16.07 Dynamics Fall 2004 Lecture D12 -Relative Motion using Translating /Rotating Axes In the previous lecture, we related the motion experienced by two observers in relative translational motion with respect to each other. In this lecture, we will extend this relation to our third type of observer. That is, observers who accelerate and rotate with respect to each other As a matter of illustration, let us consider a very simple situation, in which a particle at rest with respect to the fixed frame y, 1. e. UA=0, aA=0, is observed by an observer, B, who is standing at the center of a turn table. The table rotates with a constant angular velocity of Q2 rad /s TA B入A Assuming that the platform only rotates about its center and does not move translationally, the position of B will not change, and therefore uB=0 and aB=0. It is clear that relative to b, a will not be at rest. In fact, B will see A rotating about b with a constant angular velocity of -Q. Thus, we can easily calculate the motion of A as observed by B. Expressing the velocity and acceleration in local polar coordinates(rotating with a'y), we will have (UA/B)xy=(TAB)ry'=-SrA/B eg (aAB)ry=(uA/B)ry' (v4/B)2y er=-S-TA/B e, 厂A The notation,(.)r'y, is used to indicate the velocity, or acceleration, experienced by an observer that rotates with the axes a'y. In other words, the time derivatives are taken assuming that the directions a'y' as seen by a rotating observer, do not change. For illustration purposes, we note that UA/B=TA/B=0 and aA/B=UA/B=0. Here, UA/B and aA/B denote the relative velocity and acceleration of A with respect to B, experienced by a non-rotating observer(this was the situation considered in the last lecture). We could, in fact, write (uA/B)ry and(aA/B)ry instead of u A/B and aA/B, but we will often omit the subscripts
J. Peraire 16.07 Dynamics Fall 2004 Version 1.5 Lecture D12 - Relative Motion using Translating/Rotating Axes In the previous lecture, we related the motion experienced by two observers in relative translational motion with respect to each other. In this lecture, we will extend this relation to our third type of observer. That is, observers who accelerate and rotate with respect to each other. As a matter of illustration, let us consider a very simple situation, in which a particle at rest with respect to the fixed frame xy, i.e. vA = 0, aA = 0, is observed by an observer, B, who is standing at the center of a turn table. The table rotates with a constant angular velocity of Ω rad/s. Assuming that the platform only rotates about its center and does not move translationally, the position of B will not change, and therefore vB = 0 and aB = 0. It is clear that relative to B, A will not be at rest. In fact, B will see A rotating about B with a constant angular velocity of −Ω. Thus, we can easily calculate the motion of A as observed by B. Expressing the velocity and acceleration in local polar coordinates (rotating with x ′y ′ ), we will have (vA/B)x′y′ = (r˙ A/B)x′y′ = −ΩrA/B eθ , and (aA/B)x′y ′ = (v˙ A/B)x′y ′ = − (vA/B) 2 x′y ′ rA/B er = −Ω 2 rA/B er . The notation, ( · )x′y′ , is used to indicate the velocity, or acceleration, experienced by an observer that rotates with the axes x ′y ′ . In other words, the time derivatives are taken assuming that the directions x ′y ′ , as seen by a rotating observer, do not change. For illustration purposes, we note that vA/B = r˙ A/B = 0 and aA/B = v˙ A/B = 0. Here, vA/B and aA/B denote the relative velocity and acceleration of A with respect to B, experienced by a non-rotating observer (this was the situation considered in the last lecture). We could, in fact, write (vA/B)xy and (aA/B)xy instead of vA/B and aA/B, but we will often omit the subscripts 1
when the axes considered are fixed. Also, we note that TA/B=(TA/B)ry since this is an instantaneous measurement and is not related to whether the axes rotate or not At this point, it should be clear that the expressions derived in the last lecture, which includes only trans- lational relative motion, are not applicable when rotating observers are involved Note Angular velocity and angular acceleration vectors In the two dimensional example presented above, we have dealt with the angular velocity as if it were a scalar quantity. Here, we will introduce the concept of angular velocity and acceleration vectors. We shall ee that, especially in three dimensions, it makes a lot of sense to think of these quantities as vectors. Let us consider a rigid body, which is spinning about an axis C-C with an angular velocity of @2 rad /s We consider a unit vector, ec, along the direction of the axis, and define the angular velocity vector, n2,as a vector having magnitude Q2 and direction ec. Thus Q=Qec Note that the convention between the direction of rotation and that of n2 is determined by the right hand rule. If the body were to rotate in the direction opposite to that shown in the diagram, then we would simply have Q=-nec The angular velocity vector is useful to express the velocity, due to rotation, of any point A in the rigid body. Let r be the position vector of A relative to an origin point, O, located on the axis of rotation. It turns out that the velocity of A, u, can be simply expressed as 9×r First, we note that, when the body spins, A describes a circular trajectory around the axis of radius d rsin, where o is the angle between r and the axis of rotation. Since the body is spinning at a rate of Q2 rad /s, the magnitude of the velocity vector will be v= Qd= Or sin p. The direction of the velocity vector
when the axes considered are fixed. Also, we note that rA/B = (rA/B)x′y′ since this is an instantaneous measurement and is not related to whether the axes rotate or not. At this point, it should be clear that the expressions derived in the last lecture, which includes only translational relative motion, are not applicable when rotating observers are involved. Note Angular velocity and angular acceleration vectors In the two dimensional example presented above, we have dealt with the angular velocity as if it were a scalar quantity. Here, we will introduce the concept of angular velocity and acceleration vectors. We shall see that, especially in three dimensions, it makes a lot of sense to think of these quantities as vectors. Let us consider a rigid body, which is spinning about an axis C − C with an angular velocity of Ω rad/s. We consider a unit vector, eC , along the direction of the axis, and define the angular velocity vector, Ω, as a vector having magnitude Ω and direction eC . Thus, Ω = ΩeC . Note that the convention between the direction of rotation and that of Ω is determined by the right hand rule. If the body were to rotate in the direction opposite to that shown in the diagram, then we would simply have Ω = −ΩeC. The angular velocity vector is useful to express the velocity, due to rotation, of any point A in the rigid body. Let r be the position vector of A relative to an origin point, O, located on the axis of rotation. It turns out that the velocity of A, v, can be simply expressed as v = Ω × r . (1) First, we note that, when the body spins, A describes a circular trajectory around the axis of radius d = r sinϕ, where ϕ is the angle between r and the axis of rotation. Since the body is spinning at a rate of Ω rad/s, the magnitude of the velocity vector will be v = Ωd = Ωr sinϕ. The direction of the velocity vector 2
u will be tangent to the trajectory at A, which means that it will be perpendicular to r and ec. Thus recalling the definition of the vector product, we can verify that the velocity given by(1)does indeed have the right direction orientation and magnitude e the angu Since the angular acceleration is the derivative of a vector, it will measure both the changes in magnitude and the changes in direction of the angular velocity vector Time derivative of a fixed vector in a rotating frame We consider a reference frame a'y'z' rotating with an angular velocity S with respect to a fixed frame ryz Let V be any vector, which is constant relative to the frame a'y'z'. That is, the vector components in the r'y'z' frame do not change, and, as a consequence, V rotates as if it were rigidly attached to the frame In the absolute frame, the time derivative will be equal to (V+△V)-v =9×V which can be interpreted as the velocity of the tip of vector v. The above expression applies to any vector hich is rigidly attached to the frame a'y'z. In particular, it applies to the unit vectors i', j and k Therefore. we have that g×t dk g×k′ Time derivative of a vector in a rotating frame: Coriolis'theorem Now let V be an arbitrary vector (e.g. velocity, magnetic field, force, etc.), which is allowed to change in both the fixed zyz frame and the rotating z'y'z' frame. The vector V can be expressed, in the r'y'a'frame
v will be tangent to the trajectory at A, which means that it will be perpendicular to r and eC. Thus, recalling the definition of the vector product, we can verify that the velocity given by (1) does indeed have the right direction, orientation and magnitude. We also define the angular acceleration vector as, Ω˙ = dΩ dt . Since the angular acceleration is the derivative of a vector, it will measure both the changes in magnitude and the changes in direction of the angular velocity vector. Time derivative of a fixed vector in a rotating frame We consider a reference frame x ′y ′ z ′ rotating with an angular velocity Ω with respect to a fixed frame xyz. Let V be any vector, which is constant relative to the frame x ′y ′ z ′ . That is, the vector components in the x ′y ′ z ′ frame do not change, and, as a consequence, V rotates as if it were rigidly attached to the frame. In the absolute frame, the time derivative will be equal to dV dt = lim ∆t→0 (V + ∆V ) − V ∆t = Ω × V , which can be interpreted as the velocity of the tip of vector V . The above expression applies to any vector which is rigidly attached to the frame x ′y ′ z ′ . In particular, it applies to the unit vectors i ′ , j ′ and k ′ . Therefore, we have that di ′ dt = Ω × i ′ dj ′ dt = Ω × j ′ dk ′ dt = Ω × k ′ . Time derivative of a vector in a rotating frame: Coriolis’ theorem Now let V be an arbitrary vector (e.g. velocity, magnetic field, force, etc.), which is allowed to change in both the fixed xyz frame and the rotating x ′y ′ z ′ frame. The vector V can be expressed, in the x ′y ′ z ′ frame, 3
as, V=Vri+ vyj+vik If we now consider the time derivative of v, as seen by the fixed frame, we have dv Vr i+vy j+V2sk'+Vri+Vy'j+ vik =(V)x'y2+×v Here, Vr/i+ Vyj'+V2k'=(v)x'yz, is the time derivative of the vector V as seen by the rotating frame Hence, for this derivative, the vectors i', j' and k remain unchanged. The term Vr,i+vv i'+v,i'= V(g×i)+V(92×j)+v2(g×k)=× V is the change in v due to the rotation. The expression (2)above is known as Coriolis'theorem. Given an arbirary vector, it relates the derivative of that vector as seen by a fixed frame with the derivative of the same vector as seen by a rotating frame. Symbolically, we can write We will often omit the notation( )xyz when the derivative is taken with respect to the fixed frame Relative Motion using translating/rotating axes Here, we consider the relationship between the motion seen by an observer b that may be accelerating as well as rotating, and the motion seen by a a fixed observer O. Let ab be the acceleration of B with respect to O, and let n2 and n2 denote the angular velocity and angular acceleration, respectively, of the frame x'y'z rigidly attached to B. The vectors i, j and k are the unit vectors corresponding to the fixed frame ayz, and i', jand k are the unit vectors corresponding to the rotating frame r'y'z' k A/ A Position vector The position vector of A with respect to the fixed frame can be written as
as, V = Vx′ i ′ + Vy′j ′ + Vz ′k ′ . If we now consider the time derivative of V , as seen by the fixed frame, we have V˙ = dV dt = V˙ x′ i ′ + V˙ y′j ′ + V˙ z ′k ′ + Vx′ ˙i ′ + Vy′ ˙j ′ + Vz ′k˙ ′ = (V˙ )x′y′z ′ + Ω × V . (2) Here, V˙ x′ i ′ + V˙ y′j ′ + V˙ z ′k ′ = (V˙ )x′y′z ′ is the time derivative of the vector V as seen by the rotating frame. Hence, for this derivative, the vectors i ′ , j ′ and k ′ remain unchanged. The term Vx′ ˙i ′ + Vy ′ ˙i ′ + Vz ′ ˙i ′ = Vx′ (Ω × i ′ ) + Vy′ (Ω × j ′ ) + Vz ′ (Ω × k ′ ) = Ω × V is the change in V due to the rotation. The expression (2) above is known as Coriolis’ theorem. Given an arbirary vector, it relates the derivative of that vector as seen by a fixed frame with the derivative of the same vector as seen by a rotating frame. Symbolically, we can write, ˙ ( )xyz = ˙ ( )x′y ′z ′ + Ω × ( ) . We will often omit the notation ( )xyz when the derivative is taken with respect to the fixed frame. Relative Motion using translating/rotating axes Here, we consider the relationship between the motion seen by an observer B that may be accelerating as well as rotating, and the motion seen by a a fixed observer O. Let aB be the acceleration of B with respect to O, and let Ω and Ω˙ denote the angular velocity and angular acceleration, respectively, of the frame x ′y ′ z ′ rigidly attached to B. The vectors i, j and k are the unit vectors corresponding to the fixed frame xyz, and i ′ , j ′ and k ′ are the unit vectors corresponding to the rotating frame x ′y ′ z ′ . Position vector The position vector of A with respect to the fixed frame can be written as, rA = rB + rA/B . (3) 4
Note that this is an instantaneous concept, and therefore it is immaterial whether the observer B is rotatin or not. In other words, the above expression is valid at any given instant. Of course, if we choose to express TA and rB in ayz components, and TA/B in r'y'z' components, then we will have to make sure that the proper coordinate transformation is done before the components of the vectors are added. Also, since the orientation of x'y'z changes in time, this transformation will depend on the instant considered Velocity vector Differentiating (3) with respect to time, we have TB+(AB)xy+92×rAB B+(vA/B)x'ytx+2×T Here, we have used Coriolis theorem(2)to calculate the derivative of rA/B, i.e. rA/B=(TA/B)ry2/+n tA/B. In the above expression, UA and uB are the velocities of A and B, respectively, relative to the fixed frame. The term(UA/B)ry2 is the velocity of A measured by the rotating observer, B. Finally, n is the angular velocity of the rotating frame, and rA/B is the relative position vector of A with respect to B Acceleration vector Differentiating(4)once again, and making use of Coriolis'theorem, e.g. d(UAB)xy2/dt=(uA/B)xy2/+ &X(UAB)ry2, we obtain the following expression for the acceleration, A=UA= UB+(UA/Bry'2/+X(uA/B)r'y'2' +sxTM/B+2×(vA/B)x'y2+92×(xrAB) aB+aM/B)xy2+29×(v4/B)x'yx+gxrA/B+92×(xrA/B) Here, aA and aB are the accelerations of A and B, respectively, observed by xyz. The term(aA/B)2'y'2 is the acceleration of A measured by an observer B that rotates with the axes a'y' z. The term 2n X(ua/B)ry'z' Coriolisacceleration. The term S X rA/B is due to the change in &, and, finally, the terr &x(Q xrA/B)is called centripetal acceleration. It can be easily seen that this acceleration always point towards the axis of rotation, and is orthogonal to n(when trying to show this, you may find the vector identity, AX(BXC)=(A. C)B-(A. B)C, introduced in the Review Notes on Vectors, applied to 2×(s2×TA/B)=(2·TA/B)-(92·)rA/B, useful) A/B
Note that this is an instantaneous concept, and therefore it is immaterial whether the observer B is rotating or not. In other words, the above expression is valid at any given instant. Of course, if we choose to express rA and rB in xyz components, and rA/B in x ′y ′ z ′ components, then we will have to make sure that the proper coordinate transformation is done before the components of the vectors are added. Also, since the orientation of x ′y ′ z ′ changes in time, this transformation will depend on the instant considered. Velocity vector Differentiating (3) with respect to time, we have, vA = v˙ A = r˙ B + (r˙ A/B)x′y′z ′ + Ω × rA/B = vB + (vA/B)x′y′z ′ + Ω × rA/B . (4) Here, we have used Coriolis’ theorem (2) to calculate the derivative of rA/B, i.e. r˙ A/B = (r˙ A/B)x′y ′z ′ + Ω × rA/B. In the above expression, vA and vB are the velocities of A and B, respectively, relative to the fixed frame. The term (vA/B)x′y ′z ′ is the velocity of A measured by the rotating observer, B. Finally, Ω is the angular velocity of the rotating frame, and rA/B is the relative position vector of A with respect to B. Acceleration vector Differentiating (4) once again, and making use of Coriolis’ theorem, e.g. d(vA/B)x′y′z ′/dt = (v˙ A/B)x′y′z ′ + Ω × (vA/B)x′y′z ′ , we obtain the following expression for the acceleration, aA = v˙ A = v˙ B + (v˙ A/B)x′y′z ′ + Ω × (vA/B)x′y′z ′ + Ω˙ × rA/B + Ω × (vA/B)x′y′z ′ + Ω × (Ω × rA/B) = aB + (aA/B)x′y′z ′ + 2Ω × (vA/B)x′y′z ′ + Ω˙ × rA/B + Ω × (Ω × rA/B) . (5) Here, aA and aB are the accelerations of A and B, respectively, observed by xyz. The term (aA/B)x′y ′z ′ is the acceleration of A measured by an observer B that rotates with the axes x ′y ′ z ′ . The term 2Ω × (vA/B)x′y′z ′ is called Coriolis’ acceleration. The term Ω˙ × rA/B is due to the change in Ω, and, finally, the term Ω × (Ω × rA/B) is called centripetal acceleration. It can be easily seen that this acceleration always points towards the axis of rotation, and is orthogonal to Ω (when trying to show this, you may find the vector identity, A × (B × C) = (A · C)B − (A · B)C, introduced in the Review Notes on Vectors, applied to Ω × (Ω × rA/B) = (Ω · rA/B)Ω − (Ω · Ω)rA/B, useful). 5