1. Copyright 2004 Ken Greenebaum Introduction to Interactive Sound Synthesis Lecture 22:Physical Modeling II Ken Greenebaum

2. Copyright 2004 Ken Greenebaum Automobile Synthesizer Like my synth? Like my synth?

3. Copyright 2004 Ken Greenebaum Automobile Synthesizer Does it sound realistic? Does it sound realistic?

4. Copyright 2004 Ken Greenebaum Automobile Synthesizer What sounds are being synthesized? What sounds are being synthesized? Using what algorithms? Using what algorithms?

5. Copyright 2004 Ken Greenebaum Automobile Video

6. Copyright 2004 Ken Greenebaum Assignment 7 Questions? Questions? Demos Demos

7. Copyright 2004 Ken Greenebaum Physical Modeling Basics We will examine the simplest objects: We will examine the simplest objects: String String Rod Rod Complex objects Complex objects May be modeled May be modeled Finite element analysis Finite element analysis May be simulated May be simulated Modal synthesis Modal synthesis

8. Copyright 2004 Ken Greenebaum Similar? What do the following have in common: What do the following have in common: Tensioned strings Tensioned strings Rods Rods Tuned pipes Tuned pipes Tubular bells Tubular bells

9. Copyright 2004 Ken Greenebaum Simplest models One dimensional models One dimensional models Simple to understand Simple to understand Rich Rich Efficient to implement Efficient to implement

10. Copyright 2004 Ken Greenebaum Multidimensional models Get very complicated very quickly Get very complicated very quickly We only understand the simplest 2D We only understand the simplest 2D Round (drumhead) Round (drumhead) Square Square Recent thesis for L shaped Recent thesis for L shaped

11. Copyright 2004 Ken Greenebaum Spring Mass Models Perry’s baseball: Perry’s baseball: Baseball of mass m Baseball of mass m Spring constant k Spring constant k Displacement y Displacement y Total losses r Total losses r Air resistance Air resistance Spring losses Spring losses Gravitational constant g Gravitational constant g

12. Copyright 2004 Ken Greenebaum Newtonian Physics Spring constant represents: Spring constant represents: Force per unit distance needed to compress/expand spring Force per unit distance needed to compress/expand spring Newton’s 2 nd Law: Newton’s 2 nd Law: Force = Mass * Acceleration Force = Mass * Acceleration Velocity is the change of position wrt time: Velocity is the change of position wrt time: v=dy/dt v=dy/dt Acceleration is the change of velocity wrt time: Acceleration is the change of velocity wrt time: a = dv/dt or d 2 y/dt 2 a = dv/dt or d 2 y/dt 2 What is jerk? What is jerk?

13. Copyright 2004 Ken Greenebaum Accounting for all the forces: -ky – mg – rv = ma -ky – mg – rv = ma

14. Copyright 2004 Ken Greenebaum Spring Mass Equation -ky – mg – rv = ma -ky – mg – rv = ma 1st term: force due to spring displacement 1st term: force due to spring displacement NOTE: this works in both expansion and compression NOTE: this works in both expansion and compression 2 nd term: force due to gravity 2 nd term: force due to gravity 3 rd term: loss due to resistance to motion (v is velocity) 3 rd term: loss due to resistance to motion (v is velocity)

15. Copyright 2004 Ken Greenebaum Cheat! I mean simplify: I mean simplify: We already combined all sources of loss We already combined all sources of loss Let’s drop the gravity term Let’s drop the gravity term Gravity weak compared to the restoring force of the spring Gravity weak compared to the restoring force of the spring Especially for springs stiff enough to move an object fast Especially for springs stiff enough to move an object fast Objects need to oscillate fast to create audible sound! Objects need to oscillate fast to create audible sound!

16. Copyright 2004 Ken Greenebaum Algebra Start with: Start with: -ky – mg – rv = ma -ky – mg – rv = ma Drop gravity term: Drop gravity term: -ky – rv = ma -ky – rv = ma Put in terms of position: Put in terms of position: -ky – r*dy/dt = m* d 2 y/dt 2 -ky – r*dy/dt = m* d 2 y/dt 2 Or: Or: d 2 y/dt 2 +r/m * dy/dt + k/m *y = 0 d 2 y/dt 2 +r/m * dy/dt + k/m *y = 0

17. Copyright 2004 Ken Greenebaum Solving for position Ball will oscillate at rate based on: Ball will oscillate at rate based on: Mass and spring constant Mass and spring constant Makes sense (consider golf ball) Makes sense (consider golf ball) Motion will exponentially decay based on Motion will exponentially decay based on r/2m r/2m Overdamped system will just droop Overdamped system will just droop (r / 2m) 2 > k /m (r / 2m) 2 > k /m

18. Copyright 2004 Ken Greenebaum Spring Mass Example Example sound familiar? Example sound familiar?

19. Copyright 2004 Ken Greenebaum Sound Familiar? It should It should It is just our damped sinusoid It is just our damped sinusoid But now physically inspired! But now physically inspired!

20. Copyright 2004 Ken Greenebaum Moving On We derived the spring mass eq as a warm-up We derived the spring mass eq as a warm-up Next comes the tensioned string Next comes the tensioned string What happens when we pluck a string? What happens when we pluck a string?

21. Copyright 2004 Ken Greenebaum Plucking a string We know: We know: Ends of string must not move Ends of string must not move Pluck causes a displacement Pluck causes a displacement Released displacement causes oscillation Released displacement causes oscillation

22. Copyright 2004 Ken Greenebaum Modeling a string Easy! Just use our spring mass model Easy! Just use our spring mass model String a bunch of masses together with springs: String a bunch of masses together with springs: Well. No. Well. No. Why Not? Why Not?

23. Copyright 2004 Ken Greenebaum Need a different solution Our spring model models compression Our spring model models compression Longitudinal wave Longitudinal wave Not the transverse wave we are interested in Not the transverse wave we are interested in

24. Copyright 2004 Ken Greenebaum The ideal string Differential eq for string: Differential eq for string: d 2 y / d x 2 = d 2 y / d t 2 * 1 / c 2 d 2 y / d x 2 = d 2 y / d t 2 * 1 / c 2 Assumptions: Assumptions: ‘Balls’ can only move transversely ‘Balls’ can only move transversely Up and Down Up and Down Not compress like the previous illustration Not compress like the previous illustration Balls have limited displacement Balls have limited displacement Corresponds to stiffness of string (wire) Corresponds to stiffness of string (wire)

25. Copyright 2004 Ken Greenebaum What does it mean? d 2 y / d x 2 = d 2 y / d t 2 * 1 / c 2 d 2 y / d x 2 = d 2 y / d t 2 * 1 / c 2 Acceleration of any segment up & down Acceleration of any segment up & down Equal to a constant * curvature of string Equal to a constant * curvature of string Constant c :speed of wave motion on the string Constant c :speed of wave motion on the string Function of the string’s: Function of the string’s: Tension Tension Mass per unit length Mass per unit length

26. Copyright 2004 Ken Greenebaum Solving the equation Could discretize in time and space Could discretize in time and space Nice solution for homogeneous string Nice solution for homogeneous string Given Given Constant mass and tension along string Constant mass and tension along string y (x, t ) = y l (t +x /c )+y r (t –x /c ) y (x, t ) = y l (t +x /c )+y r (t –x /c )

27. Copyright 2004 Ken Greenebaum Sum of traveling waves y (x, t ) = y l (t +x /c )+y r (t –x /c ) y (x, t ) = y l (t +x /c )+y r (t –x /c ) Any vibration of the string is sum of Any vibration of the string is sum of Left and Right traveling waves Left and Right traveling waves

28. Copyright 2004 Ken Greenebaum Sum of traveling waves y (x, t ) = y l (t +x /c )+y r (t –x /c ) y (x, t ) = y l (t +x /c )+y r (t –x /c ) Holding x constant Holding x constant Looking at one location Looking at one location Allowing time t to progress Allowing time t to progress y l moves left y l moves left y r moves right y r moves right

29. Copyright 2004 Ken Greenebaum Sum of traveling waves y (x, t ) = y l (t +x /c )+y r (t –x /c ) y (x, t ) = y l (t +x /c )+y r (t –x /c ) Holding time t constant Holding time t constant Varying position x Varying position x Can determine displacement at any pt by Can determine displacement at any pt by Adding the left and right displacements Adding the left and right displacements y l and y r y l and y r

30. Copyright 2004 Ken Greenebaum Don’t believe in traveling waves? Lets try a rope trick! Lets try a rope trick! OK, what happened? OK, what happened?

31. Copyright 2004 Ken Greenebaum Rope Trick Should have seen Should have seen Traveling wave Traveling wave Wave reflects at either end Wave reflects at either end Reflections are the inverse of the initial wave Reflections are the inverse of the initial wave

32. Copyright 2004 Ken Greenebaum Delay lines We can model an ideal string using delay lines: We can model an ideal string using delay lines: -1 models perfect reflection at fret -1 models perfect reflection at fret -0.9 models slight loss where bridge couples to instrument and radiates sound -0.9 models slight loss where bridge couples to instrument and radiates sound

33. Copyright 2004 Ken Greenebaum Wave guide model The 2 delay lines together are wave guide The 2 delay lines together are wave guide Combined length: Combined length: Roundtrip time for wave Roundtrip time for wave Total delay: Total delay: Period (in samples) of string oscillation Period (in samples) of string oscillation Sum of contents of 2 delay lines Sum of contents of 2 delay lines Displacement of the string Displacement of the string Difference of the contents of 2 delay lines Difference of the contents of 2 delay lines Velocity of the string Velocity of the string

34. Copyright 2004 Ken Greenebaum Pluck Load the wave guide with: Load the wave guide with: (sum represents displacement) (sum represents displacement)

35. Copyright 2004 Ken Greenebaum Strike Load the wave guide with: Load the wave guide with: (difference represents velocity) (difference represents velocity)