A simple expansion based HPM for damped nonlinear oscillator

Abstract

A new amplitude expansion-based homotopy perturbation method (AE-HPM) is used to analyze the nonlinear behavior of a damped oscillator. The proposed method employs a simple amplitude expansion concept in addition to the traditional homotopy perturbation method to determine the solution and amplitude frequency relationship for the damped system. Three examples with linear damping are considered to demonstrate the simplicity, efficiency, and effectiveness of the proposed approach. The study considers a quintic oscillator, a strongly nonlinear oscillator characterized by cubic nonlinearity combined with a harmonic restoring force and a microelectromechanical system (MEMS). Analytical results obtained by the AE-HPM reveal that the amplitude of oscillation decays exponentially with the damping parameter, while the nonlinear stiffness terms strongly affect the frequency response. Comparative analysis is carried out for the amplitude with the corresponding numerical results and shows that the proposed method achieves superior accuracy, faster convergence and broader applicability with the absolute error remaining below 0.1 over the entire time interval considered for weak to strong damping effect while the traditional HPM method fails. Therefore, the proposed method provides a simple, powerful and reliable analytical tool for investigating nonlinear damped oscillatory systems.

Keywords

homotopy perturbation method damped nonlinear oscillator amplitude expansion quintic oscillator

1. Introduction

In most real-world problems, oscillatory motion is inevitably followed by the energy dissipation. The damping also alters the dynamic response of the systems thus causing the decays of the amplitude with time and changes that are characterized by the entire system. As a result, damped nonlinear oscillators are a more useful and meaningful model of the system compared to the undamped idealized oscillators. The rate and the role of energy loss is governed by the nature of the damping, both nonlinear and linear, and the nonlinear nature of the functions thus affecting stability of oscillators, frequency response and long-term dynamics. With a diversity of behaviors, such as periodic, quasi-periodic and even chaotic oscillations, possible depending on the level of damping strength and its functional structure.

Complete understanding of damped nonlinear oscillators is significant in the variety of fields of science and engineering. They characterize significant incidence in mechanical and structural systems e.g., vibration suppression, energy dissipation and damping in elastic members. In the same way, they describe nonlinear circuits and coupled oscillatory networks, which decay with time, in electrical and electromechanical systems. Engineering systems having vibration mitigation issues are usually modeled as damped nonlinear oscillatory and the dissipation of energy and behavioral dependence on amplitude are central. To improve structural performance and durability, vibration mitigation is important because too much vibration may lead to noise, exhaustion and loss of functionality. The most popular methods of passive control, including damping treatments and locally resonant structures or periodic structures are also especially desirable because of their simplicity and reliability. Recent observations^1–4 indicate that well considered structural changes can easily absorb vibrational energy in a wide frequency spectrum without reducing the load bearing capacity.

However, the presence of nonlinearity and damping in such systems makes analytical solutions exceedingly difficult. To address this, a variety of approximate analytical and semi-analytical methods have been developed over the last few decades such as the classical perturbation techniques,^5–8 the AG method,^9,10 the harmonic balance based averaging approach,¹¹ the Krylov–Bogoliubov–Mitropolskii (KBM) method,^12–16 the differential transform method,^17–19 Struble’s technique²⁰ and various forms of the homotopy perturbation method (HPM).^21,22 Although the solution obtained by these methods perform satisfactorily for systems with small damping effects and small amplitudes, they face difficulties in accurately representing the behavior of strongly nonlinear oscillators with significant damping. Their applicability is generally confined to weak nonlinearity and the corresponding solution procedures are often complex. Therefore, there remains a necessity for more robust and efficient analytical techniques capable of addressing large amplitude oscillations and significant damping.

The Homotopy Perturbation Method (HPM),²³ introduced in the late 1990s, is a semi-analytical approach for solving nonlinear differential equations that combines homotopy concepts with perturbation techniques while avoiding the need for small physical parameters. Over time, HPM has evolved through several important refinements, notably the Li–He modified HPM,²⁴ which has demonstrated improved convergence and accuracy in complex nonlinear systems such as MEMS²⁵ and vibrating strings. The selection of a good initial guess²⁶ and the introduction of a new enhanced form²⁷ are two important recent improvements of the Homotopy Perturbation Method (HPM). The traditional homotopy perturbation method, in particular, fails completely to represent damping effects.²² To overcome this shortcoming, recently J. H. He introduced three-expansion-based HPM²¹ for damped nonlinear systems.

In this changing paradigm, the amplitude expansion-based HPM (AE-HPM) is an expansion on current work that is more suited to amplitude dependent nonlinear dynamics. One of the most important aspects that determine the success of the HPM and its variations is the choice of a correct initial guess. It has recently been demonstrated that a well-designed initial approximation²⁶ can dramatically improve convergence and accuracy, especially of strongly nonlinear and damped oscillatory systems. The amplitude-expanded zero-order solution in AE-HPM is built to reflect important physical characteristics, such as decay-induced damping and amplitude dependence and therefore, offers a good basis to the higher-order solution.

The current paper introduces the AE-HPM to successfully examine three varieties of nonlinear oscillators with the force of damping. This method combines the idea of the classical homotopy perturbation technique and an expansion procedure in amplitude to provide an effective, precise and easy to implement solution process of the nonlinear system with linear damping.

2. Basic idea of the expansion based HPM

Consider a nonlinear damped oscillator

\ddot{x} + ω_{0}^{2} x + μ \dot{x} + g (x) = 0,

(1)

where dots over the variables represent differentiation with respect to

t

. The initial conditions for the system are given as

x (0) = A, \dot{x} (0) = 0 .

(2)

The homotopy formulation corresponding to Eq. (1) can be expressed as follows:

(1 - p) (\ddot{x} + ω_{0}^{2} x) + p (\ddot{x} + ω_{0}^{2} x + μ \dot{x} + g (x)) = 0 .

(3)

The solution of Eq. (1) is chosen as

x = x_{0} + p x_{1} + p^{2} x_{2} + p^{3} x_{3} + \dots \dots \dots,

(4)

and frequency is chosen as

ω_{0}^{2} = ω^{2} - p ω_{1} - p^{2} ω_{2} - p^{3} ω_{3} + \dots \dots \dots .

(5)

Substituting the series for

x

and

ω_{0}^{2}

into Eq. (3), the zero-order linear Equation is obtained as

{\ddot{x}}_{0} + ω^{2} x_{0} = 0; x_{0} (0) = A, {\dot{x}}_{0} (0) = 0 .

(6)

Due to the presence of damping, the system no longer oscillates with a constant amplitude; instead, its amplitude gradually decays over time. Hence, the solution of Eq. (6) for AE-HPM is expressed through an amplitude expansion as

x_{0} = A (1 + p c_{1} (t) + p^{2} c_{2} (t) + p^{3} c_{3} (t) + \dots \dots \dots) \cos ω t .

(7)

It is obvious that, in the limiting case as $p \to 0$ , the system naturally returns to its undamped state, where $A (1 + p c_{1} (t) + p^{2} c_{2} (t) + p^{3} c_{3} (t) + \dots \dots \dots) \to A$ and $ω \to ω_{0}$ . By substituting Eqs. (4), (5), and (7) into Eq. (3) and expanding in powers of $p$ , the coefficients of identical powers of $p$ are equated on both sides, yielding a system of linear DE for the successive approximations of the solution. The first two equations of this system can be written as follows:

{\ddot{x}}_{1} + ω^{2} x_{1} = A ω_{1} \cos ω t - A {\ddot{c}}_{1} \cos ω t - f (A \cos ω t) + 2 {\dot{c}}_{1} ω A \sin ω t + μ ω A \sin ω t; x_{1} (0) = 0, {\dot{x}}_{1} (0) = 0,

(8)

{\ddot{x}}_{2} + ω^{2} x_{2} = x_{1} ω_{1} - A {\ddot{c}}_{2} \cos ω t - μ {\dot{x}}_{1} + c_{1} ω_{1} A \cos ω t + ω_{2} A \cos ω t - {\dot{c}}_{1} μ A \cos ω t - x_{1} \dot{f} (A \cos ω t) - c_{1} \dot{f} (A \cos ω t) + 2 {\dot{c}}_{2} ω A \sin ω t + c_{1} μ ω Asin ω t; x_{2} (0) = 0, {\dot{x}}_{2} (0) = 0 .

(9)

The system of equations is then solved sequentially to get the values of $x_{1}$ , $x_{2}, \dots$ along with $ω_{1}$ , $ω_{2}, \dots$ . For $p \to 1$ we get the solution of Eq. (1) as

x = x_{0} + x_{1} + x_{2} + x_{3} + \dots \dots \dots,

(10)

and the frequency of the oscillator is

ω^{2} = ω_{0}^{2} + ω_{1} + ω_{2} + ω_{3} + \dots \dots \dots .

(11)

3. Examples

3.1. Exmple-1

Consider a quintic oscillator with linear damping,²⁸ described by

\ddot{x} + ω_{0}^{2} x + μ \dot{x} + q x^{5} = 0 .

(12)

The initial conditions are given as $x (0) = A, \dot{x} (0) = 0$ .

By constructing the homotopy on Eq. (12) and substituting the series for $x$ , $ω_{0}^{2}$ and $x_{0}$ from Eqs. (4), (5) and (7) respectively, then expanding the obtained equation in a power series of $p$ and equating the coefficients of identical powers of $p$ from both sides, we obtain

{\ddot{x}}_{1} + ω^{2} x_{1} = A ω_{1} \cos ω t - A {\ddot{c}}_{1} \cos ω t - q {(A \cos ω t)}^{5} + 2 {\dot{c}}_{1} ω A \sin ω t + μ ω A \sin ω t; x_{1} (0) = 0, {\dot{x}}_{1} (0) = 0,

(13)

{\ddot{x}}_{2} + ω^{2} x_{2} = x_{1} ω_{1} - A {\ddot{c}}_{2} \cos ω t - μ {\dot{x}}_{1} + c_{1} ω_{1} A \cos ω t + ω_{2} A \cos ω t - {\dot{c}}_{1} μ A \cos ω t - 5 q x_{1} {(A \cos ω t)}^{4} - 5 q c_{1} {(A \cos ω t)}^{5} + 2 {\dot{c}}_{2} ω A \sin ω t + c_{1} μ ω A \sin ω t; x_{2} (0) = 0, {\dot{x}}_{2} (0) = 0 .

(14)

Eq. (13) can be expressed as

{\ddot{x}}_{1} + ω^{2} x_{1} = (A ω_{1} - A {\ddot{c}}_{1} - \frac{5}{8} q A^{5}) \cos ω t + (2 {\dot{c}}_{1} ω A + μ ω A) \sin ω t - \frac{5}{16} q A^{5} \cos 3 ω t - \frac{1}{16} q A^{5} \cos 5 ω t; x_{1} (0) = 0, {\dot{x}}_{1} (0) = 0 .

(15)

To remove the secular term from Eq. (15) we choose

A ω_{1} - A {\ddot{c}}_{1} - \frac{5}{8} q A^{5} = 0,

(16)

and

2 {\dot{c}}_{1} ω A + μ ω A = 0 .

(17)

Solving Eq. (16) and Eq. (17) we found

c_{1} = - \frac{μ t}{2},

(18)

ω_{1} = \frac{5}{8} q A^{4} .

(19)

Using the values of $c_{1}$ and $ω_{1}$ in Eq. (15) and solving we get

x_{1} = \frac{5}{128 ω^{2}} q A^{5} (\cos 3 ω t - \cos ω t) + \frac{1}{384 ω^{2}} q A^{5} (\cos 5 ω t - \cos ω t) .

(20)

Now putting the values of $x_{1}, c_{1}$ and $ω_{1}$ in Eq. (14) and solving in a similar manner, we get

c_{2} = \frac{μ t q A^{4}}{48 ω^{2}} + \frac{μ^{2} t^{2}}{8},

(21)

ω_{2} = - \frac{65 q^{2} A^{8}}{1536 ω^{2}} - \frac{5}{4} μ t q A^{4} - \frac{μ^{2}}{4},

(22)

x_{2} = (- \frac{5 q^{2} A^{9}}{3072 ω^{4}} - \frac{25 μ t q A^{5}}{256 ω^{2}}) (\cos 3 ω t - \cos ω t) + (\frac{5 q^{2} A^{9}}{3072 ω^{4}} - \frac{5 μ t q A^{5}}{768 ω^{2}}) (\cos 5 ω t - \cos ω t) + \frac{95 q^{2} A^{9}}{294912 ω^{4}} (\cos 7 ω t - \cos ω t) + \frac{q^{2} A^{9}}{98304 ω^{4}} (\cos 9 ω t - \cos ω t) + \frac{15 q μ A^{5}}{256 ω^{3}} (\sin 3 ω t - 3 \sin ω t) + \frac{5 q μ A^{5}}{2304 ω^{3}} (\sin 5 ω t - 5 \sin ω t) .

(23)

2nd approximate solution can be expressed as

x = A (e^{- \frac{μ t}{2}} - \frac{q A^{4}}{24 ω^{2}} e^{- 3 μ t} - \frac{49 q^{2} A^{8}}{147456 ω^{4}}) \cos ω t + \frac{5 q A^{5}}{128 ω^{2}} (e^{- \frac{5 μ t}{2}} - \frac{q A^{4}}{24 ω^{2}}) \cos 3 ω t + \frac{q A^{5}}{384 ω^{2}} (e^{- \frac{5 μ t}{2}} + \frac{5 q A^{4}}{8 ω^{2}}) \cos 5 ω t + \frac{95 q^{2} A^{9}}{294912 ω^{4}} \cos 7 ω t + \frac{q^{2} A^{9}}{98304 ω^{4}} \cos 9 ω t - \frac{215 q μ A^{5}}{1152 ω^{3}} \sin ω t + \frac{15 q μ A^{5}}{256 ω^{3}} \sin 3 ω t + \frac{5 q μ A^{5}}{2304 ω^{3}} \sin 5 ω t .

(24)

Following the concept of enhanced HPM²⁷ and non-perturbative approach,²⁹ the frequency expression can be expressed as

ω^{2} = ω_{0}^{2} - \frac{μ^{2}}{4} + \frac{5 q A^{4}}{8} e^{- 2 μ t} - \frac{65 q^{2} A^{8}}{1536 ω^{2}} .

(25)

To test the stability, we can write

ω^{2} = ω_{0}^{2} - \frac{μ^{2}}{4} + p (\frac{5 q A^{4}}{8} e^{- 2 μ p t} - \frac{65 q^{2} A^{8}}{1536 ω^{2}}); p \in [0, 1] .

(26)

Using the perturbation concept, we can expand the frequency of the oscillator

ω^{2} = Ω_{0}^{2} + p Ω_{1} + p^{2} Ω_{2} + \dots \dots \dots .

(27)

By a simple calculation we obtain

Ω_{0}^{2} = ω_{0}^{2} - \frac{μ^{2}}{4},

(28)

Ω_{1} = \frac{5 q A^{4}}{8} e^{- 2 μ t} - \frac{65 q^{2} A^{8}}{1536 (ω_{0}^{2} - \frac{μ^{2}}{4})} .

(29)

Now choosing $p = 1$ , we can approximate the frequency

ω = \sqrt{ω_{0}^{2} - \frac{μ^{2}}{4} + \frac{5 q A^{4}}{8} e^{- 2 μ t} - \frac{65 q^{2} A^{8}}{1536 (ω_{0}^{2} - \frac{μ^{2}}{4})}} .

(30)

The stability conditions for the solution are thus $μ > 0$ and $ω_{0}^{2} - \frac{μ^{2}}{4} + \frac{5 q A^{4}}{8} e^{- 2 μ t} - \frac{65 q^{2} A^{8}}{1536 (ω_{0}^{2} - \frac{μ^{2}}{4})} > 0 .$

3.2. Example-2

To verify the accuracy and effectiveness of the present AE-HPM, consider another strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force with linear damping³⁰ as

\ddot{x} + ω_{0}^{2} x + μ \dot{x} + a x^{3} + b \sin x = 0,

(31)

subject to the initial conditions

x (0) = A, \dot{x} (0) = 0

By expanding the sine term using the Maclaurin series, the governing Eq. (31) can be approximated as

\ddot{x} + ω_{0}^{2} x + μ \dot{x} + a x^{3} + b (x - \frac{x^{3}}{6} + \frac{x^{5}}{120}) = 0 .

(32)

The present AE-HPM is then employed to obtain an analytical solution for this strongly nonlinear system and, obtain

x (t) = A k_{1} \cos ω t + k_{3} \cos 3 ω t + k_{5} \cos 5 ω t + k_{7} \cos 7 ω t + k_{9} \cos 9 ω t + l_{1} \sin ω t + l_{3} \sin 3 ω t,

(33)

with

k_{1} = e^{- \frac{μ t}{2}} - \frac{a A^{2}}{32 ω^{2}} e^{- 2 μ t} + \frac{b A^{2}}{192 ω^{2}} e^{- 2 μ t} - \frac{b A^{4}}{2880 ω^{2}} e^{- 3 μ t} - \frac{a^{2} A^{4}}{1024 ω^{4}} + \frac{a b A^{4}}{3072 ω^{4}} - \frac{b^{2} A^{4}}{3686 ω^{4}} - \frac{19 a b A^{6}}{1474560 ω^{4}} + \frac{19 b^{2} A^{6}}{8847360 ω^{4}} - \frac{49 b^{2} A^{8}}{2123366400 ω^{4}},

(34)

k_{3} = \frac{a A^{3}}{32 ω^{2}} e^{- \frac{3 μ t}{2}} - \frac{b A^{3}}{192 ω^{2}} e^{- \frac{3 μ t}{2}} + \frac{b A^{5}}{3072 ω^{2}} e^{- \frac{5 μ t}{2}} - \frac{a b A^{7}}{9830 ω^{4}} + \frac{b^{2} A^{7}}{589824 ω^{4}} - \frac{b^{2} A^{9}}{8847360 ω^{4}},

(35)

k_{5} = \frac{b A^{5}}{46080 ω^{2}} e^{- \frac{5 μ t}{2}} + \frac{a^{2} A^{5}}{1024 ω^{4}} - \frac{a b A^{5}}{3072 ω^{4}} + \frac{b^{2} A^{5}}{36864 ω^{4}} + \frac{31 a b A^{7}}{147450 ω^{4}} - \frac{31 b^{2} A^{7}}{8847360 ω^{4}} + \frac{b^{2} A^{9}}{8847360 ω^{4}},

(36)

k_{7} = \frac{a b A^{7}}{491520 ω^{4}} - \frac{b^{2} A^{7}}{2949120 ω^{4}} + \frac{19 b^{2} A^{9}}{849346560 ω^{4}},

(37)

k_{9} = \frac{b^{2} A^{9}}{1415577600 ω^{4}},

(38)

l_{1} = - \frac{9 a A^{3} μ}{128 ω^{3}} + \frac{3 b A^{3} μ}{256 ω^{3}} - \frac{43 b A^{5} μ}{27648 ω^{3}},

(39)

l_{3} = \frac{b A^{5} μ}{55296 ω^{3}},

(40)

and the frequency

ω

can be expressed as

ω = \sqrt{ω_{0}^{2} + b - \frac{μ^{2}}{4} + \frac{3 a A^{2}}{4} e^{- μ t} - \frac{b A^{2}}{8} e^{- μ t} + \frac{b A^{4}}{192} e^{- 2 μ t} + ω_{s}},

(41)

where

ω_{s} = - \frac{3 a^{2} A^{4}}{128 (ω_{0}^{2} - \frac{μ^{2}}{4})} + \frac{a b A^{4}}{128 (ω_{0}^{2} - \frac{μ^{2}}{4})} - \frac{b^{2} A^{4}}{1536 (ω_{0}^{2} - \frac{μ^{2}}{4})} - \frac{a b A^{6}}{1920 (ω_{0}^{2} - \frac{μ^{2}}{4})} + \frac{b^{2} A^{6}}{11520 (ω_{0}^{2} - \frac{μ^{2}}{4})} - \frac{13 b^{2} A^{8}}{4423680 (ω_{0}^{2} - \frac{μ^{2}}{4})} .

(42)

The stability conditions for the solution are thus $μ > 0$ and $ω^{2} > 0 .$

3.3. Example-3

Consider the following dimensionless damped microelectromechanical system (MEMS) oscillator³¹:

\ddot{x} + ω_{0}^{2} x + μ \dot{x} - \frac{K}{1 - x} = 0, x < 1,

(43)

subject to the initial conditions

x (0) = \dot{x} (0) = 0 .

(44)

To facilitate the analytical treatment, we introduce a transformation analogous to that proposed by He et al.,³¹ defined as

x (t) = A - u (t),

(45)

where

A < 1

denotes the oscillation amplitude of

u (t)

. The parameter

A

is determined from the condition as

A = \frac{ω_{0}^{2} - \sqrt{ω_{0}^{4} - 4 K ω_{0}^{2}}}{2 ω_{0}^{2}},

(46)

subject to the constraint

K < \frac{ω_{0}^{2}}{4}

. This requirement is consistent with the pull-in stability condition

K < K^{*}

for a prescribed value of

ω_{0}

. Substituting the transformation given in Eq. (45) into Eq. (43), while incorporating the relation obtained in Eq. (46), leads to the following governing equation expressed in terms of

u (t)

\ddot{u} + ω_{0}^{2} u + μ \dot{u} - \frac{K u}{(1 - A) (1 - A + u)} = 0,

(47)

with the associated initial conditions

u (0) = A, \dot{u} (0) = 0 .

(48)

Applying the present method, the solution is obtained in the form

x (t) = A - A (e^{- \frac{μ t}{2}} + \frac{r_{3}}{8 A ω^{2}}) \cos ω t + \frac{r_{3}}{8 ω^{2}} \cos 3 ω t,

(49)

where the frequency

ω

is obtained by

ω = \sqrt{ω_{0}^{2} - \frac{r_{1}}{A}},

(50)

and the coefficients

r_{1}

and

r_{3}

are expressed as

r_{1} = \frac{2 K (\sqrt{2 A - 1} (π - (π + 2) A) - 4 {(A - 1)}^{2} \tan^{- 1} (\sqrt{2 A - 1}))}{π A (A - 1) \sqrt{2 A - 1}},

(51)

r_{3} = - \frac{2 K (2 A (11 A^{2} - 24 A + 12) + 3 π (3 A^{3} - 11 A^{2} + 12 A - 4))}{3 π A^{3} (A - 1)} - \frac{8 K (A - 1) (A^{2} - 8 A + 4) \tan^{- 1} (\sqrt{2 A - 1})}{π A^{3} \sqrt{2 A - 1}} .

(52)

The stability conditions are similar as before and are given by $μ > 0$ and $ω^{2} > 0 .$

4. Results and discussions

Several figures are presented below to demonstrate the accuracy of the AE-HPM. The results obtained by this method are compared with the corresponding numerical solutions derived using the classical fourth-order Runge–Kutta scheme. The figures illustrate the time-domain response, phase-plane behavior and the influence of the damping parameter $μ$ on stability for various parameter values of the oscillators considered.

Figures 1 and 2 correspond to the parameter set $A = 0.1, ω_{0} = 1.0, μ = 0.05, q = 1.0$ . Figure 1 shows the displacement–time response, while Figure 2 presents the associated phase-plane trajectory for the quintic oscillator. In both cases, the AE-HPM results exhibit excellent agreement with the numerical solutions, confirming the present method’s reliability and precision.

Figure 1.

Comparison of the present solution with corresponding numerical solution of Eq. (12) for the parameter values $A = 0.1, ω_{0} = 1.0, μ = 0.05, q = 1.0$ .

Figure 2.

Comparison of the phase plane obtained by present method and numerical method (RK4) of Eq. (12) for the parameter values $A = 0.1, ω_{0} = 1.0, μ = 0.05, q = 1.0$ .

For Figures 3 and 4, the parameters are chosen as $A = 0.5, ω_{0} = 2.0, μ = 0.25, q = 1.0$ for Eq. (12). The larger value of damping in this case causes the amplitude to decay more rapidly with time than the previous case. This case demonstrates that the AE-HPM is efficient to capture the nonlinear system’s dynamics with high accuracy, maintaining strong consistency with the Runge–Kutta solutions.

Figure 3.

Comparison of the present solution with corresponding numerical solution of Eq. (12) for the parameter values $A = 0.5, ω_{0} = 2.0, μ = 0.25, q = 1.0$ .

Figure 4.

Comparison of the phase plane obtained by present method and numerical method (RK4) of Eq. (12) for the parameter values $A = 0.5, ω_{0} = 2.0, μ = 0.25, q = 1.0$ .

Figures 5 and 6 correspond to $A = 1.0, ω_{0} = 4.0, μ = 0.5, q = 1.0$ for Eq. (12), representing a further increase in damping strength and amplitude value. As expected, the oscillations diminish faster than previous two cases and once again, the AE-HPM solutions show remarkable coincidence with the numerical results. These comparisons confirm that the amplitude AE-HPM provides a simple, powerful and accurate analytical tool for analyzing damped nonlinear oscillatory systems.

Figure 5.

Comparison of the present solution with corresponding numerical solution of Eq. (12) for the parameter values $A = 1.0, ω_{0} = 4.0, μ = 0.5, q = 1.0$ .

Figure 6.

Comparison of the phase plane obtained by present method and numerical method (RK4) of Eq. (12) for the parameter values $A = 1.0, ω_{0} = 4.0, μ = 0.5, q = 1.0$ .

In Figure 7, the parameters are chosen as $A = 1.5, ω_{0} = 3.0, μ = 0.5, a = 0.5, b = 1.0$ for Eq. (31). The AE-HPM results show excellent agreement with the numerical solutions, demonstrating the reliability and accuracy of the proposed method.

Figure 7.

Comparison of the present solution with corresponding numerical solution of Eq. (31) for the parameter values $A = 1.5, ω_{0} = 3.0, μ = 0.5, a = 0.5, b = 1.0$ .

In Figures 8 and 9 correspond to the damped microelectromechanical system for the parameter sets $ω_{0} = 1.0, μ = 0.10, K = 0.05$ and $ω_{0} = 1.0, μ = 0.25, K = 0.05$ respectively. In both cases, the AE-HPM solution exhibit excellent agreement with the numerical results, thereby confirming the accuracy and reliability of the proposed method.

Figure 8.

Comparison of the present solution with corresponding numerical solution of Eq. (43) for the parameter values $ω_{0} = 1.0, μ = 0.10, K = 0.05$ .

Figure 9.

Comparison of the present solution with corresponding numerical solution of Eq. (43) for the parameter values $ω_{0} = 1.0, μ = 0.25, K = 0.05$ .

Finally, Figure 10 presented to show the stability performance concerns of the restriction discussed earlier. The figure plots the frequency response against the initial amplitude. The figure depicts the frequency response as a function of the initial amplitude, highlighting how the condition $ω^{2} > 0$ governs the system behavior for different values of the damping parameter $μ$ and the parameters $a$ and $b$ . The darker portions above each curve display stable states symbolized by S, whereas the white regions below the outline denote unstable region symbolized by U. The schematic representation clearly distinguishes between stable and unstable regions depending on the initial amplitude.

Figure 10.

Depicts the behavior of the damping parameter $μ$ on the stability of Eq. (31) for ω₀ = 2.0, a = 1.0, b = 0.5.

5. Conclusion

The AE-HPM has been used successfully in the current research to study dynamic behavior of three different damped oscillators including the quintic nonlinear oscillator under linear damping. The technique offers approximate solutions for the dynamics of the oscillators considered and their corresponding phase-plane responses, showing excellent agreement with the numerical solutions obtained by using the fourth-order Runge–Kutta method. The comparison with numerical results for different sets of parameters demonstrates that the AE-HPM effectively captures both the amplitude response over time and the nonlinear characteristics of the damped oscillator. Even if we increase the damping coefficient and amplitude ( $A \leq 1$ ), the proposed approach maintains high accuracy, confirming its robustness and reliability. Overall, the obtained results validate that the AE-HPM offers a simple, efficient and accurate analytical framework for studying the nonlinear oscillators with damping effects. The present approach can be extended to other classes of nonlinear dynamical systems, providing valuable insights into their response and stability characteristics. The AE-HPM can be extended to other nonlinear oscillatory systems. It may be applied to MEMS oscillators, as well as to damped and parametrically excited systems such as the Mathieu cubic–quintic Duffing oscillator.

Footnotes

ORCID iDs

Nazmul Sharif

Helal Uddin Molla

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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