Question: How to solve by eigen function expansion method for this euler bernoulli beam having fixed end at both ends?


 

NULL

Restart:

pde := diff(w(x, t), x $ 4) + diff(w(x, t), t $ 2)/c^2 = 0;

diff(diff(diff(diff(w(x, t), x), x), x), x)+(diff(diff(w(x, t), t), t))/c^2 = 0

(1)

test1 := w(x, t) = W(x)*cos(omega*t); pde1 := expand((eval(pde, test1))/cos(omega*t))

w(x, t) = W(x)*cos(omega*t)

 

diff(diff(diff(diff(W(x), x), x), x), x)-W(x)*omega^2/c^2 = 0

(2)

ode := algsubs(omega^2/c^2 = alpha^4, pde1); sol := dsolve(ode); sol := convert(sol, trig)

-W(x)*alpha^4+diff(diff(diff(diff(W(x), x), x), x), x) = 0

 

W(x) = _C1*exp(-alpha*x)+_C2*exp(alpha*x)+_C3*sin(alpha*x)+_C4*cos(alpha*x)

 

W(x) = _C1*(cosh(alpha*x)-sinh(alpha*x))+_C2*(cosh(alpha*x)+sinh(alpha*x))+_C3*sin(alpha*x)+_C4*cos(alpha*x)

(3)

L_collection := [indets(sol, specfunc({cos, cosh, sin, sinh}))[]]

[cos(alpha*x), cosh(alpha*x), sin(alpha*x), sinh(alpha*x)]

(4)

``

temp := collect(sol, [sinh, cosh, sin, cos])

W(x) = (-_C1+_C2)*sinh(alpha*x)+(_C1+_C2)*cosh(alpha*x)+_C3*sin(alpha*x)+_C4*cos(alpha*x)

(5)

CL := [coeffs(rhs(temp), L_collection)]NULL

[-_C1+_C2, _C1+_C2, _C3, _C4]

(6)

R:=[seq(cat(_D,i)=CL[i], i=1..nops(CL))]

[_D1 = -_C1+_C2, _D2 = _C1+_C2, _D3 = _C3, _D4 = _C4]

(7)

MX:= subs((rhs=lhs)~(R), temp);

W(x) = _D1*sinh(alpha*x)+_D2*cosh(alpha*x)+_D3*sin(alpha*x)+_D4*cos(alpha*x)

(8)

boundary_condition_1 := simplify(rhs(eval(diff(MX, `$`(x, 1)), x = 0))/alpha = 0)

_D1+_D3 = 0

(9)

boundary_condition_2 := simplify(rhs(eval(MX, x = 0)) = 0)

_D2+_D4 = 0

(10)

boundary_condition_3 := simplify(rhs(eval(diff(MX, `$`(x, 1)), x = L))/alpha = 0)

cosh(alpha*L)*_D1+sinh(alpha*L)*_D2+cos(alpha*L)*_D3-sin(alpha*L)*_D4 = 0

(11)

boundary_condition_4 := simplify(rhs(eval(MX, x = L)) = 0)

_D1*sinh(alpha*L)+_D2*cosh(alpha*L)+_D3*sin(alpha*L)+_D4*cos(alpha*L) = 0

(12)

boundary_condition_solve := solve([boundary_condition_1, boundary_condition_2, boundary_condition_3, boundary_condition_4]); MX := eval(MX, boundary_condition_solve)

Warning, solutions may have been lost

 

{L = L, _D1 = -_D3, _D2 = -_D4, _D3 = _D3, _D4 = _D4, alpha = 0}, {L = 0, _D1 = -_D3, _D2 = -_D4, _D3 = _D3, _D4 = _D4, alpha = alpha}, {L = L, _D1 = 0, _D2 = 0, _D3 = 0, _D4 = 0, alpha = alpha}

 

Error, invalid input: eval received ({L = L, _D1 = -_D3, _D2 = -_D4, _D3 = _D3, _D4 = _D4, alpha = 0}, {L = 0, _D1 = -_D3, _D2 = -_D4, _D3 = _D3, _D4 = _D4, alpha = alpha}, {L = L, _D1 = 0, _D2 = 0, _D3 = 0, _D4 = 0, alpha = alpha}), which is not valid for its 2nd argument, eqns

 

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Download Free_Vibration_Euler_Bernouli_Beam_1.mw

I have tried to solve these 4 equations to get the characteristic equation and finally the solution of the PDE.  But it shows some error. Can you please help with this issue?

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