Further examples of harmonic oscillation

Examples:
- Calculation of damping of harmonic oscillator;
- Forced harmonic oscillator;
- 2 coupled pendulums;
- Superposition of oscillations
Technical Topics: Better understanding of sine/cosine
AESL:
- Some DIVISIONs are optional;
- combining operators with indentation;
- direct connection

diff --git a/Analog Engine Example Applications.odt b/Analog Engine Example Applications.odt
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diff --git a/Analog Engine Scripting Language.odt b/Analog Engine Scripting Language.odt
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diff --git a/Analog Engine Technical Topics.odt b/Analog Engine Technical Topics.odt
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diff --git a/scripts/EP1 10.01 Damped Harmonic Oscillator.LACE b/scripts/EP1 10.01 Damped Harmonic Oscillator.LACE
new file mode 100644 (file)
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+++ b/scripts/EP1 10.01 Damped Harmonic Oscillator.LACE
@@ -0,0 +1,19 @@
+# Damped_Oscillator
+# phi'' = -(S*phi + D/m*phi')
+
+coefficient.1(-1) -> -phi0   # Initial Amplitude
+coefficient.3     -> S     # SpringForce
+coefficient.4     -> D/m   # Damping linear to speed
+
+iintegrate phi'' -> -phi'
+invert -phi' -> phi'
+cmultiply phi', D/m -> D/m*phi'
+
+iintegrate -phi' -> phi
+  IC: -phi0
+cmultiply S, phi -> S*phi
+
+isum 10:S*phi, D/m*phi' -> -(Sphi+D/mphi')
+assign -(Sphi+D/mphi') -> phi''
+
+output(y) -> out.y

diff --git a/scripts/EP1 10.02 Forced Harmonic Oscillator.LACE b/scripts/EP1 10.02 Forced Harmonic Oscillator.LACE
new file mode 100644 (file)
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+++ b/scripts/EP1 10.02 Forced Harmonic Oscillator.LACE
@@ -0,0 +1,33 @@
+# Forced Damped_Harmonic Oscillator
+# x'' + 2*delta*x' + omega.0^2*x = A cos (omega.f t)
+
+# generating oscillation with omega.f
+# f'' = -omega.f^2*f
+coefficient.1(-1) -> -f0      # amplitude
+coefficient.2     -> omega^2
+iintegrate f'' -> -f'
+iintegrate -f' -> f
+  IC: -f0
+cmultiply f, omega^2 -> omega^2*f
+invert omega^2*f -> -omega^2*f
+assign -omega^2*f -> f''
+invert f -> -f
+output f -> y
+
+# forced damped harmonic oscillation
+# x'' = -(2*delta*x' + omega.0^2*x) + A cos (omega.f t)
+coefficient.5(-1) -> -x0       # amplitude of oscillation
+coefficient.6     -> 2*delta   # attenuation
+coefficient.7     -> omega.0^2 # Eigenfrequency of this oscillator, ^2
+
+iintegrate x'' -> -x'
+invert -x' -> x'
+cmultiply x', 2*delta -> 2*delta*x'
+
+iintegrate -x' -> x
+  IC: -x0
+cmultiply x, omega.0^2 -> omega.0^2*x
+isum 2*delta*x', omega.0^2*x, -f -> -(2*delta*x'+omega.0^2*x)-f
+assign -(2*delta*x'+omega.0^2*x)-f -> x''
+
+output(x) -> x

diff --git a/scripts/EP1 10.03 Coupled Pendulums.LACE b/scripts/EP1 10.03 Coupled Pendulums.LACE
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@@ -0,0 +1,34 @@
+# 2 coupled pendulums
+# x1'' = -k1*x1 + k*(x2-x1)
+# x2'' = -k2*x2 - k*(x2-x1)
+# xi are the displacement of the pendulum to its resting position
+
+coefficient.1 -> k1
+coefficient.2 -> k2
+coefficient.3 -> k
+coefficient.5(-1) -> -x10 # initial displacement x1
+coefficient.6(-1) -> -x20 # initial displacement x2
+
+iintegrate x1'' -> -x1'
+iintegrate -x1' -> x1
+  IC: -x10
+
+iintegrate x2'' -> -x2'
+iintegrate -x2' -> x2
+  IC: -x20
+
+cmultiply x1, k1 -> k1*x1
+cmultiply x2, k2 -> k2*x2
+
+invert x1 -> -x1
+isum x2, -x1 -> -(x2-x1)
+cmultiply -(x2-x1), k -> -k*(x2-x1)
+
+isum k1*x1, -k*(x2-x1) -> -k1*x1+k*(x2-x1)
+assign -k1*x1+k*(x2-x1) -> x1''
+invert -k*(x2-x1) -> k*(x2-x1)
+isum k2*x2, k*(x2-x1) -> -k2*x2-k*(x2-x1)
+assign -k2*x2-k*(x2-x1) -> x2''
+
+output x1 -> out.x
+output x2 -> out.y

diff --git a/scripts/Superposition of oscillations.LACE b/scripts/Superposition of oscillations.LACE
new file mode 100644 (file)
index 0000000..b25f7e4
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+++ b/scripts/Superposition of oscillations.LACE
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+# Superposition of the oscillation of two undamped harmonic oscillators
+# x1'' = -omega1^2*x1
+# x2'' = -omega2^2*x2
+
+coefficient.1(-1) -> -x10 # amplitude of 1
+coefficient.2     -> omega1^2
+coefficient.5(-1) -> -x20 # amplitude of 2
+coefficient.6     -> omega2^2
+
+# 1st oscillator
+iintegrate x1'' -> -x1'
+iintegrate -x1' -> x1
+  IC: -x10
+cmultiply x1, omega1^2 -> omega1^2*x1
+invert omega1^2*x1 -> -omega1^2*x1
+assign -omega1^2*x1 -> x1''
+output x1 -> out.x
+
+# 2nd oscillator
+iintegrate x2'' -> -x2'
+iintegrate -x2' -> x2
+  IC: -x20
+cmultiply x2, omega2^2 -> omega2^2*x2
+invert omega2^2*x2 -> -omega2^2*x2
+assign -omega2^2*x2 -> x2''
+output x2 -> out.y
+
+# Sum
+isum x1, x2 -> -(x1+x2)
+invert -(x1+x2) -> x1+x2
+output x1+x2 -> out.z

diff --git a/scripts/TP1 04.12 Harmonic Oscillator.AESL b/scripts/TP1 04.12 Harmonic Oscillator.AESL
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-IDENTIFICATION DIVISION
-PROGRAM-ID HarmonicOscillator
-VERSION 20240201
-COMMENT A mass m is subject to a force F=-k*r.
-COMMENT What is the trajectory if the mass starts at position (a,0,0)?
-COMMENT How much time does it take to pass through zero?
-COMMENT What is the trajectory if it starts at (a,0,0) with velocity (0,v0,0)?
-COMMENT m*x'' = -k*x
-COMMENT m*y'' = -k*y (z can be set to 0).
-
-ENVIRONMENT DIVISION
-ENGINE Anabrid-THAT
-TIMEBASE 1ms
-REQUIRES COEFFICIENT 4, INTEGRATOR 4, INVERTER 2
-
-DATA DIVISION
-OUTPUT OUTPUT.X x
-OUTPUT OUTPUT.Y y
-COEFFICIENT.1 A     # (a,0,0)
-COEFFICIENT.2 K/M_x # k/m for x
-COEFFICIENT.3 V0    # (0,v0,0)
-COEFFICIENT.4 K/M_y # k/m for y, identical to k/m for x
-
-PROGRAM DIVISION
--1 -> COEFFICIENT.A -> -a # has to be negative because x' is negative
-+1 -> COEFFICIENT.V0 -> v0 # has to be positive because y'' is positive
-
-x'' -> INTEGRATOR -> -x'
--x', IC:-a -> INTEGRATOR -> x
-x -> COEFFICIENT.K/M_x -> k/m*x
-k/m*x -> INVERTER -> -k/m*x = x''
-
-y'', IC:v0 -> INTEGRATOR -> -y'
--y' -> INTEGRATOR -> y
-y -> COEFFICIENT.K/M_y -> k/m*y
-k/m*y -> INVERTER -> -k/m*y = y''
-
-OPERATION DIVISION
-MODE REPEAT
-OP-TIME 100 ms