PES 1110 Fall 2013, Spendier Lecture 41/Page 1 Today: - HW 10 due next lecture, Wedensday - Quiz 6 end of class - Damped Simple Harmonic Motion (15.8) - Forced (Driven) Oscillation and Resonance (15.9) Damped Simple Harmonic Motion (15.8) A swinging bell left to itself will eventually stop oscillating due to damping forces (air resistance and friction at the point of suspension). The decrease in amplitude caused by dissipative forces is called damping, and the corresponding motion is called damped oscillation. Simples case: simple harmonic motion with fictional damping force In this case the damping force is proportional to the velocity of the oscillating body: Fx bvx ... negative b/c force is always in opposite direction to velocity units of b [kg/s] dx vx ..... velocity dt b = constant that describes the strength of the damping force Then the net force on the body is: F ma kx bvx ma x dx d2x m 2 dt dt This new expression has an extra term. Solving this equation is something you will learn in differential equations. For now, we will down the general solution (without proof) kx b x (t ) xm ebt /( 2 m ) cos w ' t f (oscillation with little damping) angular frequency: w ' decay constant: k b2 2 m 4m b 2m The damped motion differs from the undamped motion in to ways: 1) The amplitude is not a constant, but decreases with time: A(t ) xm ebt /( 2 m ) , because of the decreasing exponential. 2) The angular frequency ω' is no longer equal to k / m but is somewhat smaller, hence is a decreased angular frequency. If the damping constant b goes to zero, then the solution reduces back to the undamped solution. PES 1110 Fall 2013, Spendier Lecture 41/Page 2 Plotting damped harmonic motion: Example 2: In a damped oscillator with m = 0.250 kg, k = 85 N/m, and b = 0.070 kg/s, what is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles? Forced (Driven) Oscillation and Resonance (15.9) A damped oscillator left to itself will eventually stop moving altogether. But we can maintain a constant-amplitude oscillation by applying a force that varies with time in a periodic or cyclic way, with a definite period and frequency. For example, you can keep your friend on a playground swing swinging with a constant amplitude by giving him/her a little push once each cycle. We call this additional force driving force. Natural frequency: System is simply displaced from equilibrium and then left alone, in which case the system, oscillates with a natural frequency w' k b2 2 m 4m Forced or driven oscillation: Apply a periodically varying driving force with angular frequency ωd to a damped harmonic oscillator PES 1110 Fall 2013, Spendier Lecture 41/Page 3 Resonance: When ωd = ω, a system, is said to be "resonant" or undergoing "resonance". This is the point where the most efficient transfer of energy occurs between driving force and oscillator. DEMO: Harmonic resonator One oscillating mass on rod sets another mass on rod oscillating in resonance. The apparatus consists of two sets of masses on light springy vertical rods. In a set, masses are all the same, but rods differ in length. The two sets are weakly coupled by a horizontal bar. When ,mass 1a oscillates, mass 1b starts to oscillate in resonance but 2a, 2b, 3a, and 3b do not oscillate. g l This works because mass 1a and 1b have the same natural frequency! Angular frequency for simple pendulum: w We also know that a small driving force can cause huge damage in this case - Earthquake can make building sway. Some buildings of a particular height collapse while others with different height do not. ωd = ω for the once which collapse! ωd =diver, low frequency vibration of earth ω = natural, resonant frequency of building - Tacoma Narrows Bridge colapse: http://www.youtube.com/watch?v=j-zczJXSxnw The Tacoma Narrows Bridge collapsed four months after it was opened for traffic. The maon span was 2800 ft long and 39 ft wide, with 8 ft high steel stiffening girders on both sides. The maximum amplitude of the torsional vibration was 35º, the frequency was about 0.2 Hz. MIT Physics Demo - Tuning Forks: Resonance & Beat Frequency http://video.mit.edu/watch/tuning-forks-resonance-a-beat-frequency-11447/ http://phet.colorado.edu/en/simulation/resonance Interactive online simulation tool, CU Boulder
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