The mathematical model of an analogous electrical system for the
The mathematical model of an analogous electrical system for the following mechanical system using the force-current analogy is (i – current, v – voltage, L – Inductance, C – Capacitance)
A. <span class="math-tex">\(\frac{1}{{{C_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt = {L_{M2}}\frac{{d{i_x}}}{{dt}} + \frac{1}{{{C_{K2}}}}\smallint {i_x}dt,\;{v_F}\left( t \right) = {L_{M1}}\frac{{d{i_y}}}{{dt}} + \frac{1}{{{C_{K3}}}}\smallint {i_y}dt + \frac{1}{{{C_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt\)</span>
B. <span class="math-tex">\(\frac{1}{{{L_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt = {L_{M2}}\frac{{d{i_x}}}{{dt}} + \frac{1}{{{L_{K2}}}}\smallint {i_x}dt,\;{v_F}\left( t \right) = {L_{M1}}\frac{{d{i_y}}}{{dt}} + \frac{1}{{{L_{K3}}}}\smallint {i_y}dt + \frac{1}{{{L_{K1}}}}\smallint \left( {{i_y} - {i_x}} \right)dt\)</span>
C. <span class="math-tex">\(\frac{1}{{{C_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt = {L_{M2}}\frac{{d{v_x}}}{{dt}} + \frac{1}{{{C_{K2}}}}\smallint {v_x}dt,\;{i_F}\left( t \right) = {L_{M1}}\frac{{d{v_y}}}{{dt}} + \frac{1}{{{C_{K3}}}}\smallint {v_y}dt + \frac{1}{{{C_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt\)</span>
D. <span class="math-tex">\(\frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt = {C_{M2}}\frac{{d{v_x}}}{{dt}} + \frac{1}{{{L_{K2}}}}\smallint {v_x}dt,\;{i_F}\left( t \right) = {C_{M1}}\frac{{d{v_y}}}{{dt}} + \frac{1}{{{L_{K3}}}}\smallint {v_y}dt + \frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt\)</span>
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Right Answer is: D
SOLUTION
Mathematical model:
A mathematical model of a dynamic system is defined as a set of differential equations that represent the dynamics of the system accurately.
A mathematical model is not unique to a given system.
A system may be represented in many different ways. Therefore it may have many mathematical models
Analysis:
The given model is
The differential equations are
\(F\left( t \right) = {M_1}\frac{{dy\left( t \right)}}{{dt}} + {K_1}\smallint \left( {y\left( t \right) - x\left( t \right)} \right)dt + {K_3}\smallint y\left( t \right)dt \ldots \ldots \ldots .\;\left( 1 \right)\)
\({M_2}\frac{{dx\left( t \right)}}{{dt}} + {K_2}\smallint x\left( t \right)dt + {K_1}\smallint \left( {x\left( t \right) - y\left( t \right)} \right)dt = 0\)
\({M_2}\frac{{dx\left( t \right)}}{{dt}} + {K_2}\smallint x\left( t \right)dt = {K_1}\smallint \left( {y\left( t \right) - x\left( t \right)} \right)dt \ldots \ldots \ldots .\left( 2 \right)\)
The mathematical model for various systems is given in the below tabular form.
|
Mechanical translational system |
Mechanical rotational system |
Current Analogous system |
Voltage Analogous system |
|---|---|---|---|
|
F (Force) |
T (Torque) |
I (current) |
V (Voltage) |
|
M (Mass) |
J (Inertia) |
C (Capacitance) |
L (Inductance) |
|
B (Friction constant) |
B |
1/R |
R (Resistance) |
|
K (Spring constant) |
K |
1/L |
1/C |
|
x(t) (Velocity) |
ω |
V |
I |
Force- Current Analogy:
Compare the elements in the above table
Substitute them in the differential equations (1) and (2).
\({i_F}\left( t \right) = {C_{M1}}\frac{{d{v_y}}}{{dt}} + \frac{1}{{{L_{K3}}}}\smallint {v_y}dt + \frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt\)
\(\frac{1}{{{L_{K1}}}}\smallint \left( {{v_y} - {v_x}} \right)dt = {C_{M2}}\frac{{d{v_x}}}{{dt}} + \frac{1}{{{L_{K2}}}}\smallint {v_x}dt\;\)
