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Bearing.m
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Bearing.m
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classdef Bearing
%BEARING
%
% written by:
% Geraldo Rebouças
% - Geraldo.Reboucas@ntnu.no OR
% - gfs.reboucas@gmail.com
%
% Postdoctoral Fellow at:
% Norwegian University of Science and Technology, NTNU
% Department of Marine Technology, IMT
% Marine System Dynamics and Vibration Lab, MD Lab
% https://www.ntnu.edu/imt/lab/md-lab
%
properties
X; % [-], Radial load factor
Y; % [-], Axial load factor
a; % [-], Exponent of the life equation
C; % [N], Basic dynamic load rating
e; % [-], Force ratio factor
L_10; % [M], Basic rating life
name(1, :) string; % [-], Bearing designation
type(1, :) string {mustBeMember(type, ["CARB", "CRB", "SRB", "TRB", "none"])} = "none";
% CARB: CARB toroidal roller bearing
% CRB: Cylindrical Roller Bearing
% SRB: Spherical Roller Bearing
% TRB: Tapered Roller Bearing
x; % [mm], bearing axial position w. r. t. shaft/pin
K_x; % [N/m], Translational stiffness, x axis
K_y; % [N/m], Translational stiffness, y axis
K_z; % [N/m], Translational stiffness, z axis
K_alpha; % [N-m/rad], Torsional stiffness, x axis (rot.)
K_beta; % [N-m/rad], Torsional stiffness, y axis
K_gamma; % [N-m/rad], Torsional stiffness, z axis
C_x; % [N-s/m], Translational damping, x axis
C_y; % [N-s/m], Translational damping, y axis
C_z; % [N-s/m], Translational damping, z axis
C_alpha; % [N-m-s/rad], Torsional damping, x axis (rot.)
C_beta; % [N-m-s/rad], Torsional damping, y axis
C_gamma; % [N-m-s/rad], Torsional damping, z axis
D; % [mm], Outer diameter
d; % [mm], Inner diameter
B; % [mm], Thickness
end
methods
function obj = Bearing(varargin)
default = {'name' , 'no_name', ...
'type' , 'none', ...
'x' , 0.0, ...
'D' , 0.0, ...
'd' , 0.0, ...
'B' , 0.0, ...
'a' , 3.0, ...
'C' , 0.0, ...
'e' , 0.0, ...
'X' , 0.0, ...
'Y' , 0.0, ...
'L_10' , 1.0, ...
'K_x' , 0.0, ...
'K_y' , 0.0, ...
'K_z' , 0.0, ...
'K_alpha', 0.0, ...
'K_beta' , 0.0, ...
'K_gamma', 0.0, ...
'C_x' , 0.0, ...
'C_y' , 0.0, ...
'C_z' , 0.0, ...
'C_alpha', 0.0, ...
'C_beta' , 0.0, ...
'C_gamma', 0.0};
default = scaling_factor.process_varargin(default, varargin);
obj.name = default.name;
obj.type = default.type;
obj.D = default.D;
obj.d = default.d;
obj.B = default.B;
obj.a = default.a;
obj.C = default.C;
obj.e = default.e;
obj.X = default.X;
obj.Y = default.Y;
obj.L_10 = default.L_10;
obj.x = default.x;
obj.K_x = default.K_x;
obj.K_y = default.K_y;
obj.K_z = default.K_z;
obj.K_alpha = default.K_alpha;
obj.K_beta = default.K_beta;
obj.K_gamma = default.K_gamma;
obj.C_x = default.C_x;
obj.C_y = default.C_y;
obj.C_z = default.C_z;
obj.C_alpha = default.C_alpha;
obj.C_beta = default.C_beta;
obj.C_gamma = default.C_gamma;
end
function tab = disp(obj)
if(isempty(obj))
disp("\t0x0 empty Bearing object")
else
tab_set = {"Type", "-+-", "-", obj.type;
"Translational Stiffness", "K_x", "N/m", obj.K_x;
"Translational Stiffness", "K_y", "N/m", obj.K_y;
"Translational Stiffness", "K_z", "N/m", obj.K_z;
"Rotational Stiffness (rot. axis)", "K_alpha", "N-m/rad", obj.K_alpha;
"Rotational Stiffness", "K_beta", "N-m/rad", obj.K_beta;
"Rotational Stiffness", "K_gamma", "N-m/rad", obj.K_gamma;
"Translational Damping", "C_x", "N-s/m", obj.C_x;
"Translational Damping", "C_y", "N-s/m", obj.C_y;
"Translational Damping", "C_z", "N-s/m", obj.C_z;
"Rotational Damping (rot. axis)", "C_alpha", "N-m-s/rad", obj.C_alpha;
"Rotational Damping", "C_beta", "N-m-s/rad", obj.C_beta;
"Rotational Damping", "C_gamma", "N-m-s/rad", obj.C_gamma;
"Inner diameter", "d", "mm", obj.d;
"Outer diameter", "D", "mm", obj.D;
"Width", "B", "mm", obj.B;
"Exponent of the life equation", "a", "-", obj.a;
"Basic dynamic load rating" , "C", "N", obj.C;
};
Parameter = tab_set(:, 1);
Symbol = tab_set(:, 2);
Unit = tab_set(:, 3);
Value = tab_set(:, 4:end);
n = numel(obj);
tab_val = cell(n + 2, 1);
tab_val{1} = table(Parameter, Symbol);
for idx = 1:n
tab_val{idx + 1} = table(Value(:, idx), 'variableNames', obj(idx).name);
end
tab_val{idx + 2} = table(Unit);
tab = [tab_val{:}];
if(nargout == 0)
disp(tab);
clear tab;
end
end
end
function data = export2struct(obj)
warning('off', 'MATLAB:structOnObject');
data = struct(obj);
warning('on', 'MATLAB:structOnObject');
end
end
%% Calculation:
methods
function val = series_association(obj)
n = numel(obj);
if(n < 2)
warninig("Only one bearing.");
val = obj;
else
nam = "series";
kx = 0.0; ky = 0.0; kz = 0.0;
ka = 0.0; kb = 0.0; kg = 0.0;
cx = 0.0; cy = 0.0; cz = 0.0;
ca = 0.0; cb = 0.0; cg = 0.0;
for idx = 1:nargin
nam = join([nam obj(idx).name], " / ");
kx = kx + 1.0/obj(idx).K_x;
ky = ky + 1.0/obj(idx).K_y;
kz = kz + 1.0/obj(idx).K_z;
ka = ka + 1.0/obj(idx).K_alpha;
kb = kb + 1.0/obj(idx).K_beta;
kg = kg + 1.0/obj(idx).K_gamma;
cx = cx + 1.0/obj(idx).C_x;
cy = cy + 1.0/obj(idx).C_y;
cz = cz + 1.0/obj(idx).C_z;
ca = ca + 1.0/obj(idx).C_alpha;
cb = cb + 1.0/obj(idx).C_beta;
cg = cg + 1.0/obj(idx).C_gamma;
end
if(1.0/kx == inf), kx = 0.0; else, kx = 1.0/kx; end
if(1.0/ky == inf), ky = 0.0; else, ky = 1.0/ky; end
if(1.0/kz == inf), kz = 0.0; else, kz = 1.0/kz; end
if(1.0/ka == inf), ka = 0.0; else, ka = 1.0/ka; end
if(1.0/kb == inf), kb = 0.0; else, kb = 1.0/kb; end
if(1.0/kg == inf), kg = 0.0; else, kg = 1.0/kg; end
if(1.0/cx == inf), cx = 0.0; else, cx = 1.0/cx; end
if(1.0/cy == inf), cy = 0.0; else, cy = 1.0/cy; end
if(1.0/cz == inf), cz = 0.0; else, cz = 1.0/cz; end
if(1.0/ca == inf), ca = 0.0; else, ca = 1.0/ca; end
if(1.0/cb == inf), cb = 0.0; else, cb = 1.0/cb; end
if(1.0/cg == inf), cg = 0.0; else, cg = 1.0/cg; end
val = Bearing('name' , nam , 'type' , obj(1).type, ...
'K_x' , kx , 'K_y' , ky , 'K_z' , kz,...
'K_alpha', ka , 'K_beta', kb , 'K_gamma', kg, ...
'C_x' , cx , 'C_y' , cy , 'C_z' , cz,...
'C_alpha', ca , 'C_beta', cb , 'C_gamma', cg, ...
'D' , obj(1).D, 'd' , obj(1).d , 'B' , obj(1).B);
end
end
function val = parallel_association(obj)
n = numel(obj);
if(n < 2)
warning("Bearing:parallel_assoc", "Only one bearing.");
val = obj;
else
nam = "parallel";
kx = 0.0; ky = 0.0; kz = 0.0;
ka = 0.0; kb = 0.0; kg = 0.0;
cx = 0.0; cy = 0.0; cz = 0.0;
ca = 0.0; cb = 0.0; cg = 0.0;
for idx = 1:n
nam = join([nam obj(idx).name], " / ");
kx = kx + obj(idx).K_x;
ky = ky + obj(idx).K_y;
kz = kz + obj(idx).K_z;
ka = ka + obj(idx).K_alpha;
kb = kb + obj(idx).K_beta;
kg = kg + obj(idx).K_gamma;
cx = cx + obj(idx).C_x;
cy = cy + obj(idx).C_y;
cz = cz + obj(idx).C_z;
ca = ca + obj(idx).C_alpha;
cb = cb + obj(idx).C_beta;
cg = cg + obj(idx).C_gamma;
end
val = Bearing('name' , nam , 'type' , obj(1).type, ...
'K_x' , kx , 'K_y' , ky , 'K_z' , kz,...
'K_alpha', ka , 'K_beta', kb , 'K_gamma', kg, ...
'C_x' , cx , 'C_y' , cy , 'C_z' , cz,...
'C_alpha', ca , 'C_beta', cb , 'C_gamma', cg, ...
'D' , obj(1).D , 'd' , obj(1).d , 'B' , obj(1).B);
end
end
function mat = stiffness_matrix(obj)
n = numel(obj);
mat = zeros(6, 6, n);
for idx = 1:n
mat(:, :, idx) = diag([obj(idx).K_x ...
obj(idx).K_y ...
obj(idx).K_z ...
obj(idx).K_alpha ...
obj(idx).K_beta ...
obj(idx).K_gamma]);
end
end
function mat = damping_matrix(obj)
n = numel(obj);
mat = zeros(6, 6, n);
for idx = 1:n
mat(:, :, idx) = diag([obj(idx).C_x ...
obj(idx).C_y ...
obj(idx).C_z ...
obj(idx).C_alpha ...
obj(idx).C_beta ...
obj(idx).C_gamma]);
end
end
function P = dynamic_equiv_load(obj, F_R, F_A)
cond = F_R./F_A <= obj.e;
P = (F_R.*obj.X(1) + F_A.*obj.Y(1)).*( cond) + ...
(F_R.*obj.X(2) + F_A.*obj.Y(2)).*(~cond);
end
function D = damage_calculation(obj, F_R, F_A, speed, time_step)
%DAMAGE_CALCULATION performs fatigue damage calculation for a
% Bearing object. F_R and F_A are the radial and axial forces
% at the Bearing, which rotates with velocity speed in [1/min.].
%
% Dynamic equivalent load:
P = obj.dynamic_equiv_load(F_R, F_A);
[N, P_edges] = ISO_6336.LDD(P, speed, time_step);
sum_term = N.*power(P_edges, obj.a);
D = power(obj.L_10*obj.C, -obj.a)*sum(sum_term);
end
function D = damage_analysis(obj, data)
F_A = data.load.ov_001.values;
F_y = data.load.ov_002.values;
F_z = data.load.ov_003.values;
F_R = sqrt(F_y.^2 + F_z.^2);
% [rad/s] to [1/min.]:
speed = data.load.ov_017.values.*30.0/pi;
D = obj.damage_calculation(F_R, F_A, speed, data.time_step);
end
function show_LDD(obj, data)
F_A = data.load.ov_001.values;
F_y = data.load.ov_002.values;
F_z = data.load.ov_003.values;
F_R = sqrt(F_y.^2 + F_z.^2);
% [rad/s] to [1/min.]:
speed = data.load.ov_017.values.*30.0/pi;
% Dynamic equivalent load:
P = obj.dynamic_equiv_load(F_R, F_A);
[N, P_edges] = ISO_6336.LDD(P, speed, data.time_step);
time_duration = length(speed)*data.time_step;
histogram('binEdges' , N*obj.L_10*3.6e3/time_duration, ...
'binCounts' , P_edges(2:end)*1.0e-3, ...
'displayStyle', 'stairs', ...
'lineWidth' , 2.0);
end
function show_histogram(obj, data)
F_A = data.load.ov_001.values;
F_y = data.load.ov_002.values;
F_z = data.load.ov_003.values;
F_R = sqrt(F_y.^2 + F_z.^2);
% [rad/s] to [1/min.]:
speed = data.load.ov_017.values.*30.0/pi;
% Dynamic equivalent load:
P = obj.dynamic_equiv_load(F_R, F_A);
[N, P_edges, speed_edges] = ISO_6336.LDD(P, speed, data.time_step);
N = diff(N)*60.0./(speed_edges.*data.time_step);
N(isnan(N)) = 0;
histogram('binEdges' , fliplr(P_edges), ...
'binCounts' , fliplr(N), ...
'displayStyle', 'stairs', ...
'lineWidth' , 2.0);
end
function [a, b] = Weibull(obj, data)
F_A = data.load.ov_001.values;
F_y = data.load.ov_002.values;
F_z = data.load.ov_003.values;
F_R = sqrt(F_y.^2 + F_z.^2);
% Dynamic equivalent load:
P = obj.dynamic_equiv_load(F_R, F_A);
wbull = fitdist(abs(P) + eps, 'weibull');
a = wbull.a; % scale param.
b = wbull.b; % shape param.
end
end
methods(Static)
function data_analysis(tag)
tag = upper(tag);
switch(tag)
case 'SRB'
type = 'spherical roller bearings';
DD_ref = 1030.0; dd_ref = 710.0;
BB_ref = 315.0; CC_ref = 11164.0;
idx = 1;
case 'CRB1'
type = 'cylindrical roller bearing (single row)';
DD_ref = 1220.0; dd_ref = 1000.0;
BB_ref = 128.0; CC_ref = 3690.0;
idx = 2;
case 'CRB2'
type = 'cylindrical roller bearing (double row, full complement)';
DD_ref = 600.0; dd_ref = 400.0;
BB_ref = 272.0; CC_ref = 5500.0;
idx = 3;
case 'CARB'
type = 'barrel-shaped and toroidal roller bearings';
DD_ref = 1030.0; dd_ref = 710.0;
BB_ref = 236.0; CC_ref = 8800.0;
idx = 4;
case 'TRB1'
type = 'tapered roller bearing (single row)';
DD_ref = 550.0; dd_ref = 410.0;
BB_ref = 86.0; CC_ref = 1467.0;
idx = 5;
case 'ALL'
tmp = {'SRB', 'CRB1', 'CRB2', 'CARB', 'TRB1'};
for idx = 1:numel(tmp)
Bearing.data_analysis(tmp{idx});
end
return;
end
fig_dim = {'units' , 'centimeters', ...
'position', [5.0 5.0 23.0 16.0]};
font_style = {'fontName' , 'Times', ...
'fontSize' , 12.0 , ...
'interpreter', 'LaTeX'};
%% Loading data:
load(sprintf('data\\data_bearing%02d', idx));
name = {'b', 'C', 'd', 'D', 'Dpw', 'Dw', 'e', 'Lw', 'X1', 'X2', 'Y1', 'Y2', 'Z'};
BB(BB == 0.0) = nan; CC(CC == 0.0) = nan;
dd(dd == 0.0) = nan; DD(DD == 0.0) = nan;
Dpw(Dpw == 0.0) = nan; Dw(Dw == 0.0) = nan;
Lw(Lw == 0.0) = nan; ZZ(ZZ == 0.0) = nan;
data = [BB; CC; dd; DD; Dpw; Dw; ee; Lw; X1; X2; Y1; Y2; ZZ]';
% idx_B = find(BB); idx_C = find(CC);
% index = intersect(idx_B, idx_C, 'stable');
% idx_d = find(dd); idx_D = find(DD);
% index = intersect(index, intersect(idx_d, idx_D, 'stable'), 'stable');
% idx_Dpw = find(Dpw); idx_Dw = find(Dw);
% index = intersect(index, intersect(idx_Dpw, idx_Dw, 'stable'), 'stable');
% idx_e = find(ee); idx_Lw = find(Lw);
% index = intersect(index, intersect(idx_e, idx_Lw, 'stable'), 'stable');
% idx_X1 = find(X1); idx_X2 = find(X2);
% index = intersect(index, intersect(idx_X1, idx_X2, 'stable'), 'stable');
% idx_Y1 = find(Y1); idx_Y2 = find(Y2);
% index = intersect(index, intersect(idx_Y1, idx_Y2, 'stable'), 'stable');
% idx_ZZ = find(ZZ);
% index = intersect(index, idx_ZZ, 'stable');
% Removing empty columns:
data_var = var(data, 'omitnan');
data_var(isnan(data_var)) = -1;
data_var(data_var < eps) = -1;
idx_0 = find(data_var < 0);
clear(name{idx_0});
data(:, idx_0) = [];
name(:, idx_0) = [];
% Removing Zero elements:
% index = sort(intersect(intersect(idx_Dw, idx_Dpw, 'stable'), idx_ZZ, 'stable'));
% data = data(index, :);
% Normalizing:
data = data./max(data);
% Creating table:
tab = array2table(data);
tab.Properties.VariableNames = name;
if(strcmpi(tag, 'CRB1'))
% idx_Lw = find(Lw(index));
% data = data(idx_Lw, :);
% %
% idx_0 = (var(data) < eps);
% clear(name{idx_0});
% data(:, idx_0) = [];
% name(:, idx_0) = [];
%
% % Updating table:
% tab = array2table(data);
% tab.Properties.VariableNames = name;
mod_2 = @(b, x)(b(1) + b(2).*x(:, 1));
fitLw = fitnlm(tab.b, tab.Lw, mod_2, ones(2, 1), 'varNames', {'D', 'L_w'});
figure(fig_dim{:});
subplot(211)
hold on;
scatter(tab.b, tab.Lw, 'filled');
plot(tab.b, fitLw.feval(tab.b), ...
'lineStyle', '-', ...
'lineWidth', 2.0);
xlabel('$B$, [-]' , font_style{:});
ylabel('$L_{we}$, [-]', font_style{:});
legend({'data', 'fit'}, 'location', 'best');
subplot(212)
hold on;
scatter(tab.b, 35948.0*power(tab.Lw*max(Lw), 8.0/9.0), 'filled');
scatter(tab.b, 35948.0*power(fitLw.feval(tab.b)*max(Lw), 8.0/9.0), 'filled');
xlabel('$B$, [-]' , font_style{:});
ylabel('$K$, [N/mm$^{8/9}$]', font_style{:});
title(type);
legend({'K-form.', 'K-fit'}, 'location', 'best');
fig_axes = findobj(gcf, 'Type', 'Axes');
set(fig_axes, 'fontName', 'Times', 'fontSize', 12.0);
set(fig_axes, 'box' , 'on');
end
%% Correlation analysis:
figure(fig_dim{:});
subplot(141)
plot(tab.d, tab.C, 'ko');
xlabel('$d$, [-]', font_style{:})
ylabel('$C$, [-]', font_style{:})
subplot(142)
plot(tab.D, tab.C, 'ko');
xlabel('$D$, [-]', font_style{:})
title(type);
subplot(143)
plot(tab.b, tab.C, 'ko');
xlabel('$B$, [-]', font_style{:})
subplot(144)
plot(tab.Dw, tab.C, 'ko');
xlabel('$D_w$, [-]', font_style{:})
fig_axes = findobj(gcf, 'Type', 'Axes');
set(fig_axes, 'fontName', 'Times', ...
'fontSize', 12.0);
set(fig_axes, 'box' , 'on');
warning('off', 'MATLAB:polyfit:RepeatedPointsOrRescale');
figure(fig_dim{:});
corrplot(tab);
title(type);
fig_axes = findobj(gcf, 'Type', 'Axes');
set(fig_axes, 'fontName', 'Times', ...
'fontSize', 12.0);
set(fig_axes, 'box' , 'on');
warning('on', 'MATLAB:polyfit:RepeatedPointsOrRescale');
%% Regression analysis:
x_1ref = DD_ref./max(DD);
x_2ref = BB_ref./max(BB);
x_1 = tab.D;
x_2 = tab.b;
y = tab.C;
mod_3 = @(b, x)(b(1).*x(:, 2).^2 + ...
b(2).*x(:, 1).*x(:, 2));
b0 = ones(2, 1);
fitC3 = fitnlm([x_1 x_2], y, mod_3, b0, ...
'varNames', {'D', 'B', 'C'});
fprintf('%s:\n', type);
fprintf('C_3 ~ %+.3e*B^2 %+.3e*D*B\t with R^2 = %.3e\n', fitC3.Coefficients.Estimate, fitC3.Rsquared.Adjusted);
fprintf('C_3: (REF.) = %.3e\t(FIT.) = %.3e [N]\n', CC_ref, max(CC)*fitC3.feval(x_1ref, x_2ref));
figure(fig_dim{:});
hold on;
scatter3(x_1, x_2, y, 'filled');
scatter3(x_1, x_2, fitC3.feval(x_1, x_2), 'filled');
xlabel('$D$, [-]', font_style{:})
ylabel('$B$, [-]', font_style{:})
zlabel('$C$, [-]', font_style{:})
grid on;
box on;
set(gca, 'zScale', 'log');
title(type);
view(45.0, 45.0);
legend({'data', 'fit'}, 'location', 'best');
fig_axes = findobj(gcf, 'Type', 'Axes');
set(fig_axes, 'fontName', 'Times', ...
'fontSize', 12.0);
set(fig_axes, 'box' , 'on');
end
end
end