From: Permondes Date: Sun, 22 Dec 2024 19:46:56 +0000 (+0100) Subject: Further examples of harmonic oscillation X-Git-Url: https://permondes.de/gitweb/Analog_Engine.git/commitdiff_plain/refs/heads/main?hp=e25ae3805c3f6ab36385c2d04bbf148736184386 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 index 0a43759..d26398f 100644 Binary files a/Analog Engine Example Applications.odt and b/Analog Engine Example Applications.odt differ diff --git a/Analog Engine Scripting Language.odt b/Analog Engine Scripting Language.odt index 50ef933..8ec8b44 100644 Binary files a/Analog Engine Scripting Language.odt and b/Analog Engine Scripting Language.odt differ diff --git a/Analog Engine Technical Topics.odt b/Analog Engine Technical Topics.odt index 28d4ff1..f8423c0 100644 Binary files a/Analog Engine Technical Topics.odt and b/Analog Engine Technical Topics.odt differ diff --git a/Potentials and Motion.odt b/Potentials and Motion.odt new file mode 100755 index 0000000..cf28c35 Binary files /dev/null and b/Potentials and Motion.odt differ diff --git a/scripts/EP1 10.01 Damped Harmonic Oscillator.LACE b/scripts/EP1 10.01 Damped Harmonic Oscillator.LACE new file mode 100644 index 0000000..0f03c01 --- /dev/null +++ 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 index 0000000..6e51b64 --- /dev/null +++ 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 new file mode 100644 index 0000000..5ef0517 --- /dev/null +++ b/scripts/EP1 10.03 Coupled Pendulums.LACE @@ -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/Pendulum.LACE b/scripts/Pendulum.LACE new file mode 100644 index 0000000..c5e8ab2 --- /dev/null +++ b/scripts/Pendulum.LACE @@ -0,0 +1,37 @@ +# Pendulum: Comparing Harmonic Oscillator with 4th order Taylor approximation + +### Taylor 4 approximation: phi'' = - g/r * (phi - 1/6 * phi^3) +# fortunately, the components of orders 0, 2 and 4 are zero + +coefficient.1 -> g/r +coefficient.2 -> 1/6 # set to 0,167 or 0 (for harmonic solution) +coefficient.3(-1) -> -phi0 + +iintegrate phi'' -> -phi' +iintegrate -phi' -> phi + IC: -phi0 + +multiply phi, phi -> phi^2 +multiply phi^2, phi -> phi^3 +invert phi^3 -> -phi^3 + +cmultiply 1/6, -phi^3 -> -1/6*phi^3 +isum phi, -1/6*phi^3 -> -(phi-1/6*phi^3) +cmultiply -(phi-1/6*phi^3), g/r -> -g/r*(phi-1/6*phi^3) +assign -g/r*(phi-1/6*phi^3) -> phi'' + +output phi -> out.x + +### Harmonic oscillator: phi'' = - g/r * phi +coefficient.5 -> g/r_h # identical to g/r +coefficient.7(-1) -> -phi0_h # identical to -phi0 + +iintegrate phi_h'' -> -phi_h' +iintegrate -phi_h' -> phi_h + IC: -phi0_h +invert phi_h -> -phi_h + +cmultiply -phi_h, g/r_h -> -g/r*phi_h +assign -g/r*phi_h -> phi_h'' + +output phi_h -> out.y diff --git a/scripts/Superposition of oscillations.LACE b/scripts/Superposition of oscillations.LACE new file mode 100644 index 0000000..b25f7e4 --- /dev/null +++ b/scripts/Superposition of oscillations.LACE @@ -0,0 +1,31 @@ +# 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 deleted file mode 100644 index 2c0e512..0000000 --- a/scripts/TP1 04.12 Harmonic Oscillator.AESL +++ /dev/null @@ -1,40 +0,0 @@ -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 diff --git a/scripts/TP1 04.12 Harmonic Oscillator.LACE b/scripts/TP1 04.12 Harmonic Oscillator.LACE new file mode 100644 index 0000000..89b0723 --- /dev/null +++ b/scripts/TP1 04.12 Harmonic Oscillator.LACE @@ -0,0 +1,31 @@ +# Harmonic Oscillator +# A mass m is subject to a force F=-k*r. +# What is the trajectory if the mass starts at position (a,0,0)? +# How much time does it take to pass through zero? +# What is the trajectory if it starts at (a,0,0) with velocity (0,v0,0)? +# Compare to the case including gravity on earth in x direction +# m*x'' = -k*x + g +# m*y'' = -k*y (z can be set to 0). + +coefficient.1(-1) -> -a # (a,0,0), has to be negative because x' is negative +coefficient.2 -> k/m_x # k/m for x +coefficient.3(+1) -> v0 # (0,v0,0), has to be positive because y'' is positive +coefficient.4 -> k/m_y # k/m for y, identical to k/m for x +coefficient.5(-1) -> -g # g + +iintegrate x'' -> -x' +iintegrate -x' -> x + IC: -a +cmultiply (k/m_x, x) -> k/m*x +isum (k/m*x, -g) -> -k/m*x+g +assign (-k/m*x+g) -> x'' + +iintegrate y'' -> -y' + IC: v0 +iintegrate -y' -> y +cmultiply (k/m_y, y) -> k/m*y +invert (k/m*y) -> -k/m*y +assign (-k/m*y) -> y'' + +output x -> out.x +output.y -> out.y diff --git a/scripts/TP1 07.19 Falling Mass.SALSA b/scripts/TP1 07.19 Falling Mass.SALSA deleted file mode 100644 index 3f3dc6b..0000000 --- a/scripts/TP1 07.19 Falling Mass.SALSA +++ /dev/null @@ -1,55 +0,0 @@ -# Wiring mache ich von links nach rechts !!! - -...IDENTIFICATION DIVISION -...PROGRAM-ID FallingParticle -...VERSION 20240225 -...COMMENT Calculate the deflection from the vertical caused by the Earth's rotation of a particle falling freely from rest from a height h. -...COMMENT Differential Equations: -...COMMENT x''=-bz'+ay' # x-axis is along latitude, directed to east -...COMMENT y''=-ax' # y-axis is along longitude, directed to north -...COMMENT z''=-g+ax' # z-axis is perpendicular to the surface of earth -...COMMENT g: gravitational acceleration = 9,81 m/s² -...COMMENT a: 2*omega*sin(phi) -...COMMENT b: 2*omega*cos(phi) -...COMMENT omega: rotation velocity of the earth = 2*pi/day -...COMMENT phi: Latitude of location (0-90°) -...COMMENT Initial Condition: z(0)=h -...COMMENT The full solution requires 6 INTEGRATORs, Anabrid-THAT just has 5. The deflection to longitude (y) is neglegible and can be omitted (marked #*). -...COMMENT It could also be solved in a separated algorithm omitting x. - -...ENVIRONMENT DIVISION -...TIMEBASE 1ms - -...DATA DIVISION -SET COEFFICIENT.1 TO AY # 2*2pi/day*sin(phi) -SET COEFFICIENT.2 TO AX # = AY -SET COEFFICIENT.3 TO B # 2*2pi/day*cos(phi) -SET COEFFICIENT.4 TO G # gravitational acceleration = 9,81 m/s² -SET COEFFICIENT.5 TO H # height h -SET OUTPUT.X TO x -SET OUTPUT.Y TO y -SET OUTPUT.Z TO z - -INITIALIZE H by -1 TO -h # same as COMPUTE -1 TIMES H TO -h -INITIALIZE G by +1 TO g - -INTEGRATE 1*-bz',1*ay' TO -x' # Input is x'' -INTEGRATE -x' TO x - -INTEGRATE y'' TO -y' -INTEGRATE -y' TO y - -INTEGRATE z'' TO -z' -INTEGRATE -z', IC:-h, LIMIT:(z >= 0) TO z - --x' * AX -> -ax' --ax' = y'' --y' * AY -> -ay' -INVERT -ay' TO ay' --z' * B -> -bz' -ADD 1*-ax', 1*g TO -g+ax' --g+ax' = z'' - -...OPERATION DIVISION -...MODE REPEAT -...OP-TIME 7,3ms