Ordinary Differential Equations And Their Application: Modeling: Free Oscillations Resonance And Electric Circuits
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1) The document discusses ordinary differential equations that model free oscillations of a spring-mass system and electric circuits. 2) It derives the differential equation for simple harmonic motion of a mass attached to a spring as well as cases with damping. 3) Kirchhoff's voltage law is used to derive differential equations for simple R-L and R-C circuits driven by external voltages. 4) As an example, the differential equation for a series R-C circuit is solved analytically to find expressions for charge and current over time.
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Dr. Rajnish Kumar will be giving a refresher course on simple harmonic motion at Patliputra University in Patna. Simple harmonic motion occurs when a restoring force is directly proportional to displacement from equilibrium, such as the force exerted by a spring obeying Hooke's law. The key characteristics of simple harmonic motion are that it is periodic, with the displacement following a simple sine curve. The motion can be viewed as the projection of uniform circular motion onto a diameter.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONS
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
(S.h.m & waves).(reflection & refrection).(interferenc) prove s & important ...
1. The document discusses several physics concepts including simple harmonic motion, waves, reflection, refraction, and interference.
2. Key formulas for simple harmonic motion include equations for total energy, angular frequency, and period.
3. Rules for summing multiple simple harmonic motions are described.
4. Formulas are derived for amplitude reflection and transmission coefficients in both transverse magnetic and transverse electric modes.
Physics formulas list
The document provides a list of physics formulas across various topics in mechanics, electricity, thermodynamics, and more. It begins with an introduction on studying physics and understanding concepts through visualization of problems. The bulk of the document then lists key formulas in different areas of physics, providing the formulas and brief explanations. It encourages readers to derive the formulas themselves and find the joy in solving problems independently.
Phys111_lecture14.ppt
This document summarizes a physics lecture on oscillatory motion and periodic motion. It discusses topics like simple harmonic motion using spring-mass systems, the differential equation of motion, energy of SHM, pendulums, and torsional pendulums. It provides recommendations for elective courses in areas like astronomy, biology, and optics based on student interests emerging from Phys 111.
simple harmonic motion
1. simple harmonic motion
and simple pendulum, relation with uniform motion
2. damped harmoic motion
and discuss its three cases
3. driving force
Transfer function and mathematical modeling
Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
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Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
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EMF unit-1 ELECTROSTATICS: Coulomb’s Law, Gauss’s Law
EMF ELECTROSTATICS:
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Engineering Physics
1. Oscillations and waves can be free, damped, or forced. Free oscillations follow the differential equation of motion for simple harmonic motion.
2. Springs can be connected in series or parallel configurations. Springs in series have an equivalent spring constant that is the reciprocal of the sum of the reciprocals of the individual spring constants. Springs in parallel have an equivalent spring constant equal to the sum of the individual spring constants.
3. Complex notation can represent oscillations using a complex number with real and imaginary parts or using polar coordinates with magnitude and phase angle, providing an alternative representation of simple harmonic motion.
Oscillations
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L7%20AC.pdf
1) The document discusses power in AC circuits, including instantaneous power, average power, and RMS power.
2) Instantaneous power is the product of instantaneous voltage and current. Average power is the average value of instantaneous power over one cycle. RMS power is equal to the voltage multiplied by the current.
3) The power factor is defined as the ratio of average power to RMS power and represents the fraction of power that is actually used. It is equal to the cosine of the phase difference between voltage and current.
AC Circuit Power Analysis.pdf
The document discusses power analysis of AC circuits. It defines instantaneous power as the product of instantaneous voltage and current at a point in time. Average power is defined as the average of instantaneous power over one period. Average power is important because power meters measure average power. For a sinusoidal voltage and current, average power is equal to one-half the product of the rms voltage and current multiplied by the cosine of the phase difference between voltage and current. Resistive circuits absorb power continuously, while reactive circuits absorb no average power. The document provides examples of calculating instantaneous and average power in AC circuits.
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Introduction to oscillations and simple harmonic motion
Introduction to oscillations and simple harmonic motion
Periodic Motion KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK.ppt
Periodic Motion KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK.ppt
Damped and undamped motion differential equations.pptx
Damped and undamped motion differential equations.pptx
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONS
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONS
(S.h.m & waves).(reflection & refrection).(interferenc) prove s & important ...
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EMF unit-1 ELECTROSTATICS: Coulomb’s Law, Gauss’s Law
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Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Ordinary Differential Equations And Their Application: Modeling: Free Oscillations Resonance And Electric Circuits
- 1. 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
- 2. Oscillation Of A Spring Concider 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.
- 3. 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 𝒅𝒙 𝒅𝒕 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 𝒎 𝒅 𝟐 𝒙 𝒅𝒕 𝟐 = 𝒎𝒈 − 𝒌 𝒆 + 𝒙 − 𝒄 𝒅𝒙 𝒅𝒕 + 𝒇 𝒕 At The Equilibrium Position B, mg=ke
- 4. Hence, 𝒎 𝒅 𝟐 𝒙 𝒅𝒕 𝟐 = −𝒌𝒙 − 𝒄 𝒅𝒙 𝒅𝒕 + 𝑭 𝒕 𝒅 𝟐 𝒙 𝒅𝒕 𝟐 + 𝒄 𝒎 𝒅𝒙 𝒅𝒕 + 𝒌 𝒎 𝒙 = 𝑭 𝒕 Let 𝒄 𝒎 =2 𝝀 And 𝒌 𝒎 =𝝎 𝟐 𝐝 𝟐 𝐱 𝐝𝐭 𝟐 +2𝛌 𝐝𝐱 𝐝𝐭 + 𝛚 𝟐 x = 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
- 5. 𝐝 𝟐 𝐱 𝐝𝐭 𝟐 + 𝛚 𝟐 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= 𝟐𝝅 𝝎 = 𝟐𝝅 𝒎 𝒌 Frequency = 𝝎 𝟐𝝅 = 𝟏 𝟐𝝅 𝒌 𝒎
- 6. Free Damped Oscillations If The External Force F(t) Is Absent And Damping Is Present Then Eq. Reduces To d2x dt2 +2λ 𝐝𝐱 𝐝𝐭 + 𝛚 𝟐 𝐱 = 𝟎 Forced Undamped Oscillation If An External Periodic Force F(t)= Q𝑪𝒐𝒔 𝒏𝒕 Is Applied To The Support Of The Spring And Damping Force Is Negligible Then Eq. Reduces To d2x 𝐝𝐭 𝟐 + 𝛚 𝟐 x = 𝛠 𝐂𝐨𝐬 𝐧𝐭
- 7. 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= 𝑑𝑄 𝑑𝑡 or Q= ∫I dt Voltage Drop Across Resistance (R)= RI Voltage Drop Across Inductance (L)=L 𝑑𝐼 𝑑𝑡 Voltage Drop Across Capacitance (C)= 𝑄 𝑐 Or 1 𝑐 ∫I dt
- 8. R-L Circuit: The Figure Shows A Simple R-l Circuit Applying Kirchhoff’s Voltage Law To The Circuit, RI + L 𝑑𝐼 𝑑𝑇 = E(t) The Differential Equation Is 𝑑𝐼 𝑑𝑡 + 𝑅 𝐿 I= 𝐸(𝑡) 𝐿 Which Is Linear In I .
- 9. R-C Circuit: The Figure Show A Simple R-C Circuit Applying Kirchoff’s Voltage Law To Circuit RI + 𝑄 𝐶 = E(t) R 𝑑𝑄 𝑑𝑡 + 𝑄 𝐶 = E(t) (I= 𝑑𝑄 𝑑𝑡 ) 𝑑𝑄 𝑑𝑡 + 1 𝑅𝐶 Q = 1 𝑅 E(T) which is linear in Q
- 10. 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 + 𝑸 𝑪 = E(t) R 𝒅𝑸 𝒅𝒕 + 𝑸 𝑪 = E(t) (I= 𝒅𝑸 𝒅𝒕 ) 𝒅𝑸 𝒅𝒕 + 𝟏 𝑹𝑪 Q = 𝟏 𝑹 E(T) which is linear in Q
- 11. Comparing With 𝒅𝑸 𝒅𝒕 + P(t)Q =Q(t) P(t)= 𝟏 𝑹𝑪 , Q(t)= 𝑬 𝑹 I.F=e ⌠ P(t) dt =e ⌠ 𝟏 𝑹𝑪 dt =e 𝒕 𝑹𝑪 Hence, Solution Is Q(I.F)= ⌠Q(t) (I.F) dt +𝒄 𝟏 Q= ⌠ E/R et/Rc dt+𝒄
- 12. =E/R ⌠ Et/Rc dt+𝒄 𝟏 =E/R (E t/Rc/1/RC) +𝒄 𝟏 =Ece T/Rc+𝒄 𝟏 =Ec +𝒄 𝟏 e T/Rc At t=0, Q=0 0=EC+𝒄 𝟏 𝒄 𝟏= -EC
- 13. hence, Q= EC(1-e -t/RC ) now, I= 𝒅𝑸 𝒅𝒕 = 𝒅 𝒅𝒕 EC(1-e -t/RC ) =EC(0- E -t/RC (-1/RC)) =E/R e -t/RC