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Roughmar
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I'm going to repost https://www.physicsforums.com/showthread.php?t=534598" in the maths subforum, that got no answers and guessed I'd try my luck here.
Roughmar said:
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Roughmar said:First and foremost, I decided with putting this in the math sub-forum since I believe my problem has to do with the general maths of this.
I'm trying to study the dynamics of an object. For a whole lot of reasons, it is to my advantage to study those dynamics on a object fixed axis.
For that, I use an Earth fixed axis and subsequently apply a rotation.
I'll use a specific example I found in a known paper:
I'm using a Z-X-Y order for such a rotation, giving me this matrix, expressed in roll, pitch and yaw components:
R =
[itex]\left[cos\psi cos\theta - sin\phi sin\psi sin\theta , -cos\phi sin \psi , cos\psi sin\theta + cos\theta sin\phi sin\psi\right][/itex]
[itex]\left[cos\theta sin\psi + cos\psi sin\phi sin\theta , cos\phi cos\psi , sin\psi sin\theta - cos\psi cos\theta sin\phi\right][/itex]
[itex]\left[-cos\phi sin\theta , sin\phi , cos\phi cos\theta\right][/itex]
(am not really proficient with coding here, so bear with me with the previous and following "matrix", if someone can provide a code example for a matrix I'd be more than willing to make this more elegant)
Now, the components of angular velocity in the body frame are p,q and r.
These values are related to the derivatives of the roll, pitch and yaw angles according to
[itex]\left[p,q,r\right]^{T} =\left[cos\theta , 0 , -cos\phi sin\theta ; 0 , 1 , sin\phi;sin\theta , 0 , cos\phi\cos\theta\right]\left[\dot{\phi}, \dot{\theta},\dot{\psi}\right]^{T}[/itex]
...
My question is why.
How is this last matrix calculated? I've been looking at this for the last 4 hours and reached no conclusion whatsoever.
The angular acceleration of an object on a fixed axis can be determined by calculating the torque acting on the object and dividing it by the moment of inertia of the object. This can be represented by the equation α = τ/I, where α is the angular acceleration, τ is the torque, and I is the moment of inertia.
The moment of inertia is a measure of an object's resistance to rotational motion. It plays a crucial role in solving the dynamics of an object on a fixed axis as it is used in calculating the angular acceleration, angular velocity, and angular displacement of the object.
Yes, the dynamics of an object on a fixed axis can be solved using Newton's laws of motion. The first law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. The second law relates the net force acting on an object to its acceleration, and the third law states that for every action, there is an equal and opposite reaction. These laws can be applied to rotational motion by using torque instead of force and angular acceleration instead of linear acceleration.
The distribution of mass affects the dynamics of an object on a fixed axis because it influences the object's moment of inertia. Objects with more mass concentrated farther away from the axis of rotation will have a higher moment of inertia and will be more difficult to accelerate. On the other hand, objects with mass distributed closer to the axis of rotation will have a lower moment of inertia and will be easier to accelerate.
Understanding the dynamics of an object on a fixed axis is essential in various real-life applications, including engineering, sports, and everyday objects. For example, engineers use this knowledge to design structures such as bridges and buildings that can withstand external forces and rotational motion. In sports, athletes use their understanding of the dynamics of an object on a fixed axis to enhance their performance, such as in figure skating and gymnastics. Everyday objects such as bicycles, cars, and even doors also rely on the principles of rotational motion and the dynamics of an object on a fixed axis.