1 简介

粒子在倒锥内部摩擦滑动的模拟和动画

2 完整代码

%% Particle sliding on the inside of an inverted cone
% Simulation and animation of a particle sliding on the inside of an
% inverted cone with projections.
%
%%
clear ; close all ; clc
%% Parameters
m = 1;              % Mass                      [kg]
c = 0.05;           % Drag coefficient          [-]
g = 9.81;           % Gravity                   [m/s2]
parameters = [m c g];
%% Initial conditions
r0   = 2;           % Initial radial position   [m]
th0  = 0;           % Initial orientatio/n      [rad]
dr0  = 0;           % Initial radial speed      [m/s]
dth0 = 1;           % Initial angular speed     [rad/s]
z0 = [r0 th0 dr0 dth0];
%% Simulation
tf  = 30;                       % Final time                [s]
fR  = 30;                       % Frame rate                [fps]
time   = linspace(0,tf,tf*fR);  % Time                      [s]
% Integration
[tout,xout] = ode45(@(t,z) particle(t,z,parameters),time,z0);
% Retrieving states
r   = xout(:,1);
th  = xout(:,2);
% Coordinates
z = r;              % r=z. Cone angle = 45 deg.
x = r.*cos(th);
y = r.*sin(th);
%% Animation
color = cool(5); % Colormap
gray_color = [200 200 200]/255;
L = 2.1;            % Distance form center
XYmin =-2;
XYmax = 4;
Zmin = -0.5;
Zmax = 2.5;
steptick = 1;
figure
% set(gcf,'Position',[50 50 1280 720])  % YouTube: 720p
% set(gcf,'Position',[50 50 854 480])   % YouTube: 480p
set(gcf,'Position',[50 50 640 640])     % Social
hold on ; grid on ; axis equal   
set(gca,'xlim',[XYmin XYmax],'ylim',[XYmin XYmax],'zlim',[Zmin Zmax])
set(gca,'CameraPosition',[-27.0446  -27.4492   14.0294])
set(gca,'XTick',[XYmin XYmax],'YTick',[XYmin XYmax],'ZTick',[Zmin Zmax])
set(gca,'XTickLabel',[],'YTickLabel',[],'ZTickLabel',[])
set(gca,'FontName','Verdana','FontSize',18)
title(["Particle sliding on the"; "inside of an inverted cone";"Projections"])
% Create and open video writer object
v = VideoWriter('particle_cone_projections.mp4','MPEG-4');
v.Quality = 100;
v.FrameRate = fR;
open(v);
% Generating frames
for i=1:length(time)
    cla
    R = r0;     % Radius
    H = r0;     % Height
    N = 50;     % Number of points
    [xCy, yCy, zCy] = cylinder([0 R], N);
    m = mesh(xCy, yCy, H*zCy);
    set(m,'edgealpha',0,'facecolor',color(1,:),'facealpha',0.3)
    % Main particle
    plot3(x(1:i),y(1:i),z(1:i),'Color',color(5,:),'LineWidth',2)
    plot3(x(i),y(i),z(i),'ko','MarkerFaceColor',color(5,:),'MarkerSize',10)
    % Projections
    plot3(x(1:i),y(1:i),Zmin*ones(i,1),'Color',gray_color,'LineWidth',2)
    plot3(x(i),y(i),Zmin,'o','Color','k','MarkerFaceColor',gray_color,'MarkerSize',10)
    plot3(XYmax*ones(i,1),y(1:i),z(1:i),'Color',gray_color,'LineWidth',2)
    plot3(XYmax,y(i),r(i),'o','Color','k','MarkerFaceColor',gray_color,'MarkerSize',10)
    plot3(x(1:i),XYmax*ones(i,1),z(1:i),'Color',gray_color,'LineWidth',2)
    plot3(x(i),XYmax,z(i),'o','Color','k','MarkerFaceColor',gray_color,'MarkerSize',10)
    frame = getframe(gcf);
    writeVideo(v,frame);
end
close(v);
%% Auxiliary function
function dz = particle(~,z,dados)
    % Parametes
    m       = dados(1);
    c       = dados(2);
    g       = dados(3);
    % States
    r       = z(1);
%     th      = z(2);
    dr      = z(3);
    dth     = z(4);
    % State Equations
    dz(1,1) = dr;
    dz(2,1) = dth;
    dz(3,1) = (-m*g + m*r*dth^2 - 2*c*dr)/(2*m);
    dz(4,1) = (-2*m*dr*dth - c*r*dth)/(m*r);
end

3 仿真结果

4 参考文献

[1]周玲. "关于"滑动摩擦"教学动画模拟的尝试." 中学物理教学参考 7(1998).

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