Determining the angular and linear accelerations of a Rigid Body

In summary, the conversation discussed working with the governing equation for a rotating, translating rigid body and how to solve for the unknowns of \alpha and \vec{a_c}. Two possible approaches were mentioned, the principle of superposition and using moments of inertia, to determine the body's rotations and translations.
  • #1
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Hello, all!

I am currently working with the governing equation of a rotating, translating rigid body. That is:

[tex] \vec{a_i} = \vec{a_c} + \vec{\alpha} \times \vec{r} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r}} \right)[/tex]

Where [tex]a_i[/tex] is the linear acceleration of some point on the body [tex]i[/tex], [tex]a_c[/tex] is the linear acceleration of the centroid of the body, [tex]\alpha[/tex] is the angular acceleration of the body, [tex]\omega[/tex] is the angular velocity of the body, and [tex]r[/tex] is the vector from the centroid to the point [tex]i[/tex].

Given some [tex]a_1[/tex] and [tex]a_2[/tex], how can I rework a system of two of those equations such that I can back out [tex]\alpha[/tex] (related to omega by the differential) and [tex]a_c[/tex]? That is:
[tex] \vec{a_1} = \vec{a_c} + \vec{\alpha} \times \vec{r_1} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_1}} \right)[/tex]
[tex]
\vec{a_2} = \vec{a_c} + \vec{\alpha} \times \vec{r_2} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_2}} \right)
[/tex]

To give another way of viewing the problem: If I mount two accelerometers on a body and then rotate and translate the body in space, how can I determine what those body rotations and translations are based off the readings of my two accelerometers? I should only need two accelerometers (read: two equations) because there are only two unknowns, [tex]a_c[/tex] and [tex]\alpha[/tex].

Thank you for the help!
 
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  • #2


Hello!

It seems like you are working with the equations of motion for a rigid body, which is a very interesting topic in mechanics. To solve the system of equations and determine the unknowns, you can use the principle of superposition. This principle states that the total acceleration of a body is the sum of its individual accelerations. So, in your case, you can add the two equations together and eliminate the \vec{a_c} term, since it appears in both equations. This will leave you with a single equation with only two unknowns, \alpha and \vec{\omega}. You can then use this equation to solve for the unknowns and determine the body's rotations and translations.

Another approach you can take is to use the concept of moments of inertia. This involves calculating the moments of inertia of the body about its centroid and then using them to determine the angular acceleration \alpha. From there, you can use the equations of motion to solve for the linear acceleration of the centroid \vec{a_c}.

I hope this helps! Let me know if you have any other questions. Good luck with your research!
 
  • #3


I can offer some insight into determining the angular and linear accelerations of a rigid body. Firstly, it is important to understand that the governing equation you have provided is known as the "Euler's equation of motion" and is derived from Newton's second law of motion. This equation relates the linear acceleration of a point on the body to the angular acceleration and velocity of the body, as well as the position vector from the centroid to the point.

To solve for the unknowns \alpha and a_c, we can use the two given equations and eliminate the unknowns \omega and \vec{a_c}. This can be done by taking the cross product of both equations with \vec{r_1} and \vec{r_2}, respectively. This will result in two equations with only \alpha and a_c as unknowns. From there, we can solve for these unknowns using basic algebraic manipulation.

In terms of using accelerometers to determine these unknowns, it is important to ensure that the accelerometers are mounted at known positions on the body and that their readings are accurate and calibrated. From there, the readings can be used to determine the linear accelerations at those points, which can then be plugged into the equations to solve for \alpha and a_c.

It is also worth noting that this approach assumes that the rigid body is undergoing pure rotational and translational motion without any external forces acting on it. If there are external forces present, additional equations and measurements may be needed to accurately determine the unknowns.

I hope this helps in your research and understanding of rigid body dynamics. Best of luck!
 

1. What is a rigid body and why is it important to determine its angular and linear accelerations?

A rigid body is an object that maintains its shape and size, even when subjected to external forces. It is important to determine its angular and linear accelerations because it allows us to understand how the object will move and how it will respond to different forces.

2. How do you calculate the angular acceleration of a rigid body?

The angular acceleration of a rigid body can be calculated by dividing the change in angular velocity by the change in time. This can be represented by the formula: α = (ω₂ - ω₁) / (t₂ - t₁), where α is the angular acceleration, ω is the angular velocity, and t is the time.

3. What is the difference between angular and linear acceleration?

Angular acceleration refers to the rate of change of the angular velocity of an object, while linear acceleration refers to the rate of change of the linear velocity of an object. Angular acceleration is measured in radians per second squared, while linear acceleration is measured in meters per second squared.

4. How do you determine the linear acceleration of a rigid body?

The linear acceleration of a rigid body can be calculated by dividing the change in linear velocity by the change in time. This can be represented by the formula: a = (v₂ - v₁) / (t₂ - t₁), where a is the linear acceleration, v is the linear velocity, and t is the time.

5. What factors can affect the angular and linear accelerations of a rigid body?

The angular and linear accelerations of a rigid body can be affected by various factors such as the mass, shape, and distribution of mass of the object, as well as the magnitude and direction of external forces acting on it. Friction and air resistance can also impact the accelerations of a rigid body.

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