Help with finding the particular solution to this difference equation

[Tygra]

At the moment, just the wind load.

Actually, it is the bending moments. The bending stiffness is the $Ci$ value $EIc/h$. Refer back to the post where I scanned the page from the book.

Here is how it looks in the software. This is just a practise frame. Its not what I am doing for my personal project

 

[mario99]

Thing is the rotation for the single cantilever beam will not work for a muiltistorey frame. However, using some approximate methods for deflection I have managed to come up with a value for $\theta B$. Try using a value of 0.00010289 rads.

Does this answer make sense to you? I will trust you that it is correct and will give you a taste of the idea. The idea of numerical analysis is to divide the interval (length of the beam) $0 < x < L$ by some number $n$ into subintervals. In other words, you will have many interior points around the boundary points.

If $\theta_0 = 0$ and $\theta_L = 0.00010289$ are the boundary conditions, then $\theta_1, \theta_2, \theta_3,\cdot\cdot\cdot, \theta_{i-1}$ are the interior functions that we want to find. And $x_1,x_2,x_3,\cdot\cdot\cdot,x_{i-1}$ are the interior points for $\theta_1, \theta_2, \theta_3,\cdot\cdot\cdot, \theta_{i-1}$ respectively.

The larger the number $n$, the better the approximation gets. But increasing the number $n$ will in return cost us to solve more equations.

Let us choose $n = 10$, then the length $h$ of $1$ subinterval is equal to $h = \displaystyle \frac{x_{10} - x_0}{n} = \frac{L - 0}{10}$. It depends on the length of the beam, say, $L = 8 \ \text{m}$, then $\displaystyle h = \frac{8 - 0}{10} = 0.8$

The interior points are:

$x_1 = 0.8$
$x_2 = 1.6$
$x_3 = 2.4$
$x_4 = 3.2$
$x_5 = 4.0$
$x_6 = 4.8$
$x_7 = 5.6$
$x_8 = 6.4$
$x_9 = 7.2$

While $x_0 = 0$ and $x_{10} = 8$ are the boundary points.

Now go back to the difference equation and focus on the variables that have the index $i$, one of them is $x_i$. We need to know it to continue our analysis.

 

[Tygra]

Sorry for the late reply, Mario. I think the forum has been having some technical issues

 

[Tygra]

I can't seem to post any latex

 

[blamocur]

Which $\LaTeX$? You need to use the $\KaTeX$ dialect. What did you try?

 

[Tygra]

I hope Im following you correctly.

So, lets use the structure I posted in #22.

The height is 12m.

Because we want the points to land on a joint (joints are 3m apart) let n=8

xi = $(12-0)/8 = 1.5$

 

[Tygra]

Sorry guys I am really having problems posting

 

[mario99]

Sorry for the late reply, Mario. I think the forum has been having some technical issues

It's OK. It happened to me too a few months ago.

I hope Im following you correctly.

So, lets use the structure I posted in #22.

The height is 12m.

Because we want the points to land on a joint (joints are 3m apart) let n=8

$(12-0)/8 = 1.5$

Yeah, we can do this.

View attachment 38177

But your difference equation is confusing me now. The difference equation must have both $\theta_i$ and $x_i$.

And from where did you get $r^{x_i}$? Are you sure about this? Or maybe the $x_i$ is inside $V_i$?

Or maybe $V_i + V_{i+1} = x_i + x_{i+1}$!

 

[Tygra]

But your difference equation is confusing me now. The difference equation must have both $\theta_i$ and $x_i$.

And from where did you get $r^{x_i}$? Are you sure about this? Or maybe the $x_i$ is inside $V_i$?

Or maybe $V_i + V_{i+1} = x_i + x_{i+1}$!

The parameter $V$ is the shear force created by the wind. $V_i$ is the shear for the storey in question; $V_i+1$ is the shear in the storey above.

Look at this diagram:



If we are looking at the first storey, $V_i = 35 kN$, $V_i+1 = 25 kN$. So, $V$ cannot be $x_i$. I believe $x_i$ is the vertical distance up the frame.

I read somewhere that difference equations are identical to differential equations, other than that difference equations represents sequences. Sometimes you get differential equations that is a function of $x$ , but has not $x$ in the expression.

Mario, did you read through the scanned page of the book I posted in post #15? Did you not make any sense of it? It talks about finding the particular solution to the diffrence equation.

And I substituted $x_i$ in the equation, $r^x_i$ because you said to focus in the variable that have $i$ in it.

 

[mario99]

Yes, I did read it, but it is ambiguous. It has written the difference equation for $\theta$ with the assumption of the story height, stiffness, the fixed ratio $r$, and shear forces have specific properties. So, using $x_i$ to solve the problem will be invalid because the difference equation is using something completely different than the length (or height) of the beam. Then, the book said that if the numerator on the right side is a polynomial, a particular solution may be obtained similar to the complementary solution. After all this information, it is still ambiguous what is the independent variable. But since it focused on the numerator on the right side, the only way to get a particular solution to this difference equation is to change it to a polynomial. But how? We don't know and I think that we will never know.

I think that a second way to solve your problem is to write everything from scratch. If you want to solve your problem with a difference equation, then write down all the original differential equations related to the problem and derive it step by step in a clear manner. Or you can post the problem of the beam or whatever it is in a new post and I am sure that professor Khan will figure out a better solution to it than the difference equation since he has already taught this subject at university.

If you cannot solve your problem by the difference equation, I am sure there is a different way to solve it. You have just to search.

What is the scope of my knowledge in this area? I know how to convert a second order differential equation to a difference equation and solve the difference equation by the Finite Difference Method which I have already explained it to you: dividing the interval of the original differential equation by a number $n$. Also I can find the complementary and particular solutions for a small number of difference equations when every variable there is stated clearly.

Good luck Tygra.