Modeling: free oscillations resonance and Electric circuits
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Transcript of Modeling: free oscillations resonance and Electric circuits
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Gandhinagar Institute
of Technology Advanced Engineering Mathematics
(2130002) Active Learning Assignment
Topic Name:-“Ordinary Differential Equations And Their Application: Modeling: Free Oscillations
Resonance And Electric Circuits” Guided By:- Prof. Jayesh Patel Name:- Jani Parth U. (150120119051)
Branch:- Mechnical Div:- A-3
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Oscillation Of A SpringConcider a Spring Suspended Vertically From A Fixed Point Support. Let a Mass m
Attached To The Lower End A Of Spring Stretches The Spring By A Length e Called Elongation And Comes To Rest At B. This Position Is Called Static Equilibrium.
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Now,The Mass Is Set In Motion From The Equilibrium Position . Let At Any Time t The Mass Is At P Such That BP=x. The Mass m Experience The Following Force.
i. Gravitational force mg acting downwards.
ii. Restoring force k (e + x) due to displacement of the spring acting upwards
iii. Damping (frictional or resistance)force c of the medium opposing the motion (action upwards)
iv. External force F(t) considering the downwards direction as positive
By Newton’s Second Law, The Differential Eqution Of The Motion Of The Mass M Is
At The Equilibrium Position B,
mg=ke
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Hence,
Let =2 And =
+2x = F(t)
Let Us Consider The Different Cases Of Motion.
Free Oscillation If The External Force F(t) Is Absent And Damping Force Is Negligible Then Eq. Reduces To
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x = 0 Free Oscillation eq.
Which Represents The Equation Of Simple Harmonic Motion.Hence, The Motion Of The Mass M Is Simple Harmonic Motion.
Time Period
Frequency
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Free Damped Oscillations If The External Force F(t) Is Absent And Damping Is Present Then Eq. Reduces To
Forced Undamped Oscillation If An External Periodic Force F(t)= Q Is Applied To The Support Of The Spring And Damping Force Is Negligible Then Eq. Reduces To
x =
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Modelling Of Electrical Circuits
Kirchhoff’s Voltage Law: The Algebraic Sum Of The Voltage Drops In Any closed Circuit Is Equal To The
Resultant E.M.F. In The Electric Circuit
Fundamental Relations:
The Current I Is The Rate Of Change Of Charge Q Thus I= or Q= ∫I dt
Voltage Drop Across Resistance (R)= RI
Voltage Drop Across Inductance (L)=L
Voltage Drop Across Capacitance (C)= Or ∫I dt
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R-L Circuit:The Figure Shows A Simple R-l Circuit
Applying Kirchhoff’s Voltage Law To The Circuit,RI + L = E(t)
The Differential Equation Is
+ I= Which Is Linear In I .
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R-C Circuit:The Figure Show A Simple R-C Circuit
Applying Kirchoff’s Voltage Law To Circuit
RI + = E(t)R + = E(t) (I= + Q = E(T) which is linear in Q
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Example 1. A Circuit Consisting of Resistance R And a Condenser Of Capacity C Is Connected In Series With A Voltage E. Assuming That There Is No Charge On Condenser At T=0, Find The Value Of Current I, Charge Q At Any Time T.Solution : The Differential Equation For R-c Circuit Is RI + = E(t)
R + = E(t) (I=
+ Q = E(T) which is linear in Q
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Comparing With + P(t)Q =Q(t)
P(t)= , Q(t)=
I.F=e ⌠ P(t) dt
=e ⌠ dt =e Hence, Solution Is Q(I.F)= ⌠Q(t) (I.F) dt +
Q= ⌠ E/R et/Rc dt+
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=E/R ⌠ Et/Rc dt+
=E/R (E t/Rc/1/RC) +
=Ece T/Rc+
=Ec + e T/Rc
At t=0, Q=0
0=EC+
= -EC
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hence, Q= EC(1-e -t/RC )
now, I=
= EC(1-e -t/RC )
=EC(0- E -t/RC (-1/RC))
=E/R e -t/RC
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