# Coulomb and static friction with ModelingToolkit.jl and DifferentialEquations.jl

**URL:** <https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802>\
**Category:** Modelling & Simulations\
**Tags:** physics, symbolic\
**Created:** [November 8, 2020, 9:26pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802 "2020-11-08T21:26:01Z")\
**Posts on this page:** 15\
**Page:** 1

<div class="post-metadata">

**Author:** ![contradict](https://avatars.discourse-cdn.com/v4/letter/c/ac91a4/32.png) [@contradict](https://discourse.julialang.org/u/contradict)\
**Post date:** [November 8, 2020, 9:26pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/1 "2020-11-08T21:26:01Z")

</div>

I am trying to create a model of a damped oscillator with friction. It works fine with only Coulomb friction, but all my attempts to add a static friction term have failed so far. Below is the most promising, but I can’t figure out why it fails.

The failure itself would be good to understand, but I would also like to understant the expresion in the traceback. It looks like my expression, but has an implementation of `sign(x(t))` in it. I would love to be able to use that, but when I add anything with a `Boolean` to my expression, I get `ERROR: TypeError: non-boolean (Term{Real}) used in boolean context `

What is going on here? What else should I try? I have seen this problem solved by adding constraints that lock motion until the force is large enough to overcome static friction when the velocity falls to zero, Is there a way to do that?

```julia
using ModelingToolkit, DifferentialEquations

@parameters t k M σs σd vmin
@variables x(t) v(t)
@derivatives D'~t

tsgn(x) = tanh(x*10)
Ht(x) = (1 + tsgn(x)) / 2

eqs = [
    D(x) ~ v,
    D(v) ~ (Ht(abs(v) - vmin) * (k * x + σd * tsgn(v)) +
            Ht(vmin - abs(v)) * Ht(k * abs(x) - σs) * (k * x - tsgn(x) * σs)) / M
]

sys = ODESystem(eqs, t)

u0 = [x=>0, v=>10]

p = [k=>4.0, M=>0.25, σd=>0.1, σs=>1.0, vmin=>0.01]

tspan = (0.0, 50.0)

prob = ODEProblem(sys, u0, tspan, p, jac=true)

sol = solve(prob, Rodas4P());

plot(sol, vars=(0,2))

```

This results in the following stack trace when creating the `ODEProblem`.

```julia
ERROR: Failed to apply rule (*)(~(~x)) => if length(~(~x)) > N
        (*)((*)((~(~x))[1:N]...) * (*)((~(~x))[N + 1:end]...))
    else
        nothing
    end on expression 2.5 * (1 + tanh(10 * ((-1 * abs(v⦗t⦘)) + vmin))) * ((k * x⦗t⦘) + (-1 * tanh(10 * x⦗t⦘) * σs)) * (1 + (-1 * (tanh(10 * ((k * abs(x⦗t⦘)) + (-1 * σs))) ^ 2))) * ifelse(signbit(x⦗t⦘), -1, 1) * k
Stacktrace:
 [1] (::SymbolicUtils.Rule{Term{Any},SymbolicUtils.var"#term_matcher#46"{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}},ModelingToolkit.var"#453#455"{Int64}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rule.jl:142
 [2] (::SymbolicUtils.Rewriters.Chain)(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:59
 [3] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:140
 [4] (::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:128
 [5] iterate at ./generator.jl:47 [inlined]
 [6] collect_to!(::Array{Term{Number},1}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Int64, ::Int64) at ./array.jl:732
 [7] collect_to_with_first!(::Array{Term{Number},1}, ::Term{Number}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Int64) at ./array.jl:710
 [8] _collect(::Array{Term{Number},1}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Base.EltypeUnknown, ::Base.HasShape{1}) at ./array.jl:704
 [9] collect_similar at ./array.jl:628 [inlined]
 [10] map(::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}, ::Array{Term{Number},1}) at ./abstractarray.jl:2162
 [11] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:138
 [12] (::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:128
 [13] iterate at ./generator.jl:47 [inlined]
 [14] _collect(::Array{Term,1}, ::Base.Generator{Array{Term,1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Base.EltypeUnknown, ::Base.HasShape{1}) at ./array.jl:699
 [15] collect_similar at ./array.jl:628 [inlined]
 [16] map(::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}, ::Array{Term,1}) at ./abstractarray.jl:2162
 [17] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:138
 [18] (::SymbolicUtils.Rewriters.Fixpoint{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:101
 [19] unflatten_long_ops(::Num, ::Int64) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:97
 [20] unflatten_long_ops at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:91 [inlined]
 [21] #62 at ./operators.jl:875 [inlined]
 [22] iterate at ./generator.jl:47 [inlined]
 [23] collect_to!(::Array{Int64,2}, ::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}, ::Int64, ::Int64) at ./array.jl:732
 [24] collect_to_with_first!(::Array{Int64,2}, ::Int64, ::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}, ::Int64) at ./array.jl:710
 [25] collect(::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}) at ./array.jl:691
 [26] _build_function(::ModelingToolkit.JuliaTarget, ::Array{Num,2}, ::Array{Term{Real},1}, ::Vararg{Any,N} where N; conv::ModelingToolkit.ODEToExpr, expression::Type{T} where T, checkbounds::Bool, linenumbers::Bool, multithread::Nothing, headerfun::typeof(ModelingToolkit.addheader), outputidxs::Nothing, convert_oop::Bool, force_SA::Bool, skipzeros::Bool, fillzeros::Bool, parallel::ModelingToolkit.SerialForm, kwargs::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:254
 [27] build_function(::Array{Num,2}, ::Vararg{Any,N} where N; target::ModelingToolkit.JuliaTarget, kwargs::Base.Iterators.Pairs{Symbol,Any,NTuple{5,Symbol},NamedTuple{(:conv, :expression, :checkbounds, :linenumbers, :parallel),Tuple{ModelingToolkit.ODEToExpr,DataType,Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:48
 [28] generate_jacobian(::ODESystem, ::Array{Term{Real},1}, ::Array{Sym{ModelingToolkit.Parameter{Real}},1}; simplify::Bool, sparse::Bool, kwargs::Base.Iterators.Pairs{Symbol,Any,NTuple{4,Symbol},NamedTuple{(:expression, :checkbounds, :linenumbers, :parallel),Tuple{DataType,Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:67
 [29] ODEFunction{true,F,TMM,Ta,Tt,TJ,JVP,VJP,JP,SP,TW,TWt,TPJ,S,TCV} where TCV where S where TPJ where TWt where TW where SP where JP where VJP where JVP where TJ where Tt where Ta where TMM where F(::ODESystem, ::Array{Term{Real},1}, ::Array{Sym{ModelingToolkit.Parameter{Real}},1}, ::Array{Int64,1}; version::Nothing, tgrad::Bool, jac::Bool, eval_expression::Bool, sparse::Bool, simplify::Bool, kwargs::Base.Iterators.Pairs{Symbol,Any,Tuple{Symbol,Symbol,Symbol},NamedTuple{(:checkbounds, :linenumbers, :parallel),Tuple{Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:147
 [30] ODEProblem{true,tType,isinplace,P,F,K,PT} where PT where K where F where P where isinplace where tType(::ODESystem, ::Array{Pair{Num,Int64},1}, ::Tuple{Float64,Float64}, ::Array{Pair{Num,Float64},1}; version::Nothing, tgrad::Bool, jac::Bool, checkbounds::Bool, sparse::Bool, simplify::Bool, linenumbers::Bool, parallel::ModelingToolkit.SerialForm, eval_expression::Bool, kwargs::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:277
 [31] #ODEProblem#206 at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:240 [inlined]
 [32] top-level scope at REPL[17]:1
caused by [exception 1]
MethodError: no method matching *(::Term{Real}, ::Symbol)
Closest candidates are:
  *(::Any, ::Any, ::Any, ::Any...) at operators.jl:538
  *(::LightGraphs.LinAlg.Noop, ::Any) at /home/russel/.julia/packages/LightGraphs/HsNig/src/linalg/graphmatrices.jl:226
  *(::ChainRulesCore.One, ::Any) at /home/russel/.julia/packages/ChainRulesCore/UayCG/src/differential_arithmetic.jl:98
  ...
Stacktrace:
 [1] (::ModelingToolkit.var"#453#455"{Int64})(::Base.ImmutableDict{Symbol,Any}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rule.jl:274
 [2] macro expansion at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/utils.jl:11 [inlined]
 [3] (::SymbolicUtils.var"#34#35"{ModelingToolkit.var"#453#455"{Int64}})(::Base.ImmutableDict{Symbol,Any}, ::Int64) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rule.jl:139
 [4] (::SymbolicUtils.var"#loop#47"{SymbolicUtils.var"#34#35"{ModelingToolkit.var"#453#455"{Int64}}})(::SymbolicUtils.LL{Array{Any,1}}, ::Base.ImmutableDict{Symbol,Any}, ::SymbolicUtils.LL{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:96
 [5] (::SymbolicUtils.var"#45#48"{SymbolicUtils.LL{Array{Any,1}},SymbolicUtils.LL{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}}})(::Base.ImmutableDict{Symbol,Any}, ::Int64) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:101
 [6] (::SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}})(::SymbolicUtils.var"#45#48"{SymbolicUtils.LL{Array{Any,1}},SymbolicUtils.LL{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}}}, ::SymbolicUtils.LL{Array{Any,1}}, ::Base.ImmutableDict{Symbol,Any}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:74
 [7] (::SymbolicUtils.var"#loop#47"{SymbolicUtils.var"#34#35"{ModelingToolkit.var"#453#455"{Int64}}})(::SymbolicUtils.LL{Array{Any,1}}, ::Base.ImmutableDict{Symbol,Any}, ::SymbolicUtils.LL{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:100
 [8] (::SymbolicUtils.var"#45#48"{Term{Number},Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}})(::Base.ImmutableDict{Symbol,Any}, ::Int64) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:101
 [9] (::SymbolicUtils.var"#literal_matcher#42"{typeof(*)})(::SymbolicUtils.var"#45#48"{Term{Number},Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}}, ::Term{Number}, ::Base.ImmutableDict{Symbol,Any}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:10
 [10] (::SymbolicUtils.var"#loop#47"{SymbolicUtils.var"#34#35"{ModelingToolkit.var"#453#455"{Int64}}})(::Term{Number}, ::Base.ImmutableDict{Symbol,Any}, ::Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:100
 [11] (::SymbolicUtils.var"#term_matcher#46"{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}})(::SymbolicUtils.var"#34#35"{ModelingToolkit.var"#453#455"{Int64}}, ::Tuple{Term{Number}}, ::Base.ImmutableDict{Symbol,Any}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/matchers.jl:105
 [12] (::SymbolicUtils.Rule{Term{Any},SymbolicUtils.var"#term_matcher#46"{Tuple{SymbolicUtils.var"#literal_matcher#42"{typeof(*)},SymbolicUtils.var"#segment_matcher#44"{SymbolicUtils.Segment{typeof(SymbolicUtils.alwaystrue)}}}},ModelingToolkit.var"#453#455"{Int64}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rule.jl:137
 [13] (::SymbolicUtils.Rewriters.Chain)(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:59
 [14] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:140
 [15] (::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:128
 [16] iterate at ./generator.jl:47 [inlined]
 [17] collect_to!(::Array{Term{Number},1}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Int64, ::Int64) at ./array.jl:732
 [18] collect_to_with_first!(::Array{Term{Number},1}, ::Term{Number}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Int64) at ./array.jl:710
 [19] _collect(::Array{Term{Number},1}, ::Base.Generator{Array{Term{Number},1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Base.EltypeUnknown, ::Base.HasShape{1}) at ./array.jl:704
 [20] collect_similar at ./array.jl:628 [inlined]
 [21] map(::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}, ::Array{Term{Number},1}) at ./abstractarray.jl:2162
 [22] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:138
 [23] (::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:128
 [24] iterate at ./generator.jl:47 [inlined]
 [25] _collect(::Array{Term,1}, ::Base.Generator{Array{Term,1},SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}}, ::Base.EltypeUnknown, ::Base.HasShape{1}) at ./array.jl:699
 [26] collect_similar at ./array.jl:628 [inlined]
 [27] map(::SymbolicUtils.Rewriters.PassThrough{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}}, ::Array{Term,1}) at ./abstractarray.jl:2162
 [28] (::SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:138
 [29] (::SymbolicUtils.Rewriters.Fixpoint{SymbolicUtils.Rewriters.Walk{:post,SymbolicUtils.Rewriters.Chain,false}})(::Term{Number}) at /home/russel/.julia/packages/SymbolicUtils/K1sGK/src/rewriters.jl:101
 [30] unflatten_long_ops(::Num, ::Int64) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:97
 [31] unflatten_long_ops at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:91 [inlined]
 [32] #62 at ./operators.jl:875 [inlined]
 [33] iterate at ./generator.jl:47 [inlined]
 [34] collect_to!(::Array{Int64,2}, ::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}, ::Int64, ::Int64) at ./array.jl:732
 [35] collect_to_with_first!(::Array{Int64,2}, ::Int64, ::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}, ::Int64) at ./array.jl:710
 [36] collect(::Base.Generator{Array{Num,2},Base.var"#62#63"{typeof(ModelingToolkit.unflatten_long_ops),ModelingToolkit.var"#476#495"{Dict{Any,Any}}}}) at ./array.jl:691
 [37] _build_function(::ModelingToolkit.JuliaTarget, ::Array{Num,2}, ::Array{Term{Real},1}, ::Vararg{Any,N} where N; conv::ModelingToolkit.ODEToExpr, expression::Type{T} where T, checkbounds::Bool, linenumbers::Bool, multithread::Nothing, headerfun::typeof(ModelingToolkit.addheader), outputidxs::Nothing, convert_oop::Bool, force_SA::Bool, skipzeros::Bool, fillzeros::Bool, parallel::ModelingToolkit.SerialForm, kwargs::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:254
 [38] build_function(::Array{Num,2}, ::Vararg{Any,N} where N; target::ModelingToolkit.JuliaTarget, kwargs::Base.Iterators.Pairs{Symbol,Any,NTuple{5,Symbol},NamedTuple{(:conv, :expression, :checkbounds, :linenumbers, :parallel),Tuple{ModelingToolkit.ODEToExpr,DataType,Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/build_function.jl:48
 [39] generate_jacobian(::ODESystem, ::Array{Term{Real},1}, ::Array{Sym{ModelingToolkit.Parameter{Real}},1}; simplify::Bool, sparse::Bool, kwargs::Base.Iterators.Pairs{Symbol,Any,NTuple{4,Symbol},NamedTuple{(:expression, :checkbounds, :linenumbers, :parallel),Tuple{DataType,Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:67
 [40] ODEFunction{true,F,TMM,Ta,Tt,TJ,JVP,VJP,JP,SP,TW,TWt,TPJ,S,TCV} where TCV where S where TPJ where TWt where TW where SP where JP where VJP where JVP where TJ where Tt where Ta where TMM where F(::ODESystem, ::Array{Term{Real},1}, ::Array{Sym{ModelingToolkit.Parameter{Real}},1}, ::Array{Int64,1}; version::Nothing, tgrad::Bool, jac::Bool, eval_expression::Bool, sparse::Bool, simplify::Bool, kwargs::Base.Iterators.Pairs{Symbol,Any,Tuple{Symbol,Symbol,Symbol},NamedTuple{(:checkbounds, :linenumbers, :parallel),Tuple{Bool,Bool,ModelingToolkit.SerialForm}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:147
 [41] ODEProblem{true,tType,isinplace,P,F,K,PT} where PT where K where F where P where isinplace where tType(::ODESystem, ::Array{Pair{Num,Int64},1}, ::Tuple{Float64,Float64}, ::Array{Pair{Num,Float64},1}; version::Nothing, tgrad::Bool, jac::Bool, checkbounds::Bool, sparse::Bool, simplify::Bool, linenumbers::Bool, parallel::ModelingToolkit.SerialForm, eval_expression::Bool, kwargs::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:277
 [42] #ODEProblem#206 at /home/russel/.julia/packages/ModelingToolkit/iu31N/src/systems/diffeqs/abstractodesystem.jl:240 [inlined]
 [43] top-level scope at REPL[17]:1

```

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 9, 2020, 12:39am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/2 "2020-11-09T00:39:32Z")

</div>

It looks like `simplify` failed, and @shashi would be the one to take a look at it. Thanks for the example: that should be very useful.

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 9, 2020, 12:43am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/3 "2020-11-09T00:43:41Z")

</div>

> [@contradict](#):
>
> I add anything with a `Boolean` to my expression, I get `ERROR: TypeError: non-boolean (Term{Real}) used in boolean context`

You can use `IfElse.ifelse(x,y,z)` as a stand-in for the standard `Core.ifelse` and it’ll be something that is symbolically recognized. We may need to fix a few derivative rules to do this.

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<div class="post-metadata">

**Author:** ![contradict](https://avatars.discourse-cdn.com/v4/letter/c/ac91a4/32.png) [@contradict](https://discourse.julialang.org/u/contradict)\
**Post date:** [November 9, 2020, 3:21am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/4 "2020-11-09T03:21:02Z")

</div>

Ah, `IfElse` is a great trick, I didn’t know about that one.

I almost got things working using that idea and then I realized this is not a good model of friction anyway, since it sets `D(v) ~ 0`, not `v ~ 0`, when the static friction becomes important. I need to think more about what I really mean here. Maybe I can do something fancy with callbacks.

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<div class="post-metadata">

**Author:** ![contradict](https://avatars.discourse-cdn.com/v4/letter/c/ac91a4/32.png) [@contradict](https://discourse.julialang.org/u/contradict)\
**Post date:** [November 9, 2020, 4:27am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/5 "2020-11-09T04:27:31Z")

</div>

OK, I got the bad model working with IfElse.

```julia

isign(x) = IfElse.ifelse(signbit(x), -1, 1)

eqs = [
    D(x) ~ v,
    D(v) ~ IfElse.ifelse(signbit(abs(v) - vmin),
                         IfElse.ifelse(signbit(k*abs(x) - σs),
                                       0,
                                       -k * x + isign(x) * σs),
                         -k * x - σd * isign(v)) / M
]

```

When I try to express this more naturally with `<`, I get

```julia
ERROR: MethodError: no method matching isless(::Term{Number}, ::Sym{Real})

```

`SymbolicUtils` only defines `isless` for arguments of matching type, Is there some way to turn that `Term{Number}`into a `Term{Real}`?

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 9, 2020, 4:31am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/6 "2020-11-09T04:31:55Z")

</div>

> [@contradict](#):
>
> `SymbolicUtils` only defines `isless` for arguments of matching type, Is there some way to turn that `Term{Number}` into a `Term{Real}` ?

This is probably worth an issue.

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<div class="post-metadata">

**Author:** ![contradict](https://avatars.discourse-cdn.com/v4/letter/c/ac91a4/32.png) [@contradict](https://discourse.julialang.org/u/contradict)\
**Post date:** [November 9, 2020, 4:57am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/7 "2020-11-09T04:57:45Z")

</div>

Hmm, [this](https://github.com/JuliaSymbolics/SymbolicUtils.jl/blob/master/src/methods.jl#L113) already looks pretty well thought out. `Number` was explicitly skipped for all comparisons besides equality, and it does allow `isless` for `T<:Real`. Looking at the tests, I think this was to leave out `Complex`.

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<div class="post-metadata">

**Author:** ![shashi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/shashi/32/1824_2.png) [@shashi](https://discourse.julialang.org/u/shashi)\
**Post date:** [November 9, 2020, 1:14pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/8 "2020-11-09T13:14:56Z")

</div>

It does not define it on `Number` because there’s no way to compare complex numbers but maybe we should allow Number x Real and Real x Number

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<div class="post-metadata">

**Author:** ![shashi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/shashi/32/1824_2.png) [@shashi](https://discourse.julialang.org/u/shashi)\
**Post date:** [November 9, 2020, 1:15pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/9 "2020-11-09T13:15:31Z")

</div>

[https://github.com/SciML/ModelingToolkit.jl/pull/648](https://github.com/SciML/ModelingToolkit.jl/pull/648) this should fix the issue in the OP

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<div class="post-metadata">

**Author:** ![contradict](https://avatars.discourse-cdn.com/v4/letter/c/ac91a4/32.png) [@contradict](https://discourse.julialang.org/u/contradict)\
**Post date:** [November 9, 2020, 5:40pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/10 "2020-11-09T17:40:23Z")

</div>

That works, thanks! I’m surprised how similar the two solutions are, I would have expected the smooth approximation to have a much more noticeable effect.

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<div class="post-metadata">

**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [November 12, 2020, 10:00pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/11 "2020-11-12T22:00:54Z")

</div>

The above code does not seem to be working in Win10 Julia 1.5.2. Tried to add ModelingToolkit#master to get latest fixes. I always get the following error in the **eqs** definition:

```julia
ERROR: MethodError: no method matching signbit(::Operation)

```

Thank you.

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 12, 2020, 10:31pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/12 "2020-11-12T22:31:15Z")

</div>

`Operation` shouldn’t exist anymore. Check `]st`: do you have MTK v4? This is a very recent release and a very recent fix.

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<div class="post-metadata">

**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [November 12, 2020, 11:28pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/13 "2020-11-12T23:28:03Z")

</div>

@ChrisRackauckas, thank you. After a lengthy package maintenance/hygiene work, managed to update from MTK v3.1 to v3.20.1. The code runs with no errors now.

PS: How can we update to v4?

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [November 13, 2020, 12:52am UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/14 "2020-11-13T00:52:32Z")

</div>

You probably need some updates which may not be released yet. Some downstream stuff are still in the release process (it’s that new still)

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<div class="post-metadata">

**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [November 13, 2020, 7:24pm UTC](https://discourse.julialang.org/t/coulomb-and-static-friction-with-modelingtoolkit-jl-and-differentialequations-jl/49802/15 "2020-11-13T19:24:47Z")

</div>

Thanks for sharing this very interesting physics problem. One observation regarding the friction model equations, if you will.  
One could define the static friction region as `k*abs(x) < σs*N` where the spring force is cancelled (N is the normal force). And the kinematic friction region over `k*abs(x) >= σs*N`. The equations simplify then to:

```julia
const g = 9.8; # m/s^2
eqs = [
    D(x) ~ v,
    D(v) ~
        IfElse.ifelse(signbit.(k*abs(x) - σs*M*g), 0, -k*x -σd*M*g*isign(v)) / M,
]

```

Also, the equations using the smooth step function trick would then become:

```julia
eqs1 = [
    D(x) ~ v,
    D(v) ~
        ( Ht(k*abs(x) - σs*M*g) * (-k*x - σd*M*g*tsgn(v)) ) / M,
]

```

Best regards.