Fourth order differential equation

[Fayz]

Homework Statement

I have this physics mathematical problem : (see link in comment)


EI(∂⁴u)/(∂x⁴) =f ......(1)


The boundary conditions are: ∂²u/∂x² =0 and EI ∂³u/∂x³ =±F

where E is Young’s modulus, I is second moment of area, f is force per unit length applied to the beam and F is the force applied to the edges and the ± applies the the left and right edges.

The upper load placed on the chip is modeled as two point forces while the loads exerted by the pins of the chip are modeled as (localised) Hookean springs as shown in figure [1 ](in the link). Thus the equations become:

EI ∂⁴u/∂x⁴ =−F[δ(x−B)+δ(x+B)]−k₂ u(0)δ(x) .......(2)

with boundary conditions :

EI∂³u/∂x³ =−k₁ u(−A), x=−A .....(3)

EI∂³u/∂x³ =k₁u(A), x=A ......(4)

∂²u/∂x² =0, x=±A .....(5)


Show that calculating the displacements can be reduced to the problem of solving a set of linear equations.

could you help please

Homework Equations

for the full question see link:
4shared.com/file/rdVHwO6j/strains_in_silicon_chip.html

The Attempt at a Solution

Trying to answer:

x axis:
−A __(I)____−B __(II)___0 __(III)_______B _____(IV)____A

•In area I , II , III and IV equation (2) become ZERO: EI∂ ⁴ u/ ∂x ⁴ =0 solving this equation we get a linear equation system:

u_I =a₁ x ³ +a₂ x ² +a₃ x+a₄
u_II =b₁ x ³ +b₂ x ² +b₃ x+b₄
u_III =c ₁ x ³ +c₂ x ² +c₃ x+c₄
u_IV =d₁ x ³ +d₂ x ² +d₃ x+d₄
which has 16 unknowns (a 's,b 's,c 's and d 's)

using the doundary conditions and integration equation (1) aroud -B and B we could find 12 of the unknowns.

After finding them what shuold I do.

Please help.

 

[mark726]

Dear student,

Thank you for posting your question. It seems like you are on the right track in solving this problem. To simplify the problem and reduce it to a set of linear equations, you can use the boundary conditions and the integration of equation (1) to find the remaining unknowns.

First, plug in the boundary conditions for ∂2u/∂x2 =0 into the equations for uI and uIV. This will give you two equations with four unknowns (a1, a2, d1, d2). Similarly, use the boundary conditions for ∂3u/∂x3 =±F to find two more equations with four unknowns (b1, b2, c1, c2).

Next, integrate equation (1) over the range of -B to B. This will give you two more equations with four unknowns (a3, a4, b3, b4). Finally, use the boundary conditions for ∂3u/∂x3 =−k1 u(−A) and ∂3u/∂x3 =k1u(A) to find the remaining two equations with four unknowns (c3, c4, d3, d4).

You should now have a total of eight equations with eight unknowns, which can be solved using any standard method for solving a system of linear equations (e.g. Gaussian elimination or matrix inversion).

I hope this helps. Good luck with your problem!