Bouc wen model in simscape model

I am trying to build the Bouc wen model in simscape.

But when I run the model in simulink, the error message shows up.

Found nonlinear expression involving time derivatives. |In bouc_wen_model (line 33)

component bouc_wen_model 

    nodes
    r = foundation.mechanical.translational.translational;
    c = foundation.mechanical.translational.translational;
    end

  parameters
    fy = {1800000,'N'};     %yield force
    uy = {0.01,'m'};        %yield displacement
    a = 0.1;          %post yield ratio  
    n = 2;
    beta = 0.1;
    gama = 0.9;
  end

  branches
  f : r.f -> c.f;
  end

  variables
    
    x = { value = { 0, 'm'}, priority = priority.high };    % Deformation
    f = {0,'N'};
    v = {0,'m/s'};  
    z = {value = { 0, '1'}, priority = priority.high};
    
  end
 
    equations
        
        f == a*(fy/uy)*x+(1-a)*fy*z;
        z.der == (1-((abs(z))^n)*(beta+sign(x.der*z)*gama))*x.der/uy;
        v == r.v-c.v;      
        v == x.der;
        
    end
end

Answer 1

[Lokesh]

Hi,

I see that you are encountering an issue with your Simscape model.The error message "Found nonlinear expression involving time derivatives" indicates that there is an issue with the way time derivatives are used in your Simscape equations.

In your case, the problematic line seems to be:

z.der == (1-((abs(z))^n)*(beta+sign(x.der*z)*gama))*x.der/uy; 

It appears that you are trying to include a nonlinear function of the derivative "x.der" and the variable "z" in the expression for "z.der". Simscape requires that the derivatives be isolated and not involved in nonlinear expressions. To fix the issue, you can introduce an intermediate variable to represent the nonlinear term and then differentiate that variable.

Here's an example of how you might restructure your equations:

component bouc_wen_model
nodes
r = foundation.mechanical.translational.translational;
c = foundation.mechanical.translational.translational;
end
parameters
fy = {1800000,'N'};     %yield force
uy = {0.01,'m'};        %yield displacement
a = 0.1;                %post yield ratio  
n = 2;
beta = 0.1;
gama = 0.9;
end
branches
f : r.f -> c.f;
end
variables
x = { value = { 0, 'm'}, priority = priority.high };    % Deformation
f = {0,'N'};
v = {0,'m/s'};  
z = {value = { 0, '1'}, priority = priority.high};      % Hysteresis variable
w = {0, '1'}; % Intermediate variable for nonlinear term
end

equations
f == a*(fy/uy)*x + (1-a)*fy*z;

% Define the nonlinear term using an intermediate variable 'w'
w == (1-((abs(z))^n)*(beta+gama*sign(x.der*z)));

% Now, differentiate 'w' to find 'z.der'
z.der == w*x.der/uy;

v == r.v - c.v;      
v == x.der;
end
end

In this revised code,an intermediate variable w is used to capture the nonlinear term involving "z" and "x.der". The derivative "z.der" is then computed using "w" and "x.der/uy", which avoids nonlinear operations on the derivative itself.

Please refer to the following MathWorks documentation to know more about ‘der’ in simscape:

• https://www.mathworks.com/help/simscape/lang/der.html

I hope this resolves your query.

Best Regards,

Lokesh

[Alexander S]

Thank you for this response, the code you posted still has the same error due to the nonlinear expression sign(x.der). This could be solved by replacing x.der with v.

component bouc_wen_hysteresis

nodes
        r = foundation.mechanical.translational.translational; % r:left
        c = foundation.mechanical.translational.translational; % c:right
    end
    parameters
        fy = {1800000,'N'};     %yield force
        uy = {0.01,'m'};        %yield displacement
        a = 0.1;                %post yield ratio
        n = 2;
        beta = 0.1;
        gama = 0.9;
    end
    branches
        f : r.f -> c.f;
    end
    variables
        x = { value = { 0, 'm'}, priority = priority.high };    % Deformation
        f = {0,'N'};
        v = {0,'m/s'};
        z = {value = { 0, '1'}, priority = priority.high};      % Hysteresis variable
        w = {0, '1'}; % Intermediate variable for nonlinear term
    end
    equations
        f == a*(fy/uy)*x + (1-a)*fy*z;
        
        % Define the nonlinear term using an intermediate variable 'w'
        w == (1-((abs(z))^n)*(beta+gama*sign(v*z)));
        % Now, differentiate 'w' to find 'z.der'
        z.der == w*x.der/uy;
        v == r.v - c.v;
        v == x.der;
    end
    end

[Alexander S]

This works now.