Desperate solve 3rd order differentail equation

[Hugh_Struct]

Homework Statement

Hi,

I am trying to complete my MSc in Structural Engineering and being an engineer my maths sometimes let's me down. Please help me to solve this 3rd order differential equation...it relates to solving the later torsional buckling of a beam. Here goes!

So far i am at this point

(3) GI_t dr/dx - EI_w d³r/dx³ = M² / (PI²EI_z / L²) dr/dx

I assume there is a homogeneous solution to this where i could use the following boundary conditions

(v)o = (v)L = 0

(r)o = (r)L = 0

(d²r/dx²)0 = (d²r/dx²) = 0

Homework Equations

The problem begins with these two equations...

(1) EI_z d²v/dv² = -M r(X)

and

(2) GI_t dr/dx - EI_w d³r/dx³ = M dv/dx

Trial solutions of

(4) v(x) = w sin (PI x /L) and r(x) = r sin (PI x /L)

so differentiating (4) you can obtain

(5) v(x) = M / (PI²EI_z / L²) r(x)

The Attempt at a Solution

Apparently substituting equation (5) into equation (2) and using the boundary conditions you can resolve the problem to find
3...below is the solution
M = Square root [ (PI²EI_z / L²) * (GI_t + PI²EI_w / L²)]

I say apparently as i can only manage the substation part! I know you could also differentiate equation (2) to a 4th order differential and then substitute d^2v/dv^2 of equation (1) but i really need to know the other method.

I am SO sorry about the awful text...first time poster...long time reader!

Please help!

Thanks

Hugh

Much better to check this link out...page 2 and half of page 3 is what I'm after!

https://noppa.tkk.fi/noppa/kurssi/rak-54.3600/luennot/Rak-54_3600_lateral-torsional_buckling.pdf"

 

[Ray Vickson]

Homework Statement

Hi,

I am trying to complete my MSc in Structural Engineering and being an engineer my maths sometimes let's me down. Please help me to solve this 3rd order differential equation...it relates to solving the later torsional buckling of a beam. Here goes!

So far i am at this point

(3) GI_t dr/dx - EI_w d³r/dx³ = M² / (PI²EI_z / L²) dr/dx

I assume there is a homogeneous solution to this where i could use the following boundary conditions

(v)o = (v)L = 0

(r)o = (r)L = 0

(d²r/dx²)0 = (d²r/dx²) = 0

Homework Equations

The problem begins with these two equations...

(1) EI_z d²v/dv² = -M r(X)

and

(2) GI_t dr/dx - EI_w d³r/dx³ = M dv/dx

Trial solutions of

(4) v(x) = w sin (PI x /L) and r(x) = r sin (PI x /L)

so differentiating (4) you can obtain

(5) v(x) = M / (PI²EI_z / L²) r(x)

The Attempt at a Solution

Apparently substituting equation (5) into equation (2) and using the boundary conditions you can resolve the problem to find
3...below is the solution
M = Square root [ (PI²EI_z / L²) * (GI_t + PI²EI_w / L²)]

I say apparently as i can only manage the substation part! I know you could also differentiate equation (2) to a 4th order differential and then substitute d^2v/dv^2 of equation (1) but i really need to know the other method.

I am SO sorry about the awful text...first time poster...long time reader!

Please help!

Thanks

Hugh

Much better to check this link out...page 2 and half of page 3 is what I'm after!

https://noppa.tkk.fi/noppa/kurssi/rak-54.3600/luennot/Rak-54_3600_lateral-torsional_buckling.pdf"

If GI_t, EI_w, M² / (PI²EI_z / L²) are constants, your equation is of the form A d³r/dx³ = B dr/dx, where A = GI_t and B = GI_t - M² / (PI²EI_z / L²) are constants. Letting y = dr/dx, you have a second-order ODE A d^2 y/dx^2 = B y, which has a sinusoidal solution if B > 0 (assuming A > 0) or an exponential solution if B < 0.

RGV

 

[Hugh_Struct]

Thanks RGV,

That makes sense I'm trying to use the 2nd O.D.E A d² y/dx² = B y together with the B.C's to solve the equation. I will post it if i can get it...I may be a while!

Yes all the other terms are constants.

Hugh