From d5d81670a99c17f8567e446b901978ee74cdf9e0 Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Sun, 11 Sep 2022 18:55:44 -0400 Subject: [PATCH 1/7] Start of Translational --- src/Mechanical/Mechanical.jl | 1 + src/Mechanical/Translational/Translational.jl | 22 +++ src/Mechanical/Translational/components.jl | 112 ++++++++++++++++ src/Mechanical/Translational/utils.jl | 16 +++ test/Mechanical/translational.jl | 126 ++++++++++++++++++ 5 files changed, 277 insertions(+) create mode 100644 src/Mechanical/Translational/Translational.jl create mode 100644 src/Mechanical/Translational/components.jl create mode 100644 src/Mechanical/Translational/utils.jl create mode 100644 test/Mechanical/translational.jl diff --git a/src/Mechanical/Mechanical.jl b/src/Mechanical/Mechanical.jl index f58b6e667..2e87392b9 100644 --- a/src/Mechanical/Mechanical.jl +++ b/src/Mechanical/Mechanical.jl @@ -6,5 +6,6 @@ module Mechanical using ModelingToolkit include("Rotational/Rotational.jl") +include("Translational/Translational.jl") end diff --git a/src/Mechanical/Translational/Translational.jl b/src/Mechanical/Translational/Translational.jl new file mode 100644 index 000000000..38323d904 --- /dev/null +++ b/src/Mechanical/Translational/Translational.jl @@ -0,0 +1,22 @@ +""" +Library to model 1-dimensional, translational mechanical systems +""" +module Translational + +using ModelingToolkit, Symbolics, IfElse +using ...Blocks: RealInput, RealOutput + +@parameters t +D = Differential(t) + +export Port +include("utils.jl") + +export Body, Spring +export SpringDamperBoundary, ForcedBoundary +include("components.jl") + +#export Torque +#include("sources.jl") + +end diff --git a/src/Mechanical/Translational/components.jl b/src/Mechanical/Translational/components.jl new file mode 100644 index 000000000..f43347984 --- /dev/null +++ b/src/Mechanical/Translational/components.jl @@ -0,0 +1,112 @@ + +@enum BoundaryType NoBoundary SpringDamperBoundary ForcedBoundary + +#Note: boundary is an argument because this is a compile time constant +function Body(boundary = NoBoundary; name, x0 = 0.0, v0 = 0.0, mass = 1.0, g=0.0, upper_limit = +Inf, lower_limit = -Inf, k=1e9, d=1e6) + @named T = Port(v0 = v0, f0 = mass*g) + @parameters mass=mass g=g + @variables begin + x(t) = x0 + dx(t) = v0 + ddx(t) = 0.0 + end + + if boundary != NoBoundary + @variables begin + f_floor(t) = 0.0 + f_ceil(t) = 0.0 + end + @parameters begin + ubnd = upper_limit + lbnd = lower_limit + k=k + d=d + end + end + + eqs = [ + T.v ~ dx + + D(x) ~ dx + D(dx) ~ ddx + ] + + if boundary == NoBoundary + feqs = [ + ddx * mass ~ mass*g + T.f + ] + return compose(ODESystem([eqs; feqs], t, [x, dx, ddx], [mass, g]; name = name), T) + elseif boundary == SpringDamperBoundary + + # let + Δceil = x - ubnd + Δfloor = lbnd - x + + feqs = [ + ddx * mass ~ mass*g + T.f - f_floor - f_ceil + f_ceil ~ ifelse(x < ubnd, 0.0, Δceil*k + dx*d) + f_floor ~ ifelse(x > lbnd, 0.0, Δfloor*k + dx*d) + ] + return compose(ODESystem([eqs; feqs], t, [x, dx, ddx, f_ceil, f_floor], [mass, g, ubnd, lbnd, k, d]; name = name), T) + elseif boundary == ForcedBoundary + + # let + f = mass*g + T.f # external forces + + feqs = [ + ddx * mass ~ f - f_floor - f_ceil + 0 ~ ifelse((x >= ubnd), + ( x ) - ( ubnd ), # make x=>ubnd, solve for f_ceil to make this true + ( f_ceil ) - ( 0 ) + ) + 0 ~ ifelse((x <= lbnd) , + ( x ) - ( lbnd ), # make x=>lbnd, solve for f_floor to make this true + ( f_floor ) - ( 0 ) + ) + ] + return compose(ODESystem([eqs; feqs], t, [x, dx, ddx, f_ceil, f_floor], [mass, g, ubnd, lbnd]; name = name), T) + end + + + +end + + +function Spring(;name, k = 1e3, delta0 = 0.0) + @parameters k=k + @variables begin + x(t) = delta0 + dx(t) = 0.0 + f(t) + end + + @named T1 = Port() + @named T2 = Port() + + eqs = [ + D(x) ~ dx + dx ~ T1.v - T2.v + f ~ k*x + T1.f ~ +f + T2.f ~ -f + ] + return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), T1, T2) +end + +function Damper(; name, c=1e2) + @parameters c=c + @variables dx=0.0 + + @named T1 = Port() + @named T2 = Port() + + eqs = [ + T1.v - T2.v ~ dx + T1.f + T2.f ~ dx*c + ] + return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), T1, T2) +end + + + + diff --git a/src/Mechanical/Translational/utils.jl b/src/Mechanical/Translational/utils.jl new file mode 100644 index 000000000..8865b41c6 --- /dev/null +++ b/src/Mechanical/Translational/utils.jl @@ -0,0 +1,16 @@ +@connector function Port(; name, v0=0.0, f0=0.0) + sts = @variables begin + v(t) + f(t), [connect = Flow] + end + ODESystem(Equation[], t, sts, [], name = name, defaults = Dict(v => v0, f => f0)) +end +Base.@doc """ + Port(;name, v_int=0.0, f_int=0.0) + +1-dim. rotational flange of a shaft. + +# States: +- `v`: [m/s] velocity of the node +- `f`: [N] force entering the node +""" Port diff --git a/test/Mechanical/translational.jl b/test/Mechanical/translational.jl new file mode 100644 index 000000000..a890988e8 --- /dev/null +++ b/test/Mechanical/translational.jl @@ -0,0 +1,126 @@ +using ModelingToolkitStandardLibrary.Mechanical.Translational, ModelingToolkit, OrdinaryDiffEq, + Test +import ModelingToolkitStandardLibrary.Blocks + + +@parameters t +D = Differential(t) + +@testset "one falling body" begin + #body starts at 1m, moving up at 2m/s in gravity of 10 + @named body = Body(x0 = 1.0, v0 = 2.0, g=-10.0) + + @named model = ODESystem(body.T.f ~ 0, t, [], []; systems=[body]) + + sys = structural_simplify(model) + prob = ODEProblem(sys, [], (0, 1.0)) + sol = solve(prob, ImplicitMidpoint(), dt=0.1) + + #x = g*t^2/2 + v_int*t + x_int + #dx = g*t + v_int + + @test sol.retcode == :Success + @test sol[body.x, 1] == 1.0 + @test sol[body.dx, 1] == 2.0 + @test sol[body.x, end] ≈ -10/2 + 2.0 + 1.0 + +#= + using CairoMakie + lines(sol.t, sol[body.x]) +=# + +end + + +@testset "two bodies ForcedBoundary" begin + @named body1 = Body(ForcedBoundary; mass=100.0, x0 = 10.0, g=-10.0, lower_limit = 0.0) + @named body2 = Body(mass = 1000.0, g = -10.0) + @named spring = Spring(k=1000.0) + + eqs = [ + connect(body1.T, spring.T1) + connect(body2.T, spring.T2) + ] + + @named model = ODESystem(eqs, t, [], []; systems=[body1, body2, spring]) + + sys = structural_simplify(model) + prob = ODEProblem(sys, [], (0, 20.0)) + + # + + sol = solve(prob, ImplicitEuler(nlsolve = NLNewton(check_div = false, always_new = true, max_iter=100, relax=0.0)), + dt = 4e-4, + adaptive=false) + + @test sol(5.0; idxs=body1.x) ≈ 0.0 atol=1e-7 +end + +#= + using CairoMakie + begin + f = Figure() + a = Axis(f[1,1]) + lines!(a, sol.t, sol[body1.x]) + lines!(a, sol.t, sol[body2.x]) + # ylims!(a, -0.0001, 0.0001) + + + a = Axis(f[2,1]) + lines!(a, sol.t, sol[body1.f_ceil]) + lines!(a, sol.t, sol[body1.f_floor]) + lines!(a, sol.t, sol[spring.f]) + # ylims!(a, -100000, 0) + + a = Axis(f[3,1]) + lines!(a, sol.t, sol[body1.x] .== 0) + + f + end + + using Makie + tm = Observable(0.0) + begin + f = Figure() + a = Axis(f[1,2], aspect=DataAspect(), ) + hidedecorations!(a) + s = @lift(sol($tm)) + + m1x = @lift($s[2]) + m2x = @lift($s[4]) + + sz1 = 0.5 + lines!(a, [-sz1, sz1, sz1, -sz1, -sz1], @lift([$m1x, $m1x, $m1x+sz1*2, $m1x+sz1*2, $m1x])) + + sz = 1.0 + lines!(a, [-sz, sz, sz, -sz, -sz], @lift([$m2x, $m2x, $m2x+sz*2, $m2x+sz*2, $m2x])) + + lines!(a, [0.0, 0.0], @lift([$m1x+0.5, $m2x+1.0]), color=:red, width=2, linestyle=:dot) + + hlines!(a, 0.0, color=:gray) + hlines!(a, -10.0, color=:gray, linestyle=:dash) + + ylims!(a, -40, 15) + + a = Axis(f[1, 1], xlabel="time [s]", ylabel="position [m]") + lines!(a, sol.t, sol[body1.x]) + lines!(a, sol.t, sol[body2.x]) + + scatter!(a, tm, m1x) + scatter!(a, tm, m2x) + ylims!(a, -40, 15) + + f + end + + framerate = 30 + timestamps = range(0, 20, step=1/framerate) + + record(f, "time_animation.mp4", timestamps; + framerate = framerate) do t + tm[] = t + end +=# + + + From fe7d103d3cba0750b932ae0d770388ec90339c8c Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Tue, 13 Sep 2022 20:52:28 -0400 Subject: [PATCH 2/7] initial Body, Spring, Damper --- src/Mechanical/Translational/Translational.jl | 4 +- src/Mechanical/Translational/components.jl | 97 +++++-------------- test/Mechanical/translational.jl | 97 +------------------ 3 files changed, 29 insertions(+), 169 deletions(-) diff --git a/src/Mechanical/Translational/Translational.jl b/src/Mechanical/Translational/Translational.jl index 38323d904..c1786a7b8 100644 --- a/src/Mechanical/Translational/Translational.jl +++ b/src/Mechanical/Translational/Translational.jl @@ -4,7 +4,6 @@ Library to model 1-dimensional, translational mechanical systems module Translational using ModelingToolkit, Symbolics, IfElse -using ...Blocks: RealInput, RealOutput @parameters t D = Differential(t) @@ -12,8 +11,7 @@ D = Differential(t) export Port include("utils.jl") -export Body, Spring -export SpringDamperBoundary, ForcedBoundary +export Body, Spring, Damper include("components.jl") #export Torque diff --git a/src/Mechanical/Translational/components.jl b/src/Mechanical/Translational/components.jl index f43347984..4760c9b2e 100644 --- a/src/Mechanical/Translational/components.jl +++ b/src/Mechanical/Translational/components.jl @@ -1,74 +1,23 @@ - -@enum BoundaryType NoBoundary SpringDamperBoundary ForcedBoundary - -#Note: boundary is an argument because this is a compile time constant -function Body(boundary = NoBoundary; name, x0 = 0.0, v0 = 0.0, mass = 1.0, g=0.0, upper_limit = +Inf, lower_limit = -Inf, k=1e9, d=1e6) - @named T = Port(v0 = v0, f0 = mass*g) - @parameters mass=mass g=g +function Body(; name, x0 = 0.0, v0 = 0.0, m = 1.0, g=0.0) + @named port = Port(v0 = v0, f0 = m*g) + @parameters m=m g=g @variables begin x(t) = x0 dx(t) = v0 ddx(t) = 0.0 end - if boundary != NoBoundary - @variables begin - f_floor(t) = 0.0 - f_ceil(t) = 0.0 - end - @parameters begin - ubnd = upper_limit - lbnd = lower_limit - k=k - d=d - end - end - eqs = [ - T.v ~ dx + port.v ~ dx D(x) ~ dx D(dx) ~ ddx - ] - - if boundary == NoBoundary - feqs = [ - ddx * mass ~ mass*g + T.f - ] - return compose(ODESystem([eqs; feqs], t, [x, dx, ddx], [mass, g]; name = name), T) - elseif boundary == SpringDamperBoundary - - # let - Δceil = x - ubnd - Δfloor = lbnd - x - feqs = [ - ddx * mass ~ mass*g + T.f - f_floor - f_ceil - f_ceil ~ ifelse(x < ubnd, 0.0, Δceil*k + dx*d) - f_floor ~ ifelse(x > lbnd, 0.0, Δfloor*k + dx*d) - ] - return compose(ODESystem([eqs; feqs], t, [x, dx, ddx, f_ceil, f_floor], [mass, g, ubnd, lbnd, k, d]; name = name), T) - elseif boundary == ForcedBoundary - - # let - f = mass*g + T.f # external forces + ddx * m ~ m*g + port.f + ] - feqs = [ - ddx * mass ~ f - f_floor - f_ceil - 0 ~ ifelse((x >= ubnd), - ( x ) - ( ubnd ), # make x=>ubnd, solve for f_ceil to make this true - ( f_ceil ) - ( 0 ) - ) - 0 ~ ifelse((x <= lbnd) , - ( x ) - ( lbnd ), # make x=>lbnd, solve for f_floor to make this true - ( f_floor ) - ( 0 ) - ) - ] - return compose(ODESystem([eqs; feqs], t, [x, dx, ddx, f_ceil, f_floor], [mass, g, ubnd, lbnd]; name = name), T) - end - - + return compose(ODESystem(eqs, t, [x, dx, ddx], [m, g]; name = name), port) end @@ -80,33 +29,33 @@ function Spring(;name, k = 1e3, delta0 = 0.0) f(t) end - @named T1 = Port() - @named T2 = Port() + @named port_a = Port() + @named port_b = Port() eqs = [ D(x) ~ dx - dx ~ T1.v - T2.v + dx ~ port_a.v - port_b.v f ~ k*x - T1.f ~ +f - T2.f ~ -f + port_a.f ~ +f + port_b.f ~ -f ] - return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), T1, T2) + return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), port_a, port_b) end -function Damper(; name, c=1e2) - @parameters c=c - @variables dx=0.0 +function Damper(; name, d=1e2) + @parameters d=d + @variables dx(t)=0.0 f(t)=0.0 - @named T1 = Port() - @named T2 = Port() + @named port_a = Port() + @named port_b = Port() eqs = [ - T1.v - T2.v ~ dx - T1.f + T2.f ~ dx*c + port_a.v - port_b.v ~ dx + f ~ dx*d + port_a.f ~ +f + port_b.f ~ -f ] - return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), T1, T2) + return compose(ODESystem(eqs, t, [dx, f], [d]; name = name), port_a, port_b) end - - diff --git a/test/Mechanical/translational.jl b/test/Mechanical/translational.jl index a890988e8..b488c6983 100644 --- a/test/Mechanical/translational.jl +++ b/test/Mechanical/translational.jl @@ -1,16 +1,17 @@ using ModelingToolkitStandardLibrary.Mechanical.Translational, ModelingToolkit, OrdinaryDiffEq, Test -import ModelingToolkitStandardLibrary.Blocks + @parameters t D = Differential(t) + @testset "one falling body" begin #body starts at 1m, moving up at 2m/s in gravity of 10 @named body = Body(x0 = 1.0, v0 = 2.0, g=-10.0) - @named model = ODESystem(body.T.f ~ 0, t, [], []; systems=[body]) + @named model = ODESystem(body.port.f ~ 0, t, [], []; systems=[body]) sys = structural_simplify(model) prob = ODEProblem(sys, [], (0, 1.0)) @@ -23,104 +24,16 @@ D = Differential(t) @test sol[body.x, 1] == 1.0 @test sol[body.dx, 1] == 2.0 @test sol[body.x, end] ≈ -10/2 + 2.0 + 1.0 +end + -#= - using CairoMakie - lines(sol.t, sol[body.x]) -=# -end -@testset "two bodies ForcedBoundary" begin - @named body1 = Body(ForcedBoundary; mass=100.0, x0 = 10.0, g=-10.0, lower_limit = 0.0) - @named body2 = Body(mass = 1000.0, g = -10.0) - @named spring = Spring(k=1000.0) - eqs = [ - connect(body1.T, spring.T1) - connect(body2.T, spring.T2) - ] - @named model = ODESystem(eqs, t, [], []; systems=[body1, body2, spring]) - sys = structural_simplify(model) - prob = ODEProblem(sys, [], (0, 20.0)) - - # - - sol = solve(prob, ImplicitEuler(nlsolve = NLNewton(check_div = false, always_new = true, max_iter=100, relax=0.0)), - dt = 4e-4, - adaptive=false) - - @test sol(5.0; idxs=body1.x) ≈ 0.0 atol=1e-7 -end -#= - using CairoMakie - begin - f = Figure() - a = Axis(f[1,1]) - lines!(a, sol.t, sol[body1.x]) - lines!(a, sol.t, sol[body2.x]) - # ylims!(a, -0.0001, 0.0001) - - - a = Axis(f[2,1]) - lines!(a, sol.t, sol[body1.f_ceil]) - lines!(a, sol.t, sol[body1.f_floor]) - lines!(a, sol.t, sol[spring.f]) - # ylims!(a, -100000, 0) - - a = Axis(f[3,1]) - lines!(a, sol.t, sol[body1.x] .== 0) - - f - end - - using Makie - tm = Observable(0.0) - begin - f = Figure() - a = Axis(f[1,2], aspect=DataAspect(), ) - hidedecorations!(a) - s = @lift(sol($tm)) - - m1x = @lift($s[2]) - m2x = @lift($s[4]) - - sz1 = 0.5 - lines!(a, [-sz1, sz1, sz1, -sz1, -sz1], @lift([$m1x, $m1x, $m1x+sz1*2, $m1x+sz1*2, $m1x])) - - sz = 1.0 - lines!(a, [-sz, sz, sz, -sz, -sz], @lift([$m2x, $m2x, $m2x+sz*2, $m2x+sz*2, $m2x])) - - lines!(a, [0.0, 0.0], @lift([$m1x+0.5, $m2x+1.0]), color=:red, width=2, linestyle=:dot) - - hlines!(a, 0.0, color=:gray) - hlines!(a, -10.0, color=:gray, linestyle=:dash) - - ylims!(a, -40, 15) - - a = Axis(f[1, 1], xlabel="time [s]", ylabel="position [m]") - lines!(a, sol.t, sol[body1.x]) - lines!(a, sol.t, sol[body2.x]) - - scatter!(a, tm, m1x) - scatter!(a, tm, m2x) - ylims!(a, -40, 15) - - f - end - - framerate = 30 - timestamps = range(0, 20, step=1/framerate) - - record(f, "time_animation.mp4", timestamps; - framerate = framerate) do t - tm[] = t - end -=# From ac567e68869d6db14c4b22efd02cd1dc3ce18d42 Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Wed, 14 Sep 2022 13:02:34 -0400 Subject: [PATCH 3/7] implemented runtests --- test/runtests.jl | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/test/runtests.jl b/test/runtests.jl index 7ecd62366..7587a1744 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -20,4 +20,6 @@ using SafeTestsets @safetestset "Magnetic" begin include("Magnetic/magnetic.jl") end # Mechanical -@safetestset "Mechanical" begin include("Mechanical/rotational.jl") end +@safetestset "Mechanical Rotation" begin include("Mechanical/rotational.jl") end +@safetestset "Mechanical Translation" begin include("Mechanical/translational.jl") end + From e40ba3a6c3a9357d73f36e687d363419ad4eeff5 Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Thu, 15 Sep 2022 09:37:46 -0400 Subject: [PATCH 4/7] added Fixed and updated Spring --- src/Mechanical/Translational/components.jl | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/src/Mechanical/Translational/components.jl b/src/Mechanical/Translational/components.jl index 4760c9b2e..ff20f652c 100644 --- a/src/Mechanical/Translational/components.jl +++ b/src/Mechanical/Translational/components.jl @@ -1,3 +1,9 @@ +function Fixed(; name) + @named port = Port() + eqs = [port.v ~ 0] + return compose(ODESystem(eqs, t, [], []; name = name), port) +end + function Body(; name, x0 = 0.0, v0 = 0.0, m = 1.0, g=0.0) @named port = Port(v0 = v0, f0 = m*g) @parameters m=m g=g @@ -15,7 +21,6 @@ function Body(; name, x0 = 0.0, v0 = 0.0, m = 1.0, g=0.0) ddx * m ~ m*g + port.f ] - return compose(ODESystem(eqs, t, [x, dx, ddx], [m, g]; name = name), port) end @@ -26,7 +31,7 @@ function Spring(;name, k = 1e3, delta0 = 0.0) @variables begin x(t) = delta0 dx(t) = 0.0 - f(t) + f(t) = k*x end @named port_a = Port() From 3318ffd1418358096bfde9b0275d1eda4dce42ca Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Fri, 16 Sep 2022 23:23:09 -0400 Subject: [PATCH 5/7] Draft report comparing Position and Velocity --- docs/connections.html | 921 ++++++++++++++++++ docs/connections.jmd | 197 ++++ docs/through_across.png | Bin 0 -> 30457 bytes docs/through_across.svg | 317 ++++++ src/Mechanical/Mechanical.jl | 1 + src/Mechanical/Translational/Translational.jl | 2 +- src/Mechanical/Translational/components.jl | 51 +- .../TranslationalPosition.jl | 21 + .../TranslationalPosition/components.jl | 98 ++ .../TranslationalPosition/sources.jl | 12 + src/Mechanical/TranslationalPosition/utils.jl | 147 +++ 11 files changed, 1738 insertions(+), 29 deletions(-) create mode 100644 docs/connections.html create mode 100644 docs/connections.jmd create mode 100644 docs/through_across.png create mode 100644 docs/through_across.svg create mode 100644 src/Mechanical/TranslationalPosition/TranslationalPosition.jl create mode 100644 src/Mechanical/TranslationalPosition/components.jl create mode 100644 src/Mechanical/TranslationalPosition/sources.jl create mode 100644 src/Mechanical/TranslationalPosition/utils.jl diff --git a/docs/connections.html b/docs/connections.html new file mode 100644 index 000000000..0ee254502 --- /dev/null +++ b/docs/connections.html @@ -0,0 +1,921 @@ + + + + + + + + + + + + + + + + + + + + + +
+
+
+
+ + + +
+ +

Introduction

+

In Physical Network Acausal modeling each physical domain must define a connector to combine model components. This is somewhat analogus to real life connections that are seen in electronics (i.e. battery connected to a wire) or fluid dynamics (i.e. pump connected to a pipe), to name a couple examples. Each physical domain connector defines a minimum of 2 variables, one which is called a Through variable, and one which is called an Across variable. Both Modelica and SimScape define these variables in the same way:

+ +

However, the standard libraries differ on the selection of the Across variable for the Mechanical Translation and Rotation libraries, Modelica choosing position and angle and SimScape choosing velocity and angular velocity, respectively for Translation and Rotation. Modelica describes their decision here. In summary they would like to provide less integration in the model to avoid lossy numerical behavior, but this decision assumes the lowest order derivative is needed by the model. Numerically it is possible to define the connector either way, but there are some consequences to this decision, and therefore we will study them in detail here as they relate to ModelingToolkit.

+

Through and Across Variable Theory

+

General

+

The idea behind the selection of the through variable is that it should be a time derivative of some conserved quantity. The conserved quantity should be expressed by the across variable. In general terms the physical system is given by

+
    +

  • Energy Dissipation: $ \partial \color{blue}{across} / \partial t \cdot c_1 = \color{green}{through} $

    +
    • +

  • Flow: $ \color{green}{through} \cdot c_2 = \color{blue}{across} $

    +
    • +
+

Electrical

+

So for the Electrical domain the across variable is voltage and the through variable current. Therefore

+
    +

  • Energy Dissipation: $ \partial \color{blue}{voltage} / \partial t \cdot capacitance = \color{green}{current} $

    +
    • +

  • Flow: $ \color{green}{current} \cdot resistance = \color{blue}{voltage} $

    +
    • +
+

Translational

+

And for the translation domain, choosing velocity for the across variable and force for the through gives

+
    +

  • Energy Dissipation: $ \partial \color{blue}{velocity} / \partial t \cdot mass = \color{green}{force} $

    +
    • +

  • Flow: $ \color{green}{force} \cdot (1/damping) = \color{blue}{velocity} $

    +
    • +
+

The diagram here shows the similarity of problems in different physical domains.

+

Through and Across Variables

+

Electrical Domain Example

+

We can generate the above relationship with ModelingToolkit and the ModelingToolkitStandardLibrary using 3 blocks:

+
    +

  • Capacitor: for energy storage with initial voltage = 1V

    +
    • +

  • Resistor: for energy flow

    +
    • +

  • Ground: for energy sink

    +
    • +
+

As can be seen, this will give a 1 equation model matching our energy dissipation relationship

+ + +
+using ModelingToolkitStandardLibrary
+using ModelingToolkitStandardLibrary.Electrical, ModelingToolkit, OrdinaryDiffEq
+using CairoMakie
+
+@parameters t
+
+@named resistor = Resistor(R = 1)
+@named capacitor = Capacitor(C = 1)
+@named ground = Ground()
+
+eqs = [
+    connect(capacitor.n, resistor.p)
+    connect(resistor.n, ground.g, capacitor.p)
+    ]
+
+@named model = ODESystem(eqs, t; systems=[resistor, capacitor, ground])
+
+sys = structural_simplify(model)
+
+equations(sys)
+
+ + +
+1-element Vector{Equation}:
+ Differential(t)(capacitor₊v(t)) ~ capacitor₊i(t) / capacitor₊C
+
+ + +

The solution shows what we would expect, a non-linear disipation of voltage and releated decrease in current flow...

+ + +
+prob = ODEProblem(sys, [1.0], (0, 10.0), [])
+sol = solve(prob, ImplicitMidpoint(); dt=0.01)
+
+fig = Figure()
+ax = Axis(fig[1,1], ylabel="voltage [V]")
+lines!(ax, sol.t, sol[capacitor.v], label="sol[capacitor.v]")
+axislegend(ax)
+
+ax = Axis(fig[2,1], xlabel="time [s]", ylabel="current [A]")
+lines!(ax, sol.t, -sol[resistor.i], label="sol[resistor.i]")
+axislegend(ax)
+
+fig
+
+ + + + +

Translational Domain Example (Across Variable = velocity)

+

Now using the Translational library based on velocity, we can see the same relationship with a system reduced to a single equation, using the components:

+
    +

  • Body (i.e. moving mass): for kinetic energy storage with an initial velocity = 1m/s

    +
    • +

  • Damper: for energy flow

    +
    • +

  • Fixed: for energy sink

    +
    • +
+ + +
+module TranslationalVelocity
+    using ModelingToolkit
+    using ModelingToolkitStandardLibrary.Mechanical.Translational
+
+    @parameters t
+
+    @named damping = Damper(d = 1)
+    @named body = Body(m = 1, v0=1)
+    @named ground = Fixed()
+
+    eqs = [
+        connect(damping.port_a, body.port)
+        connect(ground.port, damping.port_b)
+        ]
+
+    @named model = ODESystem(eqs, t; systems=[damping, body, ground])
+
+    sys = structural_simplify(model)
+end
+
+sys = TranslationalVelocity.sys
+equations(sys)
+
+ + +
+1-element Vector{Equation}:
+ Differential(t)(body₊v(t)) ~ body₊f(t) / body₊m
+
+ + +

As expected we have a similar solution...

+ + +
+prob = ODEProblem(sys, [1.0], (0, 10.0), [])
+sol_v = solve(prob, ImplicitMidpoint(); dt=0.01)
+
+fig = Figure()
+ax = Axis(fig[1,1], ylabel="velocity [m/s]")
+lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v]")
+axislegend(ax)
+
+ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]")
+lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f]")
+axislegend(ax)
+
+fig
+
+ + + + +

Translational Domain Example (Across Variable = position)

+

Now, let's consider the position based approach. We can build the same model with the same components. As can be seen, we now end of up with 2 equations, because we need to relate the lower derivative (position) to force (with acceleration).

+ + +
+module TranslationalPosition
+    using ModelingToolkit
+    using ModelingToolkitStandardLibrary.Mechanical.TranslationalPosition
+
+    @parameters t
+    D = Differential(t)
+
+    # Let's define a simple body that only tracks the across and through variables...
+    function Body(; name, m, v0 = 0.0)
+        @named flange = Flange()
+        pars = @parameters m=m v0=v0
+        vars = @variables begin
+            v(t) = v0
+            f(t) = m*v0
+        end
+        eqs = [
+            D(flange.s) ~ v
+            flange.f ~ f
+
+            D(v) ~ m/f
+            ]
+        return compose(ODESystem(eqs, t, vars, pars; name = name), flange)
+    end
+
+    @named damping = Damper(d = 1)
+    @named body = Body(m = 1, v0=1)
+    @named ground = Fixed()
+
+    eqs = [
+        connect(damping.flange_a, body.flange)
+        connect(ground.flange, damping.flange_b)
+        ]
+
+    @named model = ODESystem(eqs, t; systems=[damping, body, ground])
+
+    sys = structural_simplify(model)
+end
+
+sys = TranslationalPosition.sys
+
+full_equations(sys)
+
+ + +
+2-element Vector{Equation}:
+ Differential(t)(body₊flange₊s(t)) ~ body₊v(t)
+ Differential(t)(body₊v(t)) ~ -(body₊m / (damping₊d*body₊v(t)))
+
+ + +

Suprisingly this problem fails to converge! As can be seen something is not working when compared to the position based result.

+ + +
+prob = ODEProblem(sys, [0.0, 1.0], (0, 10.0), [])
+sol_p = solve(prob, ImplicitMidpoint(); dt=0.01)
+
+fig = Figure()
+ax = Axis(fig[1,1], ylabel="velocity [m/s]")
+lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.v], label="sol[body.v] (position based)")
+lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)")
+axislegend(ax)
+
+ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]")
+lines!(ax, sol_p.t, -sol_p[TranslationalVelocity.body.f], label="sol[body.f] (position based)")
+lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)")
+axislegend(ax)
+
+fig
+
+ + + + + +
+
+

+ Published from connections.jmd + using Weave.jl v0.10.10 on 2022-09-16. +

+
+
+
+
+ + + diff --git a/docs/connections.jmd b/docs/connections.jmd new file mode 100644 index 000000000..52d327ce3 --- /dev/null +++ b/docs/connections.jmd @@ -0,0 +1,197 @@ +# Introduction +In Physical Network Acausal modeling each physical domain must define a **connector** to combine model components. This is somewhat analogus to real life connections that are seen in electronics (i.e. battery connected to a wire) or fluid dynamics (i.e. pump connected to a pipe), to name a couple examples. Each physical domain **connector** defines a minimum of 2 variables, one which is called a *Through* variable, and one which is called an *Across* variable. Both Modelica and SimScape define these variables in the same way: + +- [Modelica Connectors](https://mbe.modelica.university/components/connectors/#acausal-connection) +- [SimScape Connectors](https://www.mathworks.com/help/simscape/ug/basic-principles-of-modeling-physical-networks.html#bq89sba-6) + +However, the standard libraries differ on the selection of the Across variable for the Mechanical Translation and Rotation libraries, Modelica choosing position and angle and SimScape choosing velocity and angular velocity, respectively for Translation and Rotation. Modelica describes their decision [here](https://mbe.modelica.university/components/connectors/simple_domains/). In summary they would like to provide less integration in the model to avoid lossy numerical behavior, but this decision assumes the lowest order derivative is needed by the model. Numerically it is possible to define the connector either way, but there are some consequences to this decision, and therefore we will study them in detail here as they relate to ModelingToolkit. + +# Through and Across Variable Theory +### General +The idea behind the selection of the **through** variable is that it should be a time derivative of some conserved quantity. The conserved quantity should be expressed by the **across** variable. In general terms the physical system is given by + +- Energy Dissipation: $ \partial \color{blue}{across} / \partial t \cdot c_1 = \color{green}{through} $ +- Flow: $ \color{green}{through} \cdot c_2 = \color{blue}{across} $ + +### Electrical +So for the Electrical domain the across variable is *voltage* and the through variable *current*. Therefore + +- Energy Dissipation: $ \partial \color{blue}{voltage} / \partial t \cdot capacitance = \color{green}{current} $ +- Flow: $ \color{green}{current} \cdot resistance = \color{blue}{voltage} $ + +### Translational +And for the translation domain, choosing *velocity* for the across variable and *force* for the through gives + +- Energy Dissipation: $ \partial \color{blue}{velocity} / \partial t \cdot mass = \color{green}{force} $ +- Flow: $ \color{green}{force} \cdot (1/damping) = \color{blue}{velocity} $ + +The diagram here shows the similarity of problems in different physical domains. + +![Through and Across Variables](through_across.png) + + +# Electrical Domain Example +We can generate the above relationship with ModelingToolkit and the ModelingToolkitStandardLibrary using 3 blocks: + +- Capacitor: for energy storage with initial voltage = 1V +- Resistor: for energy flow +- Ground: for energy sink + +As can be seen, this will give a 1 equation model matching our energy dissipation relationship + +```julia +using ModelingToolkitStandardLibrary +using ModelingToolkitStandardLibrary.Electrical, ModelingToolkit, OrdinaryDiffEq +using CairoMakie + +@parameters t + +@named resistor = Resistor(R = 1) +@named capacitor = Capacitor(C = 1) +@named ground = Ground() + +eqs = [ + connect(capacitor.n, resistor.p) + connect(resistor.n, ground.g, capacitor.p) + ] + +@named model = ODESystem(eqs, t; systems=[resistor, capacitor, ground]) + +sys = structural_simplify(model) + +equations(sys) +``` + +The solution shows what we would expect, a non-linear disipation of voltage and releated decrease in current flow... + +```julia +prob = ODEProblem(sys, [1.0], (0, 10.0), []) +sol = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="voltage [V]") +lines!(ax, sol.t, sol[capacitor.v], label="sol[capacitor.v]") +axislegend(ax) + +ax = Axis(fig[2,1], xlabel="time [s]", ylabel="current [A]") +lines!(ax, sol.t, -sol[resistor.i], label="sol[resistor.i]") +axislegend(ax) + +fig +``` +# Translational Domain Example (Across Variable = velocity) +Now using the Translational library based on velocity, we can see the same relationship with a system reduced to a single equation, using the components: + +- Body (i.e. moving mass): for kinetic energy storage with an initial velocity = 1m/s +- Damper: for energy flow +- Fixed: for energy sink + +```julia +module TranslationalVelocity + using ModelingToolkit + using ModelingToolkitStandardLibrary.Mechanical.Translational + + @parameters t + + @named damping = Damper(d = 1) + @named body = Body(m = 1, v0=1) + @named ground = Fixed() + + eqs = [ + connect(damping.port_a, body.port) + connect(ground.port, damping.port_b) + ] + + @named model = ODESystem(eqs, t; systems=[damping, body, ground]) + + sys = structural_simplify(model) +end + +sys = TranslationalVelocity.sys +equations(sys) +``` + +As expected we have a similar solution... +```julia +prob = ODEProblem(sys, [1.0], (0, 10.0), []) +sol_v = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="velocity [m/s]") +lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v]") +axislegend(ax) + +ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]") +lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f]") +axislegend(ax) + +fig +``` + +# Translational Domain Example (Across Variable = position) + +Now, let's consider the position based approach. We can build the same model with the same components. As can be seen, we now end of up with 2 equations, because we need to relate the lower derivative (position) to force (with acceleration). + +```julia +module TranslationalPosition + using ModelingToolkit + using ModelingToolkitStandardLibrary.Mechanical.TranslationalPosition + + @parameters t + D = Differential(t) + + # Let's define a simple body that only tracks the across and through variables... + function Body(; name, m, v0 = 0.0) + @named flange = Flange() + pars = @parameters m=m v0=v0 + vars = @variables begin + v(t) = v0 + f(t) = m*v0 + end + eqs = [ + D(flange.s) ~ v + flange.f ~ f + + D(v) ~ m/f + ] + return compose(ODESystem(eqs, t, vars, pars; name = name), flange) + end + + @named damping = Damper(d = 1) + @named body = Body(m = 1, v0=1) + @named ground = Fixed() + + eqs = [ + connect(damping.flange_a, body.flange) + connect(ground.flange, damping.flange_b) + ] + + @named model = ODESystem(eqs, t; systems=[damping, body, ground]) + + sys = structural_simplify(model) +end + +sys = TranslationalPosition.sys + +full_equations(sys) +``` + +Suprisingly this problem fails to converge! As can be seen something is not working when compared to the position based result. + +```julia +prob = ODEProblem(sys, [0.0, 1.0], (0, 10.0), []) +sol_p = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="velocity [m/s]") +lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.v], label="sol[body.v] (position based)") +lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)") +axislegend(ax) + +ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]") +lines!(ax, sol_p.t, -sol_p[TranslationalVelocity.body.f], label="sol[body.f] (position based)") +lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)") +axislegend(ax) + +fig +``` \ No newline at end of file diff --git a/docs/through_across.png b/docs/through_across.png new file mode 100644 index 0000000000000000000000000000000000000000..758704af8137c536caf0aef2816fb351c54143a4 GIT binary patch literal 30457 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100644 index 000000000..82edb6497 --- /dev/null +++ b/docs/through_across.svg @@ -0,0 +1,317 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + current + force + stored energy =capacitance x voltage + stored energy = mass x velocity + (f) + (i) + (v) + (v) + + + Through Variable + Across Variable + + diff --git a/src/Mechanical/Mechanical.jl b/src/Mechanical/Mechanical.jl index 2e87392b9..aa2f8dc7f 100644 --- a/src/Mechanical/Mechanical.jl +++ b/src/Mechanical/Mechanical.jl @@ -7,5 +7,6 @@ using ModelingToolkit include("Rotational/Rotational.jl") include("Translational/Translational.jl") +include("TranslationalPosition/TranslationalPosition.jl") end diff --git a/src/Mechanical/Translational/Translational.jl b/src/Mechanical/Translational/Translational.jl index c1786a7b8..c9f52b004 100644 --- a/src/Mechanical/Translational/Translational.jl +++ b/src/Mechanical/Translational/Translational.jl @@ -11,7 +11,7 @@ D = Differential(t) export Port include("utils.jl") -export Body, Spring, Damper +export Body, Spring, Damper, Fixed include("components.jl") #export Torque diff --git a/src/Mechanical/Translational/components.jl b/src/Mechanical/Translational/components.jl index ff20f652c..db0825bc0 100644 --- a/src/Mechanical/Translational/components.jl +++ b/src/Mechanical/Translational/components.jl @@ -4,63 +4,58 @@ function Fixed(; name) return compose(ODESystem(eqs, t, [], []; name = name), port) end -function Body(; name, x0 = 0.0, v0 = 0.0, m = 1.0, g=0.0) - @named port = Port(v0 = v0, f0 = m*g) - @parameters m=m g=g - @variables begin - x(t) = x0 - dx(t) = v0 - ddx(t) = 0.0 +function Body(; name, v0 = 0.0, m = 1.0) + @named port = Port(v0 = v0) + pars = @parameters m=m + vars = @variables begin + v(t) = v0 + f(t) = m*v0 end eqs = [ - port.v ~ dx - - D(x) ~ dx - D(dx) ~ ddx - - ddx * m ~ m*g + port.f + port.v ~ v + port.f ~ f + D(v) ~ f/m ] - return compose(ODESystem(eqs, t, [x, dx, ddx], [m, g]; name = name), port) + return compose(ODESystem(eqs, t, vars, pars; name = name), port) end function Spring(;name, k = 1e3, delta0 = 0.0) - @parameters k=k - @variables begin - x(t) = delta0 - dx(t) = 0.0 - f(t) = k*x + pars = @parameters k=k + vars = @variables begin + Δx(t) = delta0 + f(t) = k*delta0 end @named port_a = Port() @named port_b = Port() eqs = [ - D(x) ~ dx - dx ~ port_a.v - port_b.v - f ~ k*x + D(Δx) ~ port_a.v - port_b.v + f ~ k*Δx port_a.f ~ +f port_b.f ~ -f ] - return compose(ODESystem(eqs, t, [x, dx, f], [k]; name = name), port_a, port_b) + return compose(ODESystem(eqs, t, vars, pars; name = name), port_a, port_b) end function Damper(; name, d=1e2) - @parameters d=d - @variables dx(t)=0.0 f(t)=0.0 + pars = @parameters d=d + vars = @variables v(t)=0.0 f(t)=0.0 @named port_a = Port() @named port_b = Port() eqs = [ - port_a.v - port_b.v ~ dx - f ~ dx*d + v ~ port_a.v - port_b.v + f ~ v*d + port_a.f ~ +f port_b.f ~ -f ] - return compose(ODESystem(eqs, t, [dx, f], [d]; name = name), port_a, port_b) + return compose(ODESystem(eqs, t, vars, pars; name = name), port_a, port_b) end diff --git a/src/Mechanical/TranslationalPosition/TranslationalPosition.jl b/src/Mechanical/TranslationalPosition/TranslationalPosition.jl new file mode 100644 index 000000000..9891ddbcc --- /dev/null +++ b/src/Mechanical/TranslationalPosition/TranslationalPosition.jl @@ -0,0 +1,21 @@ +""" +Library to model 1-dimensional, translational mechanical components. +""" +module TranslationalPosition + +using ModelingToolkit, Symbolics, IfElse +using ...Blocks: RealInput, RealOutput + +@parameters t +D = Differential(t) + +export Flange +include("utils.jl") + +export Fixed, Mass, Spring, Damper, IdealGear +include("components.jl") + +export Force +include("sources.jl") + +end diff --git a/src/Mechanical/TranslationalPosition/components.jl b/src/Mechanical/TranslationalPosition/components.jl new file mode 100644 index 000000000..6e9ef0550 --- /dev/null +++ b/src/Mechanical/TranslationalPosition/components.jl @@ -0,0 +1,98 @@ +""" + Fixed(;name, s0=0.0) + +Flange fixed in housing at a given position. + +# Parameters: +- `s0`: [m] Fixed offset position of housing + +# Connectors: +- `flange: 1-dim. translational flange` +""" +function Fixed(; name, s0 = 0.0) + @named flange = Flange() + @parameters s0 = s0 + eqs = [flange.s ~ s0] + return compose(ODESystem(eqs, t, [], [s0]; name = name), flange) +end + +""" + Mass(;name, m, s_start=0.0, v_start=0.0, a_start=0.0) + +Sliding mass with inertia + +# Parameters: +- `m`: [kg] Mass of sliding mass +- `s_start`: [m] Initial value of absolute position of sliding mass +- `v_start`: [m/s] Initial value of absolute linear velocity of sliding mass +- `a_start`: [m/s²] Initial value of absolute linear acceleration of sliding mass + +# States: +- `s`: [m] Absolute position of sliding mass +- `v`: [m/s] Absolute linear velocity of sliding mass (= der(s)) +- `a`: [m/s²] Absolute linear acceleration of sliding mass (= der(v)) + +# Connectors: +- `flange_a: 1-dim. translational flange on one side of mass` +- `flange_b: 1-dim. translational flange on opposite side of mass` +""" +function Mass(; name, m, s_start = 0.0, v_start = 0.0, a_start = 0.0) + @named flange_a = Flange() + @named flange_b = Flange() + @parameters m = m + sts = @variables begin + s(t) = s_start + v(t) = v_start + a(t) = a_start + end + eqs = [s ~ flange_a.s + s ~ flange_b.s + D(s) ~ v + D(v) ~ a + m * a ~ flange_a.f + flange_b.f] + return compose(ODESystem(eqs, t, sts, [m]; name = name), flange_a, flange_b) +end + +""" + Spring(;name, c, s_rel0=0.0) + +Linear 1D translational spring + +# Parameters: +- `c`: [N/m] Spring constant +- `s_rel0`: Unstretched spring length + +# Connectors: +- `flange_a: 1-dim. translational flange on one side of spring` +- `flange_b: 1-dim. translational flange on opposite side of spring` +""" +function Spring(; name, c, s_rel0 = 0.0) + @named partial_comp = PartialCompliant() + @unpack s_rel, f = partial_comp + pars = @parameters begin + c = c + s_rel0 = s_rel0 + end + eqs = [f ~ c * (s_rel - s_rel0)] + extend(ODESystem(eqs, t, [], pars; name = name), partial_comp) +end + +""" + Damper(;name, d) + +Linear 1D translational damper + +# Parameters: +- `d`: [N.s/m] Damping constant + +# Connectors: +- `flange_a: 1-dim. translational flange on one side of damper` +- `flange_b: 1-dim. translational flange on opposite side of damper` +""" +function Damper(; name, d) + @named partial_comp = PartialCompliantWithRelativeStates() + @unpack v_rel, f = partial_comp + pars = @parameters d = d + eqs = [f ~ d * v_rel] + extend(ODESystem(eqs, t, [], pars; name = name), partial_comp) +end diff --git a/src/Mechanical/TranslationalPosition/sources.jl b/src/Mechanical/TranslationalPosition/sources.jl new file mode 100644 index 000000000..3854cb82b --- /dev/null +++ b/src/Mechanical/TranslationalPosition/sources.jl @@ -0,0 +1,12 @@ +""" + Force(;name) + +Input signal acting as external force on a flange +""" +function Force(; name, use_support = false) + @named partial_element = PartialElementaryOneFlangeAndSupport2(use_support = use_support) + @unpack flange = partial_element + @named f = RealInput() # Accelerating force acting at flange (= -flange.tau) + eqs = [flange.f ~ -f.u] + return extend(ODESystem(eqs, t, [], []; name = name, systems = [f]), partial_element) +end diff --git a/src/Mechanical/TranslationalPosition/utils.jl b/src/Mechanical/TranslationalPosition/utils.jl new file mode 100644 index 000000000..e419fbfc0 --- /dev/null +++ b/src/Mechanical/TranslationalPosition/utils.jl @@ -0,0 +1,147 @@ +@connector function Flange(; name) + sts = @variables begin + s(t) + f(t), [connect = Flow] + end + ODESystem(Equation[], t, sts, [], name = name, defaults = Dict(s => 0.0, f => 0.0)) +end +Base.@doc """ + Flange(;name) + +1-dim. translational flange. + +# States: +- `s`: [m] Absolute position of flange +- `f`: [N] Cut force into the flange +""" Flange + +@connector function Support(; name) + sts = @variables begin + s(t) + f(t), [connect = Flow] + end + ODESystem(Equation[], t, sts, [], name = name, defaults = Dict(s => 0.0, f => 0.0)) +end +Base.@doc """ + Support(;name) + +Support/housing 1-dim. translational flange. + +# States: +- `s`: [m] Absolute position of the support/housing +- `f`: [N] Cut force into the flange +""" Support + +""" + PartialCompliant(;name, s_rel_start=0.0, f_start=0.0) + +Partial model for the compliant connection of two translational 1-dim. flanges. + +# Parameters: +- `s_rel_start`: [m] Initial relative distance between the flanges +- `f_start`: [N] Initial force between flanges + +# States: +- `s_rel`: [m] Relative distance (= flange_b.s - flange_a.s) +- `f`: [N] Force between flanges (= flange_b.f) +""" +function PartialCompliant(; name, s_rel_start = 0.0, f_start = 0.0) + @named flange_a = Flange() + @named flange_b = Flange() + sts = @variables begin + s_rel(t) = s_rel_start + f(t) = f_start + end + eqs = [s_rel ~ flange_b.s - flange_a.s + flange_b.f ~ f + flange_a.f ~ -f] + return compose(ODESystem(eqs, t, sts, []; name = name), flange_a, flange_b) +end + +""" + PartialCompliantWithRelativeStates(;name, s_rel_start=0.0, v_rel_start=0.0, a_rel_start=0.0, f_start=0.0) + +Partial model for the compliant connection of two translational 1-dim. flanges. + +# Parameters: +- `s_rel_start`: [m] Initial relative distance +- `v_rel_start`: [m/s] Initial relative linear velocity (= der(s_rel)) +- `a_rel_start`: [m/s²] Initial relative linear acceleration (= der(v_rel)) +- `f_start`: [N] Initial force between flanges + +# States: +- `s_rel`: [m] Relative distance (= flange_b.phi - flange_a.phi) +- `v_rel`: [m/s] Relative linear velocity (= der(s_rel)) +- `a_rel`: [m/s²] Relative linear acceleration (= der(v_rel)) +- `f`: [N] Force between flanges (= flange_b.f) +""" +function PartialCompliantWithRelativeStates(; name, s_rel_start = 0.0, v_rel_start = 0.0, + a_rel_start = 0.0, f_start = 0.0) + @named flange_a = Flange() + @named flange_b = Flange() + sts = @variables begin + s_rel(t) = s_rel_start + v_rel(t) = v_rel_start + a_rel(t) = a_rel_start + f(t) = f_start + end + eqs = [s_rel ~ flange_b.s - flange_a.s + D(s_rel) ~ v_rel + D(v_rel) ~ a_rel + flange_b.f ~ f + flange_a.f ~ -f] + return compose(ODESystem(eqs, t, sts, []; name = name), flange_a, flange_b) +end + +""" + PartialElementaryOneFlangeAndSupport2(;name, use_support=false) + +Partial model for a component with one translational 1-dim. shaft flange and a support used for textual modeling, i.e., for elementary models + +# Parameters: +- `use_support`: If support flange enabled, otherwise implicitly grounded + +# States: +- `s_support`: [m] Absolute position of support flange" +""" +function PartialElementaryOneFlangeAndSupport2(; name, use_support = false) + @named flange = Flange() + sys = [flange] + @variables s_support(t) + if use_support + @named support = Support() + eqs = [support.s ~ s_support + support.f ~ -flange.f] + push!(sys, support) + else + eqs = [s_support ~ 0] + end + return compose(ODESystem(eqs, t, [s_support], []; name = name), sys) +end + +""" + PartialElementaryTwoFlangesAndSupport2(;name, use_support=false) + +Partial model for a component with two translational 1-dim. flanges and a support used for textual modeling, i.e., for elementary models + +# Parameters: +- `use_support`: If support flange enabled, otherwise implicitly grounded + +# States: +- `s_support`: [m] Absolute position of support flange" +""" +function PartialElementaryTwoFlangesAndSupport2(; name, use_support = false) + @named flange_a = Flange() + @named flange_b = Flange() + sys = [flange_a, flange_b] + @variables s_support(t) = 0.0 + if use_support + @named support = Support() + eqs = [support.s ~ s_support + support.f ~ -flange_a.f - flange_b.f] + push!(sys, support) + else + eqs = [s_support ~ 0] + end + return compose(ODESystem(eqs, t, [s_support], []; name = name), sys) +end From 04028605bde6b2a2ecf3a8cd374d498558a9e940 Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Sat, 17 Sep 2022 06:14:00 -0400 Subject: [PATCH 6/7] fixed position bug --- docs/connections.html | 24 ++-- docs/connections.jmd | 20 ++- docs/connections.md | 239 +++++++++++++++++++++++++++++++ docs/figures/connections_2_1.png | Bin 0 -> 28110 bytes docs/figures/connections_4_1.png | Bin 0 -> 28184 bytes docs/figures/connections_6_1.png | Bin 0 -> 46573 bytes 6 files changed, 266 insertions(+), 17 deletions(-) create mode 100644 docs/connections.md create mode 100644 docs/figures/connections_2_1.png create mode 100644 docs/figures/connections_4_1.png create mode 100644 docs/figures/connections_6_1.png diff --git a/docs/connections.html b/docs/connections.html index 0ee254502..d640930bc 100644 --- a/docs/connections.html +++ b/docs/connections.html @@ -793,13 +793,13 @@

Translational Domain Example (Across Variable = velocity)

end sys = TranslationalVelocity.sys -equations(sys) +full_equations(sys) 
 1-element Vector{Equation}:
- Differential(t)(body₊v(t)) ~ body₊f(t) / body₊m
+ Differential(t)(body₊v(t)) ~ (-damping₊d*body₊v(t)) / body₊m
@@ -849,7 +849,7 @@

Translational Domain Example (Across Variable = position)

D(flange.s) ~ v flange.f ~ f - D(v) ~ m/f + D(v) ~ f/m ] return compose(ODESystem(eqs, t, vars, pars; name = name), flange) end @@ -877,11 +877,11 @@

Translational Domain Example (Across Variable = position)

 2-element Vector{Equation}:
  Differential(t)(body₊flange₊s(t)) ~ body₊v(t)
- Differential(t)(body₊v(t)) ~ -(body₊m / (damping₊d*body₊v(t)))
+ Differential(t)(body₊v(t)) ~ -((damping₊d*body₊v(t)) / body₊m)
-

Suprisingly this problem fails to converge! As can be seen something is not working when compared to the position based result.

+

As can be seen, we get exactly the same result. The only difference here is that we are solving an extra equation, which allows us to plot the body position as well.

@@ -891,26 +891,30 @@ 
Translational Domain Example (Across Variable = position)

 fig = Figure()
 ax = Axis(fig[1,1], ylabel="velocity [m/s]")
 lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.v], label="sol[body.v] (position based)")
-lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)")
+lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)", linestyle=:dash)
 axislegend(ax)
 
-ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]")
+ax = Axis(fig[2,1], ylabel="force [N]")
 lines!(ax, sol_p.t, -sol_p[TranslationalVelocity.body.f], label="sol[body.f] (position based)")
-lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)")
+lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)", linestyle=:dash)
+axislegend(ax)
+
+ax = Axis(fig[3,1], xlabel="time [s]", ylabel="position [m]")
+lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.flange.s], label="sol[body.flange.s]")
 axislegend(ax)
 
 fig
- +

Published from connections.jmd - using Weave.jl v0.10.10 on 2022-09-16. + using Weave.jl v0.10.10 on 2022-09-17.

diff --git a/docs/connections.jmd b/docs/connections.jmd index 52d327ce3..ef36bc53d 100644 --- a/docs/connections.jmd +++ b/docs/connections.jmd @@ -108,7 +108,7 @@ module TranslationalVelocity end sys = TranslationalVelocity.sys -equations(sys) +full_equations(sys) ``` As expected we have a similar solution... @@ -152,7 +152,7 @@ module TranslationalPosition D(flange.s) ~ v flange.f ~ f - D(v) ~ m/f + D(v) ~ f/m ] return compose(ODESystem(eqs, t, vars, pars; name = name), flange) end @@ -176,7 +176,7 @@ sys = TranslationalPosition.sys full_equations(sys) ``` -Suprisingly this problem fails to converge! As can be seen something is not working when compared to the position based result. +As can be seen, we get exactly the same result. The only difference here is that we are solving an extra equation, which allows us to plot the body position as well. ```julia prob = ODEProblem(sys, [0.0, 1.0], (0, 10.0), []) @@ -185,13 +185,19 @@ sol_p = solve(prob, ImplicitMidpoint(); dt=0.01) fig = Figure() ax = Axis(fig[1,1], ylabel="velocity [m/s]") lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.v], label="sol[body.v] (position based)") -lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)") +lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)", linestyle=:dash) axislegend(ax) -ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]") +ax = Axis(fig[2,1], ylabel="force [N]") lines!(ax, sol_p.t, -sol_p[TranslationalVelocity.body.f], label="sol[body.f] (position based)") -lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)") +lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)", linestyle=:dash) +axislegend(ax) + +ax = Axis(fig[3,1], xlabel="time [s]", ylabel="position [m]") +lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.flange.s], label="sol[body.flange.s]") axislegend(ax) fig -``` \ No newline at end of file +``` + + diff --git a/docs/connections.md b/docs/connections.md new file mode 100644 index 000000000..0f80bf2cc --- /dev/null +++ b/docs/connections.md @@ -0,0 +1,239 @@ +# Introduction +In Physical Network Acausal modeling each physical domain must define a **connector** to combine model components. This is somewhat analogus to real life connections that are seen in electronics (i.e. battery connected to a wire) or fluid dynamics (i.e. pump connected to a pipe), to name a couple examples. Each physical domain **connector** defines a minimum of 2 variables, one which is called a *Through* variable, and one which is called an *Across* variable. Both Modelica and SimScape define these variables in the same way: + +- [Modelica Connectors](https://mbe.modelica.university/components/connectors/#acausal-connection) +- [SimScape Connectors](https://www.mathworks.com/help/simscape/ug/basic-principles-of-modeling-physical-networks.html#bq89sba-6) + +However, the standard libraries differ on the selection of the Across variable for the Mechanical Translation and Rotation libraries, Modelica choosing position and angle and SimScape choosing velocity and angular velocity, respectively for Translation and Rotation. Modelica describes their decision [here](https://mbe.modelica.university/components/connectors/simple_domains/). In summary they would like to provide less integration in the model to avoid lossy numerical behavior, but this decision assumes the lowest order derivative is needed by the model. Numerically it is possible to define the connector either way, but there are some consequences to this decision, and therefore we will study them in detail here as they relate to ModelingToolkit. + +# Through and Across Variable Theory +### General +The idea behind the selection of the **through** variable is that it should be a time derivative of some conserved quantity. The conserved quantity should be expressed by the **across** variable. In general terms the physical system is given by + +- Energy Dissipation: $ \partial \color{blue}{across} / \partial t \cdot c_1 = \color{green}{through} $ +- Flow: $ \color{green}{through} \cdot c_2 = \color{blue}{across} $ + +### Electrical +So for the Electrical domain the across variable is *voltage* and the through variable *current*. Therefore + +- Energy Dissipation: $ \partial \color{blue}{voltage} / \partial t \cdot capacitance = \color{green}{current} $ +- Flow: $ \color{green}{current} \cdot resistance = \color{blue}{voltage} $ + +### Translational +And for the translation domain, choosing *velocity* for the across variable and *force* for the through gives + +- Energy Dissipation: $ \partial \color{blue}{velocity} / \partial t \cdot mass = \color{green}{force} $ +- Flow: $ \color{green}{force} \cdot (1/damping) = \color{blue}{velocity} $ + +The diagram here shows the similarity of problems in different physical domains. + +![Through and Across Variables](through_across.png) + + +# Electrical Domain Example +We can generate the above relationship with ModelingToolkit and the ModelingToolkitStandardLibrary using 3 blocks: + +- Capacitor: for energy storage with initial voltage = 1V +- Resistor: for energy flow +- Ground: for energy sink + +As can be seen, this will give a 1 equation model matching our energy dissipation relationship + +```julia +using ModelingToolkitStandardLibrary +using ModelingToolkitStandardLibrary.Electrical, ModelingToolkit, OrdinaryDiffEq +using CairoMakie + +@parameters t + +@named resistor = Resistor(R = 1) +@named capacitor = Capacitor(C = 1) +@named ground = Ground() + +eqs = [ + connect(capacitor.n, resistor.p) + connect(resistor.n, ground.g, capacitor.p) + ] + +@named model = ODESystem(eqs, t; systems=[resistor, capacitor, ground]) + +sys = structural_simplify(model) + +equations(sys) +``` + +``` +1-element Vector{Equation}: + Differential(t)(capacitor₊v(t)) ~ capacitor₊i(t) / capacitor₊C +``` + + + + + +The solution shows what we would expect, a non-linear disipation of voltage and releated decrease in current flow... + +```julia +prob = ODEProblem(sys, [1.0], (0, 10.0), []) +sol = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="voltage [V]") +lines!(ax, sol.t, sol[capacitor.v], label="sol[capacitor.v]") +axislegend(ax) + +ax = Axis(fig[2,1], xlabel="time [s]", ylabel="current [A]") +lines!(ax, sol.t, -sol[resistor.i], label="sol[resistor.i]") +axislegend(ax) + +fig +``` + +![](figures/connections_2_1.png) + + +# Translational Domain Example (Across Variable = velocity) +Now using the Translational library based on velocity, we can see the same relationship with a system reduced to a single equation, using the components: + +- Body (i.e. moving mass): for kinetic energy storage with an initial velocity = 1m/s +- Damper: for energy flow +- Fixed: for energy sink + +```julia +module TranslationalVelocity + using ModelingToolkit + using ModelingToolkitStandardLibrary.Mechanical.Translational + + @parameters t + + @named damping = Damper(d = 1) + @named body = Body(m = 1, v0=1) + @named ground = Fixed() + + eqs = [ + connect(damping.port_a, body.port) + connect(ground.port, damping.port_b) + ] + + @named model = ODESystem(eqs, t; systems=[damping, body, ground]) + + sys = structural_simplify(model) +end + +sys = TranslationalVelocity.sys +full_equations(sys) +``` + +``` +1-element Vector{Equation}: + Differential(t)(body₊v(t)) ~ (-damping₊d*body₊v(t)) / body₊m +``` + + + + + +As expected we have a similar solution... +```julia +prob = ODEProblem(sys, [1.0], (0, 10.0), []) +sol_v = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="velocity [m/s]") +lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v]") +axislegend(ax) + +ax = Axis(fig[2,1], xlabel="time [s]", ylabel="force [N]") +lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f]") +axislegend(ax) + +fig +``` + +![](figures/connections_4_1.png) + + + +# Translational Domain Example (Across Variable = position) + +Now, let's consider the position based approach. We can build the same model with the same components. As can be seen, we now end of up with 2 equations, because we need to relate the lower derivative (position) to force (with acceleration). + +```julia +module TranslationalPosition + using ModelingToolkit + using ModelingToolkitStandardLibrary.Mechanical.TranslationalPosition + + @parameters t + D = Differential(t) + + # Let's define a simple body that only tracks the across and through variables... + function Body(; name, m, v0 = 0.0) + @named flange = Flange() + pars = @parameters m=m v0=v0 + vars = @variables begin + v(t) = v0 + f(t) = m*v0 + end + eqs = [ + D(flange.s) ~ v + flange.f ~ f + + D(v) ~ f/m + ] + return compose(ODESystem(eqs, t, vars, pars; name = name), flange) + end + + @named damping = Damper(d = 1) + @named body = Body(m = 1, v0=1) + @named ground = Fixed() + + eqs = [ + connect(damping.flange_a, body.flange) + connect(ground.flange, damping.flange_b) + ] + + @named model = ODESystem(eqs, t; systems=[damping, body, ground]) + + sys = structural_simplify(model) +end + +sys = TranslationalPosition.sys + +full_equations(sys) +``` + +``` +2-element Vector{Equation}: + Differential(t)(body₊flange₊s(t)) ~ body₊v(t) + Differential(t)(body₊v(t)) ~ -((damping₊d*body₊v(t)) / body₊m) +``` + + + + + +As can be seen, we get exactly the same result. The only difference here is that we are solving an extra equation, which allows us to plot the body position as well. + +```julia +prob = ODEProblem(sys, [0.0, 1.0], (0, 10.0), []) +sol_p = solve(prob, ImplicitMidpoint(); dt=0.01) + +fig = Figure() +ax = Axis(fig[1,1], ylabel="velocity [m/s]") +lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.v], label="sol[body.v] (position based)") +lines!(ax, sol_v.t, sol_v[TranslationalVelocity.body.v], label="sol[body.v] (velocity based)", linestyle=:dash) +axislegend(ax) + +ax = Axis(fig[2,1], ylabel="force [N]") +lines!(ax, sol_p.t, -sol_p[TranslationalVelocity.body.f], label="sol[body.f] (position based)") +lines!(ax, sol_v.t, -sol_v[TranslationalVelocity.body.f], label="sol[body.f] (velocity based)", linestyle=:dash) +axislegend(ax) + +ax = Axis(fig[3,1], xlabel="time [s]", ylabel="position [m]") +lines!(ax, sol_p.t, sol_p[TranslationalPosition.body.flange.s], label="sol[body.flange.s]") +axislegend(ax) + +fig +``` + 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z3ZWYSrvZlwdq+decyFZEcRxi%)z{VK<>Vxw92pj-`{A=p%nulC;D?gyiulJM+1QoG zfH_mLl~@~Wzf)3j2y-VUE;V;_=yBrcl?BWxQchta3nOT04tNS46SEKYU7i@ktQ4Re zaj~)97zF^VWBk$r@;_J}8rqn!4X@0M0oyac$=gMhzWoYPEzk_d7Kb&#BfrhE>*N*} zBxt*99oe+@kTYXJG(irDHPu}9fY-OO62!T3UyF31mTR7aN_3TviPEkCOYirU`En(;S%7;}Q`@n9 zy9+QXtS~qX8UU~W;p1g0DJmA_cz~NoZn+uYLG0K1AbuGBjQv?3e4uQASdUYB`XC#Tt_r6&9Eg`Jrf~^qlNr=XyEtJP z&p{IL!iA={Zxhx0SQx(S|GB4IHwf(wXo-R{Ut-k&3asc_mVpYL-?RJd2uptLT3%)& z6fi;%*3i_J9V>5XVxj=xi=>S`$kg>%5eyR8I>sesjH7cKI%{O`u*pI-cbvQvxVvUe*kdTh2@2`6-q L8Xd{fcD(*CVo>UA literal 0 HcmV?d00001 From dc12bde5b098acdcb479218512fc62c1aab7a6c4 Mon Sep 17 00:00:00 2001 From: Brad Carman Date: Sun, 18 Sep 2022 17:46:07 -0400 Subject: [PATCH 7/7] latest Translational changes --- src/Mechanical/Translational/Translational.jl | 4 ++-- src/Mechanical/Translational/components.jl | 23 +++++++++++++++++-- src/Mechanical/Translational/sensors.jl | 11 +++++++++ 3 files changed, 34 insertions(+), 4 deletions(-) create mode 100644 src/Mechanical/Translational/sensors.jl diff --git a/src/Mechanical/Translational/Translational.jl b/src/Mechanical/Translational/Translational.jl index c9f52b004..e16cbb88d 100644 --- a/src/Mechanical/Translational/Translational.jl +++ b/src/Mechanical/Translational/Translational.jl @@ -14,7 +14,7 @@ include("utils.jl") export Body, Spring, Damper, Fixed include("components.jl") -#export Torque -#include("sources.jl") +export PositionSensor +include("sensors.jl") end diff --git a/src/Mechanical/Translational/components.jl b/src/Mechanical/Translational/components.jl index db0825bc0..21aca06cf 100644 --- a/src/Mechanical/Translational/components.jl +++ b/src/Mechanical/Translational/components.jl @@ -4,7 +4,7 @@ function Fixed(; name) return compose(ODESystem(eqs, t, [], []; name = name), port) end -function Body(; name, v0 = 0.0, m = 1.0) +function Body(; name, v0 = 0.0, m = 1.0, s0=nothing, g=nothing) @named port = Port(v0 = v0) pars = @parameters m=m vars = @variables begin @@ -15,8 +15,27 @@ function Body(; name, v0 = 0.0, m = 1.0) eqs = [ port.v ~ v port.f ~ f - D(v) ~ f/m ] + + # gravity option + if !isnothing(g) + @parameters g=g + push!(pars, g) + push!(eqs, D(v) ~ f/m+g) + else + push!(eqs, D(v) ~ f/m) + end + + # position option + if !isnothing(s0) + @parameters s0=s0 + push!(pars, s0) + + @variables s(t)=s0 + push!(vars, s) + + push!(eqs, D(s) ~ v) + end return compose(ODESystem(eqs, t, vars, pars; name = name), port) end diff --git a/src/Mechanical/Translational/sensors.jl b/src/Mechanical/Translational/sensors.jl new file mode 100644 index 000000000..ddcfe9106 --- /dev/null +++ b/src/Mechanical/Translational/sensors.jl @@ -0,0 +1,11 @@ +function PositionSensor(; name, s0=0.0) + @named port = Port() + + pars=@parameters s0=s0 + vars=@variables s(t)=s0 + eqs = [ + D(s) ~ port.v + 0 ~ p.f + ] + compose(ODESystem(eqs, t, vars, pars; name=name), port) +end \ No newline at end of file