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Neel

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How do I solve Base Excitation Problem o...

Neel 20 Maple
dynamics
September 30 2020
0 0

Is it possible to develop a code for base excitation problem where excitation data is of earthquake acceleration?

p.s. There are 12000 data available in EQ file.

How to solve by eigen function expansion...

Neel 20 Maple 2019
pde differential-equations
June 27 2020
1 26


 

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

  
> 

``


 

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?

How do I collect the coefficients from f...

Neel 20 Maple 2019
substitution collect
June 20 2020
3 4

I'm trying to arrange the constant terms of W(x). Like sinh (alpha*x) *(C2-C1).....and by defining C2-C1= D2 I want to rearrange the equation. Without this step, I'm not able to solve the equation further. I have also tried a few steps but it's not working.

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