Question 13.4: Find the torque that is needed to rotate a rectangular plate...
Find the torque that is needed to rotate a rectangular plate (edges a and b) with constant angular velocity ω about a diagonal.
The principal moments of inertia of the rectangle are already known from Example 11.7:
I_{1} = \frac{1}{12} Ma^{2}, I_{2} = \frac{1}{12} Mb^{2}, I_{3} = \frac{1}{12} M(a^{2} +b^{2}). (13.6)
The angular velocity is
ω = (ω · e_{x} )e_{x} + (ω · e_{y} )e_{y} ,i.e.,
ω = − \frac{ωb}{\sqrt{a^{2} +b^{2}}} e_{x} + \frac{ωa}{\sqrt{a^{2} + b^{2}}}e_{y}
⇒ ω_{1} =\frac{−ωb}{\sqrt{a^{2} + b^{2}}}, ω_{2} =\frac{+ωa}{\sqrt{a^{2} +b^{2}}}, ω_{3} = 0. (13.7)
Inserting (13.6) and (13.7) into the Euler equations yields
I_{1} \dot{ω}_{1} + (I_{3} − I_{2})ω_{2}ω_{3} = D_{1},I_{2} \dot{ω}_{2} + (I_{1} − I_{3})ω_{3}ω_{1} = D_{2},
I_{3} \dot{ω}_{3} + (I_{2} − I_{1})ω_{2}ω_{1} = D_{3},
and furthermore, D_{1} = 0,D_{2} = 0, and
D_{3} =\frac{−M(b^{2} −a^{2})abω^{2}}{12(a^{2} + b^{2})}.Hence, the torque is
D =\frac{−M(b^{2} − a^{2})abω^{2}}{12(a^{2} +b^{2})}e_{z}.For a = b (square), D = 0!