Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass

Abstract

In this article, variational iteration algorithm I with an auxiliary parameter is used for simple and damped mass–spring systems that undergo forced vibrations and transverse vibration of uniform and variable beams with simply supported analytical treatment. In addition, common vibration problems are classified and the values of Lagrange multipliers are identified for each type of problem. Examples are given at the end, which shows that this modification demonstrated the high efficiency and attained very good agreement in illustrated models and might be promptly reached out to other nonlinear differential and partial differential equations.

Keywords

Variational iteration algorithm I variational iteration algorithm I with an auxiliary parameter vibration problems transverse vibration mass–spring system

Introduction

Many engineering subjects such as mechanical variation or structural dynamics, heat transfer, and theory of electric circuits focus on the theory of differential equation. In the very fast advancement of nonlinear sciences, a considerable amount of research studies have been done in the area of vibration problems. Several mathematical methods such as variational iteration method,^1–3 finite element method,⁴ Adomian decomposition method,⁴ perturbation techniques,^5–10 and series techniques¹¹ have been used to handle vibration problems.¹²

The vibration of dynamical frameworks can be separated into two fundamental classes: discrete and distributed. The factors in discrete frameworks rely upon time, while factors in appropriated frameworks, for example, beams and plates, rely upon time and space. In this way, equations of motion of discrete frameworks are portrayed by ordinary differential equations, while equations of motion of distributed frameworks are depicted by partial differential equations.^13,14

The variational iteration method^15–30 has been used for analytical treatment of many nonlinear partial differential equations, autonomous, and singular ordinary differential equations. Solitary wave solutions, rational solutions, compaction solutions, and other types of solutions were found by Abdou and Soliman.¹⁶ In addition, He and colleagues^10,15,30,31 used VIM to solve linear/nonlinear vibration problems. The basic characteristic of this method is the presence of the elements of flexibility and ability in solving linear and nonlinear problems. One of the main characteristics of this method is that the exact solution or approximate solution with high accuracy can be obtained by only a few iterations. This method has a simple procedure, acceptable results, and above all, it can be practically used to a greater number of nonlinear problems.

In this study, variational iteration algorithm I with an auxiliary parameter is applied to various vibration problems including simple and damped mass–spring systems that undergo forced vibrations and transverse vibration of uniform and variable beams. This approach makes easy the computational work for solving increasingly complex vibration problems, for example, aeroelasticity and arbitrary vibrations, arising in science and engineering, and results of high degree accuracy can be obtained in few iterations as compared with classic variational iteration algorithm I.

Variational iteration algorithm I with an auxiliary parameter

Consider the following nonlinear differential equation

N y (x, t) + L y (x, t) = c (x, t)

(1)

where N and L are linear and nonlinear operators, respectively, while c is the source term. Constructing a correction function for equation (1), according to variational iteration algorithm I with an auxiliary parameter

y_{k + 1} (x, t) = y_{k} (x, t) + h \int_{0}^{t} λ [N y_{k} + L \tilde{y_{k}} - c] d τ

(2)

where $λ$ is the general Lagrange multiplier, h is the auxiliary parameter which makes sure the convergence to the exact solution. The optimal value of this auxiliary parameter is obtained using the residual function as in my previous work.³⁰

Taking the variation $δ$ on the one side as well as the other side of equation (2) with respect to $y_{k} (x, t)$

δ y_{k + 1} (x, t) = δ y_{k} (x, t) + h δ \int_{0}^{t} λ [N y_{k} + L \tilde{y_{k}} - c] d τ

(3)

where $L \tilde{y_{k}}$ is considered as a restricted variation which means $δ L \tilde{y_{k}} = 0$ .

Different forms of linear operator

Here, a general classification of vibration problems can be made as below according to the linear operator.

Case I

Here, we consider the linear operator of the following form

N = p \frac{\partial^{2}}{\partial t^{2}} + g \frac{\partial}{\partial t}

(4)

Considering equations (4) and (1) as

p \overset{\cdot\cdot}{u} (x, t) + g \dot{u} (x, t) + N u (x, t) = c (x, t)

which shows vibration with damping coefficient g and mass p. The correction functional of above equation can be constructed as

y_{k + 1} (x, t) = y_{k} (x, t) + h \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + L \tilde{y_{k}} - c] d τ

(5)

As we know, $δ u (0) = 0$ , taking variation on both the sides

\begin{array}{l} δ y_{k + 1} (x, t) = δ y_{k} (x, t) \\ + h δ \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + g \dot{y} + L \tilde{y_{k}} - c] d τ \end{array}

(6)

which yields the following stationary conditions

p \overset{\cdot\cdot}{λ} - g \dot{λ} = 0

(7a)

1 - p \dot{λ} + g λ |_{τ = t} = 0

(7b)

λ |_{τ = t} = 0

(7c)

Combining equations (7b) and (7c), equations (7a)–(7c) can be rewritten as

p \overset{\cdot\cdot}{λ} - g \dot{λ} = 0

(8a)

1 - p \dot{λ} |_{τ = t} = 0

(8b)

λ |_{τ = t} = 0

(8c)

Therefore, for this form of linear operator, the Lagrange multiplier can be obtained as

λ = \frac{1}{g} [e^{\frac{g}{p} (τ - t)} - 1]

(9)

Expanding equation (9), one can get approximate Lagrange multiplier

\begin{array}{l} \frac{1}{g} [e^{\frac{g}{p} (τ - t)} - 1] \approx \frac{1}{p} (τ - t) \\ + \frac{g}{2! p^{2}} {(τ - t)}^{2} + \frac{g^{2}}{3! p^{3}} {(τ - t)}^{3} \end{array}

(10)

Thus

λ \approx \frac{1}{p} (τ - t) + \frac{g}{2! p^{2}} {(τ - t)}^{2} + \frac{g^{2}}{3! p^{3}} {(τ - t)}^{3}

(11)

Case II

In case II, we consider the following form of linear operator

N = p \frac{\partial^{2}}{\partial t^{2}}

(12)

Considering equation (12), equation (1) can be written as

p \overset{\cdot\cdot}{y} (x, t) + L y (x, t) = c (x, t)

(13)

The correction functional for above equation can be constructed as

y_{k + 1} (x, t) = y_{k} (x, t) + h \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + L \tilde{y_{k}} - c] d τ

(14)

As we know, $δ u (0) = 0$ , taking the variation on both the sides

δ y_{k + 1} (x, t) = δ y_{k} (x, t) + h δ \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + L \tilde{y_{k}} - c] d τ

(15)

\begin{array}{l} δ y_{k + 1} (x, t) = δ y_{k} (x, t) + p λ (τ) δ {y^{'}}_{k} (τ) |_{τ = t} \\ - p λ^{'} (τ) δ y_{k} (τ) |_{τ = t} + \int_{0}^{t} p λ^{″} δ y_{k} d τ = 0 \end{array}

(16)

which yields the following stationary conditions

p \overset{\cdot\cdot}{λ} = 0

(17a)

1 - p \dot{λ} |_{τ = t} = 0

(17b)

λ |_{τ = t} = 0

(17c)

Hence, for this case of linear operator, the Lagrange multiplier can be find out as follows

λ = \frac{1}{p} (τ - t)

Case III

In this case, considering the following form of linear operator

N = p \frac{\partial^{2}}{\partial t^{2}} + g \frac{\partial}{\partial t} + m

(18)

Considering equation (18), equation (1) can be rewritten as follows

p \overset{\cdot\cdot}{y} (x, t) + g \dot{y} (x, t) + m y (x, t) + L y (x, t) = c (x, t)

(19)

The correction functional of equation (19) can be constructed as

\begin{array}{l} y_{k + 1} (x, t) = y_{k} (x, t) \\ + h \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{n} + g \dot{y} + m y + L \tilde{y_{k}} - c] d τ \end{array}

(20)

As we know, $δ y (0) = 0$ , taking variation on both the sides

\begin{array}{l} δ y_{k + 1} (x, t) = δ y_{k} (x, t) \\ + h δ \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + g \dot{y} + m y + L \tilde{y_{k}} - c] d τ \end{array}

(21)

which yields the following stationary conditions

p λ^{″} - g λ^{'} + m λ = 0

(21)

1 - p λ^{'} + g λ |_{τ = t} = 0

(22b)

λ |_{τ = t} = 0

(22c)

Combining equations (22b) and (22c), equations (22a)–(22c), we get

p λ^{″} - g λ^{'} + m λ = 0

(23a)

1 - p λ^{'} |_{τ = t} = 0

(23b)

λ |_{τ = t} = 0

(23c)

Therefore, in this case of linear operator, the Lagrange multiplier can be obtained as

λ = \frac{\sin [β (τ - t)]}{p β} e^{\frac{g}{2 p} (τ - t)}

(24)

where

β = \frac{\sqrt{4 m p - c^{2}}}{2 p}

(25)

Expanding equation (25), one can get the approximate Lagrange multiplier

\begin{array}{l} λ = \frac{\sin [β (τ - t)]}{p β} e^{\frac{g}{2 p} (τ - t)} \\ \approx [\frac{(τ - t)}{p} + \frac{g {(τ - t)}^{2}}{2 p^{2}} + \frac{1}{3! p} (\frac{g^{2}}{p^{2}} - \frac{m}{p}) {(τ - t)}^{3}] \end{array}

(26)

Thus

λ \approx [\frac{(τ - t)}{p} + \frac{g {(τ - t)}^{2}}{2 p^{2}} + \frac{1}{3! p} (\frac{g^{2}}{p^{2}} - \frac{m}{p}) {(τ - t)}^{3}]

(27)

Case IV

In this case, considering the following form of linear operator

N = p \frac{\partial^{2}}{\partial t^{2}} + m

(28)

Considering equations (28) and (1)

p \overset{\cdot\cdot}{y} (x, t) + m y (x, t) + L y (x, t) = c (x, t)

(29)

where m is the spring coefficient.

The correction function for equation (29) can be constructed as

y_{k + 1} (x, t) = y_{k} (x, t) + h \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + m y_{k} + L \tilde{y_{k}} - c] d τ

(30)

As we know, $δ u (0) = 0$ , taking variation on both the sides

\begin{array}{l} δ y_{k + 1} (x, t) = δ y_{k} (x, t) \\ + h δ \int_{0}^{t} λ [p {\overset{\cdot\cdot}{y}}_{k} + m y + L \tilde{y_{k}} - c] d τ \end{array}

(31)

which yields the following stationary conditions

p λ^{″} + m λ = 0

(32a)

1 - p λ^{'} |_{τ = t} = 0

(32b)

λ |_{τ = t} = 0

(32c)

Therefore, in this case, the Lagrange multiplier can be obtained as

λ = \frac{1}{p ω} \sin [ω (τ - t)]

(33)

where $ω$ is the circular frequency

ω = \sqrt{\frac{m}{p}}

(34)

Expanding equation (33), one can get the following approximate Lagrange multiplier

\frac{1}{p ω} \sin [ω (τ - t)] \approx \frac{(τ - t)}{p} - \frac{ω^{2} {(τ - t)}^{3}}{3! p}

(35)

Thus

λ \approx \frac{(τ - t)}{p} - \frac{ω^{2} {(τ - t)}^{3}}{3! p}

(36)

Illustrative examples

In this section, variational iteration algorithm I with an auxiliary parameter is used for simple and damped mass–spring systems that undergo forced vibrations and transverse vibration of uniform and variable beams with simply supported analytical treatment. Results gained from the new proposed algorithm are very encouraging and significant. Illustrated examples revealed the effectiveness and power of the suggested algorithm.

A simple mass–spring system

In this section, a simple mass–spring system that undergoes forced vibration is investigated using variational iteration algorithm I with an auxiliary parameter. The differential equation of this system is

p \overset{\cdot\cdot}{y} + m y = g (t)

(37)

g (t) = G_{0} \cos (φ t)

Dividing both sides of equation (37) by $p$ , we get

\overset{\cdot\cdot}{y} + ω^{2} y = {\bar{G}}_{0} \cos (φ t)

(38)

where ${\bar{G}}_{0} = G_{0} / p$ .

The Lagrange multiplier of this problem can be obtained via variation theory

λ = \frac{1}{p φ} \sin [φ (τ - t)]

(39)

Furthermore, the recurrence relation of this problem is

\begin{array}{l} y_{k + 1} (t) = y_{k} (t) \\ + \frac{h}{p φ} \int_{0}^{t} \begin{array}{l} \sin [φ (τ - t)] \\ [p y^{″} (τ) + m y (τ) - g (τ)] d τ \end{array} \end{array}

(40)

The complementary solution of this problem is given by

y_{0} = A \sin (ω t) + B \cos (ω t)

(41)

Using equation (41) as an initial guess, we obtained

\begin{array}{l} y_{1} (t) = A \sin (ω t) + B \cos (ω t) \\ + \frac{1}{ω} h \int_{0}^{t} \sin [ω (τ - t)] [- {\bar{G}}_{0} \cos (φ τ)] d τ \end{array}

(42)

\begin{array}{l} y_{1} (t) = A \sin (ω t) + B \cos (ω t) \\ + h \frac{{\bar{G}}_{0} \cos (φ t)}{ω^{2} - φ^{2}} - h \frac{{\bar{G}}_{0} \cos (ω t)}{ω^{2} - φ^{2}} \end{array}

(43)

As the last term of above equation automatically satisfies equation (41), this term will not be used. Thus, equation (43) can be written as

y_{1} (t) = A \sin (ω t) + B \cos (ω t) + h \frac{{\bar{G}}_{0} \cos (φ t)}{ω^{2} - φ^{2}}

(44)

which is the general solution of equation (37).

In order to point out the significance of utilizing the exact Lagrange multiplier as a substitute of the approximate one, the following multiplier may be considered

λ = \frac{(τ - t)}{p}

(45)

Bearing in mind equation (45), the following recurrence relation can be obtained

\begin{array}{l} y_{k + 1} (t) = y_{k} (t) \\ + h \int_{0}^{t} (τ - t) [y^{″} (τ) + ω^{2} y (τ) - \frac{g (τ)}{p}] d τ \end{array}

(46)

If we utilize equation (41) as an initial guess, we get

\begin{array}{l} y_{1} (t) = A \sin (ω t) + B \cos (ω t) \\ + h \int_{0}^{t} (τ - t) [- {\bar{G}}_{0} \cos (φ τ)] d τ \end{array}

(47a)

Which can be written in compact form as

y_{1} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} [\frac{1 - \cos (φ t)}{φ^{2}}]

(47b)

and

\begin{array}{l} y_{2} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} [\frac{1 - \cos (φ t)}{φ^{2}}] \\ + h \int_{0}^{t} (τ - t) {{\bar{G}}_{0} [\frac{ω^{2}}{φ^{2}} \begin{array}{l} + \cos (φ τ) - \frac{ω^{2} \cos (φ τ)}{φ^{2}}] \\ - {\bar{G}}_{0} \cos (φ τ) \end{array}} d τ \end{array}

(48a)

This can be written in compact form as

\begin{array}{l} y_{2} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) \\ - h {\bar{G}}_{0} \cos (φ t) (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) - h {\bar{G}}_{0} \frac{ω^{2} t^{2}}{2 φ^{2}} \end{array}

(48b)

Similarly, $y_{4}$ can be obtained

\begin{matrix} y_{4} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} (\begin{array}{l} \frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} \\ + \frac{ω^{4}}{φ^{6}} + \frac{ω^{6}}{φ^{8}} \end{array}) \\ - h {\bar{G}}_{0} \cos (φ t) (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}} + \frac{ω^{6}}{φ^{8}}) \\ - \frac{{\bar{G}}_{0} ω^{2} t^{2}}{2} h (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}}) + h \frac{{\bar{G}}_{0} ω^{4} t^{4}}{24} (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) \\ - h \frac{{\bar{G}}_{0} ω^{6} t^{6}}{720} (\frac{1}{φ^{2}}) \end{matrix}

(49)

Then

\frac{- 1}{ω^{2} - φ^{2}} = \frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}} + \frac{ω^{6}}{φ^{8}} +

(50)

The expression for $y_{n}$ can be written as

\begin{array}{l} y_{k} = A \sin (ω t) + B \cos (ω t) + h \frac{{\bar{G}}_{0} C o s (φ t)}{ω^{2} - φ^{2}} \\ - \frac{{\bar{G}}_{0}}{ω^{2} - φ^{2}} h (1 - \frac{ω^{2} t^{2}}{2} + \frac{ω^{4} t^{4}}{24} - \frac{{\bar{G}}_{0} ω^{6} t^{6}}{720} + \dots) \end{array}

(51)

As $k \to \infty$ , then the compact form can be written as

\begin{array}{l} y (t) = A \sin (ω t) + B \cos (ω t) \\ + h \frac{{\bar{G}}_{0} [\cos (φ t) - \cos (ω t)]}{ω^{2} - φ^{2}} \end{array}

(52)

Note that equation (52) is the same as equation (47b).

If we use a different approximate form of the Lagrange multiplier, $λ$ , such as

λ = \frac{(τ - t)}{p} - \frac{ω^{2} {(τ - t)}^{3}}{3! p}

(53)

the following recurrence relation can be obtained

y_{k + 1} (t) = y_{k} (t) + h \int_{0}^{t} \begin{array}{l} [(τ - t) - \frac{ω^{2} {(τ - t)}^{3}}{3!}] \\ [y^{″} (τ) + ω^{2} y (τ) - \frac{g (τ)}{p}] d τ \end{array}

(54)

Once again using equation (41) as an initial guess, we get

\begin{array}{l} y_{1} (t) = A \sin (ω t) + B \cos (ω t) \\ + h \int_{0}^{t} [(τ - t) - \frac{ω^{2} {(τ - t)}^{3}}{3!}] [- {\bar{G}}_{0} \cos (φ τ)] d τ \end{array}

(55a)

It can be written as below in a compact form

\begin{array}{l} y_{1} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) \\ - h {\bar{G}}_{0} C o s (φ t) (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) - h {\bar{G}}_{0} \frac{ω^{2} t^{2}}{2 φ^{2}} \end{array}

(55b)

and

\begin{matrix} y_{2} = A \sin (ω t) + B \cos (ω t) + h {\bar{G}}_{0} (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}} + \frac{ω^{6}}{φ^{8}}) \\ - h {\bar{G}}_{0} \cos (φ t) (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}} + \frac{ω^{6}}{φ^{8}}) \\ - \frac{{\bar{G}}_{0} ω^{2} t^{2}}{2} h (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}} + \frac{ω^{4}}{φ^{6}}) \\ + h \frac{{\bar{G}}_{0} ω^{4} t^{4}}{24} (\frac{1}{φ^{2}} + \frac{ω^{2}}{φ^{4}}) - h \frac{{\bar{G}}_{0} ω^{6} t^{6}}{720} (\frac{1}{φ^{2}}) \end{matrix}

(56)

It can be obtained without difficulty that approximate Lagrange multiplier $λ = (τ - t) / p - (ω^{2} {(τ - t)}^{3}) / 3! p$ converges rapidly than $λ = (τ - t) / p$ .

Transverse vibration of a variable coefficient beam

Transverse vibration of a variable coefficient beam that was investigated by Wazwaz¹² is studied. The differential equation of this system is

y_{t t} + μ (x) y_{x x x x} = g (x, t), μ (x) > 0, 0 < x < N, t > 0

(57)

Here, the equation of motion of the beam can be written as

y_{t t} + (1 + x) y_{x x x x} = (x^{4} + x^{3} - \frac{6}{7!}) \cos t

(58)

The initial and boundary conditions for this problem are

\begin{array}{l} y (x, 0) = \frac{6}{7!} x^{7}, 0 < x < 1, y {}_{t}{(x, 0)} = 0, 0 < x < 1 \\ y (0, t) = 0 . t > 0, y (1, t) = \frac{6}{7!} \cos t, t > 0 \end{array}

(59)

The Lagrange multiplier is

λ = \frac{1}{p} (τ - t)

and $p = 1$ for this problem. Hence, the recurrence relation can be written as

y_{k + 1} (t) = y_{k} (t) + h \int_{0}^{t} (τ - t) [\begin{array}{l} y_{t t} + (1 + x) y_{x x x x} \\ - (x^{4} + x^{3} - \frac{6}{7!}) \cos τ \end{array}] d τ

(60)

The complementary solution which can be used as an initial guess is

y_{0} (t) = \frac{6}{7!} x^{7} \cos t + (x^{3} + x^{4}) (1 - \cos t)

(61)

Using the recurrence relation (60) and the initial guess (61), we get

\begin{array}{l} y_{1} (t) = \frac{6}{7!} x^{7} \cos t + (x^{3} + x^{4}) (1 - \cos t) \\ - h (1 + x) (- 24 + 12 t^{2} + x^{3} - (- 24 + x^{3}) \cos t) \end{array}

(62a)

which can be written in a compact form as

\begin{array}{l} y_{1} (t) = - 12 h (t^{2} - 2) (1 + x) \\ + (\frac{x^{7}}{7!} - 24 h x - 24 h) \cos t \end{array}

(62b)

and

\begin{array}{l} y_{2} (t) = - 12 h (t^{2} - 2) (1 + x) + (\frac{x^{7}}{7!} - 24 h x - 24 h) \\ \cos t + 12 h (1 + x) (t^{2} - 2 + 2 \cos t) \end{array}

(63)

Equation (63) can be written in a closed form as

y_{2} (t) = \frac{x^{7}}{7!} \cos t

(64)

The exact solution to this problem is obtained in just two iterations, which shows the reliability of the proposed method.

A damped mass–spring system

In this section, a damped mass–spring system that undergoes forced vibration is investigated. The differential equation of this system is as follows

p \overset{\cdot\cdot}{y} + g \dot{y} + m y = G_{0} \cos (φ t)

(65)

The Lagrange multiplier of this system is

λ = \frac{\sin [β (τ - t)]}{p β} e^{\frac{g}{2 p} (τ - t)}

(66)

where

β = \frac{\sqrt{4 p m - g^{2}}}{2 p}

(67)

Using the above Lagrange multiplier, the recurrence relation can be written as

\begin{array}{l} y_{k + 1} (t) = y_{k} (t) + \\ \frac{h}{p β} \int_{0}^{t} e^{\frac{g}{2 p} (τ - t)} \sin [β (τ - t)] [\begin{array}{l} p y^{″} (τ) \\ + g y^{'} (τ) + m y (τ) \end{array}] d τ \end{array}

(68)

Dividing both sides of equation (65) by $p$ , we get

\overset{\cdot\cdot}{y} + 2 ς ω \dot{y} + ω^{2} y = {\bar{G}}_{0} \cos (φ t)

(69)

where

{\bar{G}}_{0} = \frac{G_{0}}{p} and \frac{g}{2 p} = ς ω

Therefore, equation (68) becomes

\begin{matrix} y_{k + 1} (t) = y_{k} (t) + \\ \frac{h}{β} \int_{0}^{t} e^{\frac{g}{2 p} (τ - t)} \sin [β (τ - t)] [y^{″} (τ) + 2 ς ω y^{'} (τ) \\ + ω^{2} y (τ) - {\bar{G}}_{0} \cos (φ τ)] d τ \end{matrix}

(70)

and

β = \sqrt{1 - ς^{2}} ω

(71)

The complementary solution which can be used as initial guess

y_{0} = [A \sin (β t) + B \cos (β t)] e^{- ς ω t}

(72)

Taking (72) as an initial approximation, we get

\begin{array}{l} y_{1} (t) = y_{0} (t) + \\ \frac{h}{β} \int_{0}^{t} e^{\frac{g}{2 p} (τ - t)} \sin [β (τ - t)] [- {\bar{G}}_{0} \cos (φ τ)] d τ \end{array}

(73)

Integrating the second term of equation (73), we obtained

\begin{matrix} y_{1} (t) = y_{0} (t) + h \frac{(ω^{2} - φ^{2}) \cos [φ t] + 2 ς ω φ \sin [φ t]}{ω^{4} + 2 (2 ς^{2} - 1) ω^{2} φ^{2} + φ^{4}} \\ + h \frac{(φ^{2} - ω^{2}) \cos [β t] e^{- ς ω t}}{ω^{4} + 2 (2 ς^{2} - 1) ω^{2} φ^{2} + φ^{4}} \\ + h \frac{ς (φ^{2} + ω^{2}) \sin [β t] e^{- ς ω t}}{[ω^{4} + 2 (2 ς^{2} - 1) ω^{2} φ^{2} + φ^{4}] \sqrt{1 - ς^{2}}} \end{matrix}

(74)

As the last two terms of above equation automatically satisfy the homogeneous equation, these terms will not be used further. The second term of the said equation can be written as

\begin{array}{l} \frac{(ω^{2} - φ^{2}) \cos [φ t] + 2 ς ω φ \sin [φ t]}{ω^{4} + 2 (2 ς^{2} - 1) ω^{2} φ^{2} + φ^{4}} \\ = \frac{\cos (φ t - ϕ)}{\sqrt{{(1 - φ^{2} / ω^{2})}^{2} + {(2 ς φ / ω)}^{2}}} \end{array}

(75)

where

\tan (ϕ) = \frac{2 ς φ / ω}{1 - φ^{2} / ω^{2}}

(76)

Thus, the exact solution of equation (65) is obtained in very first iteration

\begin{array}{l} y_{1} = [A \sin (β t) + B \cos (β t)] e^{- ς ω t} \\ + h \frac{{\bar{G}}_{0} \cos (φ t - ϕ)}{\sqrt{{(1 - φ^{2} / ω^{2})}^{2} + {(2 ς φ / ω)}^{2}}} \end{array}

(77)

Transverse vibration of a uniform beam with simply supported ends

Here, the transverse vibration of a uniform beam with simply supported ends is investigated. The differential equation of this system is

y_{t t} + g^{2} y_{x x x x} = 0, 0 < x < N, t > 0

(78)

The initial and the boundary conditions for transverse vibration of a uniform beam with simply supported ends are

\begin{array}{l} y (x, 0) = \sin \frac{π x}{N}, y {}_{t}{(x, 0)} = 0, y (0, t) = 0 \\ y (N, t) = 0, y_{x x} (0, t) = 0, y_{x x} (N, t) = 0 \\ y {}_{t}{(x, 0)} = 0, y_{x x} (0, t) = 0 \end{array}

(79)

It is easily noted that the Lagrange multiplier of this system is

λ = \frac{1}{p} (τ - t)

(80)

Using this Lagrange multiplier, the recurrence relation can be written as

y_{k + 1} (t) = y_{k} (t) + \frac{1}{p} h \int_{0}^{t} (τ - t) [y_{τ τ} + g^{2} \frac{\partial^{4} y}{\partial x^{4}}] d τ

(81)

The complementary solution, which can be use as initial guess, can be written as

y_{0} (t) = \sin \frac{π x}{N}

(82)

Using the initial guess in equation (81), we get

\begin{matrix} y_{1} (t) = \sin \frac{π x}{N} + \frac{1}{p} h \int_{0}^{t} (τ - t) [\frac{g^{2} π^{4}}{N^{4}} \sin \frac{π x}{N}] d τ \\ = \sin \frac{π x}{N} - \frac{g^{2} π^{4} t^{2}}{2 N^{4}} \sin \frac{π x}{N} \end{matrix}

(83)

and

\begin{matrix} y_{2} (t) = \sin \frac{π x}{N} - h \frac{g^{2} π^{4} t^{2}}{2 N^{4}} \sin \frac{π x}{N} \\ + \frac{1}{p} h \int_{0}^{t} (τ - t) [- \frac{g^{2} π^{4}}{N^{4}} \sin \frac{π x}{N} \\ + h g^{2} (\frac{π^{4}}{N^{4}} \sin \frac{π x}{N} - \frac{g^{2} π^{8} τ^{2}}{2 N^{8}} \sin \frac{π x}{N})] d τ \\ = \sin \frac{π x}{N} - \frac{g^{2} π^{4} t^{2}}{2! N^{4}} \sin \frac{π x}{N} + \frac{g^{4} π^{8} t^{4}}{4! N^{8}} \sin \frac{π x}{N} \end{matrix}

(84)

Hence, the expression for $y_{k}$ can be expressed as

y_{k} (t) = h \sin \frac{π x}{N} (\begin{array}{l} 1 - \frac{g^{2} π^{4} t^{2}}{2! N^{4}} + \frac{g^{4} π^{8} t^{4}}{4! N^{8}} \\ + \dots + {(- 1)}^{k} \frac{g^{2 k} π^{4 k} t^{2 k}}{(2 k)! N^{4 k}} \end{array})

(85)

Hence, we have

\lim_{k \to \infty} y_{k} (t) = \sin \frac{π x}{N} \cos \frac{π^{2} g t}{N^{2}}

(86)

which is the exact solution of transverse vibration of a uniform beam with simply supported ends.

Conclusion

In this article, the two main goals which have been archived are employing variational iteration algorithm I with an auxiliary parameter for investigating simple and damped mass–spring systems that undergo forced vibrations and transverse vibration of uniform and variable beams with simply supported analytical treatment and showing the reliability and accuracy of this method. This method makes easy the computational work for solving increasingly complex vibration problems, for example, aeroelasticity and arbitrary vibrations, arising in science and engineering, and results of high degree accuracy can be obtained in few iterations as compared with other existing methods in the literature.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hijaz Ahmad

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Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems

Abstract

Keywords

Introduction

Variational iteration algorithm I with an auxiliary parameter

Different forms of linear operator

Case I

Case II

Case III

Case IV

Illustrative examples

A simple mass–spring system

Transverse vibration of a variable coefficient beam

A damped mass–spring system

Transverse vibration of a uniform beam with simply supported ends

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References