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

Simple Pendulum L θ T mg 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

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

f = -bv where b is a constant damping coefficient 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 where b is a constant damping coefficient F=ma give: eg: green water (weak damping) For weak damping (small b), the solution is: x t A e-(b/2m)t Physics 1D03 - Lecture 35

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

Question: What is the value of b/(2m)? 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

Example: A pendulum of length 1. 0 m has an initial 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

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

x(t) t : “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

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

and hence is always negative. 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

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

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 frequency w. The amplitude can be very large if the external driving frequency is close to the “natural” frequency of the oscillator. Physics 1D03 - Lecture 35

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

So, as the amplitude, A, increases ! If the external “push” has the same frequency as the resonant frequency ω0. The driving force is said to be in resonance with the system. A 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

A 2.0 kg object attached to a spring moves without 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

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. Problems, Chapter 15 42, 43, 70 (6th ed – Chapter 13) 42, 70 Physics 1D03 - Lecture 35