2) 例2.1 图为机械位移系统。试列写质量m在外力F作用下位移y(t)的运动方程。 解: 阻尼器的阻尼力: Ft)?fdy(t)1(弹簧弹性力:
dtkF2(t)?ky(t)F
2mmdy(t)
dt2?F(t)?F1(t)?F2(t)fy(t)整理得:
2mdy(t)dy(t) dt2?fdt?ky(t)?F(t)
例2.2 如图RLC电路，试列写以ur(t)为输入量，uc(t)为输出量的网络微分方程。 解:
i(t)RL Ldi(t)dt?uc(t)?Ri(t)?ur(t)
ur(t)Curuc(t)?1c?i(t)dt
d2u例2.3 已知Rc(t)duc(t1=1,C1=1F,uc(0)=0.1v, ur(t)=1(t),LC求 dtuc2(t) ? RC)dt?uc(t)?ur(t)解：
Rduc1C1dt?uc?ur i(t)R1 R1C1sUc(s)?R1C1uc(0)?Uc(s)?Ur(s)u r(t)C1uc(t)sUc(s)?0.1?Uc(s)?Ur(s) 零初始条件下取拉氏变换：
U)?10.1c(ss(s?1)?s?1 R1C1sUc(s)?uUc(tc)(s?)1??Ue?t?0.1e?t
r(s)U c(s)U?1r(s)R1C1s?1
例2.4 如图RLC电路，试列写网络传递函数 Uc(s)/Ur(s). 参见
2LCduc(t)dt2?RCduc(t)dt?uc(t)?ur(t)
解:1) 零初始条件下取拉氏变换：
LCs2URCsU(s)?Ur(s)
i(t)RLc(s)?c(s)?Uc传递函数：
u?1r(t)Cus)?Uc(s)c(t)G(Ur(s)LCs2?RCs?1
2)变换到复频域来求。 i(s)RLSur(s)1sCuc(s)
例2.5 已知R1=1,C1=1F，1）求零状态条件下阶跃响应uc(t)；2) uc(0)=0.1v, ur(t)=1(t), 求 uc(t)；3）求脉冲响应g(t)。
解: 1）
U(s)11R i(t)G(s)?c??Ur(s)R1C1s?1s?1
10.1U(s)??(t)(t)Cucur s(s?1)s?1对上式进行拉氏反变换： 111c uc(t)?1?e?tR1C1sUc(s)?Uc(s)?Ur(s)???（前例已得） ducR1C1?uc?urdt
R1C1sUc(s)?R1C1uc(0)?Uc(s)?Ur(s)2)
sUc(s)?0.1?Uc(s)?Ur(s)U(s)1Uc(s)?r?s?1s(s?1) u c ( t ) ? 1 ? e ? t ? e ? t 3) 0 .11?1?1?t g(t)?L[G(s)]?L[]?e s?1例2.6 具有相同极点不同零点的两个系统 G(s)1，它们零初始条件下的单位阶跃响应分别为
?4s?2(s?1)(s?2),G2(s)?1.5s?2(s?1)(s?2)
c1(t)?L[?14s?2s(s?1)(s?2)]?1?2e?t?3e?2t1.5s?2 ?1?t?2tc2(t)?L[]?1?0.5e?0.5e s(s?1)(s?2)例2.7 绘出RC电路的结构图。
R1(t)Cuc(t) ur
例2.8 绘出图示双RC网络的结构图。
1
Ur(s)?Uc(s)?R1I1(s)Uc(s)?I1(s)C1s
R1RicI1(s)?21R1?[Ui(s)?U(s)]iui1i2C1uC2u0Ic(s)?I1(s)?I2(s)U(s)?Ic(s)C1sI2(s)C2sI2(s)?1R2[U(s)?Uo(s)]Uo(s)?2H3GH242G2243HGGG1H3G4G
I2(s)(?) IcUi(s)(?)1R1I1(s)I(s)1(s)Ic(s) 1C1sU(s)U(s)U(s)(b)(a)1R2(c)I2(s)
Ui(s)
例2.9
R(s)I(s)21C2sUO(s)(?)UO(s)(d)1(?)(e)(?)I1(s)1U(s)1(?)1UO(s)R1Ic(s)C1sR2I2(s)C2s(f)C(s)Gi(s)(?)GG(?)2(s)(s)G6(s)G236?(G2?G3)G63G4(s)G?G2361?G236G54G1
例2.10 结构图化简
G5G54?G5?G4
R(?)GY2(?)
(1)结构图化简方案Ⅰ
RH2?G2H2R(?)GY1(a)G1G21?G2H2?G2G3H1YR1?G2H2G1G2G3?G2G3H1?G1G2H2Y1G34G4GGGG2134GGGG214G
(2) 结构图化简方案Ⅱ 原电路
H1
?HG23R(?)YRG1G2G31?G2H2?G2G3H1YGG23HG23HG23(a)(b)
(3) 结构图化简方案Ⅲ
HG11R(?)YH2G3(1?1G1)R(?)YH1G1H2G1G3H2G3??
(a)
例2.11 双RC网络的结构图简化。
(b)
U(s) U(s) ` Ui(s)
ii1(?)I1(s)(?)1(s)U(s)(?)1R21Uo(s)R1ICC1sI2(s)C2sR(?)1(a)1(?)1(?)1C1s1C2sUo(s)R1R2R1(b)11C2sUi(s)(?)11?T1s(?)Uo(s)R2R(?)1C)（s)(c2RC12s11?T1s(?)1R21C2sUo(s)U(s)i(?)11?T1s11?T2sUo(s)
（d)(e)例2.12
iC1i1(t)R1?uo(t)?ui(t)i1uo(t)?i(t)R2u
iR1i1(t)R1?R21Cuo?上式拉氏变换：
?(i?i)dt1I1(s)R1?Uo(s)?Ui(s)Uo(s)?I(s)R2
?信号传递流程：
C1ss?(s)??1(s)R1C1s?C1uc(0)??1(s)Ui(s)?Ui(s)?Uo(s)?I1(s)?I(s)?Uo?(s)?RCs)?(s)?Cu(0)???????(11111c?1(s)R1?1[?(s)??1(s)]?uc(0)uU(s)Uic(0)i(s)?Uo(s)I1(s)-C1I(s)UO(s)UO(s)1R1
例2.13 绘制结构图对应的信号流图(1)。 1?R1C1SR2-1
Ui(s)(-)1R1I1(s)(-)1C1sU(s)(-)1R21UO(s)Ic(S)I2(s)C2s-1Ui(s)1R11C1s1R21C2(S)UUOO(s)Ic(S)U(s)I2(S)(s)-1
例2.14 绘制结构图对应的信号流图(2) 。
-1
e2