Những lần lượt phụ thuộc vào các điều kiện ban đầu và các thông số của hàng đầu, thể hiện trong a, b,, và ˙ θmin, θmax. Nếu giá trị u = cos θ ở đó φ biến mất trong phạm vi của chương động, sau đó sự tiến động sẽ được các hướng khác nhau tại θmin và θmax, và chuyển động như trong hình. . Mặt khác, nếu θ = cos-1 (b / a) ∈ [θmin, θmax], sự tiến động sẽ luôn luôn được trong cùng một hướng, mặc dù nó sẽ tăng tốc độ và làm chậm | 94 CHAPTER 4. RIGID BODY MOTION Kinematics in a rotating coordinate system We have seen that the rotations form a group. Let us describe the configuration of the body coordinate system by the position R t of a given point and the rotation matrix A t êi ê ị which transforms the canonical fixed basis inertial frame into the body basis. A given particle of the body is fixed in the body coordinates but this of course is not an inertial coordinate system but a rotating and possibly accelerating one. We need to discuss the transformation of kinematics between these two frames. While our current interest is in rigid bodies we will first derive a general formula for rotating and accelerating coordinate systems. Suppose a particle has coordinates b t yị bi t ê i t in the body system. We are not assuming at the moment that the particle is part of the rigid body in which case the bị t would be independent of time. In the inertial coordinates the particle has its position given by r t R t b t but the coordinates of b t are different in the space and body coordinates. Thus rị t Rị t bị t Rị t X A-1 t ij bj t . j The velocity is v yi riêi because the êi are inertial and therefore considered stationary so v R X ij bj Í A-1 O ij dbj t dt êi and not R 2i db i dt e i because the êi are themselves changing with time. We might define a body time derivative but it is not the velocity of the particle a even with respect to R t in the sense that physically a vector is basis independent and its derivative . KINEMATICS IN A ROTATING COORDINATE SYSTEM 95 requires a notion of which basis vectors are considered time independent inertial and which are not. Converting the inertial evaluation to the body frame requires the velocity to include the dA-1 dt term as well as the b term. b What is the meaning of this extra term . d V E jtA t Ỵbj t êi The derivative is of course V M At iA-1 t At ij - A-1 t j bj t êi . This expression has coordinates in the body frame with basis vectors from the
