Moment of Inertia


INERTIA_XX	$I_{x} = \int ρ (y^{2} + z^{2}) d V$
INERTIA_YY	$I_{y} = \int ρ (x^{2} + z^{2}) d V$
INERTIA_ZZ	$I_{z} = \int ρ (x^{2} + y^{2}) d V$
INERTIA_XY	$I_{x y} = I_{y x} = - \int ρ x y d V$
INERTIA_XZ	$I_{x z} = I_{z x} = - \int ρ x z d V$
INERTIA_YZ	$I_{y z} = I_{y x} = - \int ρ y z d V$

Analysis-Independent Design Response

For the moment of inertia, the following table shows the allowed combinations between the strategy and the items OBJ_FUNC and CONSTRAINT with C for controller and S for sensitivity-based optimization.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC	S	S	S	S
CONSTRAINT	S	S	S	S

The moments of inertia can be applied as DRESP (sensitivity-based optimization) and as VARIABLE (controller-based optimization). The moments of inertia are defined using INERTIA_XX, INERTIA_XY (INERTIA_YX), INERTIA_XZ (INERTIA_ZX), INERTIA_YY, INERTIA_YZ (INERTIA_ZY), and INERTIA_ZZ, respectively.

Mathematically, the moments of inertia about the origin of the coordinate system are given by the above integrals, which can be calculated in a global or a local coordinate system. The local coordinate system is defined using CS_REF. The volume for which the moments of inertia are calculated is defined using EL_GROUP.

Note:

Only the elements of the element group (EL_GROUP) listed in the tables of supported element types are applied in the calculation of the moments of inertia.
The product of inertia with respect to any two orthogonal axes is zero if either of the axes is an axis of symmetry.
The physical density defined in the finite element input file is used in the calculation of the moments of inertia.

The design response (DRESP) for the moment of inertia about the line through the origin, parallel to the x-axis is defined like


DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XX
 EL_GROUP = ...
 CS_REF   = ...
END_

The design response (DRESP) for the moment of inertia about the line through the origin, parallel to the y-axis is defined like


DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_YY
 EL_GROUP = ...
 CS_REF   = ...
END_

The design response (DRESP) for the moment of inertia about the line through the origin, parallel to the z-axis is defined like


DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_ZZ
 EL_GROUP = ...
 CS_REF   = ...
END_

The design response (DRESP) for the moment of inertia describing the coupling between the rotation parallel to the x-axis and the rotation parallel to the y-axis yields:


DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XY
 (Alternatively, TYPE =INERTIA_YX)
 EL_GROUP = ...
 CS_REF   = ...
END_

The design response (DRESP) for the moment of inertia describing the coupling between the rotation parallel to the x-axis and the rotation parallel to the z-axis yields:


DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XZ
 (Alternatively, TYPE = INERTIA_ZX)
 EL_GROUP = ...
 CS_REF   = ...
END_

The design response (DRESP) for the moment of inertia describing the coupling between the rotation parallel to the y-axis and the rotation parallel to the z-axis yields:


DRESP
 ID_NAME              = ...
 DEF_TYPE             = SYSTEM
 TYPE                 = INERTIA_YZ
 (Alternatively, TYPE = INERTIA_YZ)
 EL_GROUP             = ...
 CS_REF               = ...
END_

Examples of Commands

For example, the design response (DRESP) for the moment of inertia of the entire structure (ALL_ELEMENTS) about the line through the origin of the global coordinate system, parallel to the y-axis is defined like


DRESP
 ID_NAME  = DRESP_INERTIA_YY_GLOBAL
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_YY
 EL_GROUP = ALL_ELEMENTS
END_

For example, the definition of the design response (DRESP) for the moment of inertia of the substructure called EL_GROUP_2 is calculated about the line through the origin of the local coordinate system number 23, parallel to the y-axis is like the following:


DRESP
 ID_NAME  = DRESP_INERTIA_YY_LOCAL
 DEF_TYPE = SYSTEM
 TYPE     = CENTER_GRAVITY_X
 EL_GROUP = EL_GROUP_2
 CS_REF   = CS_23
END_