EliI will try to show how we can solve an Ordinary Differential Equation using Matlab solver ODE45. However, with this problem i will use GNU Octave environment to solve. First, i will model a car suspension damping system equations then solve it.
In my previous post analogies-between-elements-in-electrica ... stems-5072, I modeled Mass, Spring, Damper System with its Electric circuit equivalent. So, in this figure a mechanical damper is presented and i will model and solve it at one initial condition as well as different initial conditions for displacements and velocity.
Balance of forces in Mass, spring, damper system:
ODE45 format:
- [t,x] = ode45(@FunctionName, tspan, [xInitial; xdotInitial])
- FunctionName = [x(2), RightHandSideofAFunction]
Where
m-mass
D-Damping Ratio
K-Spring constant
F-Force
Let
Replace the balance of Force equation with these new parameters and make
This plot shows that the Suspension system overshoots [Displacement and Velocity ] initially and becomes stable after some time on different initial conditions [from -1 to 1] , I think with the Phase Plane we might have a limit circle if i extend the initial conditions for displacement. This system has local equilibrium point as seen in the Phase Plane plot.
Asymptotically Stable System
- Simulations.JPG (37.54 KiB) Viewed 3684 times
- Simulations.JPG (37.54 KiB) Viewed 3684 times
What determines the quality of the systems, is it mass, Force, Damping Coefficient, or Spring Constant. I will modify the code to plot for different Damping coefficient and i will see the differences.
Global Stable system
- Capture.JPG (44.72 KiB) Viewed 3684 times
- Capture.JPG (44.72 KiB) Viewed 3684 times
So, in choosing the type of material to make a suspension system, the parameters above are taken into account to have a durable "shokapu"
@Admin please help me to put the codes below in a good listing, its .m file
- function kkoo = damper(t,x)
- m = 1; %mass
- F = 10; %Force
- D = 0.01; %Damping Coefficient
- K = 10; %Spring Constant
- tzr = x(2); %let xdot =x(2)
- suka = 1/m.*(F - D*x(2)-K*x(1)); % Right hand side of the ODE
- kkoo = [tzr; suka];
- endfunction
- function venom
- t = 0:0.1:20;
- initial_x = 0; % different initial condition, you can put any range say [-1:1]
- initial_xdot = 0;
- for j = 1:length(initial_x)
- for k = 1:length(initial_xdot)
- cond3 = initial_xdot(k);
- cond2 = initial_x(j);
- [t,x] = ode45(@damper, t , [cond2 cond3] );
- figure 1
- h(1) = subplot(2,2,1);
- h(2) = subplot(2,2,2) ;
- h(3) = subplot(2,2,[3 4]);
- set(h(1),'NextPlot','add');
- set(h(2),'NextPlot','add');
- set(h(3),'NextPlot','add');
- grid on
- plot(h(1),t,x(:,1));
- xlabel('Time t');
- ylabel('Displacement x');
- title('Car suspension model displacement');
- grid on
- plot(h(2),t,x(:,2));
- xlabel('Time t');
- ylabel('Velocity-xdot');
- title('Car suspension model Velocity');
- grid on
- plot(h(3),x(:,1),x(:,2));
- xlabel('Displacement x');
- ylabel('Velocity-xdot');
- title('Car suspension model Phase Plane');
- endfor
- endfor
- endfunction
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