Oscillations & Waves Notes -New Syllabus PDF

Oscillations and Waves Introduction to oscillations and vibrations – Damped harmonic motion – Under damped – Critically damped – Over damped cases – Forced oscillations – Amplitude – Quality factor at resonance – Engineering applications – Automotive suspension systems – Shock absorbers – Door close...

Oscillations and Waves Introduction to oscillations and vibrations – Damped harmonic motion – Under damped – Critically damped – Over damped cases – Forced oscillations – Amplitude – Quality factor at resonance – Engineering applications – Automotive suspension systems – Shock absorbers – Door closers – LCR Circuit.Wave motion – One dimensional wave equation – Solution (Equation only) –Distinction between transverse and longitudinal waves – Transverse vibrations in a stretched string – Engineering applications – Musical string instruments. Introduction to oscillations and vibrations Periodic Motion: A motion that repeats at regular intervals of time. Examples: Spinning of earth about its own axis Revolution of earth around the sun Oscillations of a pendulum Vibrations of a tuning fork Types of periodic motion Rotatory motion: -particle completes the rotation in regular intervals. Examples: - Rotation of earth around the sun Rotatory motion of hour hand, minute hand etc. Oscillatory motion: - the particle moves to and fro with less frequency. Examples: - Bob of a pendulum Movement of swing Vibratory motion: - the particle moves to and fro with large frequency. Examples: - The particle on a vibrating string Vibrations of atoms in a solid Period (T) The time taken to repeat a periodic motion is called the period (T). Its SI unit is second. The period of vibrations of a quartz crystal is expressed in units of microseconds (10-6 s) abbreviated as μs. Frequency (ν) The number of repetitions in one second of a periodic motion is called Frequency (ν). Its unit is Hertz (Hz). 1 The relation between v and T is, 𝜗= 𝑇 Relation connecting period (T), angular velocity (ω) and Frequency (ν) The angular velocity, 𝜔 = 2𝜋𝜗, (angular frequency is the magnitude of the vector quantity angular velocity) 1 APG  𝜔 Or =𝜗 2𝜋 The period T = (1/ ν) = 2 π/ ω Oscillatory motions are periodic and bounded to the mean position. The displacement of the particles executing oscillatory motion that can be expressed in terms of sine and cosine functions are known as harmonic motion. The simplest type of harmonic motion is known as simple harmonic motion (SHM) Simple Harmonic Motion (SHM) In SHM the restoring force on the oscillating body is directly proportional to its displacement from the mean position, and is directed opposite to the displacement. Eg: small oscillations of simple pendulum, swing, loaded spring, etc. According to the statement of SHM, 𝑭 ∝ −𝒙 𝑭 = −𝒌𝒙 Where 𝒌 is the proportionality constant called Spring Constant. Applying Newton’s second law of motion, we get 𝑑2𝑥 𝑚 2 = −𝑘𝑥 𝑑𝑡 2 𝑑 𝑥 𝑘 2 =− 𝑥 𝑑𝑡 𝑚 or 𝑑2𝑥 𝑘 = − 𝑥 𝑑𝑡 2 𝑚 𝑑2𝑥 = −𝜔2 𝑥 𝑑𝑡 2 𝒌 Where 𝝎𝟐 = 𝒎 𝒅𝟐 𝒙 + 𝝎𝟐 𝒙 = 𝟎 𝒅𝒕𝟐 This is the differential equation of motion of a particle executing SHM. Solution of the equation is given by x(t) = A sin (ωt + φ) Amplitude(A): It is the magnitude of the maximum displacement of the oscillating particle Phase: The time varying quantity, (ωt + φ), is called the phase of the motion. Phase describes the state of motion at a given time. 2 APG  Phase constant (or phase angle): The constant φ is called the phase constant. The value of φ depends on the displacement and velocity of the particle at t = 0. Free oscillations are oscillations that appear in a system as a result of a single initial deviation of the system from its state of stable equilibrium. The free oscillation possesses constant amplitude and period without any external force to set the oscillation. The frequency with which the system oscillates freely at its own is called its natural frequency. Ideally, free oscillation does not undergo damping. But in all-natural systems damping is observed unless and until any constant external force is supplied to overcome damping. I hr Damped harmonic motion The motion in which the amplitude of oscillation gradually decreases and finally dies out due to dissipative forces like friction, viscosity etc. is called damped harmonic motion. Let us consider a simple harmonic oscillator system damped by viscous damping forces. The damping force is proportional to the velocity of the system. Thus, the damping force is given 𝑑𝑥 by 𝑓𝑑 = −𝑏 𝑑𝑡 Here b is a constant that depends on the medium and the shape of the body. Thus, the equation for the damped simple harmonic oscillator could be obtained by adding the damping force term to the Hooke’s law. 𝑑2𝑥 𝑑𝑥 𝑚 2 = −𝑘𝑥 − 𝑏 𝑑𝑡 𝑑𝑡 𝑑2𝑥 𝑑𝑥 𝑚 2 +𝑏 + 𝑘𝑥 = 0 𝑑𝑡 𝑑𝑡 or 𝑑2 𝑥 𝑏 𝑑𝑥 𝑘 + + 𝑥=0 𝑑𝑡 2 𝑚 𝑑𝑡 𝑚 𝑑2 𝑥 𝑑𝑥 2 + 2𝜆 + 𝜔2 𝑥 = 0 𝑑𝑡 𝑑𝑡 This is the differential equation of a DHO. Let the solution of the equation is of the form 𝑥 = 𝐴𝑒 ∝𝑡 Where α is the root of the equation. 3 APG Find α: Let us consider 𝑑2 𝑑 2 = 𝐷2 , = 𝐷 𝑑𝑡 𝑑𝑡 Then the differential equation becomes 2 𝐷2 𝑥 + 2𝜆𝐷𝑥 + 𝜔⬚ 𝑥=0 2 (𝐷2 + 2𝜆𝐷 + 𝜔⬚ )𝑥 = 0 2 Since x is never zero; the polynomial 𝐷2 + 2𝜆𝐷 + 𝜔⬚ =0 The roots of the above equation could be written as 2 −𝜆 ± √𝜆2 − 𝜔⬚ or 2 𝛼1 = −𝜆 + √𝜆2 − 𝜔⬚ 2 𝛼2 = −𝜆 − √𝜆2 − 𝜔⬚ α = −λ ± √λ2 − ω2 Hence the two roots are −𝜆 ± √𝜆2 − 𝜔 2 i.e , ∝1 = −𝜆 + √𝜆2 − 𝜔 2 and ∝2 = −𝜆 − √𝜆2 − 𝜔 2 Thus, the general solution could be written as √𝜆2 −𝝎𝟐 )𝑡 √𝜆2 −𝝎𝟐 )𝑡 𝑥 = 𝐴1 𝑒 (−𝜆+ + 𝐴2 𝑒 (−𝜆− ------------------------------(1) Where 𝐴1 𝑎𝑛𝑑 𝐴2 are constants, whose value depends on the initial conditions of motion. Now let us discuss the nature of the solution for different cases. 4 APG Case 1: Under damped case (𝝀 < 𝝎) This motion is oscillatory. The amplitude decreases exponentially with time. And damping produces two effects, increases the time period and decreases the amplitude. Case 2: Critically damped case (𝝀 = 𝝎) As time elapses, the displacement returns exponentially from the maximum value to zero. Oscillatory motion will not occur. Such a motion is called critically damped or aperiodic. Case 3: Overdamped Case (𝝀 > 𝝎) Amplitude decreases exponentially with time and motion is non-oscillatory. Such a motion is called dead-beat or aperiodic. 5 APG In under damped case the system oscillates and the amplitude gradually decreases to zero. In over damped case the system returns (exponential decay) to equilibrium without oscillating. In critically damped case the system returns to equilibrium as quickly as possible. Practical Applications 1. Underdamped Motion: Automotive Suspension Systems: When a car encounters a bump on the road, the suspension system undergoes underdamped oscillations before settling down. Engineers design the suspension system to provide a smooth ride by controlling the damping to prevent excessive bouncing. Control Systems: In control systems, underdamped responses can occur when a system is tuned for quick response but without excessive oscillations. This can be seen in systems like robotic arms or aircraft control surfaces. 2. Overdamped Motion: Door Closers: Overdamped systems are used in door closers to ensure that the door closes smoothly without oscillating back and forth. Braking Systems: Overdamped behaviour is desirable in brake systems to ensure that the vehicle comes to a smooth stop without oscillations. 3. Critically Damped Motion: Shock Absorbers: Critically damped systems are employed in shock absorbers to provide the quickest damping without overshooting or oscillating. 6 APG  Seismometers: In seismometers, critically damped systems are used to measure seismic waves accurately without oscillating excessively, providing precise data about earthquakes. II hr Forced oscillations If an external periodic force is applied on a damped harmonic oscillator, then the oscillating system is called driven or forced harmonic oscillator and the oscillations are called forced oscillations. The forces that act on the body undergoing forced oscillations are 1. A restoring force F1, that is proportional to the displacement and oppositely directed(−𝑘𝑥), 2. A damping force F2, that is proportional to the velocity but oppositely directed (−𝑏𝑣), and 3. A driving force (external periodic force), F3 = 𝐹0 𝑠𝑖𝑛𝜔"𝑡 4.. The total force acting on the body is given by F = F1 + F2 + F3. 𝐹 = −𝑘𝑥 − 𝑏𝑣 + 𝐹0 𝑠𝑖𝑛𝜔"𝑡 Hence, the differential equation becomes 𝑑2𝑥 𝑑𝑥 𝑚 2 +𝑏 + 𝑘𝑥 = 𝐹0 𝑠𝑖𝑛𝜔"𝑡 𝑑𝑡 𝑑𝑡 𝑑2 𝑥 𝑏 𝑑𝑥 𝑘 𝐹0 2 + + 𝑥 = 𝑠𝑖𝑛𝜔"𝑡 𝑑𝑡 𝑚 𝑑𝑡 𝑚 𝑚 𝑑2 𝑥 𝑑𝑥 + 2𝜆 + 𝜔2 𝑥 = 𝑓0 𝑠𝑖𝑛𝜔"𝑡 -----------------(1) 𝑑𝑡 2 𝑑𝑡 𝐹 𝑏 𝑘 Where 𝑓0 = 𝑚0 , 2𝜆 = 𝑚 and 𝜔2 =𝑚 When the external periodic force is acting, first the body will try to vibrate in its natural frequency, but very soon it reaches a steady state and vibrate with the frequency of the external force. The oscillations that result are called forced oscillations. When external periodic force is applied, both the damping and forced terms contribute to the motion of the oscillator and a tussle occurs between the contributions of both. After some initial erratic situations, ultimately, the system reaches a steady state 7 APG  Let us assume the solution of steady state as x = Asin(ω" t − θ) Then, the value of A will be 𝑓0 𝐴= √[ (𝜔 2 − 𝜔 "2 ) 2 + 4𝜆²𝜔"² If 𝜔 = 𝜔“, A= Amax 𝑓0 𝐴𝑚𝑎𝑥 = 2𝜆𝜔 If the driving frequency 𝝎" approaches the natural frequency ω, the amplitude increases and reaches the maximum value when both are nearly equal. This phenomenon is called resonance. In other words, the phenomenon of making a body oscillate with its natural frequency under the action of an external force with same frequency is called resonance. This phenomenon in which the amplitude of the driven oscillator becomes maximum at a particular frequency is called amplitude resonance and this frequency is called resonant frequency. Eg. Tuning a radio, tuning a musical instrument. Practical Applications Bridges and Buildings: Engineering structures like bridges and buildings are subjected to external forces such as wind or earthquakes. Understanding forced oscillations helps engineers design these structures to withstand and dissipate these forces efficiently. Electrical Circuits: Forced oscillations are also prevalent in electrical circuits, such as alternating current (AC) circuits, where the current oscillates due to an applied voltage. 8 APG Quality Factor (Q-factor) at Resonance Q factor for a Forced Harmonic Oscillator (Q factor at resonance) Quality factor at resonance is defined as the ratio of amplitude at resonance to the amplitude at zero driving frequency. 𝑓0 𝐴𝑚𝑎𝑥 2𝜆𝜔 𝜔 Q factor = = = = 𝜔𝜏 𝐴𝑎𝑡𝜔′′ 𝑓0 2𝜆 𝜔2 1 where τ = 2𝜆, is the relaxation time. √𝑘 𝑚 √𝑘𝑚 Q factor = = 𝑏 𝑏 𝑚 III hr Comparison of electrical and mechanical oscillators Consider an LCR circuit. It is an example for forced harmonic oscillator in which the oscillations are sustained by an alternating emf. Applied voltage V=V0 sin ωt. 𝑞 Potential difference across its plates is = 𝐶 𝑑2 𝑞 𝑑𝐼 Self-induced emf in the inductance is = 𝐿 𝑑𝑡 2 or 𝐿 𝑑𝑡 𝑑𝑞 Voltage across resistance = 𝑅 𝑑𝑡 According to Kirchhoff’s voltage law VL + VC+ VR = V 𝑑2𝑞 𝑑𝑞 𝑞 𝐿 2 +𝑅 + = 𝑉0 sin 𝜔𝑡 𝑑𝑡 𝑑𝑡 𝐶 𝑑2 𝑞 𝑅 𝑑𝑞 𝑞 𝑉0 2 + + = sin 𝜔𝑡 𝑑𝑡 𝐿 𝑑𝑡 𝐿𝐶 𝐿 Mechanical Oscillator Electrical Oscillator 𝑑2𝑥 𝑑𝑥 𝐿 𝑑2 𝑞 𝑑𝑞 1 + 𝑅 𝑑𝑡 + 𝐶 𝑞 = 𝑉0 𝑠𝑖𝑛𝜔"𝑡 2 + 2𝜆 + 𝜔2 𝑥 = 𝑓0 𝑠𝑖𝑛𝜔"𝑡 𝑑𝑡 2 𝑑𝑡 𝑑𝑡 9 APG  Mass( m) Inductance (L) Displacement (x) Charge (q) Velocity (V) Current (i) Damping coefficient (b) Electrical resistance (R) 𝑅 2𝜆= b/m 𝐿 𝐾 1 𝐾 1 1 1 𝜔 = √𝑚 , f = 2𝜋 √𝑚 𝜔 = √𝐿𝐶 , f = 2𝜋 √𝐿𝐶 𝜔 Q-factor = 2𝜆 𝐿 1 1 𝐿 Q= √ = √ 𝑅 𝐿𝐶 𝑅 𝐶 1 𝐿 𝜏 = 2𝜆 𝜏=𝑅 III hr Tutorials Waves Types of waves: (i) Mechanical waves (ii) electromagnetic waves and (iii) matter waves 10 APG Types of wave motions: (i) Transverse wave motion It is the wave motion in which every particle of the medium vibrates harmonically about its mean position in a direction perpendicular to the direction of propagation of wave. Eg. Light waves (ii) Longitudinal wave motion It is the wave motion in which every particle of the medium vibrates harmonically about its mean position in a direction parallel to the direction of propagation of wave. Eg. Sound waves. Transverse Waves Longitudinal Wave 1. In transverse waves, the particles of the In longitudinal waves, the particles of the medium vibrate at right angles to the medium vibrate parallel to the direction of direction of wave propagation. wave propagation. 2. It consists of crests and troughs It consists of compressions and rarefactions. 3. There are no pressure variations There is a pressure variation throughout the medium 4. There is no change in density of medium There is a change in the density throughout the medium 5. Example - Light waves Example - Sound wave Periodic wave –displacement equation 2𝜋 𝑢 (𝑥, 𝑡) = 𝐴 sin (𝑥 − v𝑡) 𝜆 𝑥 𝑣 or 𝑢 (𝑥, 𝑡) = 𝐴 sin 2𝜋 ( 𝜆 − 𝜆 𝑡) 𝑥 𝑢 (𝑥, 𝑡) = 𝐴 sin 2𝜋 ( − 𝜈𝑡 ) 𝜆 2𝜋 𝑢 (𝑥, 𝑡) = 𝐴 sin ( 𝑥 − 2𝜋𝜈𝑡 ) 𝜆 2𝜋 𝑢 (𝑥, 𝑡) = 𝐴 sin ( 𝑘𝑥 − 𝜔𝑡 ), where k= 𝜆 , the wave vector 11 APG  𝜔 − 2𝜋𝜈, 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 Wave equation in one dimension 2𝜋 Consider 𝑢 (𝑥, 𝑡) = 𝐴 sin (𝑥 − v𝑡) (1) 𝜆 Differentiating eqn (1) wrt x 𝜕𝑢 2𝜋 2𝜋 = 𝐴 cos (𝑥 − v𝑡) × (2) 𝜕𝑥 𝜆 𝜆 Also, 𝜕𝑢 2𝜋 −2𝜋v = 𝐴 cos (𝑥 − v𝑡) × (3) 𝜕𝑡 𝜆 𝜆 From (2) and (3) 𝜕𝑢 1 𝜕𝑢 =− (4) 𝜕𝑥 v 𝜕𝑡 Differentiating eqn (4) wrt x 𝜕 𝜕𝑢 1 𝜕 𝜕𝑢 ( )=− ( ) (5) 𝜕𝑥 𝜕𝑥 v 𝜕𝑥 𝜕𝑡 Changing the order of integration in RHS of eqn (5) 𝜕 𝜕𝑢 1 𝜕 𝜕𝑢 ( )=− ( ) (6) 𝜕𝑥 𝜕𝑥 v 𝜕𝑡 𝜕𝑥 Substituting eqn (4) in eqn (6) 𝜕 𝜕𝑢 1 𝜕 1 𝜕𝑢 ( )=− (− ) (7) 𝜕𝑥 𝜕𝑥 v 𝜕𝑡 v 𝜕𝑡 Therefore 𝜕 2𝑢 1 𝜕 2𝑢 = (8) 𝜕𝑥 2 v 2 𝜕𝑡 2 This is the one dimensional wave equation. Solution: u(x, t) = Ae±ikx e±iωt V hr (Derivation of Solution Available in Appendix 1) 12 APG Transverse vibration of a stretched string Consider a perfectly flexible inextensible uniformly stretched string by a constant tension T. Let μ be the mass per unit length of the wire Let us assume that the undisturbed portion of the string is in the x axis. And the motion is along the x-y plane. Consider the motion of an element AB of length dx as shown in figure. θ1 and θ2 are the angles the tension T makes with A and B at the axis, respectively. Since the string is vibrating transversely, we are considering only the transverse components of tension, that is T sin θ1 and T sin θ2. The resultant force Fy acting transversely is given by 𝐹𝑦 = 𝑇 sin 𝜃1 − 𝑇 sin 𝜃2 (1) When θ is very small sin 𝜃 = 𝜃 = tan 𝜃 𝐹𝑦 = 𝑇 tan 𝜃1 − 𝑇 tan 𝜃2 (2) 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝜕𝑦 tan 𝜃 = = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝜕𝑥 Equation (2) becomes 𝜕𝑦 𝜕𝑦 𝐹𝑦 = 𝑇 ( ) − 𝑇 ( ) 𝜕𝑥 𝐵 𝜕𝑥 𝐴 𝜕𝑦 𝜕𝑦 𝐹𝑦 = 𝑇 (( ) −( ) ) (3) 𝜕𝑥 𝐵 𝜕𝑥 𝐴 Using Taylor series expansion the term inside the bracket of eqn (3) becomes 13 APG  𝜕𝑦 𝜕𝑦 𝜕 2𝑦 ( ) −( ) = 𝑑𝑥 (4) 𝜕𝑥 𝐵 𝜕𝑥 𝐴 𝜕𝑥 2 Substituting (4) in (3) 𝜕 2𝑦 𝐹𝑦 = 𝑇 ( 2 𝑑𝑥) (5) 𝜕𝑥 According to Newton’s second Law of motion 𝜕 2𝑦 𝐹𝑦 = 𝑚𝑎 = 𝜇𝑑𝑥 𝜕𝑡 2 Submitting (5) to LHS of above equation 𝜕 2𝑦 𝜕 2𝑦 𝑇 ( 2 𝑑𝑥) = 𝜇𝑑𝑥 2 𝜕𝑥 𝜕𝑡 𝜕 2𝑦 𝜇 𝜕 2𝑦 = (6) 𝜕𝑥 2 𝑇 𝜕𝑡 2 Comparing (6) with wave equation 𝜕 2𝑦 1 𝜕 2𝑢 = (7) 𝜕𝑥 2 v 2 𝜕𝑡 2 1 𝜇 2 = v 𝑇 𝑇 v= √ (8) 𝜇 This is equation for velocity of transverse vibrations in a stretched string. If  is the frequency, λ is the wavelength and v is the velocity of the wave then, Velocity is given by, v = 𝜈𝜆 (9) Substituting eqn (9) in (8) 𝑇 𝜆 = √ 𝜇 14 APG Therefore the frequency 1 𝑇 = 𝜆 √𝜇 (10) This is the equation for the vibrations in a stretched string in terms of frequency. Applications String Instruments: Instruments like guitars, violins, pianos, and harps rely on transverse vibrations in stretched strings to produce sound. The pitch and timbre of the notes produced depend on factors such as the tension, length, and material of the string. VI hr Tutorials Appendix I Stretched String Frequency Let l be the length of the string and let the string vibrate in p segments. Then the length of one segment is l/p. For transverse waves, the length of each segment is λ/2. Hence, 𝑙 𝜆 = 𝑝 2 2𝑙 𝜆= (11) 𝑝 Substituting (11) in (10) 𝑝 𝑇 = √ (12) 2𝑙 𝜇 If the string is vibrating in one segment then p =1 1 𝑇 1 = √ 2𝑙 𝜇 15 APG Where 1 is the fundamental frequency of vibration and the vibration is called fundamental mode of vibration. If the string is vibrating in two segments then p = 2, Then, 2 𝑇 2 = √ = 21 2𝑙 𝜇 Where 2 is called the first over tone and the vibration is known as second mode of vibration. When p = 3 3 𝑇 3 = √ = 31 2𝑙 𝜇 3 is the second over tone and the vibration is called the third mode of vibration. In general, 𝑛 = 𝑝1 16 APG