulas can be obtained, and we consider the global reference frame. The matrix coefficients can be calculated after choosing the shape functions and the independent nodal coordinates for expressing the displacement of a point.

Conclusions and Discussions
In the classical Lagrangian formalism, to determine the equations of motion for a three-dimensional finite element, the main step of the calculation is the Lagrangian, based essentially on the kinetic energy of the deformable finite element. The Lagrangian is: where E p , W, W c are given, respectively, by the Relations (31), (33) and (34) and the kinetic energy has the expression: Lagrange's equations are: Leaving aside the other derivatives, which have a low weight in the computing economy, the term most significant in this formulation is the term kinetic energy. In this expression, there are four terms containing the vector . Note that of the three terms of kinetic energy, only two contain the vector of accelerations. The size E a2 has a single term that contains this vector and E a1 has four terms that contain the vector .. δ L and which are differentiated. The total number of differentiation to be made in this case is five.
The conclusion from these two approaches is that the number of operations to be performed when using the Gibbs-Appell equations is lower than when Lagrange's method is applied. The use of this method has the advantage that we are familiar with the calculation of kinetic energy of a solid and to determine the energy of the accelerations it is sufficient to replace the speeds, in the kinetic energy formula, with the accelerations. The use of these equations will result in an economy in the computational effort we make to determine the equations of motion for a single finite element.
An application was made for a rod within a wind pumping mechanism ( Figure 2). The mechanism proposed for the study is a mechanism with two degrees of freedom of type "kinematic chain closed by inertia" [32]. The conclusion from these two approaches is that the number of operations to be performed when using the Gibbs-Appell equations is lower than when Lagrange's method is applied. The use of this method has the advantage that we are familiar with the calculation of kinetic energy of a solid and to determine the energy of the accelerations it is sufficient to replace the speeds, in the kinetic energy formula, with the accelerations. The use of these equations will result in an economy in the computational effort we make to determine the equations of motion for a single finite element.
An application was made for a rod within a wind pumping mechanism ( Figure 2). The mechanism proposed for the study is a mechanism with two degrees of freedom of type "kinematic chain closed by inertia" [32]  Table 1 presents a comparison between the number of differentiation required when using Lagrange's method and the number of differentiation required if the Gibbs-Appell method is applied. It is noted that the number of these operations is reduced to less than half. For a large number of finite elements used, in the case of complex structures, the reduction can be significant and can reduce the computational effort. We mention that after obtaining the matrix coefficients of the system of differential equations, the next procedures in the two methods become identical, so also the necessary computer times.  5  288  120  10  528  220  15  768  320  20  1008  420  25 1248 520  Table 1 presents a comparison between the number of differentiation required when using Lagrange's method and the number of differentiation required if the Gibbs-Appell method is applied. It is noted that the number of these operations is reduced to less than half. For a large number of finite elements used, in the case of complex structures, the reduction can be significant and can reduce the computational effort. We mention that after obtaining the matrix coefficients of the system of differential equations, the next procedures in the two methods become identical, so also the necessary computer times.

Number of Finite Elements Lagrange Gibbs-Appell
Another problem that arises is the accuracy of the results. In the case of finite element analysis of a mechanism that can have large displacements, specific problems arise, very delicate, related to the accuracy of the results obtained and the correctness of the models used. During the present paper, we did not deal with these aspects, especially interesting and exciting. By themselves, these aspects define a field of research. The purpose of the paper was to provide an alternative way of writing the equations of motion, which would have the advantage of a smaller number of operations to perform.
The test mechanism on which the considerations were made, presented in Figure 2, can be considered, at this level as regards the accuracy of the results, accessible through the classical methods. In Figures 3 and 4, we presented the space of the possible positions that can be occupied by the elements of this mechanism, to justify our hypothesis. 1968 820 Another problem that arises is the accuracy of the results. In the case of finite element analysis of a mechanism that can have large displacements, specific problems arise, very delicate, related to the accuracy of the results obtained and the correctness of the models used. During the present paper, we did not deal with these aspects, especially interesting and exciting. By themselves, these aspects define a field of research. The purpose of the paper was to provide an alternative way of writing the equations of motion, which would have the advantage of a smaller number of operations to perform.
The test mechanism on which the considerations were made, presented in Figure 2, can be considered, at this level as regards the accuracy of the results, accessible through the classical methods. In Figures 3 and 4, we presented the space of the possible positions that can be occupied by the elements of this mechanism, to justify our hypothesis.   1968 820 Another problem that arises is the accuracy of the results. In the case of finite element analysis of a mechanism that can have large displacements, specific problems arise, very delicate, related to the accuracy of the results obtained and the correctness of the models used. During the present paper, we did not deal with these aspects, especially interesting and exciting. By themselves, these aspects define a field of research. The purpose of the paper was to provide an alternative way of writing the equations of motion, which would have the advantage of a smaller number of operations to perform.
The test mechanism on which the considerations were made, presented in Figure 2, can be considered, at this level as regards the accuracy of the results, accessible through the classical methods. In Figures 3 and 4, we presented the space of the possible positions that can be occupied by the elements of this mechanism, to justify our hypothesis.