由梅森增益公式可得传递函数为
\[
\frac{C_2(s)}{R_2(s)} = \frac{\sum p_i \Delta_i}{\Delta} = \frac{\dfrac{9}{0.5s+1}\left(1+\dfrac{9}{s(s+1)}\right)+81}{1+\dfrac{9}{s(s+1)}+\dfrac{9}{0.5s+1}+81+\dfrac{81}{s(0.5s+1)(s+1)}}
\]
\[
= \frac{81s(0.5s+1)(s+1)+9s(s+1)+81}{82s(0.5s+1)(s+1)+9s(s+1)+9(0.5s+1)+81}
\]
\[
= \frac{40.5s^3+130.5s^2+90s+81}{41s^3+132s^2+95.5s+90}
\]
又解 可用结构图简化的方法对结果进行验证。以 \(C_1(s)/R_1(s)\) 为例,验证过程如下:画出图2-112系统的等效结构图,如图2-114所示;经过反馈等效,可得图2-115。
图2-114 关联系统简化结构图(\(C_1(s)/R_1(s)\))
图2-115 关联系统简化结构图(\(C_1(s)/R_1(s)\))
系统的传递函数为
\[
\frac{C_1(s)}{R_1(s)} = \frac{\dfrac{9}{s(s+1)}+\dfrac{9(4.5s+9)}{0.5s+10}}{1+\left[\dfrac{9}{s(s+1)}+\dfrac{9(4.5s+9)}{0.5s+10}\right]}
\]
\[
= \frac{9(0.5s+10)+9s(s+1)(4.5s+9)}{s(s+1)(0.5s+10)+9(0.5s+10)+9s(4.5s+9)(s+1)}
\]
\[
= \frac{40.5s^3+121.5s^2+85.5s+90}{41s^3+132s^2+95.5s+90}
\]
同理,可求解\(C_1(s)/R_2(s)\),\(C_2(s)/R_1(s)\)和\(C_2(s)/R_2(s)\)。
考研参考题
2-55 设汽车缓振系统如图2-116所示,图中,\(m_1\)为车厢及架重,\(m_2\)为车轮及轮轴