Rigid body relative acceleration

In summary, a rigid body (ship) can rotate and move around 3 axes (x,y,z) around the center of gravity, but the position of the center of gravity is not known. What is known is the acceleration, velocity, and position of point A, as well as the original position vector between point A and point B. The desired information is the vertical (z-direction) displacement, velocity, and acceleration of point B. To calculate this, the relative vector between points A and B can be found using a rotation matrix. The relative velocity and acceleration can then be calculated, taking into account the rotational speed around the z-axis. However, it is important to note that the x,y, and z axes may not be fixed
  • #1
Remko

Homework Statement


Rigid body (ship) rotate and moves around 3 axis (x,y,z) around the center of gravity. The position is of the center of gravity is not known.
What is known: At a point (A) the accereleration, velocity and position and rotational acceleration, velocity and position are known (measured). And the original position vector (relative to the fixed world) from point A to a point B.

What is wanted: The vertical (z-direction) displacement, velocity and acceleration of point B.

2. The attempt at a solution
With the known angles the relative vector (r_AB) between point A a B can be easily calculated with a rotation matrix:
a6821937d5031de282a190f75312353c970aa2df


When you add the the measured position of A to this r_AB the new position of B is known.
Now the relative velocity can be calculated as
image006.gif

so the velocity v_B=v_A+ v_B/A

and the relative acceleration is:
image012.gif

So the acceleration is a_B=a_A+a_B/A.

Now the my first question is whether this is correct?
And secondly I find it strange that when you take the z-componont of the acceleration it is dependent on the rotational speed around the z-axis (because of the double cross product). Why is this? You would think this rotational speed around its axis has nothing to do with any movement or acceleration along that axis.

And also a similar problem with the velocity in z direction which is dependent on the yaw (the angle around the z axis). Which i also find strange.
 
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  • #2
I think it's because the z-axis is being rotated into the other axes that do change the z-axis acceleration. So the instantaneous (current) z-axis rotation will start changing the (moved) z-axis acceleration.
 
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  • #3
FactChecker said:
I think it's because the z-axis is being rotated into the other axes that do change the z-axis acceleration. So the instantaneous (current) z-axis rotation will start changing the (moved) z-axis acceleration.
But that would mean that the x,y and z axis are not fixed. And in my case i want that the axis are fixed to the world, because i want the know the position, velocity and acceleration of point B with regard to the ground (seabed). Does that mean i need another way to calculate my all this? Or did i misunderstood your explanation?
 
  • #4
Remko said:
But that would mean that the x,y and z axis are not fixed. And in my case i want that the axis are fixed to the world, because i want the know the position, velocity and acceleration of point B with regard to the ground (seabed). Does that mean i need another way to calculate my all this? Or did i misunderstood your explanation?
I think you understood my answer and have a good question. I'll have to think about that and I gave away all my references when I retired. I may have to leave this to someone more expert in this subject.
 
  • #5
FactChecker said:
I think you understood my answer and have a good question. I'll have to think about that and I gave away all my references when I retired. I may have to leave this to someone more expert in this subject.
Okay, Thanks anyway!
 
  • #6
I think i found the solution. I first rotate the position (r) with a rotation matrix R and then I start multiplying that with the angular velocity which is still aligned with the 'old' fixed coordinate system. So to calculate this correctly the angular velocity should be also be multiplied with the rotation matrix to get it in the same reference frame.
 

1. What is rigid body relative acceleration?

Rigid body relative acceleration refers to the change in velocity of a rigid body with respect to a reference frame. It is the acceleration of one point on the rigid body relative to another point on the same body.

2. How is rigid body relative acceleration calculated?

Rigid body relative acceleration can be calculated using the formula a = ω x r + α x r, where a is the relative acceleration, ω is the angular velocity, r is the position vector, and α is the angular acceleration.

3. What is the difference between rigid body relative acceleration and absolute acceleration?

The main difference is that rigid body relative acceleration takes into account the movement of one point on a rigid body with respect to another point on the same body, while absolute acceleration measures the change in velocity of an object with respect to an external reference frame.

4. How is rigid body relative acceleration used in physics?

Rigid body relative acceleration is used in a variety of fields, including mechanics, engineering, and robotics. It is used to analyze the motion of objects and to design mechanisms and systems that involve rigid bodies.

5. Are there any real-life examples of rigid body relative acceleration?

Yes, there are many real-life examples of rigid body relative acceleration. One common example is the movement of a car's wheels relative to its body as it turns. Another example is the movement of a pendulum's bob relative to its fixed point of suspension.

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