Nonautonomous second order reduced dynamics

NONAUT_2NDORDER_REDDYN

function [Rk] = nonAut_2ndOrder_RedDyn(I,F,THETA,PHI,Lambda,Lambda_K, M,C,Vm,Ym,l,z_k,order)

NONAUT_2NDORDER_REDDYN

This function computes the reduced dynamics coefficients at order k for harmonic i using the second order system routine.

[Rk] = NONAUT_2NDORDER_REDDYN(I,F,THETA,PHI,Lambda,Lambda_K, M,C,Vm,Ym,l,z_k,order)

I: Eigenvalue positions that are resonant

F: Multi-index positions that are resonant for order > 0, otherwise the frequency harmonic

Lambda: vector containing master spectrum

Lambda_K: sum(Lambda * K + kappa * Omega)

M: Mass matrix

C: Damping Matrix

Vm: Velocity part of the lower order and forcing RHS

Ym: Displacement part of the lower order and forcing RHS

l: SSM dimension

z_k: number of multi-indices at order k

order: current order of computation

R1: non-autonomous RD coefficients

See also: NONAUT_2NDORDER_LEADTERMS, NONAUT_2NDORDER_HIGHTERMS

Rk = zeros(l,z_k);

if order == 0

    THETA_I = THETA(:,I);

    RHS   = -THETA_I' * (Lambda_K*M*Vm +Ym);
    C_0_r = eye(size(RHS,1)); % Fix leading order coefficient to exact resonance
    Rk(I,:) = lsqminnorm(C_0_r,RHS);

else
    % unique multi indices
    if any(F)
        [F_un, ~,i_F_un] = unique(F.');

        % Loop over multi-indices that lead to resonance
        fi = 1;

        for f = F_un
            I_f = I((i_F_un == fi)); % All resonant eigenvalues for this multi - index
            THETA_f = THETA(:,I_f);
            PHI_f   = PHI(:,I_f);

            RHS   = - Lambda_K(f).*THETA_f' *M*Vm(:,f) -THETA_f'*Ym(:,f);
            C_k_r =  THETA_f' * ( C* PHI_f + M * ((Lambda_K(f) + Lambda(I_f).') .* PHI_f)); %Coefficient Matrix
            Rk(I_f,f) = lsqminnorm(C_k_r,RHS);

            % tbd: check implementation for 1:1 internal resonances!

            fi = fi +1;
        end
    end

end

end

Nonautonomous second order reduced dynamics

Contents

NONAUT_2NDORDER_REDDYN