关于2022数学建模a题的第一问我是怎么做的

2022-11-09 06:00 作者:臥煙鈴 0人读过 | 我要投稿

首先免责声明，本人是数学与应用数学专业的，没有学过大物，所以公式的推导可能有些许错误，见谅。这个东西是我九月份竞赛的时候写的不过一直没有发出来。当然最初只是因为b站的公编辑器比较便捷所以用专栏的公式编辑器来打草稿，所以写得比较粗略，随便看看就行

设浮子和振子的质量为 $m_1%EF%BC%8Cm_2%2Cm%3Dm_1%2Bm_2$ ，兴波阻尼系数为 $%5Cmu$

波浪激励力为 $fcos%5Comega%20t$ 设浮子和振子的位移为 $%5Cvec%20x(t)%20%2C%5Cvec%20x_2(t)$

惯性附加质量为 $%5Cdelta$ 以及弹簧的刚度为 $c$

阻尼器阻尼系数为 $p$ 海水密度为 $%5Crho%20$

净水恢复力 $%5Cvec%20F_%E6%B0%B4%3D-%5Crho%5Cpi%20%20g%20%20%5Cvec%20x$

弹簧的弹力 $%5Cvec%20T%3D-c%5Cvec%20x_2$

阻尼器的阻力 $%5Cvec%20f_%E9%98%BB%3D-p%5Cdot%7B%5Cvec%20x_2%7D$

振子合力 $%5Cvec%20F_%E6%8C%AF%3D-p%5Cdot%7B%5Cvec%20x_2%7D-c%5Cvec%20x_2-m_2%5Cvec%20g%3Dm_2%5Cddot%7B%5Cvec%20x%7D_2$

兴波阻尼力 $%5Cvec%20F_%E5%85%B4%3D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D$

波浪激励力 $%5Cvec%20F_%E6%B3%A2%3D(m%2B%5Cdelta)%5Cvec%20a_%E6%B3%A2%3Dfcos%5Comega%20t$

作用到浮子上的部分 $%5Cvec%20F_2%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D$

振子对浮子的压力 $%5Cvec%20F_%E5%8E%8B%3Dp%5Cdot%7B%5Cvec%20x_2%7D%2Bc%5Cvec%20x_2$

浮子合力 $%5Cvec%20F_%E6%B5%AE%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D%2Bp%5Cdot%7B%5Cvec%20x_2%7D%2Bc%5Cvec%20x_2-%5Crho%5Cpi%20%20g%20%5Cvec%20x$

于是我们有方程组

$m_1%5Cddot%7B%5Cvec%20x%7D%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D%2Bp%5Cdot%7B%5Cvec%20x_2%7D%2Bc%5Cvec%20x_2-%5Crho%5Cpi%20%20g%20%5Cvec%20x$

$%0Am_2%5Cddot%7B%5Cvec%20x%7D_2%3D-p%5Cdot%7B%5Cvec%20x_2%7D-c%5Cvec%20x_2-m_2%5Cvec%20g-m_2%5Cddot%7B%5Cvec%20x%7D$

2）

$p%3Dk%7C%5Cdot%7B%5Cvec%20x%7D_2%7C%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2Ck%3D10000$

$m_1%5Cddot%7B%5Cvec%20x%7D%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D%2Bk%7C%5Cdot%7B%5Cvec%20x%7D_2%7C%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D(%5Cdot%7B%5Cvec%20x_2%7D)%2Bc%5Cvec%20x_2-%5Crho%5Cpi%20%20g%20%5Cvec%20x$

$%0Am_2%5Cddot%7B%5Cvec%20x%7D_2%3D-k%7C%5Cdot%7B%5Cvec%20x%7D_2%7C%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D(%5Cdot%7B%5Cvec%20x_2%7D)-c%5Cvec%20x_2-m_2%5Cvec%20g-m_2%5Cddot%7B%5Cvec%20x%7D$

设浮子和振子的转动惯量为 $J_1%EF%BC%8CJ_2$ ，兴波阻尼力矩系数为 $%5Clambda$

波浪激励力矩为 $%5Cvec%20Lcos%5Comega%20t$ 设浮子和振子的角位移为 $%5Cvec%20%5Ctheta%20(t)%20%2C%5Cvec%20%5Ctheta_2(t)$

惯性附加转动惯量为 $%5Cvarepsilon$ 以及扭转弹簧的扭转刚度为 $c_T$

扭转阻尼器阻尼系数为 $k_2$ 净水恢复力矩系数为 $%5Cgamma%20$

振子的矢径 $r$

波浪鼓励力矩为 $%5Cvec%20M_%E6%B3%A2%3D%5Cvec%20Lcos%5Comega%20t%3D(J_1%2BJ_2%2B%5Cvarepsilon)%5Cddot%7B%5Cvec%20%5Ctheta%7D$

作用在浮子上的波浪鼓励力矩为 $%5Cvec%20M_2%3D%5Cfrac%7BJ_1%5Cvec%20Lcos%5Comega%20t%7D%7B(J_1%2BJ_2%2B%5Cvarepsilon)%7D$

兴波阻尼力矩为 $%5Cvec%20M_%E5%85%B4%3D-%5Clambda%5Cdot%7B%5Cvec%20%5Ctheta%7D$

净水恢复力矩为 $%5Cvec%20M_%E6%B0%B4%3D-%5Cgamma%5Cvec%20%5Ctheta$

扭转阻尼器的阻尼力矩 $%5Cvec%20M_%E9%98%BB%3D-k_2%5Cdot%7B%5Cvec%20%5Ctheta%7D_2$

扭转弹簧的力矩为 $%5Cvec%20M_T%3D-c_T%20%5Cvec%20%5Ctheta_2$

重力的力矩为 $%5Cvec%20C_F%3D-m_2%5Cvec%20gsin(%5Cvec%5Ctheta_2)$

科氏力 $%5Cvec%20F_k%3D-2m_2%5Cvec%5Ctheta_2%5Ctimes%20%5Cdot%7B%5Cvec%20x%7D$

振子的合力矩 $%5Cvec%20M_%E6%8C%AF%3D-k_2%5Cdot%7B%5Cvec%20%5Ctheta%7D_2-c_T%20%5Cvec%20%5Ctheta_2%2Bm_2%5Cvec%20gsin(%5Cvec%5Ctheta_2)-J_2%5Cddot%7B%5Cvec%20%5Ctheta%7D$

浮子合力矩 $%5Cvec%20M_%E6%B5%AE%3D%5Cfrac%7B(J_1%2BJ_2)%5Cvec%20Lcos%5Comega%20t%7D%7B(J_1%2BJ_2%2B%5Cvarepsilon)%7D-%5Clambda%5Cdot%7B%5Cvec%20%5Ctheta%7D-%5Cgamma%20%5Cvec%20%5Ctheta%2Bk_2%5Cdot%7B%5Cvec%20%5Ctheta%7D_2%2Bc_T%20%5Cvec%20%5Ctheta_2$

浮子合力 $%5Cvec%20F_%E6%B5%AE%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D%2B(%5Cdot%7B%5Cvec%20x_2%7D%2Bc%5Cvec%20x_2)cos(%5Cvec%20%5Ctheta_2)-%5Crho%5Cpi%20%20g%20%5Cvec%20x$

振子合力 $%5Cvec%20F_%E6%8C%AF%3D-p%5Cdot%7B%5Cvec%20x_2%7D-c%5Cvec%20x_2-m_2%5Cvec%20gcos(%5Cvec%20%5Ctheta_2)-2m_2%5Cvec%5Ctheta%5Ctimes%20%5Cdot%7B%5Cvec%20x%7D-m_2%5Cddot%7B%5Cvec%20x%7D$

最后我们得到方程组

$J_2%5Cddot%7B%5Cvec%20%5Ctheta%7D_2%3D-k_2%5Cdot%7B%5Cvec%20%5Ctheta%7D_2-c_T%20%5Cvec%20%5Ctheta_2%2Bm_2%5Cvec%20gsin(%5Cvec%5Ctheta_2)-J_2%5Cddot%7B%5Cvec%20%5Ctheta%7D$

$J_1%5Cddot%7B%5Cvec%20%5Ctheta%7D%3D%5Cfrac%7B(J_1%2BJ_2)%5Cvec%20Lcos%5Comega%20t%7D%7B(J_1%2BJ_2%2B%5Cvarepsilon)%7D-%5Clambda%5Cdot%7B%5Cvec%20%5Ctheta%7D-%5Cgamma%20%5Cvec%20%5Ctheta%2Bk_2%5Cdot%7B%5Cvec%20%5Ctheta%7D_2%2Bc_T%20%5Cvec%20%5Ctheta_2$

$m%5Cddot%7B%5Cvec%20x%7D%3D%5Cfrac%7Bmfcos%5Comega%20t%7D%7Bm%2B%5Cdelta%7D-%5Cmu%20%5Cdot%7B%5Cvec%20x%7D%2B(%5Cdot%7B%5Cvec%20x_2%7D%2Bc%5Cvec%20x_2)cos(%5Cvec%20%5Ctheta_2)-%5Crho%5Cpi%20%20g%20%5Cvec%20x$

$m_2%5Cddot%7B%5Cvec%20x%7D_2%3D-p%5Cdot%7B%5Cvec%20x_2%7D-c%5Cvec%20x_2-m_2%5Cvec%20gcos(%5Cvec%20%5Ctheta_2)%2B2m_2%5Cvec%5Ctheta%5Ctimes%20%5Cdot%7B%5Cvec%20x%7D-m_2%5Cddot%7B%5Cvec%20x%7D$

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