Understanding the solid property calculations

The Solid Properties Check lets you measure the solid properties of the elements in your finite element model. This article contains details about the following solid properties calculations:

• elemental mass center of gravity

• model mass center of gravity

• mass moments of inertia

• mass products of inertia

• inertia matrix and inertia tensor

Note:

This article describes in general terms the calculations performed by the Solid Properties Check command. However, some of these calculations may vary slightly depending upon the solver and element type you are using. See Solver-specific considerations for solid property calculations for more information.

Elemental mass center of gravity calculation

The software uses the following equation to calculate the center of gravity for each element:

Where

• Mele = the total mass of the element

• = the distance from the origin to the element's center of gravity

• ρ = either the nominal value for mass density of the element or the mass density of the element evaluated at a specified temperature, Temperature (TREF), or Reference Frequency.

• x, y, z = components of the mass particle

• dM = differential mass

• dV = differential volume

Note:

If you are working with Nastran, the software does not use this equation to calculate the elemental mass center of gravity for elements, such as the concentrated mass CONM2 element, where you explicitly specify a mass.

Model mass center of gravity calculation

The software uses the following equation to calculate the mass center of gravity across the sum of the elements:

Where

• = the distance from the origin to the model's center of gravity

• xi, yi, zi = distance from the origin to the ith element center of gravity

• Mi = ith mass

• Mmodel = the total mass of the model

Mass moments of inertia calculation

The software uses the following equations to calculate mass moments of inertia of an element with respect to the X-, Y-, and Z-axes:

Where

• Ixxele, Iyyele, Izzele = an element's mass moment of inertia about a reference X-X, Y-Y, and Z-Z axis, respectively

• Rxx, Ryy, Rzz = respective distances from the X-X, Y-Y, Z-Z axis to an infinitesimal particle of the element

• x, y, z = orthogonal components of R

• ρ = either the nominal value for mass density of the element or the mass density of the element evaluated at a specified temperature, Temperature (TREF), or Reference Frequency.

• dM = differential mass

• dV = differential volume

The mass moment of inertia of the entire model with respect to the X-, Y-, and Z-axes is the sum of the elemental moments of inertia with respect to the X-, Y-, and Z-axes:

Where

• Ixx, Iyy, Izz = the mass moments of inertia of the entire model with respect to the X-, Y-, and Z-axes

• Ixxi, Iyyi, Izzi = the mass moments of inertia of the ith element with respect to the X-, Y-, and Z-axes

For lumped mass elements, the software uses the parallel axis theorem to determine the elemental mass moments of inertia. The parallel axis theorem states:

Where

• = the mass moment of inertia of an individual particle about its local center of gravity X-X axis

• d = perpendicular distance from the reference X-X axis to the local center of gravity X-X axis

• M = particle mass

Mass products of inertia calculation

The software calculates mass properties of inertia of an element with respect to two perpendicular reference planes. Typically:

• Ixyele = mass product of inertia of an element about the two reference planes, with both x=constant and y=constant, respectively

• Ixzele = mass product of inertia of an element about the two reference planes, with both x=constant and z=constant, respectively

• Iyzele = mass product of inertia of an element about the two reference planes, with both y=constant and z=constant, respectively

• x, y, z = the coordinate distance from the reference plane to the center of gravity of the particle

• ρ = either the nominal value for mass density of the element or the mass density of the element evaluated at a specified temperature, Temperature (TREF), or Reference Frequency.

• dM = differential mass

• dV = differential volume

The mass product of inertia of the whole model with respect to the two reference planes is the sum of the elemental mass products of inertia with respect to the two same reference planes:

Where

• Ixy, Ixz, Iyz = mass products of inertia of the entire model with respect to the two reference planes

• Ixyi, Ixzi, Iyzi = mass products of inertia of the entire model with respect to the two reference planes

For lumped mass elements, the software uses the parallel axis theorem to determine the elemental mass products of inertia. The parallel axis theorem states:

Where

• = the mass product of inertia of an individual particle about two reference planes based at its own local center of gravity

• M = particle mass

Inertia matrix and inertia tensor calculations

The Solid Properties Check command outputs the inertia matrix I:

Where

• Ixx, Iyy, Izz = mass moments of inertia of the entire model with respect to the X-, Y-, and Z-axes

• Ixy, Ixz, Iyz = mass moments of inertia of the entire model with respect to the two reference planes

To obtain the inertia tensor, the signs of the off-diagonal terms must be changed as follows:

Learn more

Calculating solid properties

Solver-specific considerations for solid property calculations

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Understanding the solid property calculations, Simcenter 3D 2021.1 Series

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