Name: rigid body rotation in euler’s frame
Uploaded: 2024-03-07T13:22:28.000Z
Duration: 1 min 19 s
Channel: JoVE

Consider a rigid body rotating with an angular velocity of &#969; in an inertial frame of reference.
Another rotating frame is attached to the body that moves with the body with an angular velocity of &#937;.
The summation of the total moment is equal to the sum of the rate of change of angular momentum about the center of mass with respect to the rotating frame and the cross-product of the angular velocity of the object with its angular momentum.
When the angular velocity of the rotating axes is equal to the angular velocity of the body, then the moments and product of inertia with respect to rotating axes will be constant.
Recalling the scalar components of the angular momentum, the total moment equation can be expressed in scalar components.
If the rotating axes are chosen as principle axes of inertia, then the product of the inertia term vanishes, simplifying the scalar form of the total moment equation.
These are known as Euler&#39;s equations of motion for rotating bodies.

euler equations of motion

Euler Equations of Motion

Imagine a rigid body that is rotating at an angular velocity of &#969; within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of &#937;. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body&#39;s angular velocity and its angular momentum.
Now, consider a situation where the angular velocity of these rotating axes equals the angular velocity of the body itself. In such a scenario, the moments and product of inertia concerning the rotating axes will remain constant. Recalling the scalar components of the angular momentum and using these, one can express the equation for the total moment in terms of scalar components.
If one chooses the rotating axes as the principle axes of inertia, the product of the inertia term disappears. This simplification results in a more manageable scalar form of the total moment equation. These principles and equations constitute Euler&#39;s equations of motion for rotating bodies. These equations provide valuable insights into the dynamics of rotating rigid bodies, enabling us to understand and predict their behavior under various conditions.

Consider a rigid body rotating with an angular velocity of &#969; in an inertial frame of reference.
Another rotating frame is attached to the body that ...

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3-Dimensional Kinetics of a Rigid Body

KEY TERMS AND CONCEPTS

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 frame. The other part is the cross product of the body’s angular velocity and its angular momentum.

A useful simplification appears when the angular velocity of the rotating axes equals the angular velocity of the body. In that case, the moments of inertia and product of inertia about those rotating axes remain constant. The scalar components of angular momentum can then be used to write the total moment in a simpler form.

If the rotating axes are chosen as the principal axes of inertia, the product of inertia term disappears. This gives a more manageable scalar form of the total moment equation. These steps lead to Euler’s equations of motion for rotating rigid bodies, which help describe and predict how they behave under different conditions.

Rigid Body Rotation in Euler’s Frame

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 and principle axes of inertia

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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.

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...

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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...

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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

Explore how Euler’s equations use a rotating frame, angular momentum, and principal axes to describe rigid body rotation in a simpler scalar form.

Rigid Body Rotation in Euler’s Frame | JoVE K12

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Rigid Body Rotation in Euler’s Frame

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