# Inertial "moment" of a solid given an inertia tensor

#### taalf

Hi all,

I am programming some software computing inertial forces of some moving structure.

So far, the structure have been represented as a cloud of mass points. For each mass point, inertial forces are computed using some formulas like in this article: https://fr.wikipedia.org/wiki/Force_d%27inertie#Formalisation

That is: some "Euler force" and centrifugal force are computed (Coriolis forces are neglected).

The program works well and was validated.

Now, there is some need to give to structure sub-elements not only a mass, but also an inertia tensor :

____| Ixx Ixy Ixz |
I = | Iyx Iyy Iyz |
____| Izx Izy Izz |

Intuitively, I would say adding some inertia matrix to the element will lead to some "inertial" (fictitious) torque on it, as well as there are "inertial" forces exerted on mass points.

Does anyone have an idea how to model these "fictitous moments" (provided they exist) ?

#### studiot

The elements in the interia tensor are not true moments, they are moments of inertia.

The leading diagonal entries (with repeated suffixes) give the overall moI.

The off diagonal entries (with cross suffixes) give the spatial distribution of these inertias.
that is in terms of rotational inertias the spatial distribution of the masses.
These entries are called products of inertia.

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• 1 person

#### neila9876

Nice to see the presentation of that navy again...Chen's first language is not English...haha...
Hi, taalf, it seems to be differrent methods of calculation about rigid body issues, no additional force. (or say torque)

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#### taalf

Hi all,

Thanks for replies.

I will try to explain in more detail what is the problem.

Consider a moving reference frame, for example an aircraft manoeuvring in the sky. Consider a wing, whose mass is ignored. On this wing is a massive engine.

In the following chart, the engine is at point P. Its mass is m. The wing is fixed to the fuselage at point O. One is interested on the force and moment felt by the fuselage at point O due to the presence of the engine's mass. Indeed, the engine feels a acceleration in translation a, being the sum of the aircraft's acceleration, the gravitational acceleration, plus fictitious accelerations because the frame is no interial: Euler and centrifugal accelerations:

a_tot = a_aircr + a_grav + a_euler + a_centrif

with:

a_grav = 9.81 m/s
a_euler = a_ang x GP
(a_ang the angular acceleration, GP the distance between aircraft's gravity center and point P)
a_centrif = v_ang x v_ang x GP
(v_ang the angular velocity of the aircraft)

The force exerted on the engine is:

F = m.a_tot

Considering the engine as a punctual mass, the moment at point P is:

M = 0

Now, if we consider the engine as a volume with a certain inertia tensor I, the moment won't be zero:

M = ? In an inertial reference frame, it would be enough to solve the second Newton's law in rotation, that is:

M = I . a_ang

with a_ang the angular acceleration felt by the engine.

BUT, how to model this angular acceleration in a non-inertial frame? What is the "Euler" and "centrifugal" terms for the angular acceleration ?...

#### taalf

Hi all,

Thanks for replies.

I will try to explain in more detail what is the problem.

Consider a moving reference frame, for example an aircraft manoeuvring in the sky. Consider a wing, whose mass is ignored. On this wing is a massive engine.

In the following chart, the engine is at point P. Its mass is m. The wing is fixed to the fuselage at point O. One is interested on the force and moment felt by the fuselage at point O due to the presence of the engine's mass. Indeed, the engine feels a acceleration in translation a, being the sum of the aircraft's acceleration, the gravitational acceleration, plus fictitious accelerations because the frame is no interial: Euler and centrifugal accelerations:

a_tot = a_aircr + a_grav + a_euler + a_centrif

with:

a_grav = 9.81 m/s
a_euler = a_ang x GP
(a_ang the angular acceleration, GP the distance between aircraft's gravity center and point P)
a_centrif = v_ang x v_ang x GP
(v_ang the angular velocity of the aircraft)

The force exerted on the engine is:

F = m.a_tot

Considering the engine as a punctual mass, the moment at point P is:

M = 0

Now, if we consider the engine as a volume with a certain inertia tensor I, the moment won't be zero:

M = ? In an inertial reference frame, it would be enough to solve the second Newton's law in rotation, that is:

M = I . a_ang

with a_ang the angular acceleration felt by the engine.

BUT, how to model this angular acceleration in a non-inertial frame? What is the "Euler" and "centrifugal" terms for the angular acceleration ?...