Question 12.6: A rigid body consists of three mass points that are connecte...
A rigid body consists of three mass points that are connected to the z-axis by rigid massless bars (see Fig. 12.15).
(a) Find the elements of the tensor of inertia relative to the x, y, z-system.
(b) Calculate the ellipsoid of inertia with respect to the origin 0, and the moment of inertia of the entire body with respect to the axis 0a.
(a) The elements of the tensor of inertia relative to the x,y,z-system are
Θ_{xx} =\sum\limits_{i}{m_{i}(y^{2}_{i}+z^{2}_{i} )}= m_{1}(y^{2}_{1}+ z^{2}_{1})+m_{2}(y^{2}_{2}+ z^{2}_{2})+ m_{3}(y^{2}_{3}+z^{3}_{3}),
and after inserting the numerical values from Fig. 12.15, one has
Θ_{xx}= 100(144+25)+ 200(64+225)+ 150(144+196) (g cm^{2})=125.7 (kg cm^{2}).
Likewise, one obtains
Θ_{yy} = 117.5 (kg cm^{2}) and Θ_{zz} = 104.75 (kg cm^{2}).For the deviation moments of the tensor of inertia, it follows that
Θ_{xy} =− \sum\limits_{i}{m_{i}(x_{i}y_{i})}= 100(12 · 10)−200(10 · 8)+ 150(11 · 14) (g cm^{2}) = 19.1 (kg cm^{2}),
and likewise,
Θ_{xz} =−44.8 (kg cm^{2}) and Θ_{yz} = 4.800(kg cm^{2}).(b) From (a) one now immediately obtains for the ellipsoid of inertia with respect to the origin 0 (see (12.30))
Θ_{n}= Θ_{xx} cos^{2} α +Θ_{yy} cos^{2} β +Θ_{zz} cos^{2} γ+2Θ_{xy} cos α cos β +2Θ_{xz} cos α cos γ + 2Θ_{yz} cos β cos γ. (12.30)
Θ= Θ_{xx} cos^{2} α +Θ_{yy} cos^{2} β +Θ_{zz} cos^{2} γ
+2Θ_{xy} cos α cos β +2Θ_{xz} cos α cos γ + 2Θ_{yz} cos β cos γ. (12.47)
To calculate the moment of inertia Θ_{0a}, we evaluate the direction cosines with the coordinates given in Fig. 12.15,
cos α =\frac{−6}{ \sqrt{6^{2} + 8^{2} +20^{2}}}=−0.268,cos β = \frac{8}{\sqrt{6^{2} + 8^{2} +20^{2}}}= 0.358,
and
cos γ =\frac{20}{\sqrt{6^{2} +8^{2} + 20^{2}}}= 0.895.By inserting into (12.47) for the moment of inertia, we obtain
Θ_{0a} = (0.268)^{2} · 125.7+(0.358)^{2} · 117.25+(0.895)^{2} · 104.75−2(0.268)(0.358) · 19.1+ 2(0.268)(0.895) · 44.8
−2(0.358)(0.895) · 4.800 (kg cm^{2})
= 128.87 (kg cm^{2}).