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Physics presentation in wave an oscillation
- 1. PHY-401 Waves and Oscillations BSPhysics 4th Semester Dr. Ejaz Ahmed Department of Physics Abdul Wali Khan University Mardan
- 2. Lecture 7: Topicsto be Covered 1. Damped Harmonic Motion 2. Equation of Damped Harmonic Motion
- 3. Up to thispoint we have assumed that no frictional forces act on the oscillator. If this assumption held strictly, a pendulum or a mass on a spring would oscillate indefinitely with a constant mechanical energy (that is, with no loss in the amplitude of the oscillation). Damped Harmonic Motion This loss in amplitude is called damping and the motion is called damped harmonic motion. There are many causes of damping, including friction, air resistance, and internal forces. Since we observe a loss in amplitude for real oscillators, we know that this assumption is not strictly true, although it may be a good approximation for some oscillators.
- 4. Damped Harmonic Motion Inthe given figure, we compares the motion of undamped and damped oscillators. When we add a small damping force, the frequency changes by a negligible amount but the amplitude gradually decreases to zero. In many cases this decrease in amplitude can be accounted for by multiplying the equation for the undamped oscillator by an exponential function that describes the dashed curves in given figure. where 𝝉, is called the damping time constant or the mean lifetime of the oscillation. Mathematically, it is the time necessary for the amplitude to drop to Τ 𝟏 𝐞 = 𝟎. 𝟑𝟔𝟖 of its initial value, as shown in figure (b). 𝒙 𝒕 = 𝒙𝒎𝒆− ൗ 𝒕 𝝉𝒄𝒐𝒔 𝝎𝒕 + 𝝓 ----- (1) Figure: (a) Undamped oscillation, drawn for a phase constant 𝝓 of zero. (b) Damped oscillation with the same frequency as (a).
- 5. Damped Harmonic Motion Tosolve the equation of motion of this system, we have 𝑭𝒙 = 𝒎𝒂 = −𝒌𝒙 − 𝒃𝒗 Figure: A representation of a damped harmonic oscillator. We consider the oscillating body (of mass m) to be attached to a (massless) vane immersed in a fluid, in which it experiences a viscous damping force −𝒃𝒗𝒙. We do not consider sliding friction at the horizontal surface. One common type of retarding force is the one, where the force is proportional to the speed of the moving object and acts in the direction opposite the motion. This retarding force is often observed when an object moves through air. Because the retarding force can be expressed as 𝑹 = −𝜷𝒗. Where 𝜷 is a constant called the damping coefficient or damping constant that depends on the properties of the fluid and the size and shape of the vane that is immersed in the fluid and the restoring force of the system is −𝒌𝒙. 𝒎 ሷ 𝒙 + 𝜷 ሶ 𝒙 + 𝒌𝒙 = 𝟎
- 6. Solution of DampedHarmonic Oscillator Equation Dividing both sides by mass 𝒎 ሷ 𝒙 + 𝜷 ሶ 𝒙 + 𝒌𝒙 = 𝟎 ሷ 𝒙 + 𝜷 𝒎 ሶ 𝒙 + 𝒌 𝒎 𝒙 = 𝟎 Where, Ԧ 𝐹 = 𝑚 Ԧ 𝑎 Ԧ 𝐹𝑆 𝑛𝑒𝑡 + Ԧ 𝐹𝑑𝑎𝑚𝑝 = 𝑚 Ԧ 𝑎 Ԧ 𝐹𝑆 + Ԧ 𝐹𝑑 = 𝑚 Ԧ 𝑎 𝑭𝒅 = −𝜷 𝒗 𝑭𝑺 = −𝒌𝒙 −𝑘𝑥 + −𝛽𝑣 = 𝑚 Ԧ 𝑎 𝝎 = 𝒌 𝒎 𝜸 = 𝜷 𝟐𝒎 and and ሷ 𝒙 + 𝟐𝜸 ሶ 𝒙 + 𝝎𝟐 𝒌 𝒎 𝒙 = 𝟎 𝑭𝒅 𝑭𝑺
- 7. Solution of DampedHarmonic Oscillator Equation This is a characteristic differential equation, the solution of which is ሷ 𝒙 + 𝟐𝜸 ሶ 𝒙 + 𝝎𝟐 𝒌 𝒎 𝒙 = 𝟎 𝒙 = 𝒆𝝀𝒕 ሶ 𝒙 = 𝝀 𝒆𝝀𝒕 ሷ 𝒙 = 𝝀𝟐 𝒆𝝀𝒕 𝝀𝟐 𝒆𝝀𝒕 + 𝟐𝜸𝝀 𝒆𝝀𝒕 + 𝝎𝟐 𝒆𝝀𝒕 = 𝟎 𝝀𝟐 + 𝟐𝜸 𝝀 + 𝝎𝟐 = 𝟎 This is a quadratic equation, and its solution is 𝜆± = −2𝛾 ± 4𝛾2 − 4𝜔2 2 1
- 8. Solution of DampedHarmonic Oscillator Equation The quadratic equation has two roots 𝜆± = −2𝛾 ± 4𝛾2 − 4𝜔2 2 1 𝜆± = −𝛾 ± 𝛾2 − 𝜔2 𝜆± = −𝛾 ± 𝑖 𝜔2 − 𝛾2 𝑭𝒅 𝑭𝑺 𝝀+ = −𝜸 + 𝒊 𝝎𝟐 − 𝜸𝟐 𝝀− = −𝜸 − 𝒊 𝝎𝟐 − 𝜸𝟐 and For differential equations, if we have two or more than two solutions than their linear combination is its general solution 𝒙 = 𝒆𝝀𝒕 𝒙 = 𝒂 𝒆 −𝜸+𝒊 𝝎𝟐−𝜸𝟐 𝒕 + 𝒃 𝒆 −𝜸−𝒊 𝝎𝟐−𝜸𝟐 𝒕 Where a and b are constants, and its values can be calculated with initial conditions
- 9. Solution of DampedHarmonic Oscillator Equation We can take out this general solution 𝑥 = 𝑎 𝑒 −𝛾+𝑖 𝜔2−𝛾2 𝑡 + 𝑏 𝑒 −𝛾−𝑖 𝜔2−𝛾2 𝑡 Define the damping frequency Factor out 𝒆−𝜸𝒕 𝑥 = 𝑒−𝛾𝑡 𝑎 𝑒 𝑖 𝜔2−𝛾2 𝑡 + 𝑏 𝑒 −𝑖 𝜔2−𝛾2 𝑡 Combine exponential terms and the Amplitude A results, using the relation 𝑒𝑖𝜃 = 𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃 𝑥 𝑡 = 𝐴 𝑒−𝛾𝑡 cos 𝜔2 − 𝛾2 𝑡 + 𝜙 𝝎′ = 𝝎𝟐 − 𝜸𝟐 𝒙 𝒕 = 𝑨 𝒆−𝜸𝒕 𝐜𝐨𝐬 𝝎′ 𝒕 + 𝝓
- 10. Damped Harmonic Motion Thesolution to this equation, which you can verify by direct substitution is 𝒎 ሷ 𝒙 + 𝒃 ሶ 𝒙 + 𝒌𝒙 = 𝟎 𝒙 𝒕 = 𝒙𝒎 𝒆− ൗ 𝒃𝒕 𝟐𝒎 𝒄𝒐𝒔 𝝎′ 𝒕 + 𝝓 where 𝝎′ = 𝒌 𝒎 − 𝒃 𝟐𝒎 𝟐 This solution assumes that the damping constant is small, so that the quantity under the square root of 𝝎′ cannot be negative. ----- (2) 𝒙 𝒕 = 𝒙𝒎 𝒆− ൗ 𝒕 𝝉 𝒄𝒐𝒔 𝝎𝒕 + 𝝓 ----- (1) Note that Eq. (2) has the same form as Eq. (1), with the lifetime , 𝝉 = Τ 𝟐𝒎 𝒃. ➢ The greater is the damping constant b, the more quickly the amplitude of the oscillation dies out. ➢ As b approaches zero (corresponding to no damping), then 𝝉, is infinite and the amplitude remains constant.
- 11. When damping ispresent, the oscillation frequency is smaller (the period is larger). That is, damping slows down the motion, as we might expect. If 𝒃 = 𝟎 (no damping, then 𝝎′ = 𝒌 𝒎 , which is simply the angular frequency 𝝎𝒐 of the undamped motion. Damped Harmonic Motion When damping is present, 𝝎′ is slightly less than 𝝎𝒐, but in most cases of interest, the damping is sufficiently weak that 𝝎′ ≈ 𝝎𝒐. In the special case in which 𝒃 = 𝟐 𝒌𝒎 in the equation 𝝎′= 𝒌 𝒎 − 𝒃 𝟐𝒎 𝟐 gives 𝝎′ = 𝟎, so the motion decays exponentially to zero with no oscillation at all. In this case the lifetime 𝝉 has its smallest possible value, Τ 𝟏 𝝎 . This condition, called critical damping, is often the goal of mechanical engineers in designing systems in which unwanted and often harmful oscillations can be made to disappear in the shortest possible time. 𝒙 𝒕 = 𝒙𝒎 𝒆− ൗ 𝒕 𝝉 𝒄𝒐𝒔 𝝎𝒕 + 𝝓
- 12. Damped Harmonic Motion Figure:Graphs of displacement versus time for (a) an underdamped oscillator, (b) a critically damped oscillator, and (c) an overdamped oscillator. 𝝎′ = 𝒌 𝒎 − 𝒃 𝟐𝒎 𝟐 𝝎′ = 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 We have three possibilities; 𝒄 . 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 < 𝟎 𝒂 . 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 > 𝟎 𝒃 . 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 = 𝟎 Critical Damped Over Damped Under Damped
- 13. A simple pendulumthat comes to a steady state (equilibrium point) after some finite time. The oscillation of the same pendulum in vacuum becomes an undamped system since the system keeps on oscillating between the two maxima points and does not settle. Car shocks are a prime location to start. Un-damped, the damper is broken. You go over a bump and you bounce 8 or more times as you go down the road almost loose control have stop till the bouncing stops. 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 = 𝟎 Under Damped 𝒂 .
- 14. Critically damped: Ah,just right. You hit the bump and you barely feel it. The shock absorbs all the impact and then extends back out just right 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 > 𝟎 Critical Damped 𝒃 .
- 15. Automatic Door CloseSystem Over Damped 𝒄 . 𝝎𝒐 𝟐 − 𝒃 𝟐𝒎 𝟐 < 𝟎
- 16. Video Lectures toWatch 1. https://www.youtube.com/watch?v=qgTQO2LqxaM (Introduction to Exponential Decay (Damped Oscillations) 2. https://www.youtube.com/watch?v=UtkwsWZnp5o (Solution of Damped Harmonic Oscillators equation) 3. https://www.youtube.com/watch?v=qxDvW8_fm7I (Damping and Damped Harmonic Motion) 4. https://www.youtube.com/watch?v=H6wAgtvX_8w (Damping of Simple Harmonic Motion)