Can this beam PDE problem be solve analytically?

  • Thread starter monkeyboy1
  • Start date
  • Tags
    Beam Pde
In summary, the problem at hand is solving for a Euler-Bernoulli beam with a lumped mass at one end and simply supported at the other end. The beam has a linear initial velocity profile and the boundary conditions include Y(0)=0, y''(0)=0, Y(L/2)=0, and Y'''(L)+M/m*beta*Y(L)=0. The attempt at a solution involves using separation of variables and eigenfunction expansions, but there may be a different approach to solving this problem. A hint is given to consider the form of the eigenfunction expansion.
  • #1
monkeyboy1
1
0
A work problem, but more like a homework problem...

Homework Statement



Euler-Bernoulli beam with a lumped mass at x=L, simply supported at x=0 and x=L/2. The beam has a linear initial velocity profile v(x) = w*x.

Homework Equations



BCs
Y(0)=0
y''(0)=0
Y(L/2)=0
Y'''(L)+M/m*beta*Y(L)=0

The Attempt at a Solution



I can solve the simply supported case, and I can solve the cantilever with tip mass case by separation of variables and eigenfunction expansions. Can this problem be solved in a similar way? When I go at it in a straight forward manner, I only get a single eigenvalue at zero.
 
Physics news on Phys.org
  • #2
Any hints or suggestions? Hint: Look at the boundary conditions and think about the form of the eigenfunction expansion.
 

FAQ: Can this beam PDE problem be solve analytically?

Can you explain what a PDE problem is?

A PDE (partial differential equation) problem involves finding a function that satisfies a given differential equation and boundary conditions. These equations describe physical phenomena such as heat transfer, fluid flow, and electromagnetic radiation.

Is it possible to solve PDE problems analytically?

It depends on the specific problem. Some PDEs have known analytical solutions, while others do not. In general, linear PDEs are more likely to have analytical solutions than nonlinear PDEs.

What are the advantages of solving a PDE problem analytically?

Analytical solutions provide a deeper understanding of the problem and can be used to derive important insights and relationships. They also tend to be more accurate and efficient compared to numerical solutions.

Are there any limitations to solving PDE problems analytically?

Yes, there are certain types of PDEs that do not have analytical solutions, or their solutions are too complex to be practical. In these cases, numerical methods must be used to approximate the solution.

How do you determine if a PDE problem can be solved analytically?

There is no easy way to determine if a PDE problem can be solved analytically. It often requires expertise in the specific field and a thorough analysis of the problem. However, there are some techniques and heuristics that can help in making this determination.

Similar threads

Solve Castigiliano Method for Continuous Beam Problem
Replies
8
Views
2K
Engineering Multi-point Boundary Value Problems Using Finite Difference Method
Replies
3
Views
692
Closed form solution for second order beam deflection?
Replies
1
Views
2K
Solving AB Beam Reactions with Hyperstatic System
Replies
1
Views
2K
Calculate Slope at Center of Simply Supported Beam Using Castigliano's Method
Replies
1
Views
2K
Solving boundary conditions for vibrating beam
Replies
1
Views
1K
Beam with two supports + spring
Replies
1
Views
3K
Does the angle of a beam fixed at two points affect the method?
Replies
3
Views
1K
Calculate the maximum vertical point force P that this beam can sustain
Replies
1
Views
2K
Inverting Shifted Laplace function
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top