ode with unknown initial conditions

Forum|Forum|10 years ago
October 21, 2015
2 replies
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Hi everyone,

I have a problem solving an ODE. I want to calculate the deflection of a beam, but at the beginning I don't know the force and the force application Point. So I make a guess of it. With a good guess or with iterations (very much work) I can solve it. But because I have many situations like this, I want to do it automatically.

In the attached file I have simplified the problem and explained it more detailly.

Thank you,

Stephan

1_beam-deflection-mcdx.zip

Calculus_Derivatives

Best answer by Werner_E

One way of doing it would be like the following:

You may want to compare accuracy against the values you had found:

At first sight it looks like you can find an a-value for any given F.2-value within a given range and vice versa (I have chosen the values you found in the following examples)

but as soon as you lower the value of CTOL, no solutions are found

Regards, Werner

PS: Instead of "find" you could use "minerr" (especially when you think that there is no perfect exact solution and so it will find a solution with lower values of CTOL, too) but your choice of using "minimize" was wrong.

F

Fred_Kohlhepp

23-Emerald I

You're using 3.1, so I can't open your sheet (stuck in 3.0.)

I solve beam deflection by solving the differential equation [EI d^2y/dx^2 = M(x)]. You can develop the equation for moment along the beam for whatever force arrangement you have. Two additional constraints are required--if it's a cantilever these would be displacement and slope at the origin.

S

sdidam

Author

1-Visitor

That's right, this point I already have.

I have a cantilever beam, length l, fixed at x=0 [boundary conditions: w(0)=0; w'(0)=0].

The Problem is, that I don't know at the beginning of the calculation where the force application point a is and also the Force F is unknown.

So the equation for the Moment is:

M(x)=F*(a-x) from x=0 to x=a and

M(x)=0 from x=a to x=l

But I know the following:

w'(a)=0

w(a):= is a function of x.

Another point is, that the Moment of inertia is not constant. In my example I simplified it to a linear relationship of x. In my more complicated case there are multible integrals in it, so that there is no easy solution.

I can solve the Problem with iterations manually or with a good guess, but I want to do this automatically, because that costs a lot of time and I have many problems like this.

F

Fred_Kohlhepp

23-Emerald I

You can make I a function of x as well as moment. If I(x) requires integrals it will slow down computation; one way to speed that is to create a vector of I at points along the length and fit a polynomial (function "regress") to the curve.

I have problems with tension and compression along with lateral loading, so I need terms of M(x) + Faxial y(x) was well as variable I(x). They all can solve.