The antenna coil system in the previous example has an additional angular velocity Ω normal to the plane of i′ and k in Fig. 8.5, and relative to its initially oriented shaft frame 1 = φ = {C; ik}, which is turning with angular velocity ω relative to the ground frame 0 = Φ = {F;Ik} at the instant

Question

The antenna coil system in the previous example has an additional angular velocity Ω normal to the plane of [latex]\mathbf{i}^{\prime}[/latex] and k in Fig. 8.5, and relative to its initially oriented shaft frame [latex]1=\varphi=\left\{C ; \mathbf{i}_k\right\}[/latex], which is turning with angular velocity ω relative to the ground frame [latex]0=\Phi=\left\{F ; \mathbf{I}_k\right\}[/latex] at the instant shown. Find the applied torque about the center of mass required to sustain the motion of the system referred to the shaft frame [latex]2=\varphi^{\prime}=\left\{C ; \mathbf{i}_k^{\prime}\right\}[/latex] . Ignore the mass of the control shaft.

Accepted Answer

The torque about C is given by (8.45), so we must first find [latex]h_C[/latex] in (8.22) referred to the moving frame . The total angular velocity of the moving frame [latex]2=\varphi^{\prime}=\left\{C ; \mathbf{i}_k^{\prime} ight\}[/latex] fixed in the control shaft is [latex]\boldsymbol{\omega}_f \equiv \boldsymbol{\omega}_{20}=\boldsymbol{\omega}_{21} + \boldsymbol{\omega}_{10}=\boldsymbol{\Omega} + \boldsymbol{\omega}[/latex]. Hence, with reference to Fig. 8.5, referred to φ′,

[latex]\mathbf{M}_C(\beta, t)=\frac{d \mathbf{h}_C(\beta, t)}{d t}=\frac{d \mathbf{h}_r c(\beta, t)}{d t} .[/latex] (8.45)

[latex]\mathbf{h}_C(\beta, t)=\sum_{k=1}^n ho_k imes m_k \dot{ ho}_k=\mathbf{h}_r(\beta, t)[/latex] (8.22)

[latex]\omega_f=-\Omega \mathbf{j}^{\prime} + \omega\left(\sin heta \mathbf{i}^{\prime} + \cos heta \mathbf{k}^{\prime} ight)[/latex]. (8.49a)

The velocity of each coil relative to C is [latex]\dot{ ho}_k=(-1)^k \mathbf{v} + \omega_f imes ho_k[/latex], where we recall (8.48a) in which [latex]\mathbf{i} ightarrow \mathbf{i}^{\prime}[/latex]. Specifically,

[latex]\dot ho_1=-\dot ho_2=vi^\prime+\omega d \cos heta j^\prime+\Omega d k^\prime[/latex]. (8.49b)

Then (8.22) yields

[latex]\mathbf{h}_C=2 ho_1 imes m \dot{ ho}_1=2 m d^2\left(-\Omega \mathbf{j}^{\prime} + \omega \cos heta \mathbf{k}^{\prime} ight)[/latex]. (8.49c)

When θ = 0 and Ω = 0, we recover (8.48c) .

The total torque about C required to sustain the motion of the system is determined by (8.45) for [latex]h_C[/latex] given in (8.49c). But [latex]h_C[/latex] is a vector referred to a moving reference frame, so we shall need to apply (8.47). With the aid of (8.49a), (8.49c), and noting that [latex]\dot d[/latex](t) = -v(t) and [latex]\dot{ heta}=\Omega[/latex], we determine

[latex]\frac{\delta \mathbf{h}_C}{\delta t}=2 m d\left[(2 v \Omega - d \dot{\Omega}) \mathbf{j}^{\prime} + (d \dot{\omega} \cos heta - d \omega \Omega \sin heta - 2 v \omega \cos heta) \mathbf{k}^{\prime} ight][/latex],

[latex]\begin{aligned}\boldsymbol{\omega}_f imes \mathbf{h}_C &=2 m d^2\left|\begin{array}{ccc}\mathbf{i}^{\prime} & \mathbf{j}^{\prime} & \mathbf{k}^{\prime} \\omega \sin heta & -\Omega & \omega \cos heta \0 & -\Omega & \omega \cos heta\end{array} ight| \&=2 m d^2\left(-\omega^2 \sin heta \cos heta \mathbf{j}^{\prime}-\Omega \omega \sin heta \mathbf{k}^{\prime} ight) .\end{aligned}[/latex]

Thus, by (8.47), the total moment about C of all external forces exerted on the coil system by the control shaft, by gravity, and by the drive mechanism is

[latex]\mathbf{M}_C(\beta, t)=\frac{\delta \mathbf{h}_C(\beta, t)}{\delta t}+\omega_f imes \mathbf{h}_C(\beta, t)[/latex] (8.47)

[latex]\begin{aligned}\mathbf{M}_C=& 2 m d\left[\left(2 v \Omega-d \dot{\Omega} - d \omega^2 \sin heta \cos heta ight) \mathbf{j}^{\prime} ight.\&\left. + (d \dot{\omega} \cos heta - 2 d \Omega \omega \sin heta - 2 v \omega \cos heta) \mathbf{k}^{\prime} ight]\end{aligned}[/latex] (8.49d)

When Ω = 0 and θ = 0, the applied torque required to sustain the motion of the system considered initially is [latex]\mathbf{M}_C=2 m d(d \dot{\omega} - 2 v \omega) \mathbf{k}[/latex], which also follows easily from (8.45) and (8.48c) wherein now [latex]\mathbf{k}^{\prime}= \mathbf{k}[/latex] is fixed in Φ.

Question 8.4: The antenna coil system in the previous example has an addit...

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