13.14: Angular Velocity
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Angular velocity ω
It is useful to relate the rigid-body equations of motion in the space-fixed (ˆx,ˆy,ˆz) coordinate system to those in the body-fixed (ˆe1,ˆe2,ˆe3) coordinate system where the principal axis inertia tensor is defined. It was shown in appendix 19.4 that an infinitessimal rotation can be represented by a vector. Thus the time derivatives of these rotation angles can be associated with the components of the angular velocity ω, where the precession ωϕ=˙ϕ, the nutation ωθ=˙θ, and the spin ωψ=˙ψ. Unfortunately the coordinates (ϕ,θ,ψ) are with respect to mixed coordinate frames and thus are not orthogonal axes. That is, the Euler angular velocities are expressed in different coordinate frames, where the precession ˙ϕ is around the space-fixed ˆz axis measured relative to the ˆx-axis, the spin ˙ψ is around the body-fixed ˆe3 axis relative to the rotating line-of-nodes, and the nutation ˙θ is the angular velocity between the ˆz and ˆe3 axes and points along the instantaneous line-of-nodes in the ˆe3׈z direction. By reference to Figure 13.13.1 it can be seen that the components along the body-fixed axes are as given in Table 13.14.1.
Precession ˙ϕ | Nutation ˙θ | Spin ˙ψ |
---|---|---|
˙ϕ1=˙ϕsinθsinψ | ˙θ1=˙θcosψ | ˙ψ1=0 |
˙ϕ2=˙ϕsinθcosψ | ˙θ2=−˙θsinψ | ˙ψ2=0 |
˙ϕ3=˙ϕcosθ | ˙θ3=0 | ˙ψ3=ψ |
Note that the precession angular velocity ˙ϕ is the angular velocity that the body-fixed ˆe3 and ˆz׈3 axes precess around the space-fixed ˆz axis. Table 13.14.1 gives the Euler angular velocities required to calculate the components of the angular velocity ω for the body-fixed (1,2,3) axis system. Collecting the individual components of ω, gives the components of the angular velocity of the body, relative to the space-fixed axes, in the body-fixed axis system (1,2,3)
ω1=˙ϕ1+˙θ1+˙ψ1=˙ϕsinθsinψ+˙θcosψ
ω2=˙ϕ2+˙θ2+˙ψ2=˙ϕsinθcosψ−˙θsinψ
ω3=˙ϕ3+˙θ3+˙ψ3=˙ϕcosθ+˙ψ
The angular velocity of the body about the body-fixed 3-axis, ω3, is the sum of the projection of the precession angular velocity of the line-of-nodes ˙ϕ with respect to the space-fixed x-axis, plus the angular velocity ˙ψ of the body-fixed 3-axis with respect to the rotating line-of-nodes.
Similarly, the components of the body angular velocity ω for the space-fixed axis system (x,y,z) can be derived to be
ωx=˙θcosϕ+˙ψsinθsinϕ
ωy=˙θsinϕ−˙ψsinθcosϕ
ωz=˙ϕ+˙ψcosθ
Note that when θ=0 then the Euler angles are singular in that the space-fixed z axis is parallel with the body-fixed 3 axis and there is no way of distinguishing between precession ˙ϕ and spin ˙ψ, leading to ωz=ω3=˙ϕ+˙ψ. When θ=π then the z axis and 3 axis are antiparallel and ωz=˙ϕ−˙ψ=−ω3. The other special case is when cosθ=0 for which the Euler angle system is orthogonal and the space-fixed ωz=˙ϕ, that is, it equals the precession, while the body-fixed ω3=˙ψ, that is, it equals the spin. When the Euler angle basis is not orthogonal then equations ??? - ??? and ??? - ??? are needed for expressing the Euler equations of motion in either the body-fixed frame or the space-fixed frame respectively.
Equations ??? - ??? for the components of the angular velocity in the body-fixed frame can be expressed in terms of the Euler angle velocities in a matrix form as
(ω1ω2ω3)=(sinθsinψcosψ0sinθcosψ−sinψ0cosθ01)⋅(˙ϕ˙θ˙ψ)
Note that the transformation matrix is not orthogonal which is to be expected since the Euler angular velocities are about axes that do not form a rectangular system of coordinates. Similarly equations ??? - ??? for the angular velocity in the space-fixed frame can be expressed in terms of the Euler angle velocities in matrix form as
(ωxωyωz)=(0cosϕsinθsinϕ0sinϕsinθcosϕ10cosθ)⋅(˙ϕ˙θ˙ψ)