Overshoot when solving damping differential equations

In summary, the conversation discussed solving and plotting differential equations for springs with various damping ratios. The equations used included the damped frequency, natural frequency, initial velocity, initial position, and time. The resulting graph showed that when the initial velocity is positive, the value of x does not decrease but instead increases. This can be corrected by setting the initial velocity to 0.
  • #1
max1546
8
0
I am trying to solve and plot the differential equations for springs.

when the damping factors are under 1 (underdamping), I tried damping ratios of:
0.01, 0.2, 0.1, 0.4, 0.8

If I use the following equations
(wd= damped frequency, wn= natural frequency, v0= v initial, x0= x initial, t= time)

wd=wn*sqrt(1-z^2);
A=sqrt(((v0+z*wn*x0)^2+(x0*wd)^2)/(wd^2));
phi=atan((x0*wd)/(v0+z*wn*x0));
x=A*exp(-z*wn*t)*sin(wd*t+phi);

and when I use the initial conditions
wn=2, x0=1, v0=1

I get the following picture
http://imageshack.us/photo/my-images/191/24265826.jpg/
Why does the value of x not decrease and increase instead at the start? (Shouldn't the value of x not exceed initial value?)
Is there something wrong with the equation I have formed above? Or is this what usually happens when solving these spring systems?
 
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  • #2
You have a positive initial velocity, i.e. v0=1. If you set v0=0 what your expecting will occur.
 

What is overshoot in the context of solving damping differential equations?

Overshoot refers to the maximum deviation of a system's response from its steady-state value, when solving damping differential equations. It occurs when the system's response "overshoots" or exceeds the desired or expected value before reaching its steady state.

Why is overshoot important in solving damping differential equations?

Overshoot can affect the stability and accuracy of the system's response. It can also cause oscillations or "ringing" in the system, which can be undesirable in some applications.

What factors contribute to overshoot when solving damping differential equations?

Overshoot can be caused by various factors, including the initial conditions of the system, the damping ratio, and the natural frequency of the system. In general, a higher damping ratio and lower natural frequency can result in lower overshoot.

How can overshoot be minimized when solving damping differential equations?

Overshoot can be minimized by adjusting the parameters of the system, such as the damping ratio and natural frequency, or by using control techniques such as feedback control. It can also be reduced by using numerical methods with smaller time steps.

What are the potential consequences of excessive overshoot when solving damping differential equations?

If the overshoot is too large, it can cause instability in the system and lead to unpredictable or undesirable behavior. In some cases, it can also cause damage to the system or its components if the overshoot is large enough to exceed their limits.

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