mehdibgh

235 Reputation

6 Badges

6 years, 26 days

MaplePrimes Activity


These are replies submitted by mehdibgh

@mehdibaghaee Let me explain more:

@JohnS 

As you see in my previous posts, My functional has integral form and it seems the variation command couldnt perform on integral form equations. Here I did one of the terms of my equation with your recent method:

Any comment

 

 

 

 

 

 

 

 

Hoping more comments.

@JohnS I am working with Maple 2016 now, you mean Maple 2016 do not have the Michal Marvan's library? It seems it was developed for Maple 10?!!! could you please give information about how this package could help me in solving my problem.

@brian bovril 
Here I uploaded the file, H1 is a part of Hamiltonian.

restart

u := u__0(x, y, t)-z*(diff(w(x, y, t), x))

v := (1+z/R)*v__0(x, y, t)-z*(diff(w(x, y, t), y))

`φ__N` := (z-h[N]/h[p])*`φ__0N`

with(LinearAlgebra)

e__1 := Matrix([[0, 0, 0], [0, 0, 0], [e1__31, e1__31, 0]])

e__5 := Matrix([[0, 0, 0], [0, 0, 0], [e5__31, e5__31, 0]])

c__1 := Matrix([[c1__11, c1__12, 0], [c1__12, c1__22, 0], [0, 0, c1__66]])

c__2 := Matrix([[c2__11, c2__12, 0], [c2__12, c2__22, 0], [0, 0, c2__66]])

c__3 := Matrix([[c3__11, c3__12, 0], [c3__12, c3__22, 0], [0, 0, c3__66]])

c__4 := Matrix([[c4__11, c4__12, 0], [c4__12, c4__22, 0], [0, 0, c4__66]])

c__5 := Matrix([[c5__11, c5__12, 0], [c5__12, c5__22, 0], [0, 0, c5__66]])

c__5 := c__1; c__4 := c__1; c__3 := c__1; c__2 := c__1

`ε` := Matrix([[diff(u__0(x, y, t), x)], [diff(v__0(x, y, t), y)+w(x, y, t)/R], [diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x)]])-z*Matrix([[diff(w(x, y, t), x, x)], [diff(w(x, y, t), y, y)-(diff(v__0(x, y, t), y))/R], [2*(diff(w(x, y, t), x, y))-(diff(v__0(x, y, t), x))/R]])

E__1 := -Matrix([[diff(phi1(x, y, t), x)], [diff(phi1(x, y, t), y)], [diff(phi1(x, y, t), z)]])

E__5 := -Matrix([[diff(phi5(x, y, t), x)], [diff(phi5(x, y, t), y)], [diff(phi5(x, y, t), z)]])

`ϵ__1` := Matrix([[`ϵ1__11`, 0, 0], [0, `ϵ1__22`, 0], [0, 0, `ϵ1__33`]])

`ϵ__5` := Matrix([[`ϵ5__11`, 0, 0], [0, `ϵ5__22`, 0], [0, 0, `ϵ5__33`]])

f := Matrix([[f1, f2, f3]])

`σ__1` := Multiply(c__1, `ε`)-Multiply(e__1, E__1)

D__1 := Multiply(e__1, `ε`)+Multiply(`ϵ__1`, E__1)

`σ__2` := Multiply(c__2, `ε`)

`σ__3` := Multiply(c__3, `ε`)

`σ__4` := Multiply(c__4, `ε`)

`σ__5` := Multiply(c__5, `ε`)-Multiply(e__5, E__5)

D__5 := Multiply(e__5, `ε`)+Multiply(`ϵ__5`, E__5)

``

H1 := int(int(int((Multiply(Transpose(`ε`), `σ__1`))(1), z = h__1 .. h__2), y = 0 .. b), x = 0 .. a)+int(int(int((Multiply(Transpose(`ε`), `σ__2`))(1), z = h__2 .. h__3), y = 0 .. b), x = 0 .. a)+int(int(int((Multiply(Transpose(`ε`), `σ__3`))(1), z = h__3 .. h__4), y = 0 .. b), x = 0 .. a)+int(int(int((Multiply(Transpose(`ε`), `σ__4`))(1), z = h__4 .. h__5), y = 0 .. b), x = 0 .. a)+int(int(int((Multiply(Transpose(`ε`), `σ__5`))(1), z = h__5 .. h__6), y = 0 .. b), x = 0 .. a)-(int(int(int((Multiply(Transpose(E__1), D__1))(1), z = h__1 .. h__2), y = 0 .. b), x = 0 .. a))-(int(int(int((Multiply(Transpose(E__5), D__5))(1), z = h__5 .. h__6), y = 0 .. b), x = 0 .. a))

int(int((1/3)*(-(diff(diff(w(x, y, t), x), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)^2)*(-h__1^3+h__2^3)+(1/2)*((diff(u__0(x, y, t), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))-(diff(diff(w(x, y, t), x), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)+(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)*(c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))+e1__31*(diff(phi1(x, y, t), x))+e1__31*(diff(phi1(x, y, t), y))))*(-h__1^2+h__2^2)+(diff(u__0(x, y, t), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__2-h__1)+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__2-h__1)+(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*(c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))+e1__31*(diff(phi1(x, y, t), x))+e1__31*(diff(phi1(x, y, t), y)))*(h__2-h__1), y = 0 .. b), x = 0 .. a)+int(int((1/3)*(-(diff(diff(w(x, y, t), x), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)^2)*(-h__2^3+h__3^3)+(1/2)*((diff(u__0(x, y, t), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))-(diff(diff(w(x, y, t), x), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+2*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R))*(-h__2^2+h__3^2)+(diff(u__0(x, y, t), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__3-h__2)+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__3-h__2)+c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))^2*(h__3-h__2), y = 0 .. b), x = 0 .. a)+int(int((1/3)*(-(diff(diff(w(x, y, t), x), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)^2)*(-h__3^3+h__4^3)+(1/2)*((diff(u__0(x, y, t), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))-(diff(diff(w(x, y, t), x), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+2*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R))*(-h__3^2+h__4^2)+(diff(u__0(x, y, t), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__4-h__3)+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__4-h__3)+c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))^2*(h__4-h__3), y = 0 .. b), x = 0 .. a)+int(int((1/3)*(-(diff(diff(w(x, y, t), x), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)^2)*(-h__4^3+h__5^3)+(1/2)*((diff(u__0(x, y, t), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))-(diff(diff(w(x, y, t), x), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+2*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R))*(-h__4^2+h__5^2)+(diff(u__0(x, y, t), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__5-h__4)+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__5-h__4)+c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))^2*(h__5-h__4), y = 0 .. b), x = 0 .. a)+int(int((1/3)*(-(diff(diff(w(x, y, t), x), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)^2)*(-h__5^3+h__6^3)+(1/2)*((diff(u__0(x, y, t), x))*(-c1__11*(diff(diff(w(x, y, t), x), x))+c1__12*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))-(diff(diff(w(x, y, t), x), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(-c1__12*(diff(diff(w(x, y, t), x), x))+c1__22*(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R))+(-(diff(diff(w(x, y, t), y), y))+(diff(v__0(x, y, t), y))/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))+(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*c1__66*(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)+(-2*(diff(diff(w(x, y, t), x), y))+(diff(v__0(x, y, t), x))/R)*(c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))+e5__31*(diff(phi5(x, y, t), x))+e5__31*(diff(phi5(x, y, t), y))))*(-h__5^2+h__6^2)+(diff(u__0(x, y, t), x))*(c1__11*(diff(u__0(x, y, t), x))+c1__12*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__6-h__5)+(diff(v__0(x, y, t), y)+w(x, y, t)/R)*(c1__12*(diff(u__0(x, y, t), x))+c1__22*(diff(v__0(x, y, t), y)+w(x, y, t)/R))*(h__6-h__5)+(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))*(c1__66*(diff(u__0(x, y, t), y)+diff(v__0(x, y, t), x))+e5__31*(diff(phi5(x, y, t), x))+e5__31*(diff(phi5(x, y, t), y)))*(h__6-h__5), y = 0 .. b), x = 0 .. a)-(h__2-h__1)*(int(int((diff(phi1(x, y, t), x))^2*`ϵ1__11`+(diff(phi1(x, y, t), y))^2*`ϵ1__22`, y = 0 .. b), x = 0 .. a))-(h__6-h__5)*(int(int((diff(phi5(x, y, t), x))^2*`ϵ5__11`+(diff(phi5(x, y, t), y))^2*`ϵ5__22`, y = 0 .. b), x = 0 .. a))

(1)

``


 

Download ShellMaple.mwH

Any comment yet?!!!!!!

@mehdibaghaee 

Hi

@tomleslie 

Thanks yyour kind answer. Let me explain my problem more.

All I want to do is to compute the following expression by Maple to get Equation Of Motion of shell.

or

Please guide me how to do so in Maple.

All information about hamilton_eqs(H) in Maple help is as below:

Calling Sequence
hamilton_eqs(H)
Parameters
H
-
any algebraic expression representing the Hamiltonian
Description
• hamilton_eqs receives a Hamiltonian and returns a sequence with Hamilton's equations and a list with the p's and q's involved.
• Some useful conventions were adopted to represent the p's and q's. All p's and q's must appear as pn or qn where n is a positive integer, as in p1, p2, and the time dependence need not be explicit, as in pn or qn instead of pn(t) or qn(t). The Hamilton equations will be automatically returned using pn(t) or qn(t).
This function is part of the DEtools package, and so it can be used in the form hamilton_eqs(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[hamilton_eqs](..).

Hamiltonian,
n. a function H such that a given partial differential equation of first order can be rewritten as
Partial(u)/Partial(t) =-H(t, x[1], .., x[n], p[1], .., p[n])
 where the variables are all functions of the parameter t. This is a Hamilton-Jacobi type differential equation. The Hamiltonian exists for any equation
    F(x0, x1, …, xn, u, p0, …, pn) = 0,
where
             p[k]=Partial(u)/Partial(x[k])
  that does not depend explicitly on  u. The Hamiltonian canonical form is then
d(x[k])/dt = Partial(H)/Par\tial(p[k]), d(p[k])/d(t) =-Partial(H)/Partial(x[k])
.
Such systems occur in classical mechanics, in control theory and elsewhere. See also Pontryagin's maximum principle.

Hamilton's equations of motion,
n. (Mechanics) the equations

                              d(q)/dt= Partial(H)/Partial(p) , d(p)/dt= Partial(H)/Partial(q)
                            
where H is the Hamiltonian function,
               p = Partial(L)/Partial(q)
and  q  ranges over the generalized coordinates; they are equivalent to Lagrange's equations.

But unfortunately I couldnt find any analogy between my problem and what are presented in maple help.

Please help me.

Regards

 

@mehdibaghaee 

I am wondering why Maple do not have variation operator (del) although there are similarity between del and diff? is not there anyone to encounter such problem in engineering to compute Hamilton principle to get EOM of shell structure?

I think maple is unable to help me in this case and I have to do it manualy.

 

Hi  @tomleslie

This is what I do according to you:

Any other comment???

First 7 8 9 Page 9 of 9