Help with finding the particular solution to this difference equation

[Tygra]

Hi all,

I have a difference equation that is in the form:



The book that I am going through gives the complimentary solution to this equation as:



I have the values of g, r, V and C1, and also the values of beta 1 and beta 2. I just need to find the particular solution to add on to the complimentary solution. Then I can solve for the coefficients A1 and A2. I have looked high and low on the internet and cannot find anything that deals with a inhomogenious difference equation that has the form of this equation.

Please can somebody help? Even if you can not, please don't ignore this post. Please let me know your thoughts on it.

Many thanks in advance,

Tygra

 

[blamocur]

I have the values of g, r, V and C1, and also the values of beta 1 and beta 2. I just need to find the particular solution to add on to the complimentary solution. Then I can solve for the coefficients A1 and A2. I have looked high and low on the internet and cannot find anything that deals with a inhomogenious difference equation that has the form of this equation.

When you plug in all your known values into the difference equation you left with a family of equations for [imath]A_1,A_2[/imath] which have to hold for all values of [imath]i[/imath]. Pick a couple of simple values of [imath]i[/imath] (e.g., 1 and 2), and you get a system of 2 linear equations with 2 variables.

 

[Tygra]

Hi blamocur,

Thank you for responding to my question.

I am a not sure if I understood you correctly.

If I plug into to the equations i =1 and i = 2, then I get two equations with three variables. I get [imath]\theta 1[/imath], [imath]\theta2[/imath] and [imath] \theta 3[/imath]. This looks like this when plugged into the equation.

When i = 1:


When i = 2:



I have probably misunderstood you, so sorry about that.

To give you some context to the problem, the variable [imath]\theta[/imath] is for the rotations at the joints of each storey in a multi-storey bulding. Once I have [imath]\theta[/imath] at every joint I can calculate the bending moments in the columns and beams.

Thank you for your patience,

Tygra

 

[blamocur]

Use this expression from your first post:

 

[Tygra]

Ah I see, the complimentray function.

Thing is I need boundary conditions. For i = 1 [imath]\theta1[/imath] = 0, but if I use i = 2, so [imath]\theta2[/imath] I do not know what this rotation is at this joint - its not zero. The book says for the other boundary condition the sum of the moments at the top most joint = 0. So in a 4 storey building this would be at i = 4. And the complimentray function is a function for [imath]\theta[/imath], not for the moments. I learnt a university that the moment is the first derivitave of the rotation. So I think I might have to differentiate the complimentray function. What do you think?

 

[blamocur]

Actually, I was wrong -- sorry, did not pay enough attention to the problem and assumed that the complimentary function is a solution in itself Time to start thinking...

 

[Tygra]

That's fine, blamocur.

I actually posted this question on the Just Answer UK website, and two maths teacher/tutors couldn't answer it. Im not sure what level they were, but the site claims they were experts. - probably not though

Please dont leave me on here. If you cant solve it let me know. But I am hoping if you can't perhaps you have some friends in high places that can?

 

[blamocur]

Unlike Just Answer tutors I do not claim to be an expert in the field of difference equations, no do I know any experts in there. If I had to solve this in practice, and the range of [imath]i[/imath] is not too big (< 1000?) I would reduce it to a system of linear equations and see if it can be solved numerically. If [imath]1\leq i \leq n[/imath] you will have a system of [imath]n-2[/imath] linear equations plus 2 boundary conditions, which should make the system quite solvable, especially because the resulting [imath]n\times n[/imath] matrix is quite sparse.

You might also have more luck asking this question in Math StackExchange's Recurrence Equation section.

 

[Tygra]

I have tried to solve it numerically using the shooting method, but I am getting poor results. The equation is a second order difference equation. Could you claify for me if I am correct in my conversion to first order equations below?

Second Order equation

if [imath] \theta'[/imath] = [imath]z[/imath] and [imath]\theta''[/imath] = [imath]z'[/imath]





and

[imath]\theta'[/imath] = [imath]z[/imath]

Are these correct blamocur?

 

[blamocur]

Are these correct blamocur?

No clue What is [imath]\theta^\prime[/imath] considering that [imath]\theta[/imath] is a discrete variable?

 

[mario99]

Hi all,

I have a difference equation that is in the form:

View attachment 38156

The book that I am going through gives the complimentary solution to this equation as:

View attachment 38160

I have the values of g, r, V and C1, and also the values of beta 1 and beta 2. I just need to find the particular solution to add on to the complimentary solution. Then I can solve for the coefficients A1 and A2. I have looked high and low on the internet and cannot find anything that deals with a inhomogenious difference equation that has the form of this equation.

Please can somebody help? Even if you can not, please don't ignore this post. Please let me know your thoughts on it.

Many thanks in advance,

Tygra

I have an idea to approximate the solution for [imath]\theta_1, \theta_2, \theta_3,\cdot \cdot \cdot \cdot \cdot \cdot ,\theta_n[/imath] if you tell me two things. The first thing that I need to know is that this [imath]\theta[/imath] is a function of what variable? is it [imath]V[/imath]? The second thing that I need to know is that, what are the two boundary conditions?

 

[mario99]

I have an idea to approximate the solution for [imath]\theta_1, \theta_2, \theta_3,\cdot \cdot \cdot \cdot \cdot \cdot ,\theta_n[/imath] if you tell me two things. The first thing that I need to know is that this [imath]\theta[/imath] is a function of what variable? is it [imath]V[/imath]? The second thing that I need to know is that, what are the two boundary conditions?

If my idea worked, you would be able to use, for example, the values of [imath]\theta_1[/imath] and [imath]\theta_2[/imath] to find the coefficients [imath]A_1[/imath] and [imath]A_2[/imath] of:

[imath]\theta_i = A_1 \cdot \beta_1^i + A_2 \cdot \beta_2^i[/imath]

By solving this system:

[imath]\theta_1 = A_1 \cdot \beta_1^1 + A_2 \cdot \beta_2^1[/imath]

[imath]\theta_2 = A_1 \cdot \beta_1^2 + A_2 \cdot \beta_2^2[/imath]

 

[topsquark]

I have an idea to approximate the solution for [imath]\theta_1, \theta_2, \theta_3,\cdot \cdot \cdot \cdot \cdot \cdot ,\theta_n[/imath] if you tell me two things. The first thing that I need to know is that this [imath]\theta[/imath] is a function of what variable? is it [imath]V[/imath]? The second thing that I need to know is that, what are the two boundary conditions?

This is a difference equation in i. So you are solving for all [imath]\theta _i[/imath].

Have you encountered difference equations?

-Dan

 

[khansaheb]

No clue What is [imath]\theta^\prime[/imath] considering that [imath]\theta[/imath] is a discrete variable?

Looks like

θ′ = d/dx [θ] ..... and consequently

θ′′ = d/dx [θ′] = d²/dx² [θ]

 

[Tygra]

I have an idea to approximate the solution for [imath]\theta_1, \theta_2, \theta_3,\cdot \cdot \cdot \cdot \cdot \cdot ,\theta_n[/imath] if you tell me two things. The first thing that I need to know is that this [imath]\theta[/imath] is a function of what variable? is it [imath]V[/imath]? The second thing that I need to know is that, what are the two boundary conditions?

Hi Mario,

that is a good question. Knowing what the variable is of the function has been confusing me. I don't believe it is [imath]V[/imath], as this is the shear force at each joint in the multi-storey structure. I could be wrong though. Perhaps the variable is the distance from the bottom to the top of the structure which we could call [imath]x[/imath]. I say this because for the deflection and rotation of a cantilever beam the equations are a function of [imath]x[/imath]. See below:



As for the boundary conditions, [imath]\theta1[/imath] equals zero. This is because the rigid connection at the base retrains any rotation. The book gives the second boundary condition as the sum of the moments at the joint at the top of the structure = 0. This has been confusing me also, because the equation is for the rotation [imath]\theta[/imath], not for the moments! However, like I posted previously, the bending moment is the first derivative of the rotation. So my guess is, differentiate the equation and then set the last value of [imath]i[/imath] equal to zero.

Here is the page from the book. It might help you out more.




Thank you for all your help.

Tygra

 

[mario99]

The table that you have sent gives the angle at the free end which is the second boundary condition we are seeking. The general relation between the deflection [imath]v[/imath] and the angle [imath]\theta[/imath] is:

[imath]\theta_A = v'(0)[/imath]

And

[imath]\theta_B = v'(L)[/imath]


Can you find [imath]\theta_B[/imath]? Either use the table or the general relation.

 

[Tygra]

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 [imath]\theta B[/imath]. Try using a value of 0.00010289 rads.

 

[khansaheb]

Are you taking a course in "structural mechanics or "numerical analysis"?

Do you know the differential equation/s through which the expressions in the table that you quoted in the response #16?

Can you derive the equations that you quoted in response #16 - using principles of "solid mechanics"?

If this is a "project" in your structural mechanics course - you need to know the derivations!

Then you solve those equations through numerical analysis to get the "approximate" answer. (I know this because I taught structural mechanics at a university)

 

[Tygra]

Are you taking a course in "structural mechanics or "numerical analysis"?

Do you know the differential equation/s through which the expressions in the table that you quoted in the response #16?

Can you derive the equations that you quoted in response #16 - using principles of "solid mechanics"?

If this is a "project" in your structural mechanics course - you need to know the derivations!

Then you solve those equations through numerical analysis to get the "approximate" answer. (I know this because I taught structural mechanics at a university)

Hi Khansaheb,

I am actually a graduate Civil Engineer (with more emphasis on structural engineering). I graduated last year. I am not working at the moment so I am trying to go above a beyond what I have already learnt. I am doing a personal project for a seven storey building and want to compute approximate bending moments for the rigid frames of the building. To do this I need the elusive [imath]\theta[/imath] values.

We didn't do a module on structural mechanics per se, but did structural analysis and computational mechanics (which is where we learnt numerical methods).

Yes, I do know the differential equations for the expressions in the table that I quoted. You usually start with [imath]y'' = Mx/EI [/imath] and integrate once for the rotation and once more for the deflection.

The expressions in the table are for beams, not for 2D rigid frames. The beams don't experience something called shear racking, multi-storey rigid frames do. If the multi storey rigid frame just experienced bending, then I could use the expression in the table. The deflection of the rigid frames defect due to shear racking and bending.

 

[khansaheb]

compute approximate bending moments

I think you meant bending stiffness ↑(above).

Are you modelling the whole building as a column subjected to wind load/s and gravitational load?