Damped Oscillations

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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: . - PowerPoint PPT Presentation

Transcript of Damped Oscillations

  • Physics 1D03 - Lecture 35*Damped Oscillations

    (Serway 15.6-15.7)

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*Simple Pendulum L

    T mgRecall, for a simple pendulum we have thefollowing equation of motion:Which give us:Hence:Application - measuring height - finding variations in g underground resourcesor: -------------------------------------------------------------------------

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*SHM: x(t) = A cos t Motion 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.SHM and Damping

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*For weak damping (small b), the solution is:f = -bv where b is a constant damping coefficientA damped oscillator has external nonconservativeforce(s) acting on the system. A common example is a force that is proportional to the velocity.eg: green water (weak damping)A e-(b/2m)tF=ma give:

    Physics 1D03 - Lecture 35

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

    Physics 1D03 - Lecture 35

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

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*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

  • Physics 1D03 - Lecture 35*water axle grease?Eg: Strong damping (b large): there is no oscillation when: Types of DampingSince:We can have different cases for the value underthe root: >0, 0 or
  • Physics 1D03 - Lecture 35*x(t)toverdampedcritical dampingunderdamped: Overdamped, no oscillation: Underdamped, oscillations with decreasing amplitude: Critically dampedCritical damping provides the fastest dissipation of energy.

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*Quiz: If you were designing a automobile suspensions:

    Should they be:

    Underdamped

    Critically damped

    Overdamped I dont need suspension !

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*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

  • Physics 1D03 - Lecture 35*Damped Oscillations and Resonance

    (Serway 15.6-15.7)

    Forced Oscillations Resonance

    Physics 1D03 - Lecture 35

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

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*is called the natural frequency or resonant frequency of the oscillation.Newtons 2nd Law:Assume that : x = A cos (wt + f) ; then the amplitude of a drive oscillator is given by:

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*So, as the amplitude, A, increases ! If the external push has the same frequency as the resonant frequency 0. The driving force is said to be in resonance with the system.A

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*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

  • Physics 1D03 - Lecture 35*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(8t)

    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

  • Physics 1D03 - Lecture 35*

    Physics 1D03 - Lecture 35

  • Physics 1D03 - Lecture 35*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.

    Physics 1D03 - Lecture 35

    **