Confusion in coordinate transformation of acceleration vector

Question

Consider this notation for vectors: $A_{cb}^{d}$ is the $A$ vector (velocity,acceleration, angular velocity or ...) of $b$ coordinate frame w.r.t. $c$ frame presented in $d$ frame axis. I know when we want to change the observer frame of a velocity vector $v_{cb}^{d}$(changing the $c$ to another frame e.g. $e$) we should consider the kinematics between $c$ and $e$ frames like this: $$v_{eb}=v_{eo}+v_{cb}+\omega_{ec}\times r_{cb}$$ where $v_{eo}$ is the velocity of the origin of $c$ frame w.r.t. $e$ frame and $\omega_{ec}$ is the rotation vector of $c$ frame w.r.t. $e$ frame (There is also a relation for changing the acceleration observer frame which has the centrifugal, lateral and Coriolis acceleration terms in it). But when we want to change the presenter frame of a velocity vector $v_{cb}^{d}$ to another frame (e.g. $e$ frame), we only need the rotation matrix (quaternion or Euler) between $d$ and $e$ frames. $$v_{cb}^{e}=R_{d}^{e}v_{cb}^{d}$$ But when we want to change the presenter frame of an acceleration vector, we can not do the same and we have to differentiate the above equation like this: $$\dot{v}_{cb}^{e}=\dot{R}_{d}^{e}v_{cb}^{d}+R_{d}^{e} \dot{v}_{cb}^{d}$$ I can understand the mathematics here but i'm confused by concepts! Why we can't transform the presenter frame of acceleration by only a rotation matrix? We don't want to change the observer frame like the first relation, so why we have to deal with kinematics ($\dot{R}_{d}^{e}v_{cb}^{d}$ i mean)? Conceptually i believe that for transforming a vector from a frame to another frame we only need the rotation matrix. And coordinate transforming should not change the magnitude of a vector, but mathematically this is not true about the acceleration vectors! I'll be thankful if anyone can explain the concepts behind this.

Answer 1

The confusion arises because one of the reference frames is moving w.r.t the other one. In your second eq. you are transforming one vector at the seme time, so it is just a coordinate transformation. In the last equation if you want to do that with the acceleration you have to do the same $$a^e_{cb}=R^e_d a^d_{cb}$$ Notice that $a^d_{cb}\neq\dot{v}^d_{cb}$ The third eq. is telling you that the change in $v^e_{cd}$ has two parts. One is because the components of $v^d_{cb}$ are changing, and the other is because the vector bases themselves are changing.