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Neel

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@Rouben Rostamian   See here is my ...

Neel 20
August 12 2020
0 0

@Rouben Rostamian   See here is my doubt in eq(4). Why are we using such type extra boundary condition containing 'a'? We can solve it by eigenvalue problem as I did it in the paper. How to write a matrix in maple? And to be more specific, there are several boundary conditions...we need to do something else apart from this method. And if you do this problem in this way how will approach for forced vibration part?

...

Neel 20
August 11 2020
0 0

@Rouben Rostamian  sure sir I'll definately learn Latex. Actually I'm trying to build my basic knowledge based on these works. But sir whatever you have done is understandable but I'm not able to implement the Eigen value problem in the maple coding specially the Eigen function. I've also attached my maple code in my previous message. Can you just give one example for that in maple code? 

...

Neel 20
August 10 2020
0 0

@Rouben Rostamian  yeahh you're right. Actually I've done it as a analytical solution for testing purpose. If you kindly do it by taking forcing function as F= Fo*(1-x/L)*cos(omega*t) then it'll be very helpful for me. Even at the end page of the attached PDF you can see that I started doing the problem by taking this force function.

...

Neel 20
August 10 2020
0 0

@Rouben Rostamian  q(x,t)= Fo*(1-(x/L))* cos(omega*t) and initial conditions can be any linear function or you can assume any constant value...the main thing is that homogeneous solution of the equation is not matching with the analytical solution. And also, in that case, I can do it as we did it in the bar problem but unable to follow the same way. Kindly let me know if anything is required regarding this problem.

@Rouben Rostamian hello sir. Hope you...

Neel 20
August 09 2020
0 0

@Rouben Rostamian hello sir. Hope you're fine. I've successfully completed the free and force vibration wrt different support conditions. But in the case of beam analysis, I'm getting a different answer from the analytical solution. I've applied a different approach which you said before but I'm not getting the exact solution through this maple code for free vibration. How can I induce orthonormality condition and force vibration portion....I'm not getting the proper idea. Please help me.Forced_Vibration_of_Euler_Beam.pdfTransverse_Vibration_Analysis_simply_supported.mwVibration_Analysis_Using_Harmonic_Function.mw

@Rouben Rostamian  I'm able to ...

Neel 20
July 18 2020
0 0

@Rouben Rostamian  I'm able to solve this bar problem using this same type of forcing function. But I have got stuck while calculating n=1..2 and forcing function =alpha*(1-x/L) and initial conditions assumed as zero (both). Can you please tell me what I need to modify? 
 

> 

restart
> 

NULL
> 

pde := diff(w(x, t), t, t) = c^2*(diff(w(x, t), x, x))+`ϰ`(x)*cos(Omega*t)

diff(diff(w(x, t), t), t) = c^2*(diff(diff(w(x, t), x), x))+varkappa(x)*cos(Omega*t)

 (1)
> 

NULLSolution using boundary conditions u(0, t) = 0, u(L, t) = 0 and initial conditions u(x, 0) = Phi(x)
and u__t(x, 0) = psi(x). To get the complementary Solution we need to do `ϰ`(x)*cos(omega*t) = 0.

   By solving we get Homogeneous solution w__h, and then find a particular solution w__p to the forced problem.  

  Then the solution will be w = w__h+w__p.

varkappa(x)*cos(omega*t) = 0.

 (2)
> 

pde_h := diff(w(x, t), t, t) = c^2*(diff(w(x, t), x, x))

diff(diff(w(x, t), t), t) = c^2*(diff(diff(w(x, t), x), x))

 (3)
> 

NULLw(x, t) = X(x)*T(t)

w(x, t) = X(x)*T(t)

 (4)
> 

eval(pde_h, w(x, t) = X(x)*T(t)); pde_separated := %/(X(x)*T(t))

X(x)*(diff(diff(T(t), t), t)) = c^2*(diff(diff(X(x), x), x))*T(t)

  

(diff(diff(T(t), t), t))/T(t) = c^2*(diff(diff(X(x), x), x))/X(x)

 (5)
> 

eq_T := lhs(pde_separated) = -omega^2; eq_X := rhs(pde_separated) = -omega^2

(diff(diff(T(t), t), t))/T(t) = -omega^2

  

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

 (6)
> 

sol_T := dsolve(eq_T)

T(t) = _C1*sin(omega*t)+_C2*cos(omega*t)

 (7)
> 

sol_X_eigen_function1 := dsolve(eq_X)

X(x) = _C1*sin(omega*x/c)+_C2*cos(omega*x/c)

 (8)
> 

sol_X_eigen_function2 := eval(sol_X_eigen_function1, omega = lambda*c)

X(x) = _C1*sin(lambda*x)+_C2*cos(lambda*x)

 (9)
> 

NULL
> 

eval(sol_X_eigen_function2, x = 0)

X(0) = _C2

 (10)
> 

eval(sol_X_eigen_function2, x = L)

X(L) = _C1*sin(lambda*L)+_C2*cos(lambda*L)

 (11)
> 

sol_X := subs(_C2 = 0, sol_X_eigen_function2)

X(x) = _C1*sin(lambda*x)

 (12)
> 

eval(sol_X, x = L)

X(L) = _C1*sin(lambda*L)

 (13)
> 

NULL lambda*L = n*Pi for any integer "n."  So, `λ__n` = n*Pi/L, and  `ω__n` = `λ__n`*c and `λ__n`*c = n*Pi*c/L.

    #  For infinitely many solution for "X, "  X__n(x) = sin(`λ__n`*x), n = 1, 2, () .. ()

   #  To  verify the orthogonality:

> 

Int(sin(lambda[i]*x)*sin(lambda[j]*x), x = 0 .. L); eval(%, {lambda[i] = i*Pi/L, lambda[j] = j*Pi/L}); `assuming`([value(%)], [i::integer, j::integer])

Int(sin(lambda[i]*x)*sin(lambda[j]*x), x = 0 .. L)

  

0

 (14)
> 

Int(sin(lambda[i]*x)^2, x = 0 .. L); eval(%, lambda[i] = i*Pi/L); `assuming`([value(%)], [i::integer])

Int(sin(lambda[i]*x)^2, x = 0 .. L)

  

(1/2)*L

 (15)
> 

w__h := Sum((a[n]*cos(omega[n]*t)+b[n]*sin(omega[n]*t))*X[n](x), n = 1 .. infinity)

Sum((a[n]*cos(omega[n]*t)+b[n]*sin(omega[n]*t))*X[n](x), n = 1 .. infinity)

 (16)
> 

w_form := w(x, t) = (Sum(h[n]*X[n](x), n = 1 .. infinity))*cos(Omega*t)

w(x, t) = (Sum(h[n]*X[n](x), n = 1 .. infinity))*cos(Omega*t)

 (17)
> 

force_form := `ϰ`(x) = alpha*(1-x/L)

varkappa(x) = alpha*(1-x/L)

 (18)
> 

eval(pde, [w_form, force_form]); lhs(%)-rhs(%) = 0; tmp1 := simplify(%/cos(Omega*t))

-(Sum(h[n]*X[n](x), n = 1 .. infinity))*Omega^2*cos(Omega*t) = c^2*(Sum(h[n]*(diff(diff(X[n](x), x), x)), n = 1 .. infinity))*cos(Omega*t)+alpha*(1-x/L)*cos(Omega*t)

  

-(Sum(h[n]*X[n](x), n = 1 .. infinity))*Omega^2*cos(Omega*t)-c^2*(Sum(h[n]*(diff(diff(X[n](x), x), x)), n = 1 .. infinity))*cos(Omega*t)-alpha*(1-x/L)*cos(Omega*t) = 0

  

(-c^2*(Sum(h[n]*(diff(diff(X[n](x), x), x)), n = 1 .. infinity))*L-(Sum(h[n]*X[n](x), n = 1 .. infinity))*Omega^2*L-alpha*(L-x))/L = 0

 (19)
> 

eval(tmp1, diff(X[n](x),x,x) = -omega[n]^2/c^2*X[n](x) );
combine(%);NULL
tmp2 := factor(%);

(-c^2*(Sum(-h[n]*omega[n]^2*X[n](x)/c^2, n = 1 .. infinity))*L-(Sum(h[n]*X[n](x), n = 1 .. infinity))*Omega^2*L-alpha*(L-x))/L = 0

  

(Sum(L*h[n]*omega[n]^2*X[n](x)-Omega^2*L*h[n]*X[n](x), n = 1 .. infinity)-alpha*(L-x))/L = 0

  

-(alpha*L-alpha*x-(Sum(-L*X[n](x)*h[n]*(Omega-omega[n])*(Omega+omega[n]), n = 1 .. infinity)))/L = 0

 (20)
> 

op(1, lhs(tmp2)); coeff(%, X[n](x)) = 0; h_form := isolate(%, h[n])

0 = 0

  

Error, (in isolate) 0 = 0 does not contain h[n]

  
> 

w__p := subs(h_form, rhs(w_form))

Error, invalid input: subs received h_form, which is not valid for its 1st argument

  
> 

w_sol_general := w__h+w__p

Sum((a[n]*cos(omega[n]*t)+b[n]*sin(omega[n]*t))*X[n](x), n = 1 .. infinity)+w__p

 (21)

##  To determine the coefficients a__n and b__n the initial conditions have to be satisfied.

##      w(x, 0) = phi(x), therefore...NULL

> 

eval(w_sol_general, t = 0) = Sum(A[n]*X[n](x), n = 1 .. infinity); lhs(%)-rhs(%) = 0; tmp2 := combine(%)

Sum(a[n]*X[n](x), n = 1 .. infinity)+w__p = Sum(A[n]*X[n](x), n = 1 .. infinity)

  

Sum(a[n]*X[n](x), n = 1 .. infinity)+w__p-(Sum(A[n]*X[n](x), n = 1 .. infinity)) = 0

  

Sum(-A[n]*X[n](x)+a[n]*X[n](x), n = 1 .. infinity)+w__p = 0

 (22)
> 

op(1, lhs(tmp2)); isolate(%, a[n]); a_form := expand(%)

Sum(-A[n]*X[n](x)+a[n]*X[n](x), n = 1 .. infinity)

  

Sum(-A[n]*X[n](x)+a[n]*X[n](x), n = 1 .. infinity) = 0

  

Sum(-A[n]*X[n](x)+a[n]*X[n](x), n = 1 .. infinity) = 0

 (23)
> 

NULL
> 

diff(w_sol_general, t); eval(%, t = 0) = Sum(B[n]*X[n](x), n = 1 .. infinity); lhs(%)-rhs(%) = 0; tmp3 := combine(%)

Sum((-a[n]*omega[n]*sin(omega[n]*t)+b[n]*omega[n]*cos(omega[n]*t))*X[n](x), n = 1 .. infinity)

  

Sum(b[n]*omega[n]*X[n](x), n = 1 .. infinity) = Sum(B[n]*X[n](x), n = 1 .. infinity)

  

Sum(b[n]*omega[n]*X[n](x), n = 1 .. infinity)-(Sum(B[n]*X[n](x), n = 1 .. infinity)) = 0

  

Sum(b[n]*omega[n]*X[n](x)-B[n]*X[n](x), n = 1 .. infinity) = 0

 (24)
> 

op(1, lhs(tmp3)); b_form := isolate(%, b[n])

b[n]*omega[n]*X[n](x)-B[n]*X[n](x)

  

b[n] = B[n]/omega[n]

 (25)
> 

w(x, t) = subs(a_form, b_form, w_sol_general)

w(x, t) = Sum((a[n]*cos(omega[n]*t)+B[n]*sin(omega[n]*t)/omega[n])*X[n](x), n = 1 .. infinity)+w__p

 (26)
> 

w(x, t) = Sum(((-k[n]/(-Omega^2+omega[n]^2)+A[n])*cos(omega[n]*t)+B[n]*sin(omega[n]*t)/omega[n]+k[n]*cos(Omega*t)/(-Omega^2+omega[n]^2))*X[n](x), n = 1 .. infinity)

w(x, t) = Sum(((-k[n]/(-Omega^2+omega[n]^2)+A[n])*cos(omega[n]*t)+B[n]*sin(omega[n]*t)/omega[n]+k[n]*cos(Omega*t)/(-Omega^2+omega[n]^2))*X[n](x), n = 1 .. infinity)

 (27)
> 

NULL

 


 

Download Vibration_Analysis_Using_Harmonic_Function_2.mw

@Rouben Rostamian  Thank you so muc...

Neel 20
July 11 2020
0 0

@Rouben Rostamian  Thank you so much....now I've got the complete idea

@Rouben Rostamian  Today I have sol...

Neel 20
July 07 2020
0 0

@Rouben Rostamian  Today I have solved a simple problem based on force vibration. I'm attaching the PDF. Can you please go through it? This is how I want to proceed for such problem. I'm not getting the idea of converting it in maple code. Kindly help me. Forced_Vibration_analysis.pdfForced_Vibration_analysis.pdf

@Rouben Rostamian yeah it's a trial ...

Neel 20
July 06 2020
0 0

@Rouben Rostamian yeah it's a trial before starting the actual project.. I'll definitely do these two equations and will get back to you.

@Rouben Rostamian  Thank you for yo...

Neel 20
July 06 2020
0 0

@Rouben Rostamian  Thank you for your help. I'm now able to solve for different boundary conditions at least 6 conditions. But whenever I'm introducing forcing function like beta * (1-x/L)*sin(omegat)Transverse_Vibration_Analysis_Fixed_Forced.mw
 

> 

restart;
> 

Young_*Modulus = E; Moment*of*Inertia = I; density*of*Material = rho; A = area_beam_section

Young_*Modulus = E

  

Moment*of*Inertia = I

  

density*of*Material = rho

  

A = area_beam_section

 (1)
> 

NULL
> 

c = sqrt(I*E/(rho*A))

c = (I*E/(rho*A))^(1/2)

 (2)
> 

pde := diff(w(x, t), `$`(x, 4))+(diff(w(x, t), `$`(t, 2)))/c^2 = beta*(1-x/L)*Cos(omega*_f*t)

diff(diff(diff(diff(w(x, t), x), x), x), x)+(diff(diff(w(x, t), t), t))/c^2 = beta*(1-x/L)*Cos(omega*_f*t)

 (3)
> 

NULLw(x, t) = u(x)*cos(omega*t)
> 

eval(pde, w(x, t) = u(x)*cos(omega*t)); de_tmp := expand(%/cos(omega*t))

(diff(diff(diff(diff(u(x), x), x), x), x))*cos(omega*t)-u(x)*omega^2*cos(omega*t)/c^2 = beta*(1-x/L)*Cos(omega*_f*t)

  

diff(diff(diff(diff(u(x), x), x), x), x)-u(x)*omega^2/c^2 = beta*Cos(omega*_f*t)/cos(omega*t)-beta*Cos(omega*_f*t)*x/(cos(omega*t)*L)

 (4)
> 

NULL mu
> 

mu_def := omega = c*mu^2/L^2

omega = c*mu^2/L^2

 (5)
> 

de := eval(de_tmp, mu_def)

diff(diff(diff(diff(u(x), x), x), x), x)-u(x)*mu^4/L^4 = beta*Cos(c*mu^2*_f*t/L^2)/cos(c*mu^2*t/L^2)-beta*Cos(c*mu^2*_f*t/L^2)*x/(cos(c*mu^2*t/L^2)*L)

 (6)
> 

NULL
> 

dsolve(de); U := unapply(rhs(%), x)

proc (x) options operator, arrow; -L^3*Cos(c*mu^2*_f*t/L^2)*beta*(L-x)/(mu^4*cos(c*mu^2*t/L^2))+_C1*cos(mu*x/L)+_C2*exp(mu*x/L)+_C3*sin(mu*x/L)+_C4*exp(-mu*x/L) end proc

 (7)
> 

U(0) = 0, D(U)(0)=0, (D@@2)(U)(0) = a, U(L) = 0:
C_vals := solve({%}, {_C1, _C2, _C3, _C4});

{_C1 = (1/2)*L^2*(L^2*Cos(c*mu^2*_f*t/L^2)*beta-cos(c*mu^2*t/L^2)*a*mu^2)/(mu^4*cos(c*mu^2*t/L^2)), _C2 = -(1/2)*L^2*(L^2*Cos(c*mu^2*_f*t/L^2)*cos(mu)*beta*mu+L^2*Cos(c*mu^2*_f*t/L^2)*sin(mu)*beta*mu+L^2*Cos(c*mu^2*_f*t/L^2)*exp(-mu)*beta*mu-cos(c*mu^2*t/L^2)*cos(mu)*a*mu^3+cos(c*mu^2*t/L^2)*sin(mu)*a*mu^3+cos(c*mu^2*t/L^2)*exp(-mu)*a*mu^3-2*L^2*Cos(c*mu^2*_f*t/L^2)*sin(mu)*beta)/((exp(mu)-2*sin(mu)-exp(-mu))*cos(c*mu^2*t/L^2)*mu^5), _C3 = (1/2)*L^2*(2*L^2*Cos(c*mu^2*_f*t/L^2)*cos(mu)*beta*mu+L^2*Cos(c*mu^2*_f*t/L^2)*exp(mu)*beta*mu+L^2*Cos(c*mu^2*_f*t/L^2)*exp(-mu)*beta*mu-2*cos(c*mu^2*t/L^2)*cos(mu)*a*mu^3+cos(c*mu^2*t/L^2)*exp(mu)*a*mu^3+cos(c*mu^2*t/L^2)*exp(-mu)*a*mu^3-2*L^2*Cos(c*mu^2*_f*t/L^2)*exp(mu)*beta+2*L^2*Cos(c*mu^2*_f*t/L^2)*exp(-mu)*beta)/((exp(mu)-2*sin(mu)-exp(-mu))*cos(c*mu^2*t/L^2)*mu^5), _C4 = (1/2)*L^2*(L^2*Cos(c*mu^2*_f*t/L^2)*cos(mu)*beta*mu+L^2*Cos(c*mu^2*_f*t/L^2)*exp(mu)*beta*mu-L^2*Cos(c*mu^2*_f*t/L^2)*sin(mu)*beta*mu-cos(c*mu^2*t/L^2)*cos(mu)*a*mu^3+cos(c*mu^2*t/L^2)*exp(mu)*a*mu^3-cos(c*mu^2*t/L^2)*sin(mu)*a*mu^3-2*L^2*Cos(c*mu^2*_f*t/L^2)*sin(mu)*beta)/((exp(mu)-2*sin(mu)-exp(-mu))*cos(c*mu^2*t/L^2)*mu^5)}

 (8)
> 

eval(U(x), C_vals); diff(%, x); eval(%, x = L) = 0; simplify(%)

((((mu-1)*sin(mu)+cos(mu)-1)*exp(2*mu)+(-mu+2*exp(mu)-1)*sin(mu)-cos(mu)+1)*L^2*beta*Cos(c*mu^2*_f*t/L^2)-mu^3*a*cos(c*mu^2*t/L^2)*(exp(2*mu)*cos(mu)+cos(mu)-2*exp(mu)))*L/(mu^4*cos(c*mu^2*t/L^2)*(-exp(2*mu)+2*sin(mu)*exp(mu)+1)) = 0

 (9)
> 

exp(2*mu)*cos(mu)-2*exp(mu)+cos(mu) = 0; eq_tmp := isolate(%, cos(mu))

exp(2*mu)*cos(mu)+cos(mu)-2*exp(mu) = 0

  

cos(mu) = 2*exp(mu)/(exp(2*mu)+1)

 (10)
> 

simplify(1/cosh(mu)-rhs(eq_tmp))

(-2*exp(mu)*cosh(mu)+exp(2*mu)+1)/(cosh(mu)*(exp(2*mu)+1))

 (11)
> 

plot([cos(mu), 1/cosh(mu)], mu = 0 .. 5*Pi, color = [blue, red], thickness = 3, size = [500, 400])

 
> 

seq(fsolve(cos(mu)=1/cosh(mu), mu=(2*n+1)*Pi/2), n=1..5);

4.730040745, 7.853204624, 10.99560784, 14.13716549, 17.27875966

 (12)
> 

U_final = subs(C_vals, mu = mu[j], U(x))

U_final = -L^3*Cos(c*mu[j]^2*_f*t/L^2)*beta*(L-x)/(mu[j]^4*cos(c*mu[j]^2*t/L^2))+(1/2)*L^2*(L^2*Cos(c*mu[j]^2*_f*t/L^2)*beta-cos(c*mu[j]^2*t/L^2)*a*mu[j]^2)*cos(mu[j]*x/L)/(mu[j]^4*cos(c*mu[j]^2*t/L^2))-(1/2)*L^2*(L^2*Cos(c*mu[j]^2*_f*t/L^2)*cos(mu[j])*beta*mu[j]+L^2*Cos(c*mu[j]^2*_f*t/L^2)*sin(mu[j])*beta*mu[j]+L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(-mu[j])*beta*mu[j]-cos(c*mu[j]^2*t/L^2)*cos(mu[j])*a*mu[j]^3+cos(c*mu[j]^2*t/L^2)*sin(mu[j])*a*mu[j]^3+cos(c*mu[j]^2*t/L^2)*exp(-mu[j])*a*mu[j]^3-2*L^2*Cos(c*mu[j]^2*_f*t/L^2)*sin(mu[j])*beta)*exp(mu[j]*x/L)/((exp(mu[j])-2*sin(mu[j])-exp(-mu[j]))*cos(c*mu[j]^2*t/L^2)*mu[j]^5)+(1/2)*L^2*(2*L^2*Cos(c*mu[j]^2*_f*t/L^2)*cos(mu[j])*beta*mu[j]+L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(mu[j])*beta*mu[j]+L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(-mu[j])*beta*mu[j]-2*cos(c*mu[j]^2*t/L^2)*cos(mu[j])*a*mu[j]^3+cos(c*mu[j]^2*t/L^2)*exp(mu[j])*a*mu[j]^3+cos(c*mu[j]^2*t/L^2)*exp(-mu[j])*a*mu[j]^3-2*L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(mu[j])*beta+2*L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(-mu[j])*beta)*sin(mu[j]*x/L)/((exp(mu[j])-2*sin(mu[j])-exp(-mu[j]))*cos(c*mu[j]^2*t/L^2)*mu[j]^5)+(1/2)*L^2*(L^2*Cos(c*mu[j]^2*_f*t/L^2)*cos(mu[j])*beta*mu[j]+L^2*Cos(c*mu[j]^2*_f*t/L^2)*exp(mu[j])*beta*mu[j]-L^2*Cos(c*mu[j]^2*_f*t/L^2)*sin(mu[j])*beta*mu[j]-cos(c*mu[j]^2*t/L^2)*cos(mu[j])*a*mu[j]^3+cos(c*mu[j]^2*t/L^2)*exp(mu[j])*a*mu[j]^3-cos(c*mu[j]^2*t/L^2)*sin(mu[j])*a*mu[j]^3-2*L^2*Cos(c*mu[j]^2*_f*t/L^2)*sin(mu[j])*beta)*exp(-mu[j]*x/L)/((exp(mu[j])-2*sin(mu[j])-exp(-mu[j]))*cos(c*mu[j]^2*t/L^2)*mu[j]^5)

 (13)
> 

``
> 

``
> 

NULL


 

Download Transverse_Vibration_Analysis_Fixed_Forced.mw

, it results in some error. Can you please help me out?

@Rouben Rostamian  Is there any gen...

Neel 20
June 28 2020
0 0

@Rouben Rostamian  Is there any generalised step where we can change the boundary condition?

@Rouben Rostamian  Thank you so muc...

Neel 20
June 27 2020
0 0

@Rouben Rostamian  Thank you so much..Can you tell me what should I do if I want to use force vibration part ?

 

@Rouben Rostamian  fixed end condit...

Neel 20
June 27 2020
0 0

@Rouben Rostamian  fixed end conditions. I have clearly written these conditions in this file i.e. slope and deflection both are zero at two ends.

@acer  I have attached the maple fi...

Neel 20
June 27 2020
0 0

@acer  I have attached the maple file.

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