第2章转质量陀螺仪及其力学分析

1、2021-11-1012021-11-102kjihzzyyxxjjjkkhhjzzxz yz 2021-11-103mhdtdimhhdtd0dtdhmh2021-11-1041 1二自由度陀螺的二自由度陀螺的进动性进动性（spin precession）陀螺转子轴在外力矩作用下绕与外力矩相垂直的方向的转动运动，称为陀螺的“进动运动” 2021-11-1051 1二自由度陀螺的二自由度陀螺的进动性进动性（spin precession）2021-11-1061 1二自由度陀螺的二自由度陀螺的进动性进动性（spin precession）m2021-11-107织女星 北极星 1 1二自由度陀

2、螺的二自由度陀螺的进动性进动性（spin precession）2021-11-1082021-11-1092 2二自由度陀螺的稳定性二自由度陀螺的稳定性（spin stabilization） 0m0dtdihconsth 2021-11-10103 3陀螺力矩与陀螺效应陀螺力矩与陀螺效应 2021-11-1011陀螺力矩物理实质2sincciiydfa mmrdvrd drdhimdvrd drdh2sincyar322singxydmrd drdh 42gxymr hgxymh 2021-11-10123 3陀螺力矩与陀螺效应陀螺力矩与陀螺效应2021-11-10133 3陀螺力矩与陀螺

3、效应陀螺力矩与陀螺效应2021-11-10141 1单自由度陀螺感受转动的特性单自由度陀螺感受转动的特性2021-11-1015 2021-11-1016hmxd3.3.关于单自由度陀螺漂移率的解释关于单自由度陀螺漂移率的解释2021-11-1017bbbzyoxoxyz bybzbxy 2021-11-1018（1） 框架框架 bri kjiyzzyxbe)sincos()sincos(2021-11-1019 kjiyzzyxb)sincos()sincos()(sincossincos)(yzbzzybyxbx kjjjijhyzbzzybyxbxb)sincos()sincos()(2

4、021-11-1020（2） 转子转子 kjiyzzyxc)sincos()sincos()( ()(cossin)(cossin)cexeyzzzyhjijjjk 2021-11-1021 cbbhhhkjjjjjjijjhyzbzzzzybyexbxeb)sincos)()sincos)()(zjh2021-11-1022)sincos)()sincos)()(yzbzzbzzybyebyxbxebxjjhhjjhjjhxzyyzxmhhdtdhxyzzybyezyyzbzzxbxemjjjjhjj)sincos)(sincos)()sincos)(sincos)()( 2021-11-1

5、023xxzymjhj )sincos()(cossin )()()(cossin )(cossin )ebxyzebexzbzebyyzzyxjjhjjjjjjm yxjhmxmcdyjdch2021-11-1024jchch2020 jcf210jc0)cos1(0tch2021-11-1025hcdj jc20jcdjd220)1sin(1110220techt/1arctan2 2021-11-1026chk kch 2021-11-1027hdj djt dht )1 (/ttettdh0t tdhkdhtdtk0 2021-11-1028hj ttttdtdtkdtdtjh0000

6、 2021-11-1029hcdj )()()()(2shscsdssjs)(s)(s2021-11-10302200( )( )12( )1skg ssss2021-11-1031oxyzoxyz2021-11-1032000 xzyxxyzxyyyxzzzdhdthmdhhmdthmdhdt 2021-11-1033(1) (1) 只只考虑转子列写欧拉方程考虑转子列写欧拉方程 c)sin(cosczcycxcjjjh 2021-11-1034(1) (1) 只只考虑转子列写欧拉方程考虑转子列写欧拉方程 sincos2021-11-1035(sin)coscosczcycxzdjjjmdt

7、(1) (1) 只只考虑转子列写欧拉方程考虑转子列写欧拉方程cycxjjzczmjdtd)sin( (1) 2021-11-1036(2) (2) 考虑转子、内环列写欧拉动力学方程考虑转子、内环列写欧拉动力学方程 sincosbzbybxbjjjhsin)sin(cos)()(1bzczbycybxcxjjjjjjh 2021-11-1037xbycybzczbxcxmjjjjjjdtd)sin(cos)(cossin)sin()(xbzbycyczbxcxmjjjjjjcossin)(cos)sin()(2 (2) (2) 考虑转子、内环系统列写欧拉动力学方程考虑转子、内环系统列写欧拉动力学

8、方程(2) 2021-11-1038(3) (3) 考虑转子、内环、外环整个系统列写欧拉动力学方程考虑转子、内环、外环整个系统列写欧拉动力学方程yymdtdhayzyyhhhhsincos11ayayjh22222()cos(sin)sinsin()cossin(sin)sin()cossin(sin)sinycybyczbzaycybybzczaycybybzayczhjjjjjjjjjjjjjjj2021-11-10392222()cossin(sin)sin()( 2 cossin)2 sincos()cossin(sin)sin(sin)cosyycybybzayczcybybzcyb

9、ybzayczczdhdmjjjjjdtdtjjjjjjjdjjdt sincos)sin(cossin)(2sincos)(22zyczcybybzaybzbycymmjjjjjjjj (3) (3) 考虑转子、内环、外环整个系统列写欧拉动力学方程考虑转子、内环、外环整个系统列写欧拉动力学方程(3) 2021-11-1040222()cossin2()sincos(sin)cossin()(sin) cos()sincos(sin)cybybzaybzbycyczyzcxbxczcybybzxczzjjjjjjjjmmjjjjjjmdjmdt 2021-11-1041 0zm0)sin(cz

10、jdtd222()cossin2()sincos(sin )cossin()(sin ) cos()sincos(sin )cybybzaybzbycyczyzcxbxczcybybzxczzjjjjjjjjmmjjjjjjmdjmdt constjcz)sin(2021-11-1042xbxcxyaybzbycymhjjmhjjjj cos)(cossincos)(22222()cossin2()sincos(sin)cossin()(sin) cos()sincos(sin)cybybzaybzbycyczyzcxbxczcybybzxczzjjjjjjjjmmjjjjjjmdjmdt 2

11、021-11-1043 xbxcxyaybzbycymhjjmhjjjj cos)(cossincos)(22xbxcxyaybycymhjjmhjjj )()(xxyymhjmhj 2021-11-1044l 坐标变换l 转子、内环坐标系的角速度l 角动量的计算基座坐标系相对惯性空间转动2021-11-1045zimyimximmim)()()(基座坐标系相对惯性空间转动 mg2021-11-1046 基座坐标系相对惯性空间转动mg imimigcoscossincossinsincoscossinsinsin0coscos0sin010sin0coscossin0sincos0001gmc

12、： 2021-11-1047基座坐标系相对惯性空间转动coscos)(sin)(cossin)(sincos)(cos)(sinsin)(sin)(cos)()()()(coscossincossinsincoscossinsinsin0cos)()()(zimyimximzimyimximzimximzimyimximzimyimximgmgmic2021-11-1048coscos)(sin)(cossin)(sinsincos)(cos)(sinsin)(cossin)(cos)(sincoscoscos)(sin)(cossin)(sincos)(cos)(sinsin)(sin)(c

13、os)(zimyimximzimyimximzimximzimyimximzimyimximzimximgig基座坐标系相对惯性空间转动2021-11-1049 xxyymhjmhj )()()()()()()()()()(2222222222222222smsmhsjjshhsjjjhsjjjhsjjshsmsmhsjjjhsjjshhsjjshhsjjjssyxyxyxyyxxyxxyyxyyxyxyxx2021-11-1050 hs 21sjy hs 21sjx )(smx )(s )(s )(smy 2021-11-1051 21sjy 21sjx hs hs ( ) s ( ) s

14、( )ym s 2021-11-1052 2021-11-1053 )()()()(22222222hsjjshhsjjjshsjjjhsjjshsyxyxyyxxyxtjjhthhttjjhthhtnxynnyxnsin1cos11)(sin1cos11)(yxnjjh222021-11-1054 )()()()()()(2222222222hsjjshhsjjsjshsjjsjhsjjshsyxyxyyxxyxthjjthjthhjtthjjthjthhjtnyxnyynyxnxxsincos1)(sincos1)(2222222021-11-10552021-11-1056 xxyymhjmhj 4