HOw to find displacement integrating velocity?

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Riccardo
Riccardo 2024-11-7，19:41
评论： Riccardo 2024-11-10，21:33
Ran in:
Hi, i need to find the displacement of a moving object x(t) knowing its velocity as a function of the displacement itself v(x). The function i have used is the following
I imposed that dx / dt = v(x) and then I tryed to use ODE 45 to find x(t) but I always get a vector full of zeros but i can't understand where the mistake is. This is the code that I wrote:
%% data
Po=1.5; % [bar] absolute feeding pressure
P1=1.01315; % [bar] absolute atmosferic pressure
rho=2700; %[kg/m^3] projectile density
D=0.048; % [m] projectile diameter
Length=0.10; % [m] projectile length
A=(pi/4)*D.^2; % [m^2] projectile face area
m=rho*A*Length; % [kg] projectile mass
L=0.90; %[m] useful barrell length
L_d=0.4; %[m] decelleration barrel length
Vol=0.0095; %[m^3] chamber volume
gamma=1.4; % air constant
% friction forces with 0.47 friction coefficient between aluminium and% steel
f=0.47*m*9.81;
%% model
x_dot=@(t,x) sqrt((2./m).*((Po.*1e5.*Vol)./(gamma-1).*(1-(Vol./(A.*x+Vol)).^(gamma-1))-A.*x.*P1.*1e5-x.*f));
x0=0;
tspan=0:0.1:1;
[t,x]=ode45(x_dot,tspan,x0);
plot(t,x)
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Shivam Gothi
Shivam Gothi 2024-11-8，10:15
hello @Riccardo,
can you please explain in brief what does this equation do? which system dynamics does it represents ?
I am getting some non-zero imaginery solution for velocity if I put "x0=1".
This is because, I found that the value of expression inside the sqrt() function is coming out to be negative (for positive "x", take x=1 as an example).
So sqrt() applied on negative number gives imaginary value. What does the imaginery "x_dot" represent here ?
Riccardo
Riccardo 2024-11-10，18:12
hi,@Shivam Gothi
This equation should reperesent the velocity of a projectile being accelerated by an adiabatic expansion of gas.
Vol represent the volume of the chamber containing the gas, L the length of the barrel, m the mass of the projectiel and A the cross section of the barrel

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Torsten
Torsten 2024-11-10，19:09
编辑：Torsten 2024-11-10，19:36
Ran in:
The equation from above describes the stationary velocity profile over the length L. Thus replacing v by dx/dt does not make sense in this case because there is no movement of x: x is the static position over the length L.
Or do you try to compute the position of a massless particle moving in x-direction over time ? Then use
%% data
Po=1.5; % [bar] absolute feeding pressure
P1=1.01315; % [bar] absolute atmosferic pressure
rho=2700; %[kg/m^3] projectile density
D=0.048; % [m] projectile diameter
Length=0.10; % [m] projectile length
A=(pi/4)*D.^2; % [m^2] projectile face area
m=rho*A*Length; % [kg] projectile mass
L=0.90; %[m] useful barrell length
L_d=0.4; %[m] decelleration barrel length
Vol=0.0095; %[m^3] chamber volume
gamma=1.4; % air constant
% friction forces with 0.47 friction coefficient between aluminium and% steel
f=0.47*m*9.81;
%% model
x_dot=@(t,x) sqrt((2./m).*((Po.*1e5.*Vol)./(gamma-1).*(1-(Vol./(A.*x+Vol)).^(gamma-1))-A.*x.*P1.*1e5-x.*f));
x0=1e-8;
tspan=0:0.01:1;
options = odeset('Events',@(t,x)event(t,x,L));
[t,x]=ode45(x_dot,tspan,x0,options);
plot(x,t)
xlabel('Length')
ylabel('Time')
grid on
function [position,isterminal,direction] = event(t,x,L)
position = x(1)-L; % The value that we want to be zero
isterminal = 1; % Halt integration
direction = 0; % The zero can be approached from either direction
end

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