1 / 19

Damped Oscillations

Damped Oscillations . (Serway 15.6-15.7). Simple Pendulum. Recall, for a simple pendulum we have the following equation of motion:. L θ T . Which give us:. -------------------------------------------------------------------------. mg. Hence:. or: .

keenan
Download Presentation

Damped Oscillations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Damped Oscillations (Serway 15.6-15.7) Physics 1D03 - Lecture 35

  2. Simple Pendulum Recall, for a simple pendulum we have the following equation of motion: L θ T Which give us: ------------------------------------------------------------------------- mg Hence: or: Application - measuring height - finding variations in g→ underground resources Physics 1D03 - Lecture 35

  3. x t x t SHM and Damping SHM: x(t) = A cosωtMotion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time. Physics 1D03 - Lecture 35

  4. x t A damped oscillator has external nonconservative force(s) acting on the system. A common example is a force that is proportional to the velocity. f= -bv wherebis a constant damping coefficient F=ma give: eg: green water(weak damping) For weak damping (small b), the solution is: A e-(b/2m)t Physics 1D03 - Lecture 35

  5. Without damping: the angular frequency is With damping: Effectively, the frequency ω is slower with damping, and the amplitude gets smaller (decays exponentially) as time goes on: Physics 1D03 - Lecture 35

  6. Example: A mass on a spring oscillates with initial amplitude 10 cm. After 10 seconds, the amplitude is 5 cm. Question: What is the value of b/(2m)? Question: What is the amplitude after 30 seconds? Physics 1D03 - Lecture 35

  7. Example: A pendulum of length 1.0 m has an initial amplitude 10°, but after a time of1000 s it is reduced to 5°. What is b/(2m)? Physics 1D03 - Lecture 35

  8. Types of Damping Since: water We can have different cases for the value under the root: >0, 0 or <0. This leads to three types of damping ! Eg: Strong damping (b large): there is no oscillation when: axle grease? Physics 1D03 - Lecture 35

  9. : “Underdamped”, oscillations with decreasing amplitude : “Critically damped” : “Overdamped”, no oscillation x(t) overdamped critical damping Critical damping provides the fastest dissipation of energy. t underdamped Physics 1D03 - Lecture 35

  10. Quiz: If you were designing a automobile suspensions: • Should they be: • Underdamped • Critically damped • Overdamped • I don’t need suspension ! Physics 1D03 - Lecture 35

  11. Example: Show that the time rate of change of mechanical energy for a damped oscillator is given by: dE/dt=-bv2 and hence is always negative. Physics 1D03 - Lecture 35

  12. Damped Oscillations and Resonance (Serway 15.6-15.7) • Forced Oscillations • Resonance Physics 1D03 - Lecture 35

  13. Forced Oscillations A periodic, external force pushes on the mass (in addition to the spring and damping): This transfers energy into the system The frequency w is set by the machine applying the force. The system responds by oscillating at the same frequencyw. The amplitude can be very large if the external driving frequency is close to the “natural” frequency of the oscillator. Physics 1D03 - Lecture 35

  14. is called the natural frequency or resonant frequency of the oscillation. Newton’s 2nd Law: Assumethat : x = A cos(wt + f) ; then the amplitude of a drive oscillator is given by: Physics 1D03 - Lecture 35

  15. So, as the amplitude, A, increases ! If the external “push” has the same frequency as the resonant frequencyω0. The driving force is said to be in resonance with the system. A Physics 1D03 - Lecture 35

  16. Resonance occurs because the driving force changes direction at just the same rate as the “natural” oscillation would reverse direction, so the driving force reinforces the natural oscillation on every cycle. Quiz: Where in the cycle should the driving force be at its maximum value for maximum average power? • When the mass is at maximum x (displacement) • When the mass is at the midpoint (x = 0) • It matters not Physics 1D03 - Lecture 35

  17. Example • A 2.0 kg object attached to a spring moves without • friction and is driven by an external force given by: • F=(3.0N)sin(8πt) • If the force constant of the spring is 20.0N/m, determine • the period of the motion • the amplitude of the motion Physics 1D03 - Lecture 35

  18. Physics 1D03 - Lecture 35

  19. Summary • An oscillator driven by an external periodic force will oscillate with an amplitude that depends on the driving frequency. The amplitude is large when the driving frequency is close to the “natural” frequency of the oscillator. • For weak damping, the system oscillates, and the amplitude decreases exponentially with time. • With sufficiently strong damping, the system returns smoothly to equilibrium without oscillation. Problems, Chapter 1542, 43, 70(6th ed – Chapter 13)42, 70 Physics 1D03 - Lecture 35

More Related