I have got some governing equations related to beam vibration (Figure attached) . Such equations are easily solved as eigen value problem. Two of them are coupled. It has 3rd order terms also. I am feeling they are difficult to solve.
For solving I assumed,
u1(x,t) = U1(x) f(t) and u2(x,t)=U2(x) f(t)
w1(x,t) = W1(x) g(t) and w2(x,t)=W2(x) g(t)
For u1 and w1, I can assume,
U1(x)=c1 cos(αx) + c2 sin(αx) and
W1(x)=c3 cosh(βx) + c4 sinh(βx) + c5 cos(βx) + c6 sin(βx)
Than I will apply all the conditions, to get transcendental equation, its roots and than coefficients.
But, in case of the coupled PDEs, time function is not eliminating.
So can someone please tell how the problem can be tackled? In the some simplified case (If no possibility for previous case) like ratio= f(t)/g(t) is constant, than any idea for assuming U2(x) and W2(x)??
The first two decoupled equations are much simpler, and they do not include odd derivatives. I would suggest Fourier transform in spatial coordinate. You will get then and ordinary differential equation in time, with a wavenumber as a parameter. This very simple ODE has an analytical solution. Solve it and get a function in time. Then apply inverse Fourier transform to convert wave-numbers into location coordinate
Thank you for the answer, Igor. Actually my problem is with coupled pde. Domain is finite. For u2, w2, it can be taken 0.7 to 1.
Boundary conditions, at x=1, u2=w2=w2d=0 and at x=0.7 it is linearly coupled with u1 and w1. like (u1d-0.5*u2d +0.04*w2dd=0) such 3 conditions are there. for u1, w1 it is u1=w1=w1d=0 at x=0.
d refers to derivative in x.
Nikul, good evening,
Let's understand first the two decupled equations. The first equation for u1(x, t) is of second order in bot time in space. This means that two initial and two boundary conditions are needed. The boundary conditions mean that we know u1 at the ends (at x=0.7 and at x=1). These may be given functions of time only, or just constant values (e.g., zeros). The initial conditions mean that at t=0, the function u1 (x) and its time derivative du1/dt (x) are given for all x within the range [0.7, 1].
The second equation for w1 (x, t) is of second order in space and fourth order in time. So it also has two similar initial conditions: at t=0, w1 (x) and dw1/dt (x) are known for all x. However four boundary conditions are now needed (two boundary conditions at each end). At x=0.7 and at x=1, w1 and dw1/dx are specified. Again, these values may be functions of time or just constant.
Is this the case? Do you have a proper number of the initial and boundary conditions for each of the two decoupled equations?
Thank you Igor For showing interest in the equations.
For the Coupled set of Equations, at x=1, u2=w2=w2d=0 and at x=0.7, 3 conditions are there something like this " u1d-0.5*u2d +0.04*w2d=0 and second order are also there inside "
And all initial conditions are also there. They can be taken, 0 for all first time derivative and small value for displacement.
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