252
7 Mathematical Models of Functionally Graded Beams in Temperature Field
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
¨
u = x 2 (u) i − x (w) i x 2 (w) i − x (N T ) i
¨
w + εw =
1
λ 2
−
1
12
x 4 (w) i + k x
x (u) i − k x w i −
1
2
( x (w) i )
2
−
−w i x 2 (w) i
+ x 2 (u) i x (w) i + x (u) i x 2 (w) i +
3
2
( x (w) i )
2
x 2 (w) i
−
− x 2 (M T ) i − x (N T ) i x 2 (w) i + q,
(7.99)
where x , x 2 , x 4 stand for difference operators of the first, second and third orders,
respectively.
The so far described finite difference approximations are also applied to boundary
conditions (7.89)–(7.91) and initial conditions (7.92). Then, as usually, the task is
reduced to Cauchy problem regarding an evolutionary variable, and is finally solved
by the fourth-order Runge-Kutta method. The results have been validated through
application of the sixth-order Runge-Kutta method. Since the latter one requires
essentially longer computational time, the fourth-order Runge-Kutta method has
been employed in all our computations (see [168] for the motivation of our choice).
In order to study stability of flexible Euler-Bernoulli beams, we employ the method
originally proposed by Tikhonov [124], and then successfully employed by Feodos’ev for the shell problems (see [138, 158]). The formulated dynamical problems
regarding stability have been solved using the relaxation/set-up method.
Though numerous approximate methods can be employed, an important role play
iterational methods, which allow to obtain the required solutions with a high accuracy.
If the iterational process can be treated as the result of a steady state of a certain
process, then the iterational methods can be treated as the continuation methods of
solutions with respect to a parameter.
Now, giving a physical meaning to the parameter (here beam deflection), we get a
meaning of the steady-state process in time, which is analysed by the relaxation/setup method. The so far described idea is illustrated in the attached Fig. 7.16.
Its right-hand side presents a steady state of the beam centre deflection w(0.5)
versus time for the load q = 135, for the boundary conditions (7.90), and initial
conditions (7) for f i (x) = 0; we take also k x = 24 and the temperature T = 50. Its
left-hand side reports the beam deflection versus q, which has been obtained through
the employed successive solution to the equations of beam motion via the relaxation
method by increasing the load parameter q.
It should be emphasized that in our case we analyse a dynamic dissipative process.
Now, increasing the dissipation factor (damping) ε, the studied dynamical process
is more fast damped and it is set-up on the corresponding value of the static load.
Observe that a transition from a stable state into unstable state for ε = 0 appears
earlier than for ε = 0.05, and then the solution tends to its steady/static state.
The so far introduced description explains how we have studied the problem of
stability though in static context but using the dynamical approach.
This method has been widely used earlier. In particular, let us refer to the monograph [158], where numerous theorems with their proofs have been reported, as well
7 Mathematical Models of Functionally Graded Beams in Temperature Field
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
¨
u = x 2 (u) i − x (w) i x 2 (w) i − x (N T ) i
¨
w + εw =
1
λ 2
−
1
12
x 4 (w) i + k x
x (u) i − k x w i −
1
2
( x (w) i )
2
−
−w i x 2 (w) i
+ x 2 (u) i x (w) i + x (u) i x 2 (w) i +
3
2
( x (w) i )
2
x 2 (w) i
−
− x 2 (M T ) i − x (N T ) i x 2 (w) i + q,
(7.99)
where x , x 2 , x 4 stand for difference operators of the first, second and third orders,
respectively.
The so far described finite difference approximations are also applied to boundary
conditions (7.89)–(7.91) and initial conditions (7.92). Then, as usually, the task is
reduced to Cauchy problem regarding an evolutionary variable, and is finally solved
by the fourth-order Runge-Kutta method. The results have been validated through
application of the sixth-order Runge-Kutta method. Since the latter one requires
essentially longer computational time, the fourth-order Runge-Kutta method has
been employed in all our computations (see [168] for the motivation of our choice).
In order to study stability of flexible Euler-Bernoulli beams, we employ the method
originally proposed by Tikhonov [124], and then successfully employed by Feodos’ev for the shell problems (see [138, 158]). The formulated dynamical problems
regarding stability have been solved using the relaxation/set-up method.
Though numerous approximate methods can be employed, an important role play
iterational methods, which allow to obtain the required solutions with a high accuracy.
If the iterational process can be treated as the result of a steady state of a certain
process, then the iterational methods can be treated as the continuation methods of
solutions with respect to a parameter.
Now, giving a physical meaning to the parameter (here beam deflection), we get a
meaning of the steady-state process in time, which is analysed by the relaxation/setup method. The so far described idea is illustrated in the attached Fig. 7.16.
Its right-hand side presents a steady state of the beam centre deflection w(0.5)
versus time for the load q = 135, for the boundary conditions (7.90), and initial
conditions (7) for f i (x) = 0; we take also k x = 24 and the temperature T = 50. Its
left-hand side reports the beam deflection versus q, which has been obtained through
the employed successive solution to the equations of beam motion via the relaxation
method by increasing the load parameter q.
It should be emphasized that in our case we analyse a dynamic dissipative process.
Now, increasing the dissipation factor (damping) ε, the studied dynamical process
is more fast damped and it is set-up on the corresponding value of the static load.
Observe that a transition from a stable state into unstable state for ε = 0 appears
earlier than for ε = 0.05, and then the solution tends to its steady/static state.
The so far introduced description explains how we have studied the problem of
stability though in static context but using the dynamical approach.
This method has been widely used earlier. In particular, let us refer to the monograph [158], where numerous theorems with their proofs have been reported, as well