7.5 Simple Harmonic Motion; Damped Motion; Combining Waves. Objectives

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1 Objectives 1. Build a Model for an Object in Simple Harmonic Motion. 2. Analyse Simple Harmonic Motion. 3. Analyse an Object in Damped Motion. 4. Graph the Sum of Two Functions. 30 April Kidoguchi, Kenneth

2 Simple Harmonic 30 April Kidoguchi, Kenneth

3 An Ideal Simple Harmonic Oscillator Maximum Compression Equilibrium Maximum Stretch 30 April Kidoguchi, Kenneth

4 Trigonometric Function with a Varying Amplitude Suppose the displacement of an oscillating ideal mass-spring system is modelled as: y ideal (t) = cos(p/2 t) This simple harmonic oscillator model depicts a system that, once set in motion, oscillates with the same amplitude and frequency forever. A more realistic model accounts for the effects of friction with a "damping term", so that: y damped (t) = e -kt cos(p/2 t) 1 cm 0 cm -1 cm Ideal System where k is a positive real number called the damping coefficient. We can think of y damped (t) as having a t-varying amplitude, A(t) = e -kt such that the oscillation, cos(p/2 t), is bounded by + A(t), the envelope. 30 April Kidoguchi, Kenneth

5 Trigonometric Function with a Varying Amplitude y damped (t) = e -t / 2 cos(p/2 t) y 1 y ideal (t) = cos(p/2 t) A(t) = e -t/ t -1 - A(t) = -e -t/2 30 April Kidoguchi, Kenneth

6 Trigonometric Function with a Varying Amplitude Given a signal of the form: S(t) = 100 e -kt cos(2p f t) where f = 60 Hz, is the frequency in Hz (cycles per second). Present the analysis to find the value of the damping coefficient, k, that will degrade the amplitude by a factor of 2 each second. Let: S(t) = A(t) cos(2p f t), where: A(t) = 100 e -kt 30 April Kidoguchi, Kenneth

7 Trigonometric Function with a Varying Amplitude Given a signal of the form: S(t) = 100 e -kt cos(2p f t) where f = 60 Hz, is the frequency in Hz (cycles per second). Present the analysis to find the value of the damping coefficient, k, that will degrade the amplitude by a factor of 2 each second. Let: S(t) = A(t) cos(2p f t), where: A(t) = 100 e -kt We require: A( t) A( t 1) A( t) A( t 1) 1 2 A( t) A( t) 100e 100e -kt -kt -k ( t1) -kt -k S( t) 100e -ln(2) t 2 e e e e k 2 1 cos( 120pt ) t k ln(2) cos( 120pt ) 30 April Kidoguchi, Kenneth

8 Trigonometric Function with a Varying Amplitude 30 April Kidoguchi, Kenneth

9 Sums of Trigonometric Functions - Beats Consider two signals S 1 (t) and S 2 (t) where: S 1 (t) = A cos(w 1 t), and S 2 (t) = A cos(w 2 t) The sum of these two signals: S(t) = S 1 (t) + S 2 (t) = A [ cos(w 1 t), + cos(w 2 t) ] Recalling one of your favourite identities: u v S( t) cos( u) cos( v) 2cos 2 With u = w 1 t and v = w 2 t, we have: S( t) cos( w1t) cos( w2t) 2cos u - v cos 2 w w t w - w t cos Note that S(t) has a high frequency component, ½ (w 1 + w 2 ) and a low frequency component, ½ (w 1 - w 2 ) April Kidoguchi, Kenneth

10 Sums of Trigonometric Functions - Beats Let: f 1 = 440 Hz and f 2 = 400 Hz so S 1 (t) = cos(2p 440 t) = cos(880p t), and S 2 (t) = cos(2p 400 t) = cos(800p t) The sum of these two signals: S( t) cos( w t) cos( w t) cos w1 w2 tcos w1 - w2 t cos 1 2 t t 2cos p f f t p f - f t 2 cos 880p cos 40p 30 April Kidoguchi, Kenneth

11 E( t) 2cos(40 t) Sums of Trigonometric Functions - Beats p S( t) 2cos 40pt cos 880pt -E( t) -2cos(40 pt) 30 April Kidoguchi, Kenneth

= cos(2p 442 t) S(t) = S 1 (t) + S 2 (t) 1 1 2 2 S( t) cos( w t) cos( w

12 Sums of Trigonometric Functions - Beats S 1 (t) = cos(2p 440 t) S 2 (t) = cos(2p 440 t) S(t) = S 1 (t) + S 2 (t) S 1 (t) = cos(2p 440 t) S 2 (t) = cos(2p 442 t) S(t) = S 1 (t) + S 2 (t) S( t) cos( w t) cos( w t) 2cos w w t cos w -w t April Kidoguchi, Kenneth

= cos(2p 450 t) S(t) = S 1 (t) + S 2 (t) 1 1 2 2 S( t) cos( w t) cos( w

13 Sums of Trigonometric Functions - Beats S 1 (t) = cos(2p 440 t) S 2 (t) = cos(2p 445 t) S(t) = S 1 (t) + S 2 (t) S 1 (t) = cos(2p 440 t) S 2 (t) = cos(2p 450 t) S(t) = S 1 (t) + S 2 (t) S( t) cos( w t) cos( w t) 2cos w w t cos w -w t April Kidoguchi, Kenneth

14 25 m 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Trigonometric Functions with a Sinusoidal Midline Function A Double Ferris Wheel has a 30 metre rotating arm attached at its centre to a 25 metre main support. At each end of the rotating arm is attached a Ferris Wheel measuring 20 metres in diameter. It takes the rotating arm 6 minutes to complete one revolution and it takes 4 minutes for each Ferris Wheel to complete a revolution about its hub. All rotations are counterclockwise. Main Support Reference t = 0 30 April Kidoguchi, Kenneth y (0,0) 30 m At t = 0, a reference point is at the position indicated in the figure. Present the analysis to: a) find h(t), the vertical distance of the reference point above the ground as a function of t, time in minutes, and b) sketch a properly labelled graph of h(t) over one complete cycle. 10m x

15 25 m 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Trigonometric Functions with a Sinusoidal Midline Function Step 1: Analyse the motion of the hub, i.e., the appropriate end point of the rotating arm. Let H(t) be the vertical position with respect to the ground of this endpoint as a function of t. Main Support Hub t = 0 y 10m (0,0) 30 m x 30 April Kidoguchi, Kenneth

16 25 m 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Trigonometric Functions with a Sinusoidal Midline Function Step 1: Analyse the motion of the hub, i.e., the appropriate end point of the rotating arm. Let H(t) be the vertical position with respect to the ground of this endpoint as a function of t. H(t) = 15 sin(2p/6 t) + 25 = 15 sin(p/3 t) + 25 Sanity Check: H(0) = 15 sin(p/3 0) + 25 = 25 H(3/2) = 15 sin(p/3 (3/2)) + 25 = 15 sin(p/2) + 25 = 40 H(3) = 15 sin(p/3 3) + 25 = 15 sin(p) + 25 = 25 Main Support Hub t = 0 30 April Kidoguchi, Kenneth y (0,0) 30 m 10m x

17 10 m 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Trigonometric Functions with a Sinusoidal Midline Function Step 2: Analyse the motion of the reference point with respect to the wheel, i.e., ignoring the arm rotation. Let W(t) be the vertical position of the reference point on the wheel (ignoring the arm rotation) as a function of t. Reference t = 0 30 April Kidoguchi, Kenneth

18 10 m 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves Trigonometric Functions with a Sinusoidal Midline Function Step 2: Analyse the motion of the reference point with respect to the wheel, i.e., ignoring the arm rotation. Let W(t) be the vertical position of the reference point on the wheel (ignoring the arm rotation) as a function of t. W(t) = A W sin(2p/p t + f ) + M W = 10 sin(2p/4 t + 0 ) + 25 = 10 sin(p/2 t) + 25 Sanity Check: W(0) = 10 sin(p/2 0) + 25 = 25 W(1) = 10 sin(p/2 (1)) + 25 = 35 W(3) = 10 sin(p/2 3) + 25 = 10 sin(3 p/2) + 25 = 15 Reference t = 0 30 April Kidoguchi, Kenneth

Combine Steps 1 & 2 W(t) = 10 sin(p/2 t)

= 10 sin(p/2 t) + H(t) = 10 sin(p/2 t) +

19 Trigonometric Functions with a Sinusoidal Midline Function Step 3: Combine Steps 1 & 2 W(t) = 10 sin(p/2 t) + M W H(t) = 15 sin(p/3 t) + 25 To find h(t) we note that M W, the midline for W(t), is replaced by H(t) so that: h(t) = 10 sin(p/2 t) + H(t) = 10 sin(p/2 t) + 15 sin(p/3 t) April Kidoguchi, Kenneth

Chapter 11 Vibrations and Waves

Chapter 11 Vibrations and Waves If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system