2. Reference solution#
2.1. Calculation method used for the reference solution#
The numerical values are compared to the experimental measurements and to the reference solution obtained using a Matlab script.
The expression of the force of dissipation in such a device is provided by the following formula [Peckan]:
\({F}_{D}\mathrm{=}{K}_{2}x+\frac{({K}_{1}\mathrm{-}{K}_{2})x}{\sqrt{1+{(\frac{{K}_{1}x}{{P}_{y}})}^{2}}}+C\mathit{sign}(\dot{x}){∣\dot{x}\frac{x}{{x}_{\mathit{max}}}∣}^{\alpha }\)
Matlab script:
%test case for anti-seismic device clear; Close all; %—-direct calculation—- %initializing the parameters of reckoning t0 = 0; final = 1. ; step = 0.01; tspan = t0: step: tfinal; y0 = [0 0 0 0]; y0 = y0”; options = []; %direct integration [t, y] = ode23 (“functsism1”, tspan, y0, options); depl1 = y (:, 1:1); depl2 = y (:, 2:2); vit1 = y (:, 3:3); vit2 = y (:, 4:4); kk1 = 6.e6; kk2 = 0.53e6; py = 1200; c = 0.07e5; xmax = 0.03; alpha = 0.2; for tt = 1:1:length (tspan) depl21 = depl2 (tt) -depl1 (tt); vit21 = vit2 (tt) -vit1 (tt); g1n = (kk1-kk2) depl21; g1d = sqrt (1+ ((kk1/py) *del21) ^2); g1 = g1n/g1d; g2 = c*sign (vit21) (abs (vit21*depl21/xmax)) ^ alpha; g0 = kk2*depl21; f (tt) = g0 + g1 + g2; End f = f”; depl = depl2 - depl1; |
function yp = functsism1 (t, y, flag) % temporary initialization m1 = 25. ; m2 = 25. ; k1 = 1.e10; m2 = 1.e10; kk1 = 6.e6; kk2 = 0.53e6; py = 1200; c = 0.07e5; xmax = 0.03; alpha = 0.2; omega = 2*pi; % %—-direct resolution—- x0 = (0.66*sin (omega*t))/(omega*omega); depl21 = y (2) -y (1); vit21 = y (4) -y (3); g1n = (kk1-kk2) depl21; g1d = sqrt (1+ ((kk1/py) *del21) ^2); g1 = g1n/g1d; g2 = c*sign (vit21) (abs (vit21*depl21/xmax)) ^ alpha; g0 = kk2*depl21; gg = g0 + g1 + g2; %creation of state matrices u = [1 0 0 0 0; 0 1 0 0; 0 0 m1 0; 0 0 0 [m2]; a = [0 0 -1 0; 0 0 0 -1; k1 0 0 0; 0 m2 [0 0]; g = [0; 0; gg + k1*x0; -gg]; % %derivative calculation yp = -inv (u) *a*y + inv (u) *g; |
2.2. Benchmark results#
Maximum values and RMS of the relative and absolute displacements in \(B\), and of the force due to the anti-seismic device.
2.3. Uncertainty about the solution#
The excitation imposed on the mass-spring system is an approximation of the displacement imposed on the experimental device.
The uncertainty about reference solution MATLAB is low.
2.4. Bibliographical references#
PEKCAN, J.B. MANDER, M. EERI: The seismic response of a 1:3 scale model R.C. structure with elastomeric spring dampers. - Earthquake Spexctra, vol. 11, No. 2, no. 2, p.249-267 - may 1995