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This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.

Many problems related to mathematicians, physicists, biologists, chemists and engineers are formulated in differential equations whether linear or nonlinear. The solutions of a linear one can be obtained easily using some of well-established methods on the contrary with nonlinear differential Equations (NDE) that we often refuge to approximate solutions. Nonlinear oscillations had shed the interest of many scientists due to that most of the problems dealing with vibrations are nonlinear, see [

Since it is difficult to find the exact solutions of such equations, many researchers have turned their attention to obtain the approximate solutions of these problems using perturbation techniques [

Over the past two decades, many mathematicians and physicists have done their great efforts to find new mathematical tools to deal with the dynamical systems that mathematically described by nonlinear differential equations [

A max-min method is presented in [

In [

In [

HPT is used in [

In this paper, the solution of a nonlinear oscillating dynamical system is investigated. This system consists of a mass m_{1} connected with a spring of linear stiffness and with other mass m_{2} through a massless string of length l. HPT is utilized to obtain the solution of the equation of motion. This solution is graphically represented for different values of the system parameters and compared with the numerical solution of the governing equation of motion using the Runge-Kutta method [

This paper is designated as follows. In Section 2, a description of the investigated problem and the derivation of the equation of motion are presented. Section 3 sheds light on the basic idea of HPT. Section 4 is devoted to reduce the equation of motion into appropriate equation and to obtain the solution of this equation analytically using HPT. In Section 5, we are going to represent the attained solution graphically and to obtain the numerical solution using the Runge-Kutta method. The stability of the obtained solution is discussed in Section 6. Finally, the manuscript is finished with some concluding remarks.

In this section, we are going to obtain the governing equation of motion of a nonlinear oscillation system using HPT of a dynamical model. This model consists of two masses; a first one m_{1} moves horizontally in which it is attached to a spring of linear stiffness k and connected with the second mass m_{2} with a massless string of length l see (_{0}, x and y are the natural length of the given spring, the horizontal coordinate of the centroid

of m_{1} and the vertical coordinate of the centroid of m_{2}, respectively. Therefore, the potential and kinetic energies V and T of the system can be written in the forms

V = − m 2 g l 2 − x 2 + k x 2 2 , T = 1 2 ( m 1 x ˙ 2 + m 2 y ˙ 2 ) , (1)

where g is the gravitational acceleration, x is the extension of string after time t, dots denote to the differentiation with respect to time and ( x ˙ , y ˙ ) is the Cartesian velocity of the point ( x , y ) .

According to (1), the Lagrangian L = T − V has the form

L = 1 2 ( m 1 + m 2 x 2 l 2 − x 2 ) x ˙ 2 + m 2 g l 2 − x 2 − k x 2 2 . (2)

An inspection of the Lagrange’s function (2) shows that the investigated system has only one degree of freedom. Therefore, Lagrange’s equation for conservative system may be written as

d d t ( ∂ L ∂ x ˙ ) − ∂ L ∂ x = 0. (3)

Here, x and x ˙ are the generalized coordinate and velocity of the system respectively. Making use of (2) and (3) yields to the following form of the governing equation of motion

( m 1 + m 2 x 2 l 2 − x 2 ) x ¨ + m 2 l 2 x ( l 2 − x 2 ) 2 x ˙ 2 + m 2 g x ( l 2 − x 2 ) 1 2 + k x = 0 (4)

This section is devoted to illustrate HPT [

K ( u ) − f ( r ) = 0, r ∈ Ω , (5)

beside the following boundary condition

B ( u , ∂ u ∂ n ) = 0, r ∈ Γ . (6)

Here K and B represent the general differential operator and the boundary operator respectively, f ( r ) denotes a known analytical function, Γ is the boundary of a domain Ω and ∂ u ∂ n refers to differential along the normal drawn outwards from Ω .

An inspection of Equation (5), broadly speaking, the operator K can be separated into two parts; which are a linear part L and a nonlinear one N. Therefore Equation (5) can be rewritten in the form

L ( u ) + N ( u ) − f ( r ) = 0 (7)

It is worthwhile to notice that according to HPT, we can construct the homotopy v ( r , ρ ) : Ω × [ 0,1 ] → R , which satisfies

H ( v , ρ ) = ( 1 − ρ ) ( L ( v ) − L ( U ) ) + ρ ( K ( v ) − f ( r ) ) = 0 , ρ ∈ [ 0 , 1 ] (8)

or in an equivalent form as

H ( v , ρ ) = L ( v ) − L ( U ) + ρ L ( U ) + ρ ( N ( v ) − f ( r ) ) = 0, ρ ∈ [ 0 , 1 ] (9)

where ρ ∈ [ 0 , 1 ] is a homotopy parameter and U (initial guess) is an initial approximation of Equation (5), in which it satisfies the boundary conditions.

In order to investigate the solution of (8) or (9), we express about this solution as a power series of ρ as

v = v 0 + ρ v 1 + ρ 2 v 2 + ⋯ . (10)

At ρ → 1 , Equations (8) or (9) corresponds to Equation (5) and the results in the approximation to the solution of Equation (5) can be expressed as

u = lim ρ → 1 v = v 0 + v 1 + v 2 + ⋯ . (11)

It is important to note that, series (11) is convergent for more cases. Some criteria are suggested for convergence of this series, see [

Dividing both sides of (4) by m_{1} and consider that

ω 0 2 = k m 1 + R g l , R = m 2 m 1 , u = x l , | u | ≪ 1 (12)

to reduce the equation of motion (4) to a more appropriate as

[ 1 + m 2 x 2 m 1 ( l 2 − x 2 ) ] x ¨ + [ m 2 l 2 x m 1 ( l 2 − x 2 ) 2 ] x ˙ 2 + g m 2 x m 1 ( l 2 − x 2 ) 1 2 + k x m 1 = 0.

On the use of (12), the previous equation can be rewritten in the form

[ 1 + R u 2 1 − u 2 ] u ¨ + [ R u ( 1 − u 2 ) 2 ] u ˙ 2 + R g u l ( 1 − u 2 ) 1 2 + k u m 1 = 0.

Expanding the previous equation to obtain

[ 1 + R u 2 ( 1 + u 2 + ⋯ ) ] u ¨ + R u ( 1 + 2 u 2 + ⋯ ) u ˙ 2 + R g l u ( 1 + 1 2 u 2 + ⋯ ) + k u m 1 = 0 ; ( u ≪ 1 ) .

Therefore, we obtain the following equation

( 1 + R u 2 ) u ¨ + R u u ˙ 2 + ω 0 2 u + R g 2 l u 3 + ⋯ = 0. (13)

A closer look of this equation reveals that it is a second order differential equation with high nonlinearity.

The aim of this section is to obtain the approximate solution of the governing equation of motion utilizing HPT in the presence of the following initial conditions

u ( 0 ) = A , u ˙ ( 0 ) = 0. (14)

By virtue of Equations (13) and (7), the linear part L ( u ) and nonlinear one N(u) have the forms

L ( u ) = u ¨ + ω 0 2 u , (15)

N ( u ) = R u 2 u ¨ + R u u ˙ 2 + R g 2 l u 3 , (16)

where

f ( r ) = 0. (17)

Equation (8) can be rewritten in the form

H ( v , ρ ) = ( 1 − ρ ) ( L ( v ) − L ( U ) ) + ρ ( L ( v ) + N ( v ) − f ( r ) ) = 0 , ρ ∈ [ 0 , 1 ] (18)

Substituting (15)-(17) into (18) to obtain

H ( v , ρ ) = ( 1 − ρ ) ( v ¨ + ω 0 2 v − U ¨ − ω 0 2 U ) + ρ ( v ¨ + ω 0 2 v + R v 2 v ¨ + R v v ˙ 2 + R g 2 l v 3 ) = 0. (19)

Let U = 0 ; the previous equation have the form

H ( v , ρ ) = ( 1 − ρ ) ( v ¨ + ω 0 2 v ) + ρ ( v ¨ + ω 0 2 v + R v 2 v ¨ + R v v ˙ 2 + R g 2 l v 3 ) = 0. (20)

Making use of (10) and (20), then equating the coefficients of similar powers of ρ in both sides to obtain

Coefficient of ρ 0 :

v ¨ 0 + ω 0 2 v 0 = 0 , (21)

Coefficient of ρ :

v ¨ 1 + R v 0 2 v ¨ 0 + R v 0 v ˙ 0 2 + ω 0 2 v 1 + R g 2 l v 0 3 = 0 , (22)

Coefficient of ρ 2 :

v ¨ 2 + R v 0 2 v ¨ 1 + 2 R v 0 v ¨ 0 v 1 + 2 R v 0 v ˙ 0 v ˙ 1 + R v 1 v ˙ 0 2 + ω 0 2 v 2 + 3 R g 2 l v 1 v 0 2 = 0. (23)

The previous Equations (21)-(23) can be solved subsequently with the aid if the following conditions

v 0 ( 0 ) = A , v ˙ 0 ( 0 ) = 0 , v 1 ( 0 ) = 0 , v ˙ 1 ( 0 ) = 0 , v 2 ( 0 ) = 0 , v ˙ 2 ( 0 ) = 0 , (24)

to get

v 0 = A c o s ( ω 0 t ) , (25)

v 1 = A 3 R s i n ( ω 0 t ) 32 l ω 0 2 [ 2 ω 0 ( 4 l ω 0 2 − 3 g ) t + ( 4 l ω 0 2 − g ) s i n ( 2 ω 0 t ) ] , (26)

v 2 = A 5 R 2 1024 l 2 ω 0 4 { g 2 [ 22 cos ( ω 0 t ) − cos ( 3 ω 0 t ) ] sin 2 ( ω 0 t ) + 3 g 2 ω 0 t [ 8 sin ( ω 0 t ) − 3 sin ( 3 ω 0 t ) ] − 2 g ω 0 2 [ 2 l + 9 g t 2 − 6 l cos ( 2 ω 0 t ) + 4 l cos ( 4 ω 0 t ) ] cos ( ω 0 t ) + 24 g l ω 0 3 t [ sin ( ω 0 t ) + 2 sin ( 3 ω 0 t ) ] + 24 l ω 0 4 [ − 2 l + 2 g t 2 + l cos ( 2 ω 0 t ) + l cos ( 4 ω 0 t ) ] cos ( ω 0 t ) } − 16 l 2 ω 0 5 t { 3 [ 2 sin ( ω 0 t ) + sin ( 3 ω 0 t ) ] − 2 ω 0 t cos ( ω 0 t ) } . (27)

Since ρ ∈ [ 0 , 1 ] , one gets directly the desired solution when ρ → 1 in the form

u = lim ρ → 1 v = A cos ( ω 0 t ) + A 3 R sin ( ω 0 t ) 32 l ω 0 2 [ 2 ω 0 ( 4 l ω 0 2 − 3 g ) t + ( 4 l ω 0 2 − g ) sin ( 2 ω 0 t ) ] + A 5 R 2 1024 l 2 ω 0 4 { g 2 [ 22 cos ( ω 0 t ) − cos ( 3 ω 0 t ) ] sin 2 ( ω 0 t ) + 3 g 2 ω 0 t [ 8 sin ( ω 0 t ) − 3 sin ( 3 ω 0 t ) ] − 2 g ω 0 2 [ 2 l + 9 g t 2 − 6 l cos ( 2 ω 0 t ) + 4 l cos ( 4 ω 0 t ) ] cos ( ω 0 t )

+ 24 g l ω 0 3 t [ sin ( ω 0 t ) + 2 sin ( 3 ω 0 t ) ] + 24 l ω 0 4 [ − 2 l + 2 g t 2 + l cos ( 2 ω 0 t ) + l cos ( 4 ω 0 t ) ] cos ( ω 0 t ) } − 16 l 2 ω 0 5 t { 3 [ 2 sin ( ω 0 t ) + sin ( 3 ω 0 t ) ] − 2 ω 0 t cos ( ω 0 t ) } . (28)

In this section, we are going to shed light on the great accuracy of the results obtained by HPT when they are compared with the numerical results of the governing equation of motion (4) using the fourth order Runge-Kutta method [

Figures 2-4 are calculated at m 1 = 3 kg ,5 kg and m 1 = 7 kg respectively for different values of l = 0.1 m , 0.6 m and 1 m, in which their parts (a), (b) and (c) are plotted when m 2 = 0.2 kg ,0.4 kg and m 2 = 0.6 kg respectively. It is

worthwhile to notice that these drawings have periodic forms and therefore the attained solution has a stable manner.

An inspection of the corresponding parts of these figures reveals that when l increases from 0.1 m to 1 m passing the value 0.6 m; the number of oscillations decreases and the wavelength of the ripples increases while the amplitudes of these ripples remain unchanged.

When parts (a) of Figures 2-4 are generally compared to parts (b) and (c) of the same figures, we observe that when m_{2} increases from 0.2 kg to 0.6 m through the value 0.4 m; the number of oscillations increases and the wavelength of waves decreases beside the constancy of their amplitudes.

Moreover, these results are plotted in some figures for the same considered parameters; see Figures 5-7 when m 1 = 5 kg and m 2 = ( 0.2,0.4,0.6 ) kg . Figures 5-7 are calculated at l = 0.1 m , 0.6 m and 1 m respectively. It is not difficult to notice from the parts of

On the other side, this difference becomes very slightly which can be neglected as in

Tables 1-8 reveal a comparison between the results obtained by HPT with the

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.470208 | −0.460229 | 0.0212217 |

2 | 0.144114 | 0.111063 | 0.229342 |

3 | 0.239213 | 0.286819 | 0.19901 |

4 | −0.526688 | −0.556329 | 0.0562768 |

5 | 0.591097 | 0.56918 | 0.0370788 |

6 | −0.400474 | −0.314833 | 0.21385 |

7 | 0.0452104 | −0.0731896 | 2.61886 |

8 | 0.327927 | 0.423236 | 0.290642 |

9 | −0.568019 | −0.586387 | 0.0323366 |

10 | 0.564724 | 0.463692 | 0.178905 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.595921 | −0.59828 | 0.00395892 |

2 | 0.583752 | 0.593003 | 0.0158477 |

3 | −0.563693 | −0.583815 | 0.0356974 |

4 | 0.536067 | 0.570128 | 0.0635394 |

5 | −0.501313 | −0.551128 | 0.0993686 |

6 | 0.45997 | 0.525782 | 0.143079 |

7 | −0.412657 | −0.492854 | 0.194344 |

8 | 0.360058 | 0.450927 | 0.252375 |

9 | −0.302906 | −0.398435 | 0.315377 |

10 | 0.241969 | 0.333704 | 0.379119 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.445572 | −0.464127 | 0.0416424 |

2 | 0.067099 | 0.121775 | 0.814853 |

3 | 0.344045 | 0.272124 | 0.209045 |

4 | −0.584389 | −0.544756 | 0.0678198 |

5 | 0.524674 | 0.564912 | 0.0766916 |

6 | −0.198083 | −0.31593 | 0.594938 |

7 | −0.225545 | −0.0692667 | 0.692891 |

8 | 0.538462 | 0.39216 | 0.271704 |

9 | −0.576907 | −0.503061 | 0.128003 |

10 | 0.319492 | 0.326598 | 0.0222431 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | 0.0235546 | 0.0275701 | 0.170475 |

2 | −0.598121 | −0.597439 | 0.00113993 |

3 | −0.0705211 | −0.0824829 | 0.16962 |

4 | 0.592496 | 0.589777 | 0.00458767 |

5 | 0.11706 | 0.136712 | 0.167885 |

6 | −0.583162 | −0.57708 | 0.0104286 |

7 | −0.162888 | −0.189802 | 0.165227 |

8 | 0.57018 | 0.559452 | 0.018816 |

9 | 0.207727 | 0.241293 | 0.161591 |

10 | −0.553637 | −0.537039 | 0.0299789 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.108747 | −0.102759 | 0.0550631 |

2 | −0.560058 | −0.564482 | 0.0078988 |

3 | 0.312164 | 0.296315 | 0.050769 |

4 | 0.445702 | 0.46227 | 0.037171 |

5 | −0.474821 | −0.455237 | 0.0412442 |

6 | −0.272606 | −0.305748 | 0.121575 |

7 | 0.574763 | 0.560593 | 0.0246535 |

8 | 0.0639964 | 0.113599 | 0.775091 |

9 | −0.598273 | −0.599561 | 0.00215312 |

10 | 0.15289 | 0.0915716 | 0.401061 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.2203 | −0.213465 | 0.0310281 |

2 | −0.437498 | −0.447683 | 0.0232808 |

3 | 0.542499 | 0.532471 | 0.0184855 |

4 | 0.0386421 | 0.0686075 | 0.775458 |

5 | −0.571011 | −0.581362 | 0.0181276 |

6 | 0.380935 | 0.345017 | 0.094289 |

7 | 0.290291 | 0.335323 | 0.155126 |

8 | −0.595001 | −0.583698 | 0.0189959 |

9 | 0.146686 | 0.0798101 | 0.455913 |

10 | 0.48678 | 0.525629 | 0.0798072 |

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | 0.0849478 | 0.0871748 | 0.0262152 |

2 | −0.575847 | −0.574597 | 0.00217201 |

3 | −0.248074 | −0.254183 | 0.024628 |

4 | 0.505338 | 0.50056 | 0.0094552 |

5 | 0.391384 | 0.399766 | 0.0214169 |

6 | −0.394196 | −0.384207 | 0.0253417 |

7 | −0.503322 | −0.511601 | 0.0164498 |

8 | 0.251453 | 0.235423 | 0.0637503 |

9 | 0.574791 | 0.580176 | 0.00936958 |

10 | −0.0886238 | −0.0667953 | 0.246305 |

numerical ones of the governing equation of motion (4) that obtained using the fourth order Runge-Kutta method and the corresponding error percentage of HPT; for different values of parameters of the considered dynamical model. This compression shows high consistency between them which expresses the great accuracy of the obtained solutions using HPT.

Time | Numerical Results (NR) | HPT Results (HPTR) | |(HPTR − NR)/NR| |
---|---|---|---|

0 | 0.6 | 0.6 | 0 |

1 | −0.0786568 | −0.0744064 | 0.0540374 |

2 | −0.579581 | −0.58166 | 0.00358832 |

3 | 0.230568 | 0.218613 | 0.0518523 |

4 | 0.519584 | 0.52774 | 0.0156975 |

5 | −0.366616 | −0.3493 | 0.0472322 |

6 | −0.423893 | −0.44148 | 0.041489 |

7 | 0.477398 | 0.458431 | 0.0397306 |

8 | 0.29899 | 0.3281 | 0.0973608 |

9 | −0.555316 | −0.539339 | 0.0287718 |

10 | −0.153496 | −0.194521 | 0.267272 |

In this section, we investigate the stability of the governing equation of motion (13). It is obvious from the preceding section that, this investigation will be unsuccessful in view of Equation (28). Therefore, we are going to obtain a periodic solution of (13).

It should be noticed that Equation (13) is transformed into linear and nonlinear parts as indicated in Equations (15) and (16) respectively in which ω 0 denotes a natural frequency of Equation (15). It is clear that the linear part represents a simple harmonic equation. Therefore, the stability of this part depends upon the frequency ω 0 which is always positive and consequently, the represented figures have periodic forms as expected. Therefore the system is always stable.

Now, let us focus attention on the stability of a nonlinear part in which we consider a nonlinear frequency analysis. Therefore, a nonlinear frequency Ω 2 is assumed to be in the following form

Ω 2 = ω 0 2 + ρ ϖ 1 + ρ 2 ϖ 2 + ⋯ , (29)

where ϖ 1 , ϖ 2 , ⋯ are arbitrary parameters can be estimated.

According to the reported work [

Ω 2 = ω 0 2 + lim ρ → 1 ∑ i = 1 ∞ ρ i ϖ i (30)

Substitution of (29) into (20) yields

v ¨ + Ω 2 v + ρ R [ v 2 v ¨ + v v ˙ 2 + g 2 l v 3 ] − ∑ i = 1 ∞ ρ i ϖ i v = 0 (31)

Making use of (10) and (31), then equating the coefficients of like powers of ρ in both sides to obtain

Coefficient of ρ 0 :

v ¨ 0 + Ω 2 v 0 = 0, (32)

Coefficient of ρ :

v ¨ 1 + R [ v 0 2 v ¨ 0 + v 0 ( v ˙ 0 2 + g 2 l v 0 2 ) ] + Ω 2 v 1 − ϖ 1 v 0 = 0, (33)

Coefficient of ρ 2 :

v ¨ 2 + R [ v 0 2 v ¨ 1 + 2 v 0 ( v ¨ 0 v 1 + v ˙ 0 v ˙ 1 ) + v 1 ( v ˙ 0 2 + 3 g 2 l v 0 2 ) ] + Ω 2 v 2 − ϖ 1 v 1 − ϖ 2 v 0 = 0, (34)

Taking into account conditions (24), one can solve Equations (32)-(34) subsequently to get

v 0 = A c o s ( Ω t ) . (35)

It is worthy to mention that in order to get a uniform to expand solution, the terms that produce secular terms in Equations (33) and (34) must be deleted. Substituting (35) into (33) and (34), then expanding the trigonometric functions to obtain

v ¨ 1 + Ω 2 v 1 = A ( ϖ 1 − R A 2 Ω 2 ) cos ( Ω t ) + 1 8 l R A 3 ( 4 l Ω 2 − g ) [ 3 cos ( Ω t ) + cos ( 3 Ω t ) ] , v ¨ 2 + Ω 2 v 2 = 1 8 l A [ 8 l ϖ 2 + Q A R ( 16 l Ω 2 − 3 g ) ] cos ( Ω t ) + 21 4 Q A 2 R Ω 2 [ cos ( 2 Ω t ) + cos ( 4 Ω t ) ] + 1 4 l Q [ 4 l ϖ 1 − A 2 R ( 2 l Ω 2 + 3 g ) ] cos ( 3 Ω t ) + 1 8 l Q A 2 R [ ( 16 l Ω 2 − 3 g ) cos ( 5 Ω t ) + 2 l Ω 2 ( 1 + cos ( 6 Ω t ) ) ] . (36)

Omitting terms that lead to secular terms in (36) to get

ϖ 1 = 1 8 l A 2 R ( 3 g − 4 l Ω 2 ) , ϖ 2 = 1 8 l Q A R ( 3 g − 16 l Ω 2 ) . (37)

According to (37), one can write the solutions v 1 ( t ) and v 2 ( t ) of (36) in the form

v 1 = Q [ cos ( 3 Ω t ) − cos ( Ω t ) ] , v 2 = Q 8 [ ϖ 1 Ω 2 + A 2 R 2 ( 631 21 − 7 g 4 l Ω 2 ) ] cos ( Ω t ) − Q A 2 R 4 Ω sin ( Ω t ) − 7 20 Q A 2 R [ 5 cos ( 2 Ω t ) + cos ( 4 Ω t ) ] + Q 32 l Ω 2 [ 3 R A 2 ( g + 7 l Ω 2 ) − 4 l ϖ 1 ] cos ( 3 Ω t ) + R Q A 2 16 [ 1 12 ( 3 g − 16 l Ω 2 ) cos ( 5 Ω t ) − 1 35 cos ( 6 Ω t ) + 1 ] , (38)

where

Q = R A 3 16 l Ω 2 ( g − l Ω 2 ) .

Making use of (10), (35) and (38), then considering ρ → 1 , to obtain the approximate periodic solution in the form

u = A cos ( Ω t ) + Q [ cos ( 3 Ω t ) − cos ( Ω t ) ] + Q 8 [ ϖ 1 Ω 2 + A 2 R 2 ( 631 21 − 7 g 4 l Ω 2 ) ] cos ( Ω t ) − Q A 2 R 4 Ω sin ( Ω t ) − 7 20 Q A 2 R [ 5 cos ( 2 Ω t ) + cos ( 4 Ω t ) ] + Q 32 l Ω 2 [ 3 R A 2 ( g + 7 l Ω 2 ) − 4 l ϖ 1 ] cos ( 3 Ω t ) + R Q A 2 16 [ 1 12 ( 3 g − 16 l Ω 2 ) cos ( 5 Ω t ) − 1 35 cos ( 6 Ω t ) + 1 ] . (39)

An inspection of the previous solution u is given as a function of time t and has a periodic form. Therefore, the arguments of the trigonometric functions must be real values. To achieve this aim substituting (37) into (29) and considering ρ → 1 , we obtain

l [ A 2 R ( A 2 R − 4 ) − 8 ] Ω 4 + [ A 2 g R ( 3 − 7 16 A 2 R ) + 8 l ω 0 2 ] Ω 2 + 3 A 4 R 2 g 2 64 l = 0. (40)

Under the present circumstances, the stability conditions require that Ω must be taken a real and positive quantity. Therefore, the necessary and sufficient conditions for the stability have the forms

A 2 R ( A 2 R − 4 ) − 8 > 0 , A 2 g R ( 3 − 7 16 A 2 R ) + 8 l ω 0 2 < 0 (41)

To gain more insight into the existence of real roots, the distinction of (40) must be positive or becomes worthless i.e.,

64 l 2 ( ω 0 2 ) 2 + g R A 2 l ( 48 − 7 A 2 R ) ω 0 2 + 1 8 A 4 R 2 g 2 [ A 2 R ( 1 32 A 2 R − 15 ) + 84 ] ≥ 0 (42)

Therefore, one obtains the restrictions on the initial angular velocity ω 0 ¯ have the forms

ω 0 2 ≥ 1 128 l A 2 g R ( 7 A 2 R − 48 + 4 3 A 2 R ( A 2 R − 4 ) − 24 ) , (43)

or

ω 0 2 ≤ 1 128 l A 2 g R ( 7 A 2 R − 48 − 4 3 A 2 R ( A 2 R − 4 ) − 24 ) (44)

Beside the first condition in (41), the stability region requires that

ω 0 2 < 1 8 l A 2 g R ( 7 16 A 2 R − 3 ) , with ω 0 2 ≥ 1 128 l A 2 g R ( 7 A 2 R − 48 + 4 3 A 2 R ( A 2 R − 4 ) − 24 ) , (45)

or

ω 0 2 < 1 8 l A 2 g R ( 7 16 A 2 R − 3 ) with ω 0 2 ≤ 1 128 l A 2 g R ( 7 A 2 R − 48 + 4 3 A 2 R ( A 2 R − 4 ) − 24 ) (46)

Based on the above inequalities we can obtain another condition of the stability between the parameters A and R as follows: From (45), one can deduce the following inequality easily

1 16 [ 7 A 2 R − 48 + 4 3 A 2 R ( A 2 R − 4 ) − 24 ] < ( 7 A 2 R 16 − 3 )

Therefore, one gets the stability condition between A and R in the form

A 2 R ( 4 − A 2 R ) > 8 (47)

The motion of a nonlinear oscillating dynamical system is studied. HPT is used to achieve the solution of the governing equation of motion. The graphical representations of the obtained solution are represented for some different values of the physical parameters of the studied system. The numerical results of the governing equation of motion are obtained utilizing the Runge-Kutta method from fourth order and compared with the obtained ones by HPT. The comparison between them reveals high consistency in both results which emphasize the accuracy of the obtained results by HPT. The stability criteria is investigated through a fourth order equation in terms of the initial frequency ω 0 .

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The authors declare that they have no conflict of interest.

Amer, T.S., Galal, A.A. and Elnaggar, S. (2020) The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique. Applied Mathematics, 11, 1081-1099. https://doi.org/10.4236/am.2020.1111073