Advanced Engineering Dynamics

1.

1. With reference to the figure, show that the gravitational field \(\bG_\Sigma (P)\) at a point \(P\) of position \(\br\equiv \br_{OP} = r\be_r\) is given by

   \begin{equation*} \bG_\Sigma (P) = -{G \over r^2} \be_r \int_\Sigma \left[ 1 - \frac{3}{2r^2} (\bs^2 -5 (\bs\cdot\be_r)^2 ) \right] \, dm + {3G \over r^4} \int_{\Sigma} (\bs\cdot\be_r) \bs \, dm \end{equation*}

   assuming that the distance \(r \equiv | \br_{OP}|\) from \(P\) to \(O\) is much larger than the distance \(|\bs|\) from any point \(Q\) of \(\Sigma\) to \(O\) (use the expansion \((1+x)^{-3/2} = 1 -3x/2 +15x^2 /8 + \cdots\))
2. Show that \(\bG_\Sigma\) is given by the expression

   \begin{equation*} \bG_\Sigma (P) = - {G M \over r^2} \be_r + {3G \over 2r^4} (A-C) \left[ (1-3\sin^2 \phi)\be_r + 2 \sin\phi\cos\phi \be_\phi \right] \end{equation*}

   in terms of the coordinates \((r, \phi)\) of point \(P\text{,}\) and of the equatorial and polar moments of inertia \(A = I_{Ox} = I_{Oy}\) and \(C = I_{Oz}\text{.}\)
3. Deduce that the gravitational field can be expressed in terms of potential \(\pot\text{,}\) that is

   \begin{equation*} \bG_\Sigma (P) = -\nabla \pot, \qquad \pot= - {G M \over r} + {G \over 2r^3} (C-A) (3\sin^2\phi -1) \end{equation*}

   with \(\nabla = - \be_r \partial / \partial r - \be_\phi \partial / r\partial\phi\text{.}\) Assuming that the free surface of \(\Sigma\) is a surface of equipotential, show that the flattening ratio \(f\) takes the expression

   \begin{equation*} f = \frac{3}{2} \frac{(C-A)}{MR_E^2} \end{equation*}

   (The value of \(f\) can range from 1/300 for Earth to 1/10 for Saturn).

2.

Consider the small-body approximation of Example 10.2.6. Show that the gravitational effect of Earth \(\cE\) (of mass \(M\text{,}\) and mass center \(C\text{.}\) modeled as a spherical body with material symmetry on a satellite \(\cS\) (of mass \(m\text{,}\) and mass center \(G\) can be approximated as

\begin{equation*} \{\cA^g_{\cE \to \cS}\} = - {G Mm \over \rho^2} \left\{ \begin{array}{c} \bx \\ \\ {3 \over m\rho} \Big[ (B-C) c_y c_z \bx_s + (C-A) c_x c_z \by_s +(A-B) c_x c_y \bz_s \Big] \end{array} \right\}_G \end{equation*}

where \((\bx_s,\by_s,\bz_s)\) is a principal basis of inertia of \(\cS\text{,}\) \((A,B,C)\) the corresponding principal moments of inertia, and \((c_x , c_y, c_z)\) are the direction cosines of \(\bx = \br_{CG}/\rho\) (with \(\rho= |CG|\)0) on \((\bx_s,\by_s,\bz_s)\text{:}\) \(\bx = c_x\bx_s +c_y\by_s+c_z\bz_s\text{.}\) To derive this result, assume that \(\br \equiv \br_{GP}\) is much smaller in magnitude than \(\rho\) for any point \(P\) of \(\cS\text{.}\)

3.

Two bodies \(\cB_1\) and \(\cB_2\) are in relative rotation about the same axis \(\Delta\text{.}\) Body \(\cB_1\) is being pressed against \(\cB_2\) by an axial force \(P\) as shown, so that two conical surfaces characterized by kinetic friction coefficient \(\mu\) remain in contact.

Find the contact action screw \(\{ \cA_{\cB_1 \to \cB_2}^c \}\) in terms of \(P\text{,}\) \(\mu\) and the geometrical parameters \(r_1\text{,}\) \(r_2\) and \(\theta\text{.}\)

4.

Show that the resulting contact action screw \(\{ \cA^c_{\cB_1 \to \cB_2} \}\) satisfies

\begin{equation*} \bz \cdot \bM^c_{O, 1\to 2} = 0 \end{equation*}

where point \(O\) is the common center of the tori.