Dynamical analysis of a vertical excited pendulum using He’s perturbation method

Abstract

The current paper investigates the nonlinear dynamics of an excited pendulum by a crank-shaft-slider mechanism (CSSM), in which it is restricted to move vertically. He’s method of homotopy is employed to obtain the analytical solution of the governing differential equation. The numerical solution of this equation is obtained using the fourth-order Runge-Kutta method (RKM). The comparison between both solutions demonstrates a high level of consistency, which confirms the precision of the gained analytic solution. The graphical representations of these solutions are presented to reveal the influence of various parameters on the investigated motion.

Introduction

Prototypes of pendulums are highly useful e.g.^1,2 for investigational justifications in real-world engineering. Engineers also do great efforts to comprehend the mechanics of gravity, centripetal force, and inertia. The oscillatory movement of a pendulum has become one of the most examined movements in applied physics and engineering. The exact solution for a lot of systems seems to be highly complicated and, at times, unreachable^3,4 As a result, asymptotic solutions have piqued the interest of several scientists to handle a variety of nonlinear equations, like the approaches of Lindstedt-Poincaré (LP) and multiple scales (MS) ⁵ have numerous benefits when it comes to obtaining vibratory system solutions.^6–8

Also, there was the homotopy perturbation method (HPM), which dates to Ji-Huan He ⁹ and is not dependent on a minor parameter, unlike LP and MS methods. In, ¹⁰ the authors investigated the dynamical motion of a pendulum connected to a rigid rotating frame that rotates about the vertical axis. The stability has been studied, in addition to emphasizing the accuracy of the solutions by HPM. In, ¹¹ the HPM is applied to obtain the analytic approximate solution of the governing equations. The solutions were estimated in a series form. Clearly, these solutions and their behaviors depend upon the initial conditions. In, ¹² time-delayed velocity and position were considered for the investigation of an excited oscillator of Van der Pol-Duffing. The HPM and its modification were utilized to deduce an approximate solution for the studied system. It was concluded that the modification of HPM produces better results. The authors followed their studies to examine the movement of a basic pendulum that is attached to a wheel and a thin spring in. ¹³ The HPM and Laplace transforms were combined with an extended frequency to produce a regular approximate solution. Moreover, the fourth-order RKM was used to confirm the gained solution. The analytical solution for the motion of a magnetic spherical pendulum was achieved in ¹⁴ using Laplace transforms and HPM. The authors used the procedure of linearized stability to investigate stability around the fixed points. In, ¹⁵ the authors used the HPM to locate precise solutions for different forms in linear and nonlinear differential equations (DEs), while in ¹⁶ a sliding bead problem in a smooth parabola was examined using the same approach to obtain the desired solution.

The MS is absorbed in ¹⁷ to provide a modification of the HPM. Particularly for nonlinear oscillators, this adjustment is effective. The obtained solution was developed in accordance with a periodic structure for its amplitude. In, ¹⁸ the author looked at the behavior of a cubic nonlinear Duffing’s equation with time delay. To overcome the difficulty of this problem, a combination between MS and HPM was used. Two solvability criteria of nonlinear first-order DEs were achieved, one is cubic and the other is a quantic in slow and slower variables, respectively. The periodic solution of Mathieu equation time delay using HPM a version of timescales was examined in. ¹⁹ Periodic structure is generated for zero-order amplitude, in which three-time scales are taken into account along with the stability analysis. For both harmonic and non-harmonic resonance cases, the approximate periodic solutions were determined to the second perturbations’ precision.

In ²⁰ a power series solution is obtained for time less than a quarter of the period. The non-perturbative approach backs to El-Dib^21,22 to gain the simplest solution to different problems. For example, to obtain the solutions, of non-conservative Gaylord’s oscillator. The idea of this approach depends on obtaining an analogous linear system, in which it has the benefit of having coefficients that are simple to compute and that consider the impact of the original nonlinear factors. Even with periodic pressures present, this linearization procedure yields nearly precise results. Therefore, it is simple to demonstrate the correlations between frequencies and amplitudes. It must be noted that the correlation between the frequency and amplitude of the mentioned model in ²¹ was calculated using the Bessel function. Comparison between the analytical results and the numerical ones; exhibits good approbation and validates the accuracy of the analytical results.

In, ²³ the He-Laplace method is used to deduce the solutions of the oscillating system whereas it is solved with help of the variational iteration method with Laplace transformations in. ²⁴ Authors ²⁵ used a modification of HPM to get the analytical solution of some nonlinear oscillators. The HPM is applied to get amplitude-frequency relationship of the studied oscillator ²⁶ used to produce nanofibers. In, ²⁷ HPM is used to have the approximate solution of nonlinear oscillators. The energy approach is combined with Li-HPM ²⁸ to get the approximate solution of the variating system. In the study, ²⁹ the investigated system is solved by using the Li-He’s technique. The Hamilton principle establishes a governing equation for string axial vibrations with temporal and spatial damping forces. ³⁰ The fractal duffing oscillators^31–33 are solved with helping fractal derivatives, and HPM.

The literature has extensively investigated the classical parametrically excited pendulum, and the harmonic excitation of the pendulum confirmed may cause distinct types of a simple pendulum motion. This work focuses on a new version of the kinetics of a vertically excited parametric pendulum by a CSSM. In the mathematical model created for this paper, the CSSM is examined as an extension of the standard parametric pendulum given in.^34–37 The HPM is used in this work to find an asymptotic solution to the equation that governs motion. The numerical solution of the original equation is obtained by using the fourth-order RKM. When comparing the two solutions, it is clear that there is a great degree of consistency between them, which supports the accuracy of the acquired analytical solution. To demonstrate the impact of different factors on the motion under investigation, graphical representations of these solutions are provided.

Description of the problem

The goal of the current section is to obtain the equation of motion (EOM) of an excited pendulum. It is assumed that its suspension point moves vertically on the support by the CSSM, see Figure 1. The mechanism under consideration here is a rigid weightless rod of length $\mathrm{l}$ connected at the end with a mass $m$ . It is vertically excited by a CSSM. A weightless rod of length $a$ is considered to move with a stationary angular speed, while a link $b$ is used to move the suspension point of the pendulum up and down. In summary, the single mass $m$ under consideration is at the extreme of the pendulum.

Figure 1.

The dynamical model.

Let $\alpha$ is rotation angle of the pendulum that represents its generalized coordinate. To write down the system’s Lagrangian function, the coordinates of the mass can be expressed in the forms

$\begin{array}{l}x=|\sin \alpha ,\\ y=-a\cos \theta -b\cos \varphi -|\cos \alpha ,\end{array}$

(1)

where $\theta$ and $\varphi$ are the angles between $OB$ and both of $OA$ and $BA$ , respectively, as shown in Figure 1.

Making use of the trigonometric relationship between $\theta$ and $\phi$ , one can write the following form of $\cos \varphi$

$\cos \varphi ={\left(1-\frac{a^2}{b^2}{\sin }^2\theta \right)}^{1/2}.$

(2)

From 1 and 2, one can write the first derivative with respect to time $t$ as follows

$\begin{array}{l}\dot{x}=\ell \dot{\alpha }\cos \alpha ,\\ \dot{y}=a\dot{\theta }\sin \theta +\frac{a^2\dot{\theta }\sin \theta \cos \theta }{b{\left(1-\frac{a^2}{b^2}{\sin }^2\theta \right)}^{1/2}}+\ell \dot{\alpha }\sin \alpha \end{array}$

Therefore, one can write

$\dot{y}=a\dot{\theta }\left(\sin \theta +\frac{a\sin \theta \cos \theta }{b{\left(1-\frac{a^2}{b^2}{\sin }^2\theta \right)}^{1/2}}\right)+\ell \dot{\alpha }\sin \alpha$

(3)

Introducing a dimensionless variable $F$ depending on $\theta$ ³⁸ as follows

$F=\sin \theta +\frac{a\sin \theta \cos \theta }{b{\left(1-\frac{a^2}{b^2}{\sin }^2\theta \right)}^{1/2}},$

(4)

Substituting equation 4 into equation 3 it is easy to write:

$\begin{array}{l}\dot{x}=\ell \dot{\alpha }\cos \alpha ,\\ \dot{y}=a\dot{\theta }F+\ell \dot{\alpha }\sin \alpha .\end{array}$

(5)

The kinetic energy $T$ and potential energy $V$ can be computed using the information provided above as follows

$\begin{array}{l}T=\frac{1}{2}m\left[{\ell }^2{\dot{\alpha }}^2{\cos }^2\alpha +{\left(a\dot{\theta }F+\ell \dot{\alpha }\sin \alpha \right)}^2\right],\\ V=-mg\ell \cos \alpha -mgb{\left(1-\frac{a^2}{b^2}{\sin }^2\theta \right)}^{1/2}-mga\cos \theta ,\end{array}$

(6)

where $g$ denotes gravitational acceleration and dots the derivative regarding to time $t$ .

Lagrange’s equation of the examined conservative system has the form

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\alpha }}\right)-\frac{\partial L}{\partial \alpha }=G_{\alpha }^{NC}=-c{\ell }^2\dot{\alpha },$

(7)

where $L=T-V$ , $G_{\alpha }^{NC}$ is the generalized force and it has a dimension of moment, c is the coefficient of friction.

Hence, the EOM may be reduced to

$m\ell \ddot{\alpha }+ma\dot{\theta }\dot{F}\sin \alpha +mg\sin \alpha =-c\ell \dot{\alpha }.$

(8)

Now, let us insert a dimensionless time $\tau$ according to the transformation $\tau ={\omega }_{0}t$ where ${\omega }_0=\sqrt{g/\ell }$ . Therefore, on can write

$\begin{array}{l}\dot{\alpha }={\omega }_0{\alpha }^{\prime },\ddot{\alpha }={\omega }_0^2{\alpha }^{″},\\ \dot{F}={\omega }_0{F}^{\prime },\dot{\theta }={\omega }_0{\theta }^{\prime }={\omega }_0\omega ,\end{array}$

(9)

where the prime denotes to differentiation regarding $\tau$ .

Referring to the above, the EOM in 8 can be rewritten as follows

${\alpha }^{″}+(1+\frac{a}{\ell }\omega F^{\prime })\sin \alpha =0.$

(10)

Method of solution

This section’s purpose is to use the HPM to find the EOM’s asymptotic approximate solution. To realize this aim, Taylor’s series can be used to approximate $\sin \theta$ as follows

$\sin \alpha =\alpha -\frac{{\alpha }^3}{3!}+\frac{{\alpha }^5}{5!}-....$

(11)

Making use of 10 and 11 to get

${\alpha }^{″}+s\left(\alpha -\frac{{\alpha }^3}{3!}+\frac{{\alpha }^5}{5!}-...\right)=0.$

(12)

where

$s=1+\frac{a}{\ell }\omega F^{\prime }.$

(13)

Assuming that $\omega F^{\prime }$ is a constant. Consequently, $s$ is an arbitrary const.

Considering the following initial conditions of equation 12 as below

$\alpha \left(0\right)=1,{\alpha }^{\prime }\left(0\right)=0.$

(14)

According to the procedure of HPM, one can write

$H\left(\alpha ,\rho \right)=\left(1-\rho \right)\left[L\left(\alpha \right)-L\left(U\right)\right]+\rho \left[L\left(\alpha \right)+N\left(\alpha \right)\right]=0,\rho \in \left[\mathrm{0,1}\right]$

(15)

where $\rho$ is a homotopy parameter, $L\left(\alpha \right)$ and $N\left(\alpha \right)$ are the linear and nonlinear parts of equation 12, in which they may be expressed as follows

$L\left(\alpha \right)={\alpha }^{″}+s\alpha ,$

(16)

$N\left(\alpha \right)=s\left(-\frac{{\alpha }^3}{3!}+\frac{{\alpha }^5}{5!}\right),$

(17)

The substitution of 16, 17 into 15 yields

$H\left(\alpha ,\rho \right)=\left(1-\rho \right)\left[{\alpha }^{″}+s\alpha -U^{″}-sU\right]+\rho \left[{\alpha }^{″}+s\alpha +s\left(-\frac{{\alpha }^3}{3!}+\frac{{\alpha }^5}{5!}\right)\right]=0.$

(18)

At $U=0,$ the above equation has the form

$H\left(\alpha ,\rho \right)={\alpha }^{″}+s\alpha +\rho s\left(-\frac{{\alpha }^3}{3!}+\frac{{\alpha }^5}{5!}\right)=0.$

(19)

Let us express the solution of this equation as a power series of $\rho$ as follows

$\alpha ={\alpha }_0\left(t\right)+\rho {\alpha }_1\left(t\right)+{\rho }^2{\alpha }_2\left(t\right)+{\rho }^3{\alpha }_3\left(t\right)$

(20)

Substituting the series 20 into equation 19, then equating coefficients of various powers of $\rho$ to get

Order of ${\rho }^0$ :

${{\alpha }_0}^{″}+s{\alpha }_0=0.$

(21)

Order of $\rho$ :

${{\alpha }_1}^{″}+s{\alpha }_1-s\frac{{\alpha }_0^3}{3!}+s\frac{{\alpha }_0^5}{5!}=0.$

(22)

Order of ${\rho }^2$ :

${{\alpha }_2}^{″}+s{\alpha }_2+\frac{1}{24}s{\alpha }_0^4{\alpha }_1-\frac{1}{2}s{\alpha }_0^2{\alpha }_1=0.$

(23)

If the following criteria are met, then the prior Equations 21–23 can be solved

$\begin{array}{l}{\alpha }_0\left(0\right)=1,{\alpha }_0^{\prime }\left(0\right)=0,\\ {\alpha }_1\left(0\right)=0,{\alpha }_1^{\prime }\left(0\right)=0,\\ {\alpha }_2\left(0\right)=0,{\alpha }_2^{\prime }\left(0\right)=0,\end{array}$

(24)

to obtain

${\alpha }_0=\cos \left(\tau \sqrt{s}\right),$

(25)

${\alpha }_1=\frac{1}{23040}\sin \left(\tau \sqrt{s}\right)\left[1380\tau \sqrt{s}+224\sin \left(2\tau \sqrt{s}\right)-\sin \left(4\tau \sqrt{s}\right)\right],$

(26)

$\begin{array}{l}{\alpha }_2=\frac{1}{4246732800}[\left(2366782-7617600s\hspace{0.17em}{\tau }^2\right)\cos \left(\sqrt{s}\hspace{0.17em}\tau \right)-2477280\cos \left(3\sqrt{s}\hspace{0.17em}\tau \right)\\ +111840\cos \left(5\sqrt{s}\hspace{0.17em}\tau \right)-1345\cos \left(7\sqrt{s}\hspace{0.17em}\tau \right)+3\cos \left(9\sqrt{s}\hspace{0.17em}\tau \right)+10058400\sqrt{s}\hspace{0.17em}\tau \sin \left(\sqrt{s}\hspace{0.17em}\tau \right)\\ -3726000\sqrt{s}\hspace{0.17em}\tau \sin \left(3\sqrt{s}\hspace{0.17em}\tau \right)+27600\sqrt{s}\hspace{0.17em}\tau \sin \left(5\sqrt{s}\hspace{0.17em}\tau \right)].\end{array}$

(27)

Substituting (26)–(28) into (22) to obtain the required solution which is consistent with the HPM

$\begin{array}{l}\alpha =\underset{\rho \to 1}{\lim }\alpha \left(\rho ,\tau \right)=\cos \left(\tau \sqrt{s}\right)+\frac{1}{23040}\sin \left(\tau \sqrt{s}\right)[1380\tau \sqrt{s}+224\sin \left(2\tau \sqrt{s}\right)\\ -\sin \left(4\tau \sqrt{s}\right)]+\frac{1}{4246732800}[\left(2366782-7617600s\hspace{0.17em}{\tau }^2\right)\cos \left(\sqrt{s}\hspace{0.17em}\tau \right)\\ -2477280\cos \left(3\sqrt{s}\hspace{0.17em}\tau \right)+111840\cos \left(5\sqrt{s}\hspace{0.17em}\tau \right)-1345\cos \left(7\sqrt{s}\hspace{0.17em}\tau \right)+3\cos \left(9\sqrt{s}\hspace{0.17em}\tau \right)\\ +10058400\sqrt{s}\hspace{0.17em}\tau \sin \left(\sqrt{s}\hspace{0.17em}\tau \right)-3726000\sqrt{s}\hspace{0.17em}\tau \sin \left(3\sqrt{s}\hspace{0.17em}\tau \right)+27600\sqrt{s}\hspace{0.17em}\tau \sin \left(5\sqrt{s}\hspace{0.17em}\tau \right)].\end{array}$

(28)

The drawn curves of Figure 2 are calculated when $a\left(=\mathrm{0.5,1,2}\right)$ and reveals the temporal history of the solutions (31). These curves behave progressive waves with an increasing of the amplitude during the investigated time interval. The amplitudes of different waves increase with the increase of $a$ values. The corresponding curves of the solution in the plane $\left(\alpha -\dot{\alpha }\right)$ are represented in Figure 3, which describe the steady behaviors of these curves at the considered values of Figure 2.

Figure 2.

Represents the homotopy Analytic solution at $a\left(=\mathrm{0.5,1,2}\right)$ .

Figure 3.

Represents the curves of phases in the plane $\left(\alpha -\dot{\alpha }\right)$ at the values of Figure 2.

Discussion and Results

This section emphasizes the high degree of accuracy of the HPM results by contrasting them with numerical (NUM) ones generated by using the fourth-order RKM.

To compare the obtained results by using the RKM and HPM, Tables 1–3 are presented to show the error between them. For the same comparable values of $a$ , the given results in these tables match with the curves in Figures 4–6.

Table 1.

The percentage error of HPM at $a=0.5$ .

Time	Numerical results	HPM results	$\left|\frac{HPM-NV}{NV}\right|$
0	1	1	0
1	−0.0546794	−0.0553409	0.012098
2	−0.994524	−0.995946	0.00142939
3	0.163328	0.163639	0.00190623
4	0.978137	0.990184	0.0123169
5	−0.269863	−0.264594	0.0195275
6	−0.950954	−0.996447	0.0478393
7	0.372934	0.35643	0.044254
8	0.91318	1.02437	0.121764
9	−0.471269	−0.44942	0.0463627
10	−0.865109	−1.0725	0.239731

Table 2.

The percentage error of HPM at $a=1$ .

Time	Numerical results	HPM results	$\left|\frac{HPM-NV}{NV}\right|$
0	1	1	0
1	−0.510575	−0.511338	0.0014958
2	−0.503852	−0.507769	0.00777467
3	0.999971	1.00744	0.0074735
4	−0.517265	−0.513899	0.00650564
5	−0.497096	−0.546238	0.0988574
6	0.999886	1.07593	0.0760572
7	−0.523921	−0.507931	0.0305208
8	−0.490308	−0.656591	0.33914
9	0.999744	1.2954	0.295727
10	−0.530544	−0.584202	0.101138

Table 3.

The percentage error of HPM at $a=2$ .

Time	Numerical results	HPM results	$\left|\frac{HPM-NV}{NV}\right|$
0	1	1	0
1	−0.948752	−0.949142	0.000411774
2	0.798638	0.799297	0.000824708
3	−0.561673	−0.558623	0.00543005
4	0.260424	0.240618	0.076051
5	0.0717943	0.136085	0.895481
6	−0.395393	−0.5481	0.386218
7	0.672607	0.967936	0.439081
8	−0.87448	−1.36496	0.560881
9	0.983185	1.70674	0.735925
10	−0.990557	−1.96069	0.979386

Figure 4.

Illustrates the difference between the HPM and NUM solution at $a=0.5$ .

Figure 5.

Shows the variation of $\alpha \left(t\right)$ for the HPM and NUM at $a=1$ .

Figure 6.

Describes the plotted curves by HPM and NUM at $a=2$ .

In Figures 4–6, the displayed curves are calculated when $F^{\prime }=0.6,\omega =5,\mathrm{l}=0.75$ with the various values of $a=\left(\mathrm{0.5,1,2}\right)$ . The purpose of these curves is to show the difference between the approximate analytical solutions of equation of 28 (with red color) and the numerical results equation 12 (with blue color).

Conclusion

The asymptotic analytic solution of the EOM of an excited pendulum subjected to CSSM has been obtained using the HPM. The fourth-order RKM is used to achieve the numerical solution of the original governing EOM. These solutions have been compared through some graphical plots. The feedback observation is that high matching between them, which demonstrates the great accuracy of the obtained analytic solutions. The phase plane graphs of the gained solutions that have forms of closed symmetric curves shows the stability of these solutions.