1. What is 3D Moment of Inertia?

The 3D moment of inertia is a mathematical representation of an object's resistance to rotational motion about different axes in three-dimensional space. It is represented as a 3x3 matrix that captures the distribution of mass relative to the coordinate system.

2. How Does the Calculator Work?

The calculator uses the matrix of integrals to compute the 3D moment of inertia:

Where:

• \( I_{xx}, I_{yy}, I_{zz} \) — Moments of inertia about x, y, z axes
• \( I_{xy}, I_{xz}, I_{yz} \) — Products of inertia
• The matrix is symmetric (\( I_{xy} = I_{yx} \), etc.)

Explanation: The calculator processes various integrals representing mass distribution to populate the inertia matrix components.

3. Importance of 3D Moment of Inertia Calculation

Details: Accurate 3D moment of inertia calculation is crucial for analyzing rotational dynamics, stability analysis, and mechanical design of complex 3D objects and systems.

4. Using the Calculator

Tips: Enter the required integral values representing the mass distribution of your 3D object. The calculator will compute the complete 3D inertia matrix.

5. Frequently Asked Questions (FAQ)

Q1: What are the diagonal elements of the inertia matrix?
A: The diagonal elements represent moments of inertia about the principal axes (Ixx, Iyy, Izz).

Q2: What do the off-diagonal elements represent?
A: The off-diagonal elements represent products of inertia, which indicate how mass is distributed relative to the coordinate planes.

Q3: When is the inertia matrix diagonal?
A: The matrix becomes diagonal when the coordinate axes align with the principal axes of the object.

Q4: How does 3D moment of inertia differ from 2D?
A: 3D moment of inertia captures rotational resistance about all three spatial axes and their couplings, while 2D typically deals with rotation about a single axis.

Q5: What units are used for 3D moment of inertia?
A: Typically kg·m² in SI units, representing mass times distance squared for each matrix component.