| Problem statement Solution video |
DISCUSSION THREAD
Ask your questions here. You can also answer questions of others. You can learn from helping others.
DISCUSSION and HINTS
Bar OA has two components of rotation:
- One component of Ω about the fixed J-axis.
- The second component of θ_dot about the moving k-axis.
Write out the angular velocity vector ω in terms of the two components described above.
Take a time derivative of ω to get the angular acceleration α of the bar. When taking this derivative, you will need to find the time derivative of the unit vector k. How do you do this? Read back over Section 3.2 of the lecture book. There you will see: k_dot = ω x k, where ω is the total angular velocity vector of bar OA that you found above.
In this problem, when we translate from the big J hat to the local coordinate system, do we multiply the omega or whatever that symbol is by Lcosthetha + Lsinthetha?
I believe that since we are looking at the arm when theta is equal to zero degrees, big J hat and small j hat are in the same direction, therefore translation is unnecessary. This is also indicated in the problem statement when it says the XYZ and xyz axes are aligned. I might be incorrect though.
In the problem statement it says that the XYZ and xyz axes are aligned with each other when theta = 0, which it does. So J_hat and j_hat would align with each other and a direct substitution can be made.
General question with moving reference frames. Are we allowed to simplify the last term of the acceleration to -w^2r like we did with the planar frames?
I don't believe so, as the point of the double cross product is intended to account for the fact that there is rotation in 3D. The
-w^2 simplification is due to the fact that in planar we can't have that 3rd axis of rotation. That's the way I think about it. Hope that helps.
No, the following equality only holds when A and B lie in the same plane.
$\vec{\omega} \cross (\vec{\omega} \cross \vec{r}_{B/A}) = -\omega^{2} \vec{r}_{B/A}$
Refer to page 89 in the textbook for more info on the derivation.
You can assume the last term of the acceleration equation to be equal to -w^2r when the angular velocity is in terms of only one unit vector (either i, j, or k). You can think of this as being that the angular velocity is entirely in one plane. However, when the angular velocity is in terms of several unit vectors (meaning it is not entirely in one plane), you cannot make this assumption. Using the equation w x (w x r) will always give you the right answer, unlike the assuming -w^2r.
Why is it mentioned in this problem to find angular velocity and acceleration when theta equals zero? Based on the equations we are using and the given theta{dot} and theta{ddot} are constant would the found angular velocity and acceleration be the same for all theta values?
I believe it is to simplify the problem so that as it says in the last line the xyz and XYZ axis are aligned. But like you said that won't affect the equations other than not having to project/convert coordinate systems.
I believe that the magnitude of angular velocity and acceleration would be constant regardless of theta, but the values of angular velocity and acceleration in the I, J, and K directions will change as theta changes. Theta equaling 0 means you will not have to convert coordinate systems.
In general, the magnitude of the angular acceleration will not be constant (although in this problem it might be). It is not recommended that you assume that when solving.
I'm not clear on how k_dot is ω x k. I understand that's how it is in general, but in this problem, it doesn't seem to make sense. It seems to me like k is rotating with the shaft about the fixed Y axis completely independently of theta, and that therefore, k_dot should be ΩJ.
But applying ω x k gives me a different result.
Please note that ω x k is perpendicular to both J and k. For θ ≠ 0, J will have both i and j components, with those components being functions of θ. That is how θ becomes mathematically a part of k_dot.
From a physical perspective, k_dot is perpendicular to both J and k, which will be in the I direction since k_dot points in the direction of movement of the tip of the k vector. Again, for θ ≠ 0, I will have both i and j components, each depending on θ.
Please take a look at the animation on the course webpage of: https://www.purdue.edu/freeform/me274/course-material/animations/angular-acceleration-in-3d/. The problem in that animation is mathematically the same as this homework problem (although phyiscally it looks different). The 3D animation near the bottom of that webpage helps in visualizing how the ω changes in direction, leading to a non-zero angular acceleration vector.
Is it normal to solve the problem without using all of the given parameters?
The amount of information that is needed to solve a problem will depend on what one seeks from the analysis. If a different answer is needed, then the parameters needed to solve might change. This would be typical of a "real world" type of problem where the customer provides you with everything that they know when hiring you to perform an analysis, with some of which you may not use.
When we write a homework problem, we attempt to provide you with information that you will need. We would not purposely give you more than you need; however, as the problems change throughout the semesters, what is needed now may not be needed in a later semester.
In some offerings of this problem, we might also ask for the acceleration of A. For those offerings, you would need to use L. Here, only angular velocity and angular acceleration are requested, and L is not used.
Do you need to translate the k to find the angular velocity or does it not matter?
Not sure of what you are asking.
Your final answer can be in terms of either the ijk unit vectors or the IJK unit vectors. You should NOT mix ijk and IJK unit vectors in your final answer. Does this answer your question?
Can you look at this with OA being completely horizontal? If you were pretending there was someone standing on it looking to the left would they see OA slanting downwards or would they see it as horizontal?
Link OA is changing elevation with the given theta_dot and theta_double_dot, but when it comes to an observer I believe they'd see it being horizontal since theta is 0.