20 | | [[Image(Screen Shot 2017-12-15 at 3.06.47 PM.png)]] |
| 20 | [[Image(Screen Shot 2017-12-15 at 3.10.01 PM.png)]] |
| 21 | |
| 22 | Note the center of mass of the injected material is not .05,.05 and the injected mass is not 1.0. This is because of discretization error. |
| 23 | |
| 24 | We would like to find a solution for $\delta m_i$ that is close to this, but subject to the constraints involving the center of mass and the total mass injected. If we consider each zone inside the kernel, we can write the matrix equation |
| 25 | |
| 26 | {{{ |
| 27 | #!latex |
| 28 | \[ |
| 29 | \left[\begin{array}{rrrrr} |
| 30 | x_1 & x_2 & x_3 & ... & x_n \\ |
| 31 | y_1 & y_2 & y_3 & ... & y_n \\ |
| 32 | 1 & 1 & 1 & ... & 1 \\ |
| 33 | \end{array} \right] |
| 34 | \left[\begin{array}{r} |
| 35 | \delta m_1 \\ |
| 36 | \delta m_2 \\ |
| 37 | \delta m_3 \\ |
| 38 | ... \\ |
| 39 | \delta m_n \\ |
| 40 | \end{array} \right]= |
| 41 | \left[\begin{array}{r} |
| 42 | MX_0 \\ |
| 43 | MY_0 \\ |
| 44 | M \\ |
| 45 | \end{array} \right] |
| 46 | \] |
| 47 | }}} |
| 62 | |
| 63 | |
| 64 | Now for the more complicated case, we have $\delta m_i, v_i$, but the matrix would contain two blocks - so we can solve the problem in two steps. First we can solve for $\delta m_i$, and then use that solution to construct the matrix equation for $v_i$ |
| 65 | |
| 66 | {{{ |
| 67 | #!latex |
| 68 | \[ |
| 69 | \left[\begin{array}{rrrrrrrrr} |
| 70 | m_1 & ... & m_n & 0 & ... & 0 & 0 & ... & 0 \\ |
| 71 | 0 & ... & 0 & m_1 & ... & m_n & 0 & ... & 0 \\ |
| 72 | 0 & ... & 0 & 0 & ... & 0 & m_1 & ... & m_n \\ |
| 73 | 0 & ... & 0 & m_1z_1 & ... & m_nz_n & -m_1y_1 & ... & -m_ny_n\\ |
| 74 | -m_1z_1 & ... & -m_nz_n & 0 & ... & 0 & m_1x_1 & ... & m_nx_n \\ |
| 75 | m_1y_1 & ... & m_ny_n & -m_1x_1 & ... & -m_nx_n &0 & ... & 0 \\ |
| 76 | \end{array} \right] |
| 77 | \left[\begin{array}{r} |
| 78 | v_{x1} \\ |
| 79 | ... \\ |
| 80 | v_{xn} \\ |
| 81 | v_{y1} \\ |
| 82 | ... \\ |
| 83 | v_{yn} \\ |
| 84 | v_{z1} \\ |
| 85 | ... \\ |
| 86 | v_{zn} \\ |
| 87 | \end{array} \right]= |
| 88 | \left[\begin{array}{r} |
| 89 | 0 \\ |
| 90 | 0 \\ |
| 91 | 0 \\ |
| 92 | J_x \\ |
| 93 | J_y \\ |
| 94 | J_z \\ |
| 95 | \end{array} \right] |
| 96 | \] |
| 97 | }}} |