Name: angular momentum in principal axes
Uploaded: 2024-03-07T13:22:28.000Z
Duration: 1 min 9 s
Channel: JoVE

The angular momentum for a rigid body can be expressed as the integral of the cross-product of the position vector of the mass element with the cross-product of the angular velocity of the body and the position vector.
This equation can be written in terms of rectangular coordinates by choosing another set of xyz axes that are inclined arbitrarily with the reference frame.
The rectangular components of angular momentum are derived by expanding the cross-product, combining i, j, and k terms, and applying the product of inertia definition.
These equations can be simplified further by choosing the xyz axes such that they form principal axes for the rigid body.
In this case, the rectangular components of the angular momentum are expressed in terms of the principal moments of inertia about xyz axes.
Each of these components of angular momentum is independent of the other and follows the principle of conservation of angular momentum independently.

angular momentum and principle axes of inertia

Angular Momentum and Principle Axes of Inertia

The concept of angular momentum for a solid structure is illustrated as the cumulative result of the cross-product of the position vector of the mass element and the cross-product of the body&#39;s angular velocity with the position vector.
To put this equation into simpler terms, it can be reconfigured using rectangular coordinates. This involves choosing an alternative set of XYZ axes that are arbitrarily inclined with respect to the reference frame. The process of deriving the rectangular components of angular momentum involves unfolding the cross-product, merging components, and applying the definition of the product of inertia. The equations derived can be further simplified by selecting the XYZ axes in such a way that they create principal axes for the solid structure.
In this specific instance, the rectangular components of angular momentum are articulated in relation to the principal moments of inertia about the XYZ axes. Each component of angular momentum is distinct from the others and adheres independently to the principle of conservation of angular momentum. This means that each individual component does not influence the others and maintains its momentum separately. This approach provides a more comprehensive understanding of the dynamics of a rigid body in motion, enabling a more accurate prediction of its movement and behavior under various conditions.

The angular momentum for a rigid body can be expressed as the integral of the cross-product of the position vector of the mass element with the ...

angular-momentum-and-principle-axes-of-inertia

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Mechanical Engineering

Unit 1

3-Dimensional Kinetics of a Rigid Body

KEY TERMS AND CONCEPTS

angular momentum in principal axes

Angular momentum in a rigid body can be written using rectangular coordinates. The body’s angular momentum comes from the cross-product of each mass element’s position vector with the cross-product of the body’s angular velocity and that position vector. This gives a compact way to describe how a solid structure moves in rotation.

To work with the equation more easily, the XYZ axes can be chosen as a new set of rectangular axes. These axes may be inclined at any angle to the reference frame. The rectangular components of angular momentum are then found by expanding the cross-products, combining terms, and using the definition of product of inertia. Product of inertia describes how mass is distributed relative to a pair of axes.

The equations can be simplified further by choosing the XYZ axes to be principal axes of the solid structure. In that case, the rectangular components of angular momentum are written in terms of the principal moments of inertia about the XYZ axes. Each angular momentum component is separate from the others. Each one also follows the conservation of angular momentum on its own, so one component does not affect the others. This makes the motion of a rigid body easier to analyze and helps predict its behavior under different conditions.

Angular Momentum in Principal Axes

moments and product of inertia

Moment and Product of Inertia Basics

Moment of inertia and product of inertia describe how mass is distributed in a rigid body. They help show how the body responds about coordinate axes and planes. For a small mass element, the moment of inertia is found by multiplying the element's mass by the square of the shortest distance to one of the three coordinate axes. That same idea is extended to the whole body by integration. Adding the contributions of all the small mass elements gives the body's moment of inertia about that axis.

inertia tensor

Principal Axes and Moments of Inertia

The inertia tensor describes how mass is spread out in a rigid body and how that body resists rotation. It is written as a 3×3 matrix, and each entry represents a moment of inertia about a particular axis. This makes the tensor a compact way to study rotational behavior in three dimensions. The diagonal entries show the moments of inertia about the principal axes. These principal axes are the directions where the object has the least resistance to rotation. A smaller moment of inertia along...

moment of inertia about an arbitrary axis

moment-of-inertia-about-an-arbitrary-axis

Calculating Moment of Inertia on Any Axis

Moment of inertia can be calculated about any chosen axis, not just a principal axis. It depends on the mass distribution of the object along that axis. This makes it useful when an object rotates in a real machine or other setup where balance and stability matter. To find the moment of inertia about an arbitrary axis, the perpendicular distance from the axis to each mass element is needed. That distance is obtained with a cross product between the unit vector that gives the axis direction and...

angular momentum about an arbitrary axis

Angular Momentum Relative to Point P

Angular momentum relative to point P is found by adding the motion of every mass element in a rigid body. The body has mass m and a center of mass at point G. It rotates in an inertial reference frame, and the angular momentum at P comes from the cross product of each position vector and its linear momentum. To build the full expression, the velocity of each mass element is written as two parts. One part is the translational velocity of the body. The other part is the relative velocity caused...

Angular momentum in a rigid body can be written using rectangular coordinates. The body’s angular momentum comes from the cross-product of each mass element’s position vector with the cross-product of the body’s angular velocity and that position vector. This gives a compact way to describe how a solid structure moves in rotation. To work with the equation more easily, the XYZ axes can be chosen as a new set of rectangular axes. These axes may be inclined at any angle to the reference frame.

principle of impulse and moment

Impulse and Angular Momentum in Rigid Bodies

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion. For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the...

kinetic energy for a rigid body

Rigid Body Energy from Center and Rotation

Rigid body kinetic energy in planar motion can be written using the body’s center of mass and its rotation. The solid object moves in general planar motion, and its center of mass is marked at point G. To find the kinetic energy of the i-th particle, the relative velocity definition is used with the position vector rA from point A to that particle. The kinetic energy of the whole body is built from the particles. First, a scalar product is taken. Then the expression is written in integral form.

equation of motion for a rigid body

Rigid Body Motion: Force and Moment Balance

Rigid body motion is described with both translational motion and rotational motion about the center of mass, point G. The center of mass is the point where Newton's Second Law applies to translation. It gives the linear motion of the body. The same body also has rotational motion about point G. The combined moments about the center of mass equal the rate of change of angular momentum. A moment is the turning effect caused by a force, and an external force applied away from the center of mass...

euler equations of motion

Rigid Body Rotation in Euler’s Frame

Euler’s equations of motion describe how a rigid body rotates when measured in a moving frame attached to the body. The body rotates with angular velocity ω in an inertial frame, and the attached frame rotates with angular velocity Ω. Together, these two frames help describe the motion of the rigid body around its center of mass. The total moment about the center of mass is found by adding two parts. One part is the rate of change of angular momentum about the center of mass in the rotating...

torque free motion

Mars Spin Under No Torque

Torque-free motion describes the movement of a rigid body when no external torques act on it. It can be seen in places with no outside forces or friction, such as outer space. Mars is one example of this kind of motion because it rotates in space without external torque. Mars is an axisymmetric object, which means it has an axis of symmetry. That symmetry defines the rotating z-axis. The rotating frame is set so that Mars&amp;#39;s center of mass is at the origin. This setup makes the moments...

key terms and concepts

Learn how rigid body angular momentum is written in rectangular coordinates, simplified with principal axes, and linked to principal moments of inertia.

Angular Momentum in Principal Axes | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Angular Momentum in Principal Axes

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