?12??400?2?400??2???4??0.14????1??800800????????2
?1.31
???2??????n?? ??????arctg 2???1???????n??400?2?0.14????800?
??arctg2?400?1???800?? ??10.6 当阻尼比改为??0.7时 A?????12??400?2?400??2???4??0.7????1??800800????????2?0.97
?400?2?0.7???800????43?
??????arctg2?400?1????800? 即阻尼比变化时，二阶振荡系统的输出副值变小，同时相位角也变化剧烈，相位差变大。
2.12对一个可视为二阶系统的装置输入一单位阶跃函数后，测得其响应中产生了数值为1.5的第一个超调量峰值。同时测得其振荡周期为6.28s。设已知该装置的静态增益为3，试求该装置的传递函数和该装置在无阻尼固有频率处的频率响应。
解： 最大超调量
?????1?2?1?????? M?e 即 ???1.5
?0.13
1??????1?ln1.5?2 且 Td?2??d?6.28
2 ∴ ?d??n1?? ?n??2??1 6.2811??2?11??0.13?k2?S2?1.01
系统的传递函数 H?s??Y?s??2X?s?S2?n
??n?1 ?3 2S?1.01?2?2?0.13?S?11.01该装置在无阻尼固有频率处的频率响应 由H?j???Y???K? X????j??22??j????1??????n?n? ?K
??????1??2j???2??nn??K3 ?2??????0.26j?1????????2j???n??n??? ∴ H?j?n?? ?d为有阻尼固有频率 M=0.5，?d???????2?1??????2??1 T M?e???1??????1?lnM?2?0.215
2 ?d??n1?? ，∴ ?n??d1??2?1.02
S=3
2?n ∴H?s??2?S 2S?2??nS??n ? A??n??1.04?3 2S?0.44?S?1.0414?2?3?6.98 （???n时代入得）
A????1,??????90? 2? ???n???arctg??? y?t??6.98sin?1.02t?4.1解 ：?=2?m时，
?2
????? 2?单臂，Uy??RU0 4R0Sg?R??4RU0
Uy?
2?120?2?10?6Uy?*3?3?10?6(V) 4?120双臂，Uy??RU0 2R0Sg?R??2RU0
Uy?2?120?2?10?6Uy?*3?6?10?6(V) 2?120
：?=2000?m时，
单臂，Uy??RU0 4R0Sg?R??4RU0
Uy?
2?120?2000?10?6Uy?*3?3?10?3(V) 4?120双臂，Uy??RU0 2R0Sg?R??2RU0
Uy?2?120?2000?10?6Uy?*3?6?10?3(V) 2?120双臂的灵敏度比单臂的提高一倍。
4.4解：Uy??RU0 R0Sg?R??RU0
Uy?Uy?Sg?(Acos10t?Bcos100t)?Esin10000t
?Sg?AEcos10tsin10000t?Sg?BEcos100tsin10000t11SgAE(sin10010t?sin9990t)?SgBE(sin10100t?sin9900t)221100101001099909990Uy(f)?jSgAE[?(f?)??(f?)??(f?)??(f?)]42?2?2?2?1101001010099009900?jSgBE[?(f?)??(f?)??(f?)??(f?)]42?2?2?2??
4.5解：xa?(100?30cos?t?20cos3?t)(cos?ct)
?100cos2000?t?30cos1000?tcos2000?t?20cos3000?tcos2000?t
?100cos2000?t?15(cos3000?t?cos1000?t)?10(cos5000?t?cos1000?t)Xa(f)?50[?(f?10000)??(f?10000)]?7.5[?(f?10500)??(f?10500)]?7.5[?(f?9500)??(f?9500)]?5[?(f?11500)??(f?11500)]?5[?(f?8500)??(f?8500)]4.10 解：H(s)?111???3 ?s?1RCs?110s?1 H(?)?1 ?310j??1
A(?)?11?(??)2?11?(10?)?3
?(?)??arctan(??)??arctan(10?3?)
Uy?10A(1000)sin(1000t??(1000))?10?0.707sin(1000t?450)?7.07sin(1000t?45)0
4.11 解：A(?)?11?(??)2
?(?)??arctan(??) 1
??10时，
A(10)?1?(0.05?10)?0.816
?(10)??arctan(0.05?10)?26.56?
??100时，A(100)?11?(0.05?100)?0.408
?(100)??arctan(0.05?100)?78.69?
y(t)?0.5?0.816cos(10t?26.56?)?0.2?0.408cos(100t?45??78.69?)?0.408cos(10t?26.56)?0.0816cos(100t?33.69)
5.1
??
?e??t;(t?0,??0)h(t)???0;(t?0)Rx(?)??h(t)?h(t??)dt??e??te??(t??)dt??0??????e0?????e?2?tedt?2????5.2 x(t)?A1sin(?1t??1??2)?A2sin(?2t??2??2)