Basically I have to find omega by solving determinant of the following matrix:

M := Matrix(4, 4, {(1, 1) = 0, (1, 2) = -1, (1, 3) = 0, (1, 4) = 1, (2, 1) = -EI*beta^3, (2, 2) = -m*omega^2, (2, 3) = EI*beta^3, (2, 4) = -m*omega^2, (3, 1) = EI*sin(147*beta)*beta+k_r*cos(147*beta)+I*c_r*omega*cos(147*beta), (3, 2) = EI*cos(147*beta)*beta-k_r*sin(147*beta)-I*c_r*omega*sin(147*beta), (3, 3) = -EI*sinh(147*beta)*beta+k_r*cosh(147*beta)+I*c_r*omega*cosh(147*beta), (3, 4) = -EI*cosh(147*beta)*beta+k_r*sinh(147*beta)+I*c_r*omega*sinh(147*beta), (4, 1) = -EI*cos(147*beta)*beta^3+sin(147*beta)*k_h, (4, 2) = EI*sin(147*beta)*beta^3+cos(147*beta)*k_h, (4, 3) = EI*cosh(147*beta)*beta^3+sinh(147*beta)*k_h, (4, 4) = EI*sinh(147*beta)*beta^3+cosh(147*beta)*k_h}):

The remaining values are:

beta=((5000/10^12)*(omega^2))^(1/4),k_r=3.33*10^10,k_h=1.62*10^9,c_r=3.14*10^9,m=350000,L=147,EI=10^12:

What is the proper way to deal with this problem numerically. Or maybe it is even possible to get a reasonable analytical expression?

Can someone please help me to find first 5 coplex valued roots of the followind expression:

Frequency_Equation := -(2.220315292*10^21*I)*csgn(omega)*omega^4*sin(1.236117731*(omega^2)^(1/4))*cosh(1.236117731*(omega^2)^(1/4))-2.354665581*10^22*csgn(omega)*omega^3*sin(1.236117731*(omega^2)^(1/4))*cosh(1.236117731*(omega^2)^(1/4))-2.354665581*10^22*sinh(1.236117731*(omega^2)^(1/4))*csgn(omega)*omega^3*cos(1.236117731*(omega^2)^(1/4))-6.415296703*10^25*(omega^2)^(3/4)*cos(1.236117731*(omega^2)^(1/4))*cosh(1.236117731*(omega^2)^(1/4))-2.177438228*10^22*cosh(1.236117731*(omega^2)^(1/4))*(omega^2)^(3/4)*omega^2*cos(1.236117731*(omega^2)^(1/4))-5.946035576*10^21*omega^2*(omega^2)^(3/4)-6.415296702*10^25*(omega^2)^(3/4)-(3.560760000*10^24*I)*sinh(1.236117731*(omega^2)^(1/4))*omega^3*cos(1.236117731*(omega^2)^(1/4))+(3.560760000*10^24*I)*cosh(1.236117731*(omega^2)^(1/4))*omega^3*sin(1.236117731*(omega^2)^(1/4))-(2.613877239*10^21*I)*cosh(1.236117731*(omega^2)^(1/4))*(omega^2)^(3/4)*omega^3*cos(1.236117731*(omega^2)^(1/4))-(6.049258752*10^24*I)*(omega^2)^(3/4)*omega-1.907153069*10^25*sinh(1.236117731*(omega^2)^(1/4))*omega^2*sin(1.236117731*(omega^2)^(1/4))*(omega^2)^(1/4)-(2.220315292*10^21*I)*sinh(1.236117731*(omega^2)^(1/4))*csgn(omega)*omega^4*cos(1.236117731*(omega^2)^(1/4))-(6.049258753*10^24*I)*(omega^2)^(3/4)*cos(1.236117731*(omega^2)^(1/4))*cosh(1.236117731*(omega^2)^(1/4))*omega-3.500000000*10^21*cosh(1.236117731*(omega^2)^(1/4))*omega^4*sin(1.236117731*(omega^2)^(1/4))+3.500000000*10^21*sinh(1.236117731*(omega^2)^(1/4))*omega^4*cos(1.236117731*(omega^2)^(1/4))-2.156220000*10^25*sinh(1.236117731*(omega^2)^(1/4))*omega^2*cos(1.236117731*(omega^2)^(1/4))+2.156220000*10^25*cosh(1.236117731*(omega^2)^(1/4))*omega^2*sin(1.236117731*(omega^2)^(1/4));

Hi. I am trying to identify mode shapes (phi(x)) and natural frequencies  of non-uniform euler-bernoulli beam. There are number of numerical methods to solve ODE with certain boundary conditions (i.e. Runge Kutta method). Problem is that I am newbie here. I am interested in particularly first vibration mode and its frequency. Is there anyone acquainted with it and would be able to help me?  Non-unif.mw

Can someone please advise me how to solve the following for 'beta'. Solve function is not able to do that, or at least I dont know how.

-9999990000000000000000*cos(166*beta)*sinh(166*beta)*cosh(88*beta)^2-9999990000000000000000*cos(88*beta)^2*sin(166*beta)*cosh(166*beta)+9999990000000000000000*sinh(166*beta)*cos(88*beta)^2*cos(166*beta)+9999990000000000000000*cosh(88*beta)^2*cosh(166*beta)*sin(166*beta)+10000010000000000000000*cos(166*beta)*sinh(166*beta)+10000010000000000000000*sin(166*beta)*cosh(166*beta)+9999990000000000000000*sinh(88*beta)*cos(166*beta)*cosh(166*beta)*cosh(88*beta)-9999990000000000000000*sinh(88*beta)*sin(166*beta)*sinh(166*beta)*cosh(88*beta)+9999990000000000000000*sin(88*beta)*cos(88*beta)*sinh(166*beta)*sin(166*beta)+9999990000000000000000*cos(88*beta)*cos(166*beta)*sin(88*beta)*cosh(166*beta)-9980010000000000000000*cosh(88*beta)^2*sinh(166*beta)*cos(88*beta)^2*cos(166*beta)-9980010000000000000000*cosh(88*beta)^2*cosh(166*beta)*sin(166*beta)*cos(88*beta)^2+9980010000000000000000*sinh(88*beta)*cos(88*beta)^2*sin(166*beta)*sinh(166*beta)*cosh(88*beta)-9980010000000000000000*cos(88*beta)*cosh(88*beta)^2*sin(88*beta)*sin(166*beta)*sinh(166*beta)+9980010000000000000000*sinh(88*beta)*cosh(88*beta)*cosh(166*beta)*cos(88*beta)^2*cos(166*beta)+9980010000000000000000*cosh(88*beta)^2*cosh(166*beta)*cos(88*beta)*sin(88*beta)*cos(166*beta)-9980010000000000000000*cos(88*beta)*sinh(88*beta)*cos(166*beta)*sin(88*beta)*sinh(166*beta)*cosh(88*beta)+9980010000000000000000*cos(88*beta)*cosh(88*beta)*sin(88*beta)*sin(166*beta)*cosh(166*beta)*sinh(88*beta)=0

Can someone help me to solve system of equations please. I have a system of 8 complex valued equations, with 8 unknowns: _C1,_C2........_C8

Equation system looks like:

eq_system:={ -3.248046797 10 _C1 + 1.773373463 10 _C2 + (2.182313824 10 - 9.987524076 10 I) _C3 + 1.773373463 10 _C4 = -7.389056097 10 _C2- 7.389056097 10 _C4+ (4.161468365 10 + 9.092974265 10 I)_C3,

............}  its only 1st equation, others are similar.

It looks rather simple though I am not able to solve it with solve or fsolve commands. What I'm doing wrong?

solve(eq_system,{_C1,_C2,_C3,_C4,_C5,_C6,_C7,_C8});