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# How do I solve cantilever bar made up of two materials analytically (dynamic analysis)?

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## How do I solve cantilever bar made up of two materials analytically (dynamic analysis)?

(OP)
Hello Eng-Tips team,

I am trying to solve the below problem analytically ∇∙σ + 𝑏_𝑥 = 𝜌 (𝜕^2 𝑢)/(𝜕𝑡^2 ) --> (1) is the governing equation
[Note: In the above problem, b_x = 0 (no body forces)]
IC: u(x,0) = v(x,0) = 0
BC: u(0,t) =0; u'(x=1,t) = F(t)/(A*E2)
[Note: Since force (F(t)) is applied on bar that has modulus E2, I'm assuming the BC is F(t)/(A*E2)]].

I'm having trouble understanding how to solve the above problem.

Q1). Does interface conditions come into play in this problem? The displacements, stresses are continuous at the interface (Assuming it is perfectly bonded).
Q2). If I consider each portion of the bar individually (material1 or material2), how do the boundary conditions behave?

Any help would be much appreciated. Thank you!

### RE: How do I solve cantilever bar made up of two materials analytically (dynamic analysis)?

Not sure what you're trying to solve for, but I can tell you this much - F(t) is constant through both sections, so the displacement of each part is (F(t)/ A*E)*L, where L is the length of the section under consideration. The total displacement is the sum of the displacements of the 2 parts. I hope that helps.

Rod Smith, P.E., The artist formerly known as HotRod10

### RE: How do I solve cantilever bar made up of two materials analytically (dynamic analysis)?

(OP)
Hello BridgeSmith,

Thank you for the response. In this problem I'm solving for displacement only.

1. The blue color section is fixed at the left end. So, the BC is u1(x=0,t) = 0.
2. Now, if I consider the yellow section the left boundary is the right boundary of the blue section.
So, can we say u2(x=0.5,t) = u1(x=0.5, t)?

Note: u1 is the displacement of section 1,u2 is the displacement of section 2.

When I consider each of the above sections separately, I have a differential equation that I need to solve. (Governing equation (1) mentioned in the problem).
This governing equation could be solved if I impose 2 Initial and 2 Boundary conditions.
i.e,

For blue section:
∇∙σ1 + 𝑏_𝑥 = 𝜌1 (𝜕^2 𝑢1)/(𝜕𝑡^2 ) --> (1) is the governing equation
[Note: In the above problem, b_x = 0 (no body forces)]
IC: u1(x,0) = v1(x,0) = 0
BC: u1(0,t) =0; u1'(x=1,t) = ?
Note: Based of your first answer, u1'(x=0.5,t) = F(t)/(A*E1)

For yellow section:
∇∙σ2 + 𝑏_𝑥 = 𝜌2 (𝜕^2 𝑢2)/(𝜕𝑡^2 ) --> (1) is the governing equation
[Note: In the above problem, b_x = 0 (no body forces)]
IC: u2(x,0) = v2(x,0) = 0
BC: u2(0.5,t) =?; u2'(x=1,t) = F(t)/(A*E2)

Is this u2(0.5,t) = u1(0.5,t)?

I was majorly confused about how to handle the boundary conditions.

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