A method for composite transformation of inertia tensor translation and rotation

Abstract

The present invention relates to a method for compound transformation of inertia tensor translation and rotation, belonging to the field of engineering design technology. Firstly, based on the basic definition of inertia tensor, the general form of compound transformation of inertia tensor component matrix translation and rotation is derived by matrix transformation, and then conclusions under different simplified conditions are given. Finally, the correctness of the conclusion is verified by a calculation example of the synthesis of moment of inertia and product of inertia of three components of a certain aircraft. This method can adapt to the calculation requirements of inertia tensor of combined system based on the understanding of the data of each component changing over time through simple combined calculation.

Description

Technical Field

[0001] The invention belongs to the technical field of engineering design, and in particular relates to a composite transformation method of inertia tensor translation and rotation. Background Art

[0002] In aviation, aerospace and automotive engineering design, since the reference systems, installation positions and postures of the moments of inertia and products of inertia of each component are different, it is often necessary to transform and synthesize the moments of inertia (rotational inertia) and products of inertia of different components of a complex system. In particular, when the mass characteristics of a launch vehicle change over time, the inertia tensor needs to be synthesized to calculate the flight trajectory. The existing methods mainly obtain the synthesized moments of inertia and products of inertia through physical measurement and CAD model measurement. Physical measurement has high requirements on the measuring equipment, and the cycle and cost are also high; CAE model measurement requires the establishment of a complete three-dimensional model and the assignment of corresponding density. It can only synthesize static models, and cannot quickly synthesize the moments of inertia and products of inertia.

[0003] In order to realize the rapid synthesis of the moments of inertia and products of inertia of different rigid body components that change with time, it is necessary to perform coordinate transformation on them. Existing literature has given the forms of translational or rotational transformation of rigid body moments of inertia and products of inertia respectively, but there is a lack of relevant research on translational and rotational composite transformation.

[0004] Therefore, at this stage, it is necessary to design a composite transformation method of inertia tensor translation and rotation to solve the above problems. Summary of the invention

[0005] The purpose of the present invention is to provide a method for the composite transformation of inertia tensor translation and rotation, which is used to solve the technical problems existing in the above-mentioned prior art. In order to realize the rapid synthesis of the moments of inertia and products of inertia of different rigid body components that change with time, it is necessary to perform coordinate transformation on them. Existing literature has given the forms of translation or rotation transformation of rigid body moments of inertia and products of inertia respectively, but there is a lack of relevant research on composite transformation of translation and rotation.

[0006] To achieve the above object, the technical solution of the present invention is:

[0007] A composite transformation method of inertia tensor translation and rotation comprises the following steps:

[0008] S1: Starting from the basic definition of the inertia tensor, the translation and rotation transformation relationship of the inertia tensor component matrix is ​​derived using matrix transformation;

[0009] S2: Then transform the component inertia tensors in different coordinate systems to the same coordinate system.

[0010] S3: Finally, direct superposition is performed to achieve rapid synthesis of the moment of inertia and the product of inertia.

[0011] Furthermore, step S1 is specifically as follows:

[0012] Define the rectangular coordinate system O-xyz(S), the primitives in the x, y, z directions are i, j, k respectively, then the position vector of any point on the rigid body can be expressed as:

[0013] u=f T r (1)

[0014] Where:

[0015] u—position vector of any point on the rigid body;

[0016] f—the vector matrix composed of primitives of the coordinate system S, f=[i, j, k] T ;

[0017] r—coordinates of u in S, r=[x, y, z] T ;

[0018] The inertia tensor of a rigid body is defined as:

[0019]

[0020] Where:

[0021] I—rigid body inertia tensor;

[0022] u is the position vector of any point on the rigid body, u·u is the inner product, and uu is the dyadic vector;

[0023] E—unit diagonal matrix;

[0024] From equations (1) and (2), we can get:

[0025]

[0026] Where:

[0027] J—component matrix of inertia tensor I in S (inertia matrix), in the form of:

[0028]

[0029] J xx —Moment of inertia of the rigid body about the x-axis;

[0030] J yy —Moment of inertia of the rigid body about the y-axis;

[0031] J zz —Moment of inertia of the rigid body about the z-axis;

[0032] J xy —The inertia product of the rigid body relative to the x and y axes, J xy =J yx ;

[0033] J yz —The inertia product of the rigid body relative to the y and z axes, J yz =J yx ;

[0034] J zx —The inertia product of the rigid body relative to the z and x axes, J zx =J xz .

[0035] Further, step S2 is specifically as follows:

[0036] Construct two coordinate systems, where O1x1y1z1(S1) is the local coordinate system and O2x2y2z2(S2) is the global coordinate system; the position vector of O1 relative to O2 is u0, and its coordinate in S2 is r0; the position vector of any point A on the rigid body relative to O1 is u1, and its coordinate in S1 is r1; the position vector of A relative to O2 is u2, and its coordinate in S2 is r2. The coordinates of the center of mass of the rigid body under S1 are r C ;

[0037] From the geometric relationship:

[0038] u2=u0+u1 (5)

[0039] Written in coordinate system:

[0040] f2 T r2=f2 T r0+f1 T r1 (6)

[0041] The primitive vector matrix f1 of S1 can be obtained by primitive vector matrix f2 of S2 through primitive rotation:

[0042] f1=Lf2 (7)

[0043] Where L is the rotation matrix and L is the orthogonal matrix ( T Any coordinate system whose two coordinate axes are not parallel can be transformed into a coordinate system whose axes are parallel to each other through a number of primitive rotations.

[0044] Furthermore, f2 is rotated to be parallel to f1 in the order of zyx, which is achieved by three Euler angle primitive rotations. The rotation transformation matrix is:

[0045]

[0046] Substituting equation (7) into equation (6) yields the coordinate transformation relationship including translation and rotation:

[0047] r2=r0+L T r1 (9).

[0048] Furthermore, step S3 is specifically as follows:

[0049] After obtaining the coordinate transformation relationship of the rigid body position vector, consider the tensor matrix relationship under the coordinate transformation; from formula (4), it can be seen that the inertia tensor component matrix under the global coordinate system S2 is:

[0050]

[0051] Substituting equation (9) into equation (10) yields the transformed inertia tensor component matrix:

[0052]

[0053] make

[0054]

[0055]

[0056]

[0057]

[0058] Substituting equations (12) to (14) into equation (11), we get the transformation relationship:

[0059] J2=J0+L T J1L+(J 01 +J 10 ) (16)

[0060] in:

[0061] (1) When r0 = 0, that is, the origin O1 of the local coordinate system coincides with the origin O2 of the global coordinate system, this is a pure rotation transformation, and equation (16) is simplified to:

[0062] J2=L T J1L (17)

[0063] (2) When r0≠0 and L=E, it is a pure translation transformation, and the transformation relationship is the inertia translation theorem. Equation (16) is simplified to:

[0064] J2=J0+J1 (18)

[0065] (3) When r C = 0, that is, the origin O1 of the local coordinate system coincides with the center of mass of the rigid body, the translation transformation and rotation transformation are independent of each other, and equation (16) is simplified to:

[0066] J2=J0+L T J1L (19)

[0067] (4) When r0≠0 and r C When ≠0, that is, the origin O1 of the local coordinate system does not coincide with the origin O2 of the global coordinate system, and the origin O1 of the local coordinate system does not coincide with the center of mass of the rigid body, then there is a cross term in equation (16), indicating that there is a coupling effect between the translation and rotation transformations.

[0068] Compared with the prior art, the present invention has the following beneficial effects:

[0069] One of the beneficial effects of this scheme is that, firstly, based on the basic definition of the inertia tensor, the general form of the translation and rotation composite transformation of the inertia tensor component matrix is ​​derived by matrix transformation, and then the conclusions under different simplified conditions are given. Finally, the correctness of the conclusion is verified by the example of the synthesis of the moment of inertia and product of inertia of three components of a certain aircraft. This method can adapt to the calculation needs of the inertia tensor of the combined system based on the understanding of the data of each component changing over time through simple combined calculations. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 Schematic diagram of the coordinate transformation relationship of the rigid body in an embodiment of the present application.

[0071] Figure 2 Schematic diagram of primitive transformation of the coordinate system of an embodiment of the present application.

[0072] Figure 3 It is a schematic diagram of the step flow of an embodiment of the present application. DETAILED DESCRIPTION

[0073] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention, that is, the embodiments described are only part of the embodiments of the present invention, rather than all of the embodiments. The components of the embodiments of the present invention described and shown in the drawings herein can be arranged and designed in various different configurations.

[0074] Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative work are within the scope of protection of the present invention. It should be noted that relational terms such as the terms "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations.

[0075] Moreover, the terms "comprises", "includes" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or apparatus that includes a list of elements includes not only those elements, but also includes other elements not explicitly listed, or also includes elements inherent to such process, method, article or apparatus. In the absence of more restrictions, an element defined by the phrase "comprises a ..." does not exclude the presence of additional identical elements in the process, method, article or apparatus that includes the element.

[0076] The features and performance of the present invention are further described in detail below in conjunction with the embodiments.

[0077] In order to realize the rapid synthesis of the moments of inertia and products of inertia of different rigid body components that change with time, it is necessary to perform coordinate transformation on them. Existing literature has given the forms of translational or rotational transformation of rigid body moments of inertia and products of inertia respectively, but there is a lack of relevant research on translational and rotational composite transformation.

[0078] Example:

[0079] like Figure 3 As shown, a composite transformation method of inertia tensor translation and rotation is proposed, including the following steps:

[0080] S1: Starting from the basic definition of the inertia tensor, the translation and rotation transformation relationship of the inertia tensor component matrix is ​​derived using matrix transformation;

[0081] S2: Then transform the component inertia tensors in different coordinate systems to the same coordinate system.

[0082] S3: Finally, direct superposition is performed to achieve rapid synthesis of the moment of inertia and the product of inertia.

[0083] The inertia tensor is a physical quantity that describes the fixed-point rotation of a rigid body, including the moment of inertia and product of inertia information. In order to describe the inertia tensor, a rectangular coordinate system O-xyz(S) is defined, and the primitives in the x, y, and z directions are i, j, and k respectively. Then the position vector of any point on the rigid body can be expressed as:

[0084] u=f T r (1)

[0085] Where:

[0086] u—position vector of any point on the rigid body;

[0087] f—the vector matrix composed of primitives of the coordinate system S, f=[i, j, k] T ;

[0088] r—coordinates of u in S, r=[x, y, z] T ;

[0089] The inertia tensor of a rigid body is defined as:

[0090]

[0091] Where:

[0092] I—rigid body inertia tensor;

[0093] u is the position vector of any point on the rigid body, u·u is the inner product, and uu is the dyadic vector;

[0094] E—unit diagonal matrix;

[0095] From equations (1) and (2), we can get:

[0096]

[0097] Where:

[0098] J—component matrix of inertia tensor I in S (inertia matrix), in the form of:

[0099]

[0100] J xx —Moment of inertia of the rigid body about the x-axis;

[0101] J yy —Moment of inertia of the rigid body about the y-axis;

[0102] J zz —Moment of inertia of the rigid body about the z-axis;

[0103] J xy —The inertia product of the rigid body relative to the x and y axes, J xy =J yx ;

[0104] J yz —The inertia product of the rigid body relative to the y and z axes, J yz =J yx ;

[0105] J zx —The inertia product of the rigid body relative to the z and x axes, J zx =J xz .

[0106] In order to transform the inertia tensor component matrix in the local coordinate system to the global coordinate system, we first need to know the coordinate transformation relationship between the local coordinate system and the global coordinate system. Figure 1The two coordinate systems shown are O1x1y1z1(S1) as the local coordinate system and O2x2y2z2(S2) as the global coordinate system. The position vector of O1 relative to O2 is u0, and its coordinate in S2 is r0; the position vector of any point A on the rigid body relative to O1 is u1, and its coordinate in S1 is r1; the position vector of A relative to O2 is u2, and its coordinate in S2 is r2. The coordinate of the center of mass of the rigid body under S1 is r C .

[0107] Depend on Figure 1 The geometric relationship is:

[0108] u2=u0+u1 (5)

[0109] Written in coordinate system:

[0110] f2 T r2=f2 T r0+f1 T r1 (6)

[0111] The primitive vector matrix f1 of S1 can be obtained by primitive vector matrix f2 of S2 through primitive rotation:

[0112] f1=Lf2 (7)

[0113] Where L is the rotation matrix and L is the orthogonal matrix ( T L = E). Any coordinate system with two non-parallel axes can be transformed into a coordinate system with parallel axes by several primitive rotations. Here, f2 is rotated to be parallel to f1 in the order of zyx (other orders are also possible). Figure 2 The Euler angle primitive rotation is shown, and the rotation transformation matrix is:

[0114]

[0115] Substituting equation (7) into equation (6) yields the coordinate transformation relationship including translation and rotation:

[0116] r2=r0+L T r1 (9).

[0117] After obtaining the coordinate transformation relationship of the rigid body position vector, the tensor matrix relationship under the coordinate transformation is considered below. From formula (4), it can be seen that the inertia tensor component matrix under the global coordinate system S2 is:

[0118]

[0119] Substituting equation (9) into equation (10) yields the transformed inertia tensor component matrix:

[0120]

[0121] make

[0122]

[0123]

[0124]

[0125]

[0126] Substituting equations (12) to (14) into equation (11), we get the transformation relationship:

[0127] J2=J0+L T J1L+(J 01 +J 10 ) (16)

[0128] discuss:

[0129] (1) When r0 = 0, that is, the origin O1 of the local coordinate system coincides with the origin O2 of the global coordinate system, this is a pure rotation transformation, and equation (16) is simplified to:

[0130] J2=L T J1L (17)

[0131] (2) When r0≠0 and L=E, it is a pure translation transformation, and the transformation relationship is the inertia translation theorem. Equation (16) is simplified to:

[0132] J2=J0+J1 (18)

[0133] (3) When r C = 0, that is, the origin O1 of the local coordinate system coincides with the center of mass of the rigid body, the translation transformation and rotation transformation are independent of each other, and equation (16) is simplified to:

[0134] J2=J0+L T J1L (19)

[0135] (4) When r0≠0 and r C When ≠0, that is, the origin O1 of the local coordinate system does not coincide with the origin O2 of the global coordinate system, and the origin O1 of the local coordinate system does not coincide with the center of mass of the rigid body, then there is a cross term in equation (16), indicating that there is a coupling effect between the translation and rotation transformations.

[0136] Case Study:

[0137] Considering the synthesis problem of the moments of inertia and products of inertia of three components in a certain aircraft cabin, the mass characteristics of the known components in the local coordinate system of the body are shown in Table 1.

[0138] Table 1 Mass characteristics of the component body in the local coordinate system

[0139]

[0140] Assuming that the local coordinate system of the body coincides with the global coordinate system of the cabin when the three components are not installed, after installation, the local coordinate system of the body is first rotated relative to the global coordinate system, and then translated. The transformation parameters are shown in Table 2.

[0141] Table 2 Component translation and rotation transformation parameters

[0142]

[0143]

[0144] After the components are installed, the moment of inertia and product of inertia relative to the global coordinate system of the cabin are shown in Table 3, where the calculated values ​​are the calculation results of this paper, and CAD is the measurement result of the three-dimensional software (NX UG 12.0). It can be seen that the calculation results are almost consistent with the CAD measurement results, and the slight difference is mainly caused by numerical rounding errors and can be ignored. The results show that the formula for the composite transformation of the inertia tensor derived in this paper is correct and can provide a basis for the rapid synthesis of the moment of inertia and product of inertia in engineering.

[0145] Table 3 Moment of inertia and product of inertia of components in global coordinate system

[0146]

[0147] Table 3 (continued)

[0148]

[0149] The above are preferred embodiments of the present invention. Any changes made according to the technical solution of the present invention, as long as the resulting functions do not exceed the scope of the technical solution of the present invention, belong to the protection scope of the present invention.

Claims

1. A composite transformation method of inertia tensor translation and rotation, characterized in that: The following steps are involved: S1: Starting from the basic definition of the inertia tensor, the translation and rotation transformation relationship of the inertia tensor component matrix is ​​derived using matrix transformation; S2: Then transform the component inertia tensors in different coordinate systems to the same coordinate system. S3: Finally, direct superposition is performed to achieve rapid synthesis of the moment of inertia and the product of inertia; Step S1 is specifically as follows: Define the rectangular coordinate system O-xyz, denoted as S, the primitives in the x-axis, y-axis, and z-axis directions are i, j, and k respectively, then the position vector of any point on the rigid body can be expressed as: u=f T r (1) Where: u—position vector of any point on the rigid body; f—the vector matrix composed of primitives of the coordinate system S, f=[i, j, k] T ; r—coordinate of u in S, r=[x1, y2, z3] T ; The inertia tensor of a rigid body is defined as: Where: I—rigid body inertia tensor; u is the position vector of any point on the rigid body, u·u is the inner product, and uu is the dyadic vector; E—unit diagonal matrix; From equations (1) and (2), we can get: Where: J—component matrix of inertia tensor I in S, in the form of: J xx —Moment of inertia of the rigid body about the x-axis; J yy —Moment of inertia of the rigid body about the y-axis; J zz —Moment of inertia of the rigid body about the z-axis; J xy —The inertia product of the rigid body relative to the x and y axes, J xy =J yx ; J yz —The inertia product of the rigid body relative to the y and z axes, J yz =J zy ; J zx —The inertia product of the rigid body relative to the z and x axes, J zx =J xz ; Step S2 is specifically as follows: Construct two coordinate systems, where O1x1y1z1 is the local coordinate system, denoted as S1, and O2x2y2z2 is the global coordinate system, denoted as S2; the position vector of O1 relative to O2 is u0, and its coordinate in S2 is r0; the position vector of any point A on the rigid body relative to O1 is u1, and its coordinate in S1 is r1; the position vector of A relative to O2 is u2, and its coordinate in S2 is r2. The coordinates of the center of mass of the rigid body under S1 are r C ; From the geometric relationship: u2=u0+u1 (5) Written in coordinate system: The primitive vector matrix f1 of S1 can be obtained by primitive vector matrix f2 of S2 through primitive rotation: f1=Lf2 (7) Where L is the rotation matrix and L is the orthogonal matrix T L = E; any coordinate system with two non-parallel coordinate axes can be transformed into a coordinate system with parallel axes through a number of primitive rotations; Rotate f2 to be parallel to f1 in the order of zyx, which is achieved by three Euler angle primitive rotations. The rotation matrix is: Substituting equation (7) into equation (6) yields the coordinate transformation relationship including translation and rotation: r2＝r0+L T r1 (9)； Step S3 is as follows: After obtaining the coordinate transformation relationship of the rigid body position vector, consider the tensor matrix relationship under the coordinate transformation; from formula (4), it can be seen that the inertia tensor component matrix under the global coordinate system S2 is: Substituting equation (9) into equation (10) yields the transformed inertia tensor component matrix: make Substituting equations (12) to (15) into equation (11), we get the transformation relationship: J2=J0+L T J1L+(J 01 +J 10 ) (16) in: (1) When r0 = 0, that is, the origin O1 of the local coordinate system coincides with the origin O2 of the global coordinate system, this is a pure rotation transformation, and equation (16) is simplified to: J2=L T J1L (17) (2) When r0≠0 and L=E, it is a pure translation transformation, and the transformation relationship is the inertia translation theorem. Equation (16) is simplified to: J2=J0+J1 (18) (3) When r C = 0, that is, the origin O1 of the local coordinate system coincides with the center of mass of the rigid body, the translation transformation and rotation transformation are independent of each other, and equation (16) is simplified to: J2=J0+L T J1L (19) (4) When r0≠0 and r C When ≠0, that is, the origin O1 of the local coordinate system does not coincide with the origin O2 of the global coordinate system, and the origin O1 of the local coordinate system does not coincide with the center of mass of the rigid body, then there is a cross term in equation (16), indicating that there is a coupling effect between the translation and rotation transformations.

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