JP2013050814A - Analysis method of rigid body motion and elastic deformation - Google Patents

Abstract

PROBLEM TO BE SOLVED: To enable rigid body displacement and elastic deformation to be separated and analyzed even in the case of an object in which an elastic mode is expressed by unknown finite elements because of an ever-changing stiffness matrix.SOLUTION: A coordinate system using an initial center-of-gravity position and an initial inertia main axis of an object in a global coordinate system as an origin and an axis vector, respectively, is set as a local coordinate system, and a displacement of each joint of the object in the global coordinate system is set as a reference displacement of this joint. Then, setting of a new local coordinate system with a center-of-gravity position and an inertia main axis of the object after rigid body motion and elastic deformation in the local coordinate system as an origin and an axis vector, respectively, calculation of displacement due to elastic deformation of each joint, and updating for making the new local coordinate system to be a local coordinate system and making the displacement due to elastic deformation of each joint to be a reference displacement of each joint are performed until deviation of the local coordinate system between before and after the updating becomes equal to or less than a default value. The rigid body motion of the object is obtained from a coordinate value of the origin and a rotation matrix of the axis vector of the local coordinate system in the global coordinate system at that time, and the elastic deformation of the object is obtained from the deference displacement of each joint at that time.

Description

  The present invention relates to a method for analyzing rigid body motion and elastic deformation.

  Structural analysis of machine elements by a non-linear finite element method (hereinafter referred to as non-linear FEM (Finite Element Method)) has been used in a wide range of fields with the development of computers. The analysis by the non-linear FEM is also used for the analysis of the motion by translation and rotation of the object, that is, the rigid body motion, and the deformation accompanied by the distortion, that is, the elastic deformation.

  In order to visualize the analysis results of rigid body motion and elastic deformation of an object by such a nonlinear FEM, a deformation diagram may be generated using a post processor. The post processor can generate a deformation diagram in which the nodal displacement is enlarged and displayed at a constant magnification. However, in many cases, the elastic deformation of the object to be analyzed is very small compared to its rigid body motion, and even if the deformation figure in which the node displacement is enlarged at a constant magnification is seen, the deformation shape of the object can be intuitively understood. It is difficult to grasp.

  Therefore, conventionally, analysis methods such as those found in Patent Documents 1 and 2 have been proposed. In this conventional analysis method, a local coordinate system that is located near the object to be analyzed and is displaced according to the rigid body motion of the object is set, and the rigid body motion is elastically deformed in the rigid body mode in the local coordinate system. Are each expressed in an elastic mode. That is, in this conventional analysis method, the rigid body motion is expressed as the sum of the movement of the local coordinate system in the global coordinate system and the rigid body mode in the local coordinate system.

JP 2000-305922 A JP 2005-091370 A

  In such a conventional analysis method, it is possible to separate the nodal displacement caused by the rigid body motion of the object to be analyzed (hereinafter referred to as rigid body displacement) and the nodal displacement caused by the elastic deformation and display them separately. However, in such a conventional analysis method, the object to be analyzed needs to be an object expressed in a known elastic mode, and the stiffness matrix changes from moment to moment, so the elastic mode is expressed by an unknown finite element. It is not applicable to the analysis of the nonlinear transient response calculation result of the object.

  The present invention has been made in view of such circumstances, and the problem to be solved is that even in the case of an object expressed by a finite element whose elastic mode is unknown because the stiffness matrix changes from moment to moment. An object of the present invention is to provide a rigid body motion and elastic deformation analysis method capable of separately analyzing the rigid body displacement and the elastic deformation.

  In order to solve the above problems, the invention according to claim 1 as a method for analyzing rigid body motion and elastic deformation is characterized in that the object is divided by a finite element method performed by dividing the object into a finite number of elements having a node as a boundary. A method of analyzing rigid body motion and elastic deformation of the same object from the results of displacement and deformation analysis, (1) obtaining initial coordinate values of each node and mass matrix of each element in a fixed global coordinate system A first step, (2) setting a local coordinate system having an initial center of gravity position and an inertial principal axis of the object as an origin and an axis vector, and calculating an initial coordinate value of each node in the local coordinate system; And (3) a third step of calculating eigenvectors of rigid modes of the nodes from the initial coordinate values of the nodes in the mass matrix and the local coordinate system. (4) Calculate the rigid body mode displacement of the object in the local coordinate system from the eigenvector, the mass matrix and the reference displacement, after setting the displacement of each node due to elastic deformation as the reference displacement of each node. And (5) calculating the coordinate value of the center of gravity of the object after the rigid body motion and elastic deformation in the local coordinate system and the rotation matrix of the principal axis of inertia from the rigid body mode displacement and the eigenvector, respectively. Fifth step, (6) A new local coordinate system is set with the origin and axis vector as the center of gravity and inertial principal axis of the object after the rigid body motion and elastic deformation, and each node in the local coordinate system The difference between the initial coordinate value and the coordinate value after the rigid body motion and elastic deformation of each node in the new local coordinate system is used as the elastic deformation of each node. And calculating the new local coordinate system until the deviation between the local coordinate system and the new local coordinate system in the global coordinate system is equal to or less than a predetermined value. As a local coordinate system, after resetting the displacement due to elastic deformation of each node as the reference displacement of each node, the fourth to sixth steps are repeated, and the deviation is less than or equal to the predetermined value. From the coordinate value of the origin of the new local coordinate system and the rotation matrix of the axis vector in the global coordinate system at the time of becoming, the rigid body motion of the object from the displacement due to the elastic deformation of each node at the same point Each of them is asked for deformation.

  In this analysis method, when the analysis result of the nonlinear FEM is obtained, first, as the first step, the initial coordinate values of each node and the mass matrix of each element in the fixed global coordinate system are acquired. Then, as a second step, a local coordinate system is set in which the initial center of gravity position and the principal axis of inertia of the object are the origin and the axis vector, and the initial coordinate value of each node in the local coordinate system is calculated.

Subsequently, as a third step, a rigid mode eigenvector of each node is calculated from the initial value of each node in the mass matrix and the local coordinate system.
Next, the displacement of each node due to rigid body motion and elastic deformation is acquired from the analysis result of the nonlinear finite element method, and the displacement is set as the reference displacement of each node. And it transfers to the process of the convergence loop comprised by the following 4th-6th steps.

  When the process of the convergence loop is started, first, as a fourth step, the rigid mode displacement of the object in the local coordinate system is calculated from the eigenvector, the mass matrix, and the reference displacement. As a fifth step, the coordinate value of the center of gravity of the object after the rigid body motion and elastic deformation in the local coordinate system and the rotation matrix of the inertial principal axis are calculated from the rigid body mode displacement and the eigenvector. As a sixth step, a new local coordinate system is set with the origin and axis vector as the center of gravity and inertial axis of the object after rigid body motion and elastic deformation, and each local coordinate system set in the second step The deviation between the initial coordinate value of the node and the coordinate value after the rigid body motion and elastic deformation of each node in the new local coordinate system is calculated as the displacement due to the elastic deformation of each node.

  If the rigid body mode displacement of the object can be made small (≈0) at this time, the rigid body motion of the object is excluded from the new local coordinate system as translation and rotation of the center of gravity position. The displacement due to elastic deformation of each node calculated in the step does not include any displacement due to rigid body motion. However, as long as the position of the local coordinates of the object and the rotation matrix are not known, the rigid body mode displacement is not guaranteed to be very small (≈0), and the displacement due to the elastic deformation of each node calculated at this point includes the rigid body motion. There are still a few minutes left.

  Therefore, in this analysis method, the new local coordinate system is changed to the local coordinate system, and the displacement calculated by the elastic deformation of each node is set as the reference displacement of each node, and then the process of the convergence loop is performed. It is re-executed. When the process of the convergence loop is re-executed, the rigid body motion of the object is further removed from the reference displacement of each node updated in the process.

  Such re-execution of the process of the convergence loop is repeated until the deviation (deviation) between the local coordinate system in the global coordinate system and the new local coordinate system becomes equal to or less than a predetermined value. Therefore, the displacement due to elastic deformation of each node calculated immediately before exiting the convergence loop is obtained by sufficiently removing the rigid body motion of the object. That is, the calculated value of the displacement due to the elastic deformation of each node at this time almost accurately indicates the displacement due to the elastic deformation of each node. Further, the coordinate value of the origin of the new local coordinate system and the rotation matrix of the axis vector in the global coordinate system at this time indicate the translation and rotation of the rigid body motion of the object, respectively.

  Therefore, according to the above analysis method, since the stiffness matrix changes from moment to moment, even in the case of an object represented by a finite element whose elastic mode is unknown, the rigid body displacement and the elastic deformation are separated and analyzed. Can do.

The figure which shows the state of the object after an initial stage and a deformation | transformation. The figure which shows the setting aspect of the local coordinate system in one Embodiment of this invention. The flowchart of the rigid body motion and elastic deformation analysis routine applied to the embodiment. The flowchart of the rigid body motion and elastic deformation calculation routine applied to the embodiment. The figure which shows the position and shape of a connecting rod of the initial stage and predetermined time step in a global coordinate system, respectively. The position and shape of the connecting rod in the local coordinate system at the time of the end of the first convergence loop (a) without expanding the deformation, (b) showing the deformation enlarged. The position and shape of the connecting rod in the local coordinate system when exiting the convergence loop are diagrams in which (a) does not enlarge the deformation, and (b) shows the deformation enlarged.

  DESCRIPTION OF EMBODIMENTS Hereinafter, an embodiment embodying a rigid body displacement and elastic deformation analysis method of the present invention will be described in detail with reference to FIGS. Note that the analysis method of the present embodiment can be used for analysis of rigid body displacement and elastic deformation during the operation of a moving part such as a connecting rod.

  In the analysis method of the present embodiment, the rigid body displacement and elastic deformation of the object are separated and presented from the analysis result of the displacement and deformation of the object by nonlinear FEM. Analysis of deformation of an object by nonlinear FEM is performed using existing nonlinear FEM software. In the analysis of deformation of an object by nonlinear FEM, first, the object to be analyzed is divided into a finite number of elements with the nodes as boundaries, and each element such as initial coordinate values and constraint conditions of each node, elastic modulus, Poisson's ratio, etc. Enter the physical quantity and the load applied to each node. Based on this input data, the nonlinear FEM software solves the simultaneous equations formed by approximating the behavior of each element with a model, so that the coordinate values of each node and the stress acting on each element can be obtained. , Seeking every time step. The analysis result of the non-linear FEM software is processed by a post processor, and is presented as visualized data such as a deformation diagram and a Mises stress diagram.

  In the analysis method of the present embodiment, the rigid body displacement and the elastic deformation of the object to be analyzed are obtained separately from the analysis result of the nonlinear FEM. Next, the basic principle will be described using a simple model.

  Here, consider a case where the rigid body displacement and elastic deformation of an object M composed of a single element having four nodes as shown in FIG. In the same figure, the initial position and shape of the object M in the global coordinate system and the position and shape of the object M after the rigid body motion and elastic deformation in the global coordinate system are also shown. The initial coordinate value of each node in the global coordinate system is known, and the coordinate value after rigid body motion and elastic deformation of each node in the global coordinate system is obtained from the analysis result of nonlinear FEM.

  Here, consider one of the nodes (node P). Here, the initial position of the node P is represented by a vector p0 starting from the origin of the global coordinate system and ending at the initial position of the node P. Further, the displacement of the node P due to the rigid body motion and elastic deformation is represented by a vector u0 having the initial position of the node P as the start point and the rigid body motion and elastic deformation of the node P as the end point. The position of the deformed node P at this time is expressed as the sum of the vector p0 and the vector u0. Further, the position of the node P with respect to the center of gravity position of the object M in the initial stage is a vector p with the initial position of the center of gravity P of the object M as the start point and the initial position of the node P as the end point. The translation and rotation of the displacement of the center of gravity G of the object M due to elastic deformation are represented by a vector r and a rotation matrix A, and the displacement due to elastic deformation of the node P is represented by a vector u. The position after the rigid body motion and elastic deformation of the node P in the global coordinate system at this time, that is, the sum of the vector p0 and the vector u0 is expressed by the following equation. The rigid body motion of the node P is represented by “r + Ap”, and the elastic deformation thereof is represented by “Au”.

p0 + u0 = r + A (p + u) = (r + Ap) + Au

Therefore, the rigid body motion of each node in the global coordinate system and the position after elastic deformation ({p0} + {u0}) are expressed by the following equations. In the following equation, the column vector {p0} is the initial position of each node, the column vector {u0} is the displacement of each node, the column vector {r} and the matrix [A] of the rotation matrix are rigid bodies The translation vector and the rotation component of the displacement of the center of gravity G of the object M due to motion and elastic deformation, and the column vector {u} respectively represent the displacement due to elastic deformation of each node.

{P0} + {u0} = ({r} + [A] · {p}) + [A] {u}

In the above equation, “{r} + [A] {p}” represents the rigid body motion of each node, and “[A] {u}” represents the elastic deformation of each node. Therefore, the position of each node after the initial and after the rigid body motion and elastic deformation, the translation and rotation of the displacement of the gravity center position G of the object M, and even the position of each node based on the gravity center position G of the initial object M If it understands, even if the elastic mode is unknown and the displacement by elastic deformation of each node is unknown, the rigid body motion and elastic deformation of each node after a deformation | transformation can be calculated | required. That is, as shown in FIG. 2, when a local coordinate system is set in which the center of gravity G of the object M is the origin and the inertia principal axis is an axis vector, the rigid body motion of the object M is the local coordinate system on the global coordinate system. It can be obtained as translation and rotation of. The elastic deformation of the object M can be obtained as a displacement of each node on the local coordinate system.

Next, a specific procedure for analysis of rigid body displacement and elastic deformation in the present embodiment will be described. This analysis is performed according to the procedure shown in the rigid body motion and elastic deformation analysis routine of FIG.
When the analysis result by the nonlinear FEM is obtained, first, in step S100, the initial coordinate value {p0} of each node in the fixed global coordinate system is acquired from the analysis result of the nonlinear FEM. In step S101, the mass matrix [M] of the object M is acquired from the analysis result of the nonlinear FEM. The mass matrix [M] is a determinant representing the mass of each element constituting the object M.

  Next, in step S102, from the shape of the object M and the mass matrix [M], the initial coordinate value {g} of the center of gravity position of the object M in the global coordinate system, the main inertia moment, and the rotation matrix [I] of the main inertial axis are obtained. Each is calculated.

  Thereafter, in step S103, the initial coordinate value {p} of each node with respect to the initial gravity center position of the object M is calculated from the initial coordinate value {p0} of each node in the global coordinate system. This coordinate value {p} indicates the coordinate value of each node in the local coordinate system having the initial center of gravity position and the principal axis of inertia of the object M in the global coordinate system as the origin and the axis vector.

Next, in step S104, the eigenvector [Φr] of the rigid body mode of each node is calculated from the initial coordinate value {p} of each node and the mass matrix [M].
When the above processing is completed, the following steps S105 to S107 are executed for each time step of the nonlinear FEM analysis result, and the rigid body motion and elastic deformation of the object M at each time step are obtained. Then, the analysis of the rigid body motion and the elastic deformation is finished when the processing of the last time step is completed (S108: YES).

  That is, in step S105, a coordinate system in which the initial gravity center position and the inertial principal axis of the object M are set as the origin and the axis vector is set as the local coordinate system. In step S106, the displacement {u0} of each node due to the rigid body motion and elastic deformation at the corresponding time step is acquired from the analysis result of the nonlinear FEM, and the acquired displacement {u0} of each node is the reference displacement { u}. In step S107, a rigid body motion and elastic deformation calculation process for calculating the rigid body motion and elastic deformation of the object M in the time step is executed.

  The rigid body motion and elastic deformation calculation processing is performed through the rigid body motion and elastic deformation calculation routine shown in FIG. When the processing of this routine is started, first, in step S200, the rigid body mode displacement {qr} of the object M in the local coordinate system is calculated by the following equation. In the following expression, “t [Φr]” represents a transposed matrix of the eigenvector [Φr].

{Qr} = t [Φr] · [M] · {u}

Next, in step S201, the translation of the displacement of the center of gravity of the object M in the local coordinate system (deviation from the origin of the local coordinate system) from the eigenvector [Φr] of the rigid body mode and the rigid body mode displacement {qr} of each node. Minutes and rotations are calculated. In step S202, the coordinate value {ri} of the center of gravity position of the object M at the current time step and the rotation matrix [Ai] of the inertial main axis in the local time frame are calculated from the calculated translation and rotation. The

  Subsequently, in step S203, a new local coordinate system is set in which the center of gravity and the inertial main axis of the object M at the current time step are set as the origin and the axis vector. In step S204, the displacement {u '} due to the elastic deformation of each node is calculated by the following equation. In the following expression, “t [Ai]” represents a transposed matrix of the rotation matrix [Ai]. Strictly speaking, the displacement {u ′} due to elastic deformation of each node calculated here is the initial coordinate value {p} of each node in the local coordinate system, the rigid body motion of each node in the new local coordinate system, and This is a deviation from the coordinate value after elastic deformation, and includes displacement due to rigid body motion until the processing of this routine proceeds to the final stage.

{U ′} = t [Ai] ({p} + {u} − {ri}) − {p}

Subsequently, in step S205, the new local coordinate system set in step S203 is set as the local coordinate system. That is, the coordinate value {r} of the origin of the local coordinate system in the global coordinate system and the axis vector rotation matrix [A] are updated according to the following formula. In step S205, the displacement {u ′} due to the elastic deformation of each node calculated in step S204 is set as the reference displacement {u}.

{R} = {r} + [A] · {ri}
[A] = [A] · [Ai]

In subsequent step S206, whether or not the deviation of the local coordinate system before and after the update, that is, the deviation between the local coordinate system before the update in step S205 and the new local coordinate system set in step S203 is equal to or less than a predetermined value. Is confirmed. In this confirmation, the deviation between the coordinate value {r (i-1)} of the origin of the local coordinate system before update and the coordinate value {ri} of the origin of the new local coordinate system is equal to or less than the predetermined value α and is updated. This is performed depending on whether or not the deviation between the rotation vector [A (i−1)] of the axis vector of the previous local coordinate system and the rotation matrix [Ai] of the axis vector of the new local coordinate system is equal to or less than the predetermined value β. ing.

  Here, if the deviation of the local coordinate system before and after the update is not less than the predetermined value (S206: NO), the process returns to step S200. And the process of step S200-step S206 is performed again using the updated local coordinate system and reference | standard displacement {u}. In the following, the processing loop from step S200 to step S206 is referred to as a convergence loop.

  On the other hand, if the deviation of the local coordinate system before and after the update is equal to or smaller than the predetermined value (S206: YES), the process proceeds to step S207 through the convergence loop. In step S207, the coordinate value {r} of the origin of the current local coordinate system in the global coordinate system and the rotation matrix [A] of the axis vector are stored as the rigid body motion of the object M in the current time step. In step S208, the current reference displacement {u} of each node is stored as the elastic deformation of the object M at the current time step. Thereafter, the process of this routine is terminated, and the process proceeds to step S108 of the rigid body motion and elastic deformation analysis routine.

  Next, an analysis mode of rigid body motion and elastic deformation when a connecting rod connecting the piston and the crankshaft of the engine is an analysis target will be described. In FIG. 5, the position and shape of the initial connecting rod are indicated by a one-dot chain line, and the position and shape of the connecting rod after rigid body motion and elastic deformation are indicated by a solid line on the overall coordinate system. The analysis by the non-linear FEM is performed by dividing the connecting rod into a finite number of elements whose boundaries are nodes.

In this embodiment, after such a nonlinear FEM analysis result is obtained, rigid body motion and elastic deformation of the connecting rod at each time step are obtained in the following manner.
That is, when the analysis result of the nonlinear FEM is obtained, first, as the first step, the initial coordinate values {p0} of each node in the global coordinate system and the mass matrix [M] of each element are acquired. Then, as a second step, a local coordinate system is set with the initial center of gravity position and inertial principal axis of the connecting rod as the origin and the axis vector, and the initial coordinate value {p} of each node in the local coordinate system is calculated. .

Subsequently, as a third step, the eigenvector [Φr] of the rigid body mode of each node is calculated from the mass matrix [M] and the initial coordinate value {p} of each node in the local coordinate system.
Next, the displacement {u0} due to the rigid body motion and elastic deformation of each node of the connecting rod is acquired from the analysis result of the nonlinear FEM, and the displacement {u0} is set as the reference displacement {u} of each node. And it transfers to the process of the convergence loop comprised by the following 4th-6th steps.

  When the process of the convergence loop is started, as a fourth step, the rigid body mode displacement {qr} of the connecting rod in the local coordinate system is calculated from the eigenvector [Φr], the mass matrix [M] and the reference displacement {u}. Is done. Further, as a fifth step, from the rigid body mode displacement {qr} and the eigenvector [Φr], the coordinate value {ri} of the center of gravity of the object after the rigid body motion and elastic deformation in the local coordinate system and the rotation matrix [Ai] of the principal axis of inertia Are calculated respectively. As a sixth step, a new local coordinate system is set in which the center of gravity and the inertial principal axis of the connecting rod after rigid body motion and elastic deformation are the origin and the axis vector. Further, the deviation between the initial coordinate value {p} of each node in the local coordinate system set in the second step and the coordinate value after the rigid body motion and elastic deformation of each node in the new local coordinate system is Calculated as displacement {u ′} due to elastic deformation of the node.

  FIG. 6A shows the position and shape of the connecting rod in the new local coordinate system at this point, that is, when the first convergence loop process is completed. If the rigid body mode displacement {qr} of the connecting rod object can be made very small (≈0) at this time, the connecting rod rigid body motion is excluded from the new local coordinate system as translation and rotation of the center of gravity. Therefore, the displacement {u ′} due to elastic deformation of each node calculated in the sixth step does not include any displacement due to rigid body motion. However, since the position and rotation matrix of the local coordinate system of the connecting rod are unknown, there is no guarantee that the rigid body mode displacement is very small (≈0), and the displacement {u ′ } Still has some rigid body motion.

  FIG. 6B shows the position and shape of the connecting rod in the new local coordinate system at this time, with the deformation magnified 100 times. As shown in the figure, at this time, the removal of the rigid body motion is still insufficient.

  Therefore, in the present embodiment, the new local coordinate system is set to the local coordinate system, and the displacement {u ′} due to elastic deformation of each node calculated previously is set as the reference displacement {u} of each node. Thus, the process of the convergence loop is re-executed. When the process of the convergence loop is executed again, the rigid body motion of the connecting rod is further removed from the reference displacement {u} of each node updated in the process.

  The convergence loop is re-executed until the deviation of the local coordinate system before and after the update is less than or equal to the predetermined value, more specifically, the coordinate value {r} of the origin of the local coordinate system before and after the update and the rotation of the axis vector. The process is repeated until the deviation of the matrix [A] becomes equal to or less than the predetermined value. Therefore, the rigid motion of the object is almost completely removed from the displacement {u ′} due to elastic deformation of each node calculated immediately before exiting the convergence loop. Accordingly, the calculated value {u '} of the nodes due to elastic deformation immediately before exiting the convergence loop, that is, the reference displacement {u} updated in step S205 indicates the displacement due to elastic deformation of each node almost accurately. It becomes. Further, the coordinate value {r} of the origin in the global coordinate system of the new local coordinate system set immediately before exiting the convergence loop, that is, the local coordinate system updated in step S205, and the rotation matrix [A] of the axis vector thereof. Represents the translational component and the rotational component of the rigid body motion of the connecting rod, respectively.

  FIG. 7 (a) shows the position and shape of the connecting rod in the local coordinate system at this point, that is, when the convergence loop is exited. FIG. 7B shows the position and shape of the connecting rod in the local coordinate system at this time with the deformation magnified 10,000 times. Thus, the displacement due to the rigid body motion is almost completely excluded from the position and shape of the connecting rod in the local coordinate system at this time. Therefore, by looking at the position and shape of the connecting rod in the local coordinate system at this time, it is possible to easily and accurately grasp the aspect of the elastic deformation.

According to the rigid body motion and elastic deformation analysis method of the present embodiment described above, the following effects can be obtained.
(1) Since the stiffness matrix changes from moment to moment, even in the case of an object represented by a finite element whose elastic mode is unknown, the rigid body displacement and elastic deformation can be separated and analyzed.

(2) Existing FEM software can be used as it is for the analysis of the displacement and deformation of an object by FEM, which is the premise of the analysis of rigid body motion and elastic deformation.
In addition, the said embodiment can also be changed and implemented as follows.

  In the above embodiment, the transient response calculation is performed for each time step of the rigid body motion and elastic deformation of the object M. However, if only the final rigid body motion and elastic deformation of the object M need only be understood, Only the final rigid body motion and elastic deformation of the object M may be obtained. That is, in this case, in step S106 of the rigid body displacement and elastic deformation analysis routine of FIG. 3, the final displacement of each node is acquired, and the processing of the routine is performed when the processing of step S107 is completed. Will be finished.

  In the above embodiment, the rigid body motion and elastic deformation of the object are obtained from the analysis result of the nonlinear FEM, but the present invention is similarly applied to the analysis of the rigid body motion and elastic deformation of the object from the analysis result of the linear FEM. can do.

  M ... object, G ... center of gravity, P ... node.

Claims (1)

A method of analyzing rigid body motion and elastic deformation of the object from a result of analysis of displacement and deformation of the object by a finite element method performed by dividing the object into a finite number of elements having a node as a boundary,
Obtaining an initial coordinate value of each node and a mass matrix of each element in a fixed global coordinate system;
A second step of setting a local coordinate system having an initial center of gravity position and an inertial main axis of the object as an origin and an axis vector, and calculating initial coordinate values of the nodes in the local coordinate system;
A third step of calculating a rigid mode eigenvector of each node from the mass matrix and initial coordinate values of each node in the local coordinate system;
After setting the displacement of each node due to rigid body motion and elastic deformation as the reference displacement of each node,
A fourth step of calculating a rigid mode displacement of the object in the local coordinate system from the eigenvector, the mass matrix and the reference displacement;
A fifth step of calculating, from the rigid body mode displacement and the eigenvector, a coordinate value of the center of gravity position of the object after the rigid body motion and elastic deformation in the local coordinate system and a rotation matrix of the principal axis of inertia, respectively;
A new local coordinate system is set with the origin and axis vector of the center of gravity and inertial principal axis of the object after the rigid body motion and elastic deformation, and the initial coordinate value of each node in the local coordinate system and the new A sixth step of calculating a deviation from the coordinate value after the rigid body motion and elastic deformation of each node in the local coordinate system as a displacement due to elastic deformation of each node;
Thereafter, until the deviation between the local coordinate system and the new local coordinate system in the global coordinate system is equal to or less than a predetermined value, the new local coordinate system is used as the local coordinate system, and the elasticity of each node is determined. After re-setting the displacement due to deformation as the reference displacement of the respective nodes, the fourth to sixth steps are repeated, and the deviation in the global coordinate system at the time when the deviation becomes equal to or less than the predetermined value. The rigid body motion of the object is obtained from the coordinate value of the origin of the new local coordinate system and the rotation matrix of the axis vector, and the elastic deformation of the object is obtained from the displacement due to the elastic deformation of each node at the same point. Analyzing method of rigid body motion and elastic deformation.

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