Yn+1二Yn +Zn凡 n(凡 ωn +Exn凡)12 +C2凡(Vωn EVxn)]
.n+1二.n +C1[Zn(xn Eωn)+Z2 n(Vxn EVωn)12]
凡凡 凡凡
An+1二An 凡+C1[Zn(凡 ωn +Exn)+Z2 n(Vωn +EVxn)12]
x.n+1二xn +ZnVxn
ω二凡凡ωn +ZnV凡ωn
.n+1
Vx.n+1二Vxn
Vω.n+1二V凡ωn
where:
J
C1二 11(EI)n 1 + E2
Z2 (ZEI)(5-38)C2二 6 n (α GA)n n
where:
E二 Young's modulus of elasticity
I二 sectional moment of inertia
G二 shear modulus
E二 logarithmic decrement of internal shaft damping divided
by shaft vertical position
α二 cross-sectional shape factor (α二 .75 for circular cross section)
.lectromechanical Systems and Analogies
Where physical systems are so complex that mathematical solutions arenot possible， experimental techniques based on various analogies may be one type of solution. Electrical systems that are analogous to mechanicalsystems are usually the easiest， cheapest， and fastest solution to the prob-lem. The analogy between systems is a mathematical one based on the similarity of the differential equations. Thomson has given an excellent treatise on this subject in his book on vibration. Some of the highlights are given here.
A forced-damped system is shown in Figure 5-17. This system has a mass
， which is suspended on a spring主 with a spring constant and a dash pot to produce damping. The viscous damping coefficient is J
t
生V生t + JV +主 JV生t二.(t)(5-39)
A force-voltage system can be designed to represent this mechanical system as shown in Figure 5-18.
The equation representing this system when巾(t) is the voltage and repre-sents theforce， while inductance (L)， capacitance C， and resistance R
Figure 5-17. Forced vibration with viscous damping
Figure 5-18. A force-voltage system
represent themass， spring constant， and the viscous damping， respectively， can be written as follows:
J
L生t生i+ Ri + C 1 ti生t二巾 (t)(5-40)
A force-current analogy can also be obtained where the mass is repre-sented by capacitance， the spring constant by the inductance， and the resist-
ance by the conductance as shown in Figure 5-19. The system can be
represented by the following relationship:
生巾 1 J t
C 生t +G巾 + L 巾生 t二 i(t) (5-41)
Comparing all these equations shows that the mathematical relationships are all similar. These equations convey the analogous values. For conveni-ence， Table 5-1 also shows these relationships.
Forces Acting on a Rotor .earing System
There are many types of forces that act on a rotor-bearing system. The forces can be classified into three categories: (1) casing and foundationforces， (2) forces generated by rotor motion， and (3) forces applied to a rotor. Table 5-2 by Reiger is an excellent compilation of these forces.
Forces transmitted to casing and foundations. These forces can bedue to foundation instability， other nearby unbalanced machinery， pipingstrains， rotation in gravitational or magneticfields， or excitation of casing or
.able 5-1
.lectromechanical System Analogies
Mechanical .lectrical .arameters
.arameters
Force-Voltage Analogy Force-.urrent Analogy
Force (F) Voltage巾 Current i current i Voltage巾
Velocity xior V 耳 耳
Displacement x二0t V生t Charge q二0t i生t Mass Inductance L Capacitance C Dashpot J Resistance R Conductance G Coefficient Spring Constant王 Capacitance C Inductance L
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