《控制工程基础》课程作业习题

1G1?s??1?e?2s?

?st????1?2?1t?1edt?0211?2s?e1?e?2ss1?1?e?s1?e?s??2e?s?11?1?e?ss??2?????

1?e?s?s1?e?s??

例2 试求例图2-2a所示力学模型的传递函数。其中,xi?t?为输入位移,xo?t?为输出位移,k1和k2为弹性刚度,D1和D2为粘性阻尼系数。

解: 粘性阻尼系数为D的阻尼筒可等效为弹性刚度为Ds的弹性元件。并联弹簧的弹性刚度等于各弹簧弹性刚度之和,而串联弹簧弹性刚度的倒数等于各弹簧弹性刚度的倒数之和,因此,例图2-2a所示力学模型的函数方块图可画成例图2-2b的形式。

k1xiD1D2k2xo

例图2-2a 弹簧-阻尼系统

Xi?s???k1?sD1k1?sD1F?s?1k2?sD2Xo?s?

例图2-2b 系统方块图 根据例图2-2b的函数方块图,则

k1?D1s1D1?sXo?s?k1?D1sk2?D2sk2 ??1Xi?s?1?k1?D1s???D1D22D1D1D2?s????s?1??k1?D1sk2?D2sk1k2?k1k2k2?

6

例3 试求例图2-3所示电路网络的传递函数。其中,uo?t?为输出电压,ui?t?为输入电压,R1和R2为电阻,C1和C2为电容。

例图2-3 无源电路网络

解: 如例图2-3,设电流i1?t?和i2?t?为中间变量,根据基尔霍夫定律,可列出如下方程组

1?i1?t?dt?R1i2?t??C1? ? ui?t??uo?t??R1i2?t??1?i1?t??i2?t??dt??i1?t??i2?t??R2?uo?t??C2??? 消去中间变量i1?t?和i2?t?,得

R1R2C1C2d2uo?t?dt2duo?t?d2ui?t?dui?t?????R1C1?R2C2?R1C2??uo?t??R1R2C1C2?RC?RC?ui?t? 1122dtdtdt2令初始条件为零,将上式进行拉氏变换,得

R1R2C1C2s2Uo?s???R1C1?R2C2?R1C2?sUo?s??Uo?s??R1R2C1C2s2Ui?s???R1C1?R2C2?sUi?s??Ui?s? 由此,可得出系统传递函数为

Uo?s?R1R2C1C2s2??R1C1?R2C2?s?1 ?Ui?s?R1R2C1C2s2??R1C1?R2C2?R1C2?s?1例4 试求例图2-4所示有源电路网络的传递函数。其中,ui?t?为输入电压,uo?t?为

输出电压。

7

例图2-4 有源电路网络

解: 如例图2-4,设R2、R4和R5中间点的电位为中间变量uA?t?。按照复阻抗的概念,电容C上的复阻抗为

1。 Cs 根据运算放大器的特性以及基尔霍夫定律,可列出如下方程组 Ui?s?U?s????A?R1R2? ?UA?s?UA?s??Uo?s?UA?s?

???R?1R25?R4?Cs?消去中间变量UA?s?,可得

R2R4?R2R5?R4R5Cs?1U?s?R?R5R2?R5 o ??2?Ui?s?R1R4Cs?1例5 如例图2-5所示系统,ui?t?为输入电压,io(t)为输出电流,试写出系统状态空间表达式。

ui(t) R1 io(t)

2

例图2-5 电路网络 解:该系统可表示为

iL(t)L C uc(t)

R 8

di(t)?ui(t)?R1io(t)?LL?dt? ?u(t)?R1io(t)?uC(t)?[io(t)?iL(t)]R2

i?duc(t)i(t)?i(t)?C?oL?dt则

R1R2R1R2?diL(t)??iL(t)?uc(t)?ui(t)??dtL(R1?R2)L(R1?R2)L(R1?R2) ?

duc(t)R111???iL(t?)uc(t?)ui(t)?C(1R?R2)C(1R?R2)C(1R?R2)?dtio(t)?R2(R1?R2)iL(t)?1(R1?R2)uc(t)?1(R1?R2)ui(t)

可表示为

R1R2???i?L???L(R1?R2)???c?R1?u?????C(R1?R2)??R2?L(R1?R2)??iL??L(R1?R2)???????ui uc?11?????C(R1?R2)????C(R1?R2)??R1io??R2?(R?R)?12???iL?1?ui ???u(R1?R2)??c?(R1?R2)1

1.试求下列函数的拉氏变换 (1)f?t???4t?5???t???t?2??1?t? (2)f?t??sin?5t???????1?t? 3?(3)f?t?????0?t???sint

?0t?0,t????(4)f?t???4cos?2t????????5t???1?t???e?1?t? 3???6?(5)f?t??15t2?4t?6??t??1?t?2? (6)f?t??6sin?3t????????????1?t?? 4??4?(7)f?t??e?6t?cos8t?0.25sin8t??1?t?

9

(8)f?t??e?20t?2?5t??1?t???7t?2???t???3sin?3t??????????????1?t?? 2???6?2.试求下列函数的拉氏反变换

s?1(1)F?s??

?s?2??s?3?1 2s?4s(3)F?s??2

s?2s?5(2)F?s??e?s(4)F?s??

s?1s(5)F?s??

?s?2??s?1?2(6)F?s??4s2?s?4

(7)F?s??s?1 s2?9dx?t?dx?t??8x?t??1,其中x?0??1,dtdt3.用拉氏变换法解下列微分方程。

(1) (2) (3)

d2x?t?dt2?6t?0?0

dx?t??10x?t??2,其中x?0??0

dtdx?t?dx?t??100x?t??300,其中dtdtt?0?50

4.对于题图2-4所示的曲线,求其拉氏变换。

题图2-4

dy?t?dx?t?5. 某系统微分方程为3o?2yo?t??2i?3xi?t?,已知yo0??x0??0,当输人为1dtdt????(t)时,输出的终值和初值各为多少? 6. 化简下列方块图,并确定其传递函数。 (1)

10

联系客服:779662525#qq.com(#替换为@) 苏ICP备20003344号-4