Computer Sound Synthesis 2

MUS_TECH 335 Selected Topics

Physical Modeling Synthesis

Types of Physical models

• Modal (resonance) We dealt with this while

discussing filtering!

• Finite elements (mass-spring)

• Waveguides (delay-lines)

• Direct (Differential equations)

• Others

Sound ExamplesSome on-line examples:

Claude Cadoz:http://www-acroe.imag.fr/mediatheque/sonotheque/sonotheque.html

Perry Cook:http://www.cs.princeton.edu/~prc/

Julius Smith:http://ccrma.stanford.edu/~jos/pasp/Sound_Examples.html

Download and InstallPerry Cook’s stkugens!

Download is available at:

http://swiki.hfbk-hamburg.de:8888/MusicTechnology/677

Definitions

Kinetic Energy

KE = 1/2 m v2

Potential Energy

PE = 1/2 k y2

Peak KE

Peak PE

Simple Harmonic Motion

(Makes sine waves)

Definitions

Displacement

y = f(t)

Instantaneous velocity

v = Dty = y´ = dy/ dt = f´(t)

Instantaneous acceleration

a = Dtv = y´´ = d2y/ dt2 = f ´´(t)

DefinitionsPeak Kinetic Energy

ty

v

a

t

t

displacement

Peak Potential Energy

velocity

acceleration

Definitions

∫a(t) v(t) y(t) v(t) a(t)∫ d dt

d dt

At each step, integration and differentiation of simusoids causes 90-degree phase shifts

A sin (t + ø) A cos (t + ø) -A sin (t + ø) A cos (t + ø) A sin (t + ø)

Simple Harmonic Motion

y = A sin (t + ø)

Dty = A cos (t + ø)

Dt2y = -A sin (t + ø)

= -y

0 = Dt2y + y

differential equation for simple harmonic motion

Simple Harmonic Motion

0 = y + y´´ 0 = ∫ ∫ a + a

a = - ∫ ∫ a integral equation for simple harmonic motion

Simple Harmonic Motion

a = - ∫ ∫ a

-initial position

position

∫ ∫

acceleration

-1

velocity

Integrator:

unstable with DC

z-1in

out

z-1k

a

in

outLeaky Integrator:

Simple Harmonic Motion

∫

Mass-and-Spring OscillatorsFree Vibration

The summation of all forces in the system must be zero.∑ F = 0

• ky is the restoring force where k is the string constant and y is the displacement of the mass m• ma, mass • acceleration, is the inertia once set in motion

0 = ky + ma

ky ma

y

Mass-and-Spring Oscillators

0 = ky + ma0 = k/m y + y´´

0 = y + y´´ (k/m) 0 = ∫ ∫ a + a

a = - ∫ ∫ a (integral equation)

ky ma

y

Mass-and-Spring Oscillators

Free Vibration with Damping

0 = ky + cv + ma

where c is the damping coefficient

ky ma

cv

c y

Mass-and-Spring Oscillators0 = ky + cv + ma

0 = (k/m) y + (c/m) y´ + y´´y´´ = -(k/m) y - (c/m) y´

a = -(k/m) ∫ ∫ a - (c/m) ∫ a= - ∫ ∫ a - (c/m)∫ a

ky ma

Rv

ky ma

Rv

c y

Mass-and-Spring Oscillators

a = - ∫ ∫ a - (c/m) ∫ aa = - ∫ ∫ a - b ∫ a

where b = (c/m) = 1/Q = Bw/CF

∫ ∫

acceleration

-b

-initial position

position

∫ ∫

-1

velocity

Mass-and-Spring Oscillators

Damping has a range of results:

over damped

critically damped

under damped

b = 2 = (c/2m

If > 1 overdamped

= 1 critically damped

< 1 underdamped

x(t) = A e -cos(dt + ø)

d = SQRT(1- )