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Analogy in Mechanical (trans.-rot.), Electrical, Fluid, Thermal Systems
.
..
x
Disp.
(m)
x
Vel.
(m/s)
x
Acc.
(m/s2)
.
..
θ
Angular
pos.
(rad)
θ
Angular
vel.
(rad/s)
q
Charge
(Coulomb)
i
Current
(Amper)
Vf
Volume
Qf
Flow rate
(m3)
(m3/s)
Ht
Heat
Qt
Heat flow
rate
(J/s)
(Joule)
θ
Ang.
acc.
(rad/s2)
m
Mass
(kg)
k
Spring cons.
(N/m)
c
Damp. Cons.
(Ns/m)
f
Force
(N)
IG
mass m.
ineratia
(kg-m2)
L
Induc.
(Henry)
Kr
Rot. Spring
const.
(Nm/rad)
Cr
Rot. damp.
const.
Nm/(rad/s)
M
Moment
1/C
C:Capac.
(Farad)
R
Resistance
(Ohm)
If
Fluid
inertia
(kg/m4)
1/Cf
Cf:Fluid
capacitance
Rf
Fluid
resistance
(Nm)
V
Voltage
(Volt)
P
Pressure
(N/m2)
1/Ct
Ct:Thermal
capacitance
Rt
Thermal
resistor
T
Temp.
(oC)
Vf
Volume
(m3)
Qf
Flow
Rate
(m3/s)
If
Fluid
inertia
(kg/m4)
1/Cf
Cf:Fluid
capacitance
Rf
Fluid
resistance
P
Pressure
(N/m2)
D.Rowell & D.N.Wormley, System Dynamics:An Introduction, Prentice Hall, 1997
Fluid inertia in pipes:
L
If 
If:Fluid inertia, ρ: Density, L: Pipe length A: Cross section of pipe
A
Fluid reservoir capacitance:
Ad
Cf: Fluid reservoir capacitance , Ad:Cross section area of
Cf 
g
reservoir, g=9.81 m/s2
Fluid resistance of pipes in laminar flow:
128L
Rf:Resistance, µ:Viscosity, L:Pipe length, d:Pipe diameter
4
d
4Q f
Reynolds number in laminar flow<2000
Reynolds Number=
d
Rf 
Equivalent Electrical Circuit of Fluidic Systems
In this lesson, we will learn and study the modeling fluidic systems by creating equivalent
circuits. There is an analogy between the fluidic and electrical quantities.
Let’s consider Example 5.1. A fluidic system is given here. We will create the equivalent
circuit for the system by using the analogy.
Example 5.1
Qk1
pa
2
5
1
pk1
1
2
4
3
Q5
Q3
Fluid.
pn
Elec.
Vn
Rn
In
Qn
Rn
Ln
q n
Cn
Cn
pa
Fluidic System
pk1
1
Fluid.
pn
Elec.
Vn
Rn
In
Qn
Rn
Ln
q n
Cn
Cn
In the fluidic system the pumps is connected the pipe
#1.
Equivalent Electrical Circuit
Vk1
+
Va
L1
R1
q 1
The pump’s output pressure is Pk. The atmospheric
pressure is Pa. The top of the tank #1 is open to the
atmosphere.
In the electrical circuit, a supply voltage operating at
Vk-Va is connected to the resistor R1 and inductor L1.
R1 represents the resistance to a flow in the pipe #1.
L1 represents the inertia of a flow in the pipe #1.
Qk1
In the fluidic system the tank #1 is connected to
both of the pipe #1 and the pipe#2.
pa
Fluidic System
However, in the electrical system, the capacitor #1
is connected to both of the resistor R1, inductor L1
and the resistor R2, inductor L2.
1
pk1
2
1
Equivalent Electrical Circuit
C1
Vk1
+
Va
L1
R1
R2
q 1
q k 1
q 2
The other end of the capacitor #1 is connected to
the voltage Va because the tank #1 is open to the
atmospehere.
The capacity of a tank in a fluidic system
corresponds to a capacitor in an electrical circuit.
There is an external flow with the flow rate Qk1 in
the fluidic system. The current supply qk1_dot is
placed in the electrical circuit.
L2
Qk1
In the fluidic system, the pipe #2 is connected
to both of the pipe #3 and the pipe#4 at the
point A. On the other hand, in the electrical
system, the line with the resistor R2, inductor
L2 is connected to the lines the resistor R3,
inductor L3 and the resistor R4, inductor L4 at
3the point A.
pa
Fluidic System
1
pk1
2
1
4
Q3
A
There is a pressure P4 in the pipe #4 due to the placement of pipe #4 in the vertical
direction because the flow is opposite to the gravity in pipe #4. So, the voltage
supply V4 is placed in the circuit due to the analogy. The positive end of the V4 is
connected to the point A. The current produced from V4 flows in the opposite
V4
direction.
L4
R
4
Equivalent Electrical Circuit
C1
Vk1
+
Va
+
-
q 4
L1
R1
R2
q 1
q k 1
q 2
L2
L3
R3
A
q 3
Qk1
pa
Fluidic System
2
5
1
pk1
2
1
4
3
Q5
Q3
A
In the fluidic system the tank #2 is connected to both of
the pipe #1 and the pipe#2. On the other hand, in the
electrical system, the capacitor #2 is connected both of
the resistor R4, inductor L4 and the resistor R5, inductor
L5.
V4
Vk1
+
Va
L4
R4
Equivalent Electrical Circuit
C1
+
-
q 4
R2
q 1
q k 1
L5
R5
q 5
L1
R1
C2
q 2
L2
L3
R3
A
q 3
Electrical systems can be analyzed instead of fluidic systems
Dynamic (Transient) behaviour
Steady-state behaviour
By analyzing the equivalent circuit, transient or steady-state dynamic
behavior at a desired point of a circuit can be calculated.
Thus, the corresponding flow rates and pressures at a desired line of a
fluidic system are found.
Nowadays computer aided engineering (CAD/CAE) is used.
Nowadays, circuits and fluidic systems can be modeled and analyzed
by the computers.
Before the developments in computer technology, in order to analyze
fluidic systems engineers have used the equivalent circuits, which are
produced easily and cheaply.
Complex fluidic systems
experiments of circuits.
have
been
easily analyzed with the