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

Considere um corpo rígido girando com uma velocidade angular de ω em um referencial inercial.
Outra estrutura rotativa é fixada ao corpo que se move com o corpo com uma velocidade angular de Ω.
A soma do momento total é igual à soma da taxa de variação do momento angular em torno do centro de massa em relação ao referencial rotativo e o produto cruzado da velocidade angular do objeto com seu momento angular.
Quando a velocidade angular dos eixos rotativos é igual à velocidade angular do corpo, então os momentos e o produto da inércia em relação aos eixos rotativos serão constantes.
Lembrando os componentes escalares do momento angular, a equação do momento total pode ser expressa em componentes escalares.
Se os eixos rotativos são escolhidos como eixos principais de inércia, então o produto do termo de inércia desaparece, simplificando a forma escalar da equação do momento total.
Estas são conhecidas como equações de movimento de Euler para corpos em rotação.

equações de movimento de euler

Euler Equations of Motion

Imagine um corpo rígido que gira a uma velocidade angular de ω dentro de um referencial inercial. Junto com isso, imagine uma segunda estrutura rotativa que está presa ao próprio corpo. Este referencial se move junto com o corpo e possui uma velocidade angular de Ω. O momento total em torno do centro de massa é calculado somando a taxa de variação do momento angular em torno do centro de massa em relação ao referencial rotativo e o produto vetorial da velocidade angular do corpo e seu momento angular.


 Agora, considere uma situação em que a velocidade angular destes eixos rotativos é igual à velocidade angular do próprio corpo. Neste cenário, os momentos e o produto de inércia relativos aos eixos rotativos permanecerão constantes. Recordando as componentes escalares do momento angular e utilizando-as, pode-se expressar a equação do momento total em termos de componentes escalares.


 Se escolhermos os eixos rotativos como principais eixos de inércia, o produto do termo de inércia desaparece. Esta simplificação resulta em uma forma escalar mais manejável da equação do momento total. Esses princípios e equações constituem as equações de movimento de Euler para corpos em rotação. Estas equações fornecem informações valiosas sobre a dinâmica de corpos rígidos em rotação, permitindo-nos compreender e prever o seu comportamento sob diversas condições.

Equações de Movimento de Euler

Considere um corpo rígido girando com uma velocidade angular de ω em um referencial inercial.
Outra estrutura rotativa é fixada ao corpo que ...

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Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

3-Dimensional Kinetics of a Rigid Body

Cinética Tridimensional de um Corpo Rígido

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

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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...

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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...

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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...

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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 equations of motion

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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mechanical engineering

Rigid Body Rotation in Euler’s Frame

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