1 Reply Latest reply on Apr 24, 2015 11:48 AM by Diego Saba

# Positive or Negative Inertia Tensors?

When using the Mass Properties tool to calculate the Moments of Inertia, does SolidWorks use positive or negative inertia tensors?

• ###### Re: Positive or Negative Inertia Tensors?

Solidworks 2013 shows the "positive" matrix of inertia.

The Mass Properties tool should warn about it, because many people (like me some hours ago) are not even aware of the existence of the "positive" convention.

The tensor of inertia is obtained by integrating the product of density and

[ y^2+z^2    -x*y        -x*z    ]

[  -x*y      x^2+z^2    -y*z    ]

[  -x*z        -y*z      y^2+z^2 ]

and is in fact a tensor. See Moment of Inertia -- from Wolfram MathWorld.

It is called "negative" inertia matrix because of the signs of the terms off the diagonal. Notice, however, that these terms can be positive or negative.

The "positive" inertia matrix is computed using

[ y^2+z^2    x*y        x*z    ]

[    x*y      x^2+z^2    y*z    ]

[    x*z        y*z      y^2+z^2 ]

with the signs of the terms off the diagonal inverted. It is no longer a tensor and it cannot be used "as is" to obtain the angular momentum from

I*omega (product of the inertia matrix and the angular velocity vector). It does not transform like a tensor when changing system of reference and

its eigenvalues are not the principal moments of inertia.

The possibility of two different conventions can be very confusing and should definitely be warned upon. I suggest using the "negative" convention, since it is more common.