The vibration control study of a coupling beam system attached to NESs

Abstract

This study investigates the influence of nonlinear energy sinks (NESs) on the forced transverse vibration of a primary system, where the primary system is composed of two beams, boundary-supporting springs, internal supporting springs, and an elastic coupling element. The forced transverse vibration of the primary system with NESs is predicted by the Galerkin truncation method (GTM), where the reliability of the transverse vibration responses calculated by the GTM is verified by the harmonic balance method (HBM). On this basis, the influence of NESs on the forced transverse vibration of the primary system is investigated. Forced transverse vibration of the primary system with NESs is significantly influenced by the structural parameters of NESs, including their nonlinear stiffness, viscous damping, and motion mass. The complex dynamic behavior of the primary system with NESs appears in its forced transverse vibration. Under certain parameters of NESs, the targeted energy transfer phenomenon appears between the primary system and NESs. Furthermore, the influence of viscous damping of NESs on single-frequency responses of the primary system with NESs is monotonous, while the influence of nonlinear stiffness and motion mass of NESs on single-frequency responses of the primary system with NESs is nonmonotonic. A suitable combination of nonlinear stiffness and motion mass of NESs beneficially influences vibration reduction of the primary system.

Keywords

double beams forced vibration nonlinear energy sink Galerkin truncation method

Introduction

In engineering, various complex structures, such as bridges,^1–3 shafts,^4,5 and blades,^6,7 among others, are composed of beams. For most engineering applications, beams typically suffer from external excitation introduced by power equipment, the working environment, and other sources. Such external excitation can cause unexpected vibrations in beams, which may lead to serious accidents. To suppress the unwanted vibration of beams, a good understanding of the vibration characteristics of beam structures is necessary.^8–11 Then, representative vibration control devices have been proposed to suppress the vibration of beam structures.¹²

With the development of engineering, engineers noticed the nonlinear vibration of elastic structures.¹³ Then, engineers attempted to utilize nonlinear factors to suppress unwanted vibration. Gendelman et al.¹⁴ and Vakakis and Gendelman¹⁵ proposed a type of nonlinear mechanical oscillator and studied its energy-pumping characteristics. Vakakis¹⁶ established a vibration analysis model of a linear structure with a local nonlinear attachment and studied its energy pumping. Jiang et al.¹⁷ and Kerschen et al.^18,19 investigated nonlinear energy pumping in coupled oscillators experimentally and theoretically, where the targeted energy transfer phenomenon of nonlinear oscillators was discovered. Vakakis²⁰ then established the vibration analysis model of a linear lattice with a local essentially nonlinear attachment and studied its relaxation oscillations, subharmonic orbits, and chaos. Sapsis et al.²¹ investigated the energy transmissibility characteristics of structures with strong nonlinearity. Al-Shudeifat et al.²² made a kind of rotating nonlinear energy sink (NES) and studied its vibration characteristics. Vakakis²³ deeply studied and explained the physical meaning of the targeted energy transfer phenomenon of NES. Zhang²⁴ investigated the NES with an inerter and discussed the influence of the inerter on the vibration absorption of NES. Qiu et al.^25–27 employed conical and pitch springs to realize the cubic stiffness of NES, providing a design idea for NES. Sun et al.²⁸ generalized and modified the Equal-peak method for the design of a nonlinear vibration absorber for use in the vibration suppression of a nonlinear primary system. Zhang et al.²⁹ established the vibration absorption performance evaluation of NESs based on vibration transmissibility. Chen³⁰ studied the weight effect on vibration suppression of NES under a moving vertical. Lee et al.,³¹ Gatti et al.,³² and Ding and Chen³³ reviewed the engineering applications of NES in various fields, indicating that NESs have broad application prospects.

To promote the application of NESs in vibration suppression for engineering structures, Georgiades and Vakakis³⁴ established the vibration analysis model of a linear beam structure with an attached local NES and studied the vibration absorption of NES. Samani and Pellicano^35,36 studied the influence of nonlinear absorbers on the vibration suppression of beam structures under moving loads. Ahmadabadi and Khadem^37,38 studied nonlinear vibration control of the beam structure with NES and employed the NES to harvest the vibration energy of beam structures, providing a new idea for vibration energy harvest. Kani et al.^39,40 studied the influence of NESs on the vibration control of beam structures under different support conditions. Bab et al.⁴¹ employed the NES to suppress the vibration level of a rotating beam. Parseh et al.^42,43 investigated the robustness and steady-state dynamics of beam structures with the NES and different boundary conditions. Zhang et al.⁴⁴ investigated the influence of NES on the forced vibration of the axially moving beam. Fang et al.⁴⁵ studied transient nonlinear vibration and the targeted energy transfer of the beam structure coupled with a continuous bistable NES. Li et al.⁴⁶ established the vibration analysis model of an Acoustic Black Hole beam attached to nonlinear vibration absorbers and studied its broadband mitigation of flexural vibrations. Zhao et al.⁴⁷ compared two types of adjustable nonlinear vibration absorbers and made a realized model of a type of nonlinear stiffness control equipment. Chen et al.⁴⁸ and Zhang et al.⁴⁹ studied the influence of parallel NESs on vibration suppression of the beam structure, where the parallel NESs were located at the boundary or internal, respectively. Furthermore, a type of coupling nonlinear oscillator was introduced into various coupling structures, including beam-plate systems,⁵⁰ coupling beam systems,⁵¹ coupling plate systems,⁵² and cavity-plate coupling systems.⁵³ The above research mainly concentrated on the elastic structure with NESs, studying the nonlinear dynamic responses and vibration reduction of the elastic structure with NESs. The corresponding research provided theoretical support for the application of NESs in beams. However, most studies ignore the boundary rotational restraints of beams. In addition, some complex structures in engineering were composed of multiple beams, few studies investigate the influence of NESs on vibration responses of the primary system consisting of multiple beams, limiting the application of NESs in some complex structures.

Considering engineering practice and limitations in the current research, this study establishes a vibration analysis model of the primary system with NESs, where the primary system is composed of two beams, boundary-supporting springs, internal supporting springs, and an elastic coupling element. The Galerkin truncation method (GTM) is utilized to predict the forced transverse vibration of the corresponding primary system, whereas the harmonic balance method (HBM) is employed to verify transverse forced vibration responses predicted by the GTM. Based on this, the influence of NESs’ parameters on the forced transverse vibration of the system is investigated.

Formula derivation

Theoretical modeling

In marine engineering, some shafting systems can be simplified as multiple-beam systems, whereas supporting bearings of shafting systems can be simplified as supporting springs. It is worth mentioning that shafting systems are prone to deflection under the action of gravity. Engineers usually add intermediate bearings to the shafting systems to effectively suppress the deflection phenomenon, where the internal supporting springs in the shafting system help make the model closer to engineering practice. In addition to internal supporting bearings, the end of the shafting system is usually also equipped with boundary-supporting bearings, which can be simplified to boundary-support springs. Besides, the boundaries of the shafting system typically include other restrictions in addition to supporting the spring. For example, the drive shaft of the shafting is usually connected to the diesel engine, and the driven shaft is usually connected to the propeller. These connection relations introduce rotation constraints at the end of the shafting to limit its movement, so the conditions at the boundary are simplified to rotating and translational springs. Generally, boundary conditions of the primary system are typically determined for specific engineering occasions, where the stiffness of boundary springs cannot be changed in time. Thus, the vibration of the primary system needs to be controlled by introducing additional vibration control elements. Importantly, couplers are generally not installed in between shafting systems directly, which requires a certain amount of support. Therefore, the coupler is also generally employed in sync with the supporting bearing, where the coupler is generally simplified to the elastic coupling element. That is the reason for the introduction of k_I and k_E in the primary system with NESs. Against this background, Figure 1 is the model of a primary system with NESs, where the primary system is composed of two beams (Beam 1 and Beam 2), internal supporting springs, boundary-supporting springs, and an elastic coupling element. NES 1 and NES 2 are installed at Beam 1 and Beam 2, respectively. Structural parameters’ definitions of the primary system and NESs are listed in Tables 1 and 2. Additionally, for beams, u₁ (x₁,t) and u₂ (x₂,t) are their transverse vibration displacement, respectively. For NESs, u_N1 (t) and u_N2 (t) are their vibration displacement. Considering the marine engineering, the vibration excitations acting on shafting systems are mainly produced by diesel engines and propellers. Such equipment is mainly installed at the boundaries of shafting systems. Therefore, the simplified excitations of shafting systems are mainly located at their boundaries. To make this work closer to marine engineering, the vibration excitation is installed on the boundary of the primary system. The external excitation studied in this work is defined as a point harmonic excitation; its specific expression is F(x₂,t) = δ(x₂-x_F)F₀sin (ωt), where δ(.) is the Dirac function, x_F, F₀, and ω is the position, amplitude, and angle frequency of the external excitation.

Figure 1.

The model of a primary system with NESs.

Table 1.

Parameters of beams and their boundary conditions.

Parameters	Symbol (Unit)	Value
Elastic modulus	E₁/E₂ (Pa)	6.89×10¹⁰/6.89×10¹⁰
Density	ρ₁/ρ₂ (kg/m³)	2.8×10³/2.8×10³
Length	L₁/L₂ (m)	1/1
Diameter	D₁/D₂ (m)	0.016/0.02
Viscous damping	C_B1/C_B2 (Ns/m)	5/5
Additional mass	m_A (kg)	0.01
Position of the additional mass	x_A (m)	0
Position of the excitation	x_F (m)	1
Position of the internal springs	x_S1/x_S2 (m)	0.6/0.4
The amplitude of the excitation	F₀ (N)	10
Coupling stiffness	k_E (N/m)	10⁴
Internal translation stiffness	k_S1/k_S2 (N/m)	10³/10³
Boundary translational stiffness	k_L/k_I/k_R (N/m)	5×10⁴/5×10⁴/5×10⁴
Boundary rotational stiffness	K_L/K_R (Nm/rad)	10⁴/10⁴

Table 2.

Parameters of the NESs.

Parameters	Symbol (Unit)	Value
Concentrated mass of NESs	m_N1/m_N2 (kg)	0.05/0.05
Viscous damping of NESs	C_N1/C_N2 (Ns/m)	10/10
Position of the NESs	x_N1/x_N2 (m)	0.6/0.4
Nonlinear stiffness of NESs	kn_N1/kn_N2 (N/m³)	2×10⁸

According to classical vibration theory related to the beam structure, the governing equations of the single-beam structure can be derived as

ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} + δ (x_{1}) {m {\frac{\partial^{2} u_{1}}{\partial t^{2}}}_{B 1} {\frac{\partial u_{1}}{\partial t}}_{1}_{1} \frac{\partial^{4} u_{1}}{\partial {x_{1}}^{4}} (x_{1} - x_{S 1} () 1_{S 1})}_{A}

(1a)

and

ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} + C_{B 2} \frac{\partial u_{2}}{\partial t} + E_{2} I_{2} \frac{\partial^{4} u_{2}}{\partial {x_{2}}^{4}} + δ (x_{2} - x_{F} {()}_{0} \sin (ω t) {(x_{2} - x_{S 2})}_{S 22})

(1b)

where equations (1a) is the governing equation of Beam 1 while equations (1b) is the governing equation of Beam 2. S₁ and S₂ are the sectional areas of Beam 1 and Beam 2, which can be calculated by employing the diameters of Beam 1 and Beam 2.

Considering the coupling elastic coupling element is employed to connect Beam 1 and Beam 2, the restoring force acting on Beam 1 and Beam 2 introduced by the elastic coupling element should be added to the governing equations of Beam 1 and Beam 2, namely

\begin{array}{l} ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} + δ (x_{1}) m_{A} \frac{\partial^{2} u_{1}}{\partial t^{2}} + C_{B 1} \frac{\partial u_{1}}{\partial t} + E_{1} I_{1} \frac{\partial^{4} u_{1}}{\partial x_{1}^{4}} \\ + δ (x_{1} - L_{1}) k_{B} [u_{1} - u_{2} (0, t)] + δ (x_{1} - x_{S 1}) k_{S 1} u_{1} = 0 \end{array}

(2a)

and

\begin{array}{l} ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} + C_{B 2} \frac{\partial u_{2}}{\partial t} + E_{2} I_{2} \frac{\partial^{4} u_{2}}{\partial x_{2}^{4}} + δ (x_{2}) k_{E} [u_{2} - u_{1} (L_{1}, t)] \\ + δ (x_{2} - x_{F}) F_{0} \sin (ω t) + δ (x_{2} - x_{S 2}) k_{S 2} u_{2} = 0 \end{array}

(2b)

Back to the model shown in Figure 1, NES 1 and NES 2 are, respectively, installed at Beam 1 and Beam 2, where restoring forces acting on beams must be introduced by NES 1 and NES 2. Considering marine engineering, the introduction of NESs needs some additional connecting relations, which suggests that NESs cannot be directly installed on the primary system. Fortunately, the existence of internal supporting bearings of shafting systems provides a suitable installation platform for NESs, where NESs can be installed on the shafting systems through the internal supporting bearings. Against this background, to make this work closer to engineering practice, the positions of NESs should remain the same as those of the internal supporting springs. The influence of NES structural parameters on the dynamic behavior and vibration reduction effect of the primary system when the NES position is fixed. However, the vibration analysis model established in this manuscript can still be employed to study the influence of installation positions of NESs on the dynamic behavior of the primary system. One can obtain the dynamic behavior of the primary system with NESs under different locations of NESs by changing their location parameters. Considering the restoring forces motivated by NES 1 and NES 2, the governing equations of Beam 1 and Beam 2 are further derived as

\begin{array}{l} ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} + δ (x_{1}) m_{A} \frac{\partial^{2} u_{1}}{\partial t^{2}} + C_{B 1} \frac{\partial u_{1}}{\partial t} + E_{1} I_{1} \frac{\partial^{4} u_{1}}{\partial {x_{1}}^{4}} + δ (x_{1} - L_{1}) k_{E} [u_{1} - u_{2} (0, t)] \\ + δ (x_{1} - x_{S 1}) k_{S 1} u_{1} + δ (x_{1} - x_{N 1}) [k n_{N 1} {(u_{1} - u_{N 1})}^{3} + C_{N 1} (\frac{\partial u_{1}}{\partial t} - \frac{\partial u_{N 1}}{\partial t})] = 0 \end{array}

(3a)

and

\begin{array}{l} ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} + C_{B 2} \frac{\partial u_{2}}{\partial t} + E_{2} I_{2} \frac{\partial^{4} u_{2}}{\partial {x_{2}}^{4}} + δ (x_{2}) k_{E} [u_{2} - u_{1} (L_{1}, t)] + δ (x_{2} - x_{F}) F_{0} \sin (ω t) \\ + δ (x_{2} - x_{S 2}) k_{S 2} u_{2} + δ (x_{2} - x_{N 2}) [k n_{N 2} {(u_{2} - u_{N 2})}^{3} + C_{N 2} (\frac{\partial u_{2}}{\partial t} - \frac{\partial u_{N 2}}{\partial t})] = 0 \end{array}

(3b)

Obviously, the motion equations of NESs can be derived as

m_{N 1} \frac{d^{2} u_{N 1}}{d t^{2}} + C_{N 1} [\frac{d u_{N 1}}{d t} - \frac{d u_{1} (x_{N 1})}{d t}] + k n_{N 1} {[u_{N 1} - u_{1} (x_{N 1}, t)]}^{3} = 0

(4a)

and

{m \frac{d^{2} u_{N 2}}{{d t^{2}}_{N 2} [\frac{d u_{N 2}}{d t \frac{d u_{2} (x_{N 2} ())}{d t}} + k n_{N 2} [u_{N 2} - u_{2} (x_{N 2} ()) {[]}^{3}]]}}_{N 2}

(4b)

in which equations (4a) and (4b) are the motion equations of NES 1 and NES 2. Then, the boundary conditions of the primary system are derived as

x_{1} = 0 \begin{array}{l} : & {k {1_{1}}_{1} \frac{\partial^{3} u_{1}}{\partial {x_{1}}^{3}}}_{L} \begin{array}{l} , & {K {\frac{\partial u_{1}}{\partial x_{1}}}_{1}_{1} \frac{\partial^{2} u_{1}}{\partial {x_{1}}^{2}}}_{L} \end{array} \end{array}

(5a)

x_{1} = L_{1} \begin{array}{l} : & {k {1_{1}}_{1} \frac{\partial^{3} u_{1}}{\partial {x_{1}}^{3}}}_{I} \begin{array}{l} , & E_{1} I_{1} \frac{\partial^{2} u_{1}}{\partial {x_{1}}^{2}} = 0 \end{array} \end{array}

(5b)

x_{2} = 0 \begin{array}{l} : & E_{2} I_{2} \frac{\partial^{3} u_{2}}{\partial {x_{2}}^{3}} = 0 \end{array} \begin{array}{l} , & E_{2} I_{2} \frac{\partial^{2} u_{2}}{\partial {x_{2}}^{2}} = 0 \end{array}

(5c)

and

x_{2} = L_{2} \begin{array}{l} : & {k {2_{2}}_{2} \frac{\partial^{3} u_{2}}{\partial {x_{2}}^{3}}}_{R} \begin{array}{l} , & {K {\frac{\partial u_{2}}{\partial x_{2}}}_{2}_{2} \frac{\partial^{2} u_{2}}{\partial {x_{2}}^{2}}}_{R} \end{array} \end{array}

(5d)

The procedure of the GTM

In this section, the transverse vibration displacement of the primary system is expanded through the mode superposition, namely

u_{1} (x_{1}, t) = \sum_{i = 1}^{N_{1}} φ_{1 i} (x_{1}) q_{1 i} (t)

(6a)

and

u_{2} (x_{2}, t) = \sum_{j = 1}^{N_{2}} φ_{2 j} (x_{2}) q_{2 j} (t)

(6b)

in which N₁ and N₂ are truncation numbers, φ_1i (x₁) and φ_2j (x₂) are the i^th and j^th trial functions of beams while q_1i(t) and q_2i(t) are the corresponding generalized coordinates.

By utilizing the GTM, the residual equations of each beam are derived as

\int_{0}^{L_{1}} {\begin{cases} ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} + δ (x_{1}) m_{A} \frac{\partial^{2} u_{1}}{\partial t^{2}} + C_{B 1} \frac{\partial u_{1}}{\partial t} \\ + E_{1} I_{1} \frac{\partial^{4} u_{1}}{\partial {x_{1}}^{4}} + δ (x_{1} - L_{1}) k_{E} [u_{1} - u_{2} (0, t)] + δ (x_{1} - x_{S 1}) k_{S 1} u_{1} \\ + δ (x_{1} - x_{N 1}) [k n_{N 1} {(u_{1} - u_{N 1})}^{3} + C_{N 1} (\frac{\partial u_{1}}{\partial t} - \frac{\partial u_{N 1}}{\partial t})] \end{cases}} ψ_{1 m} (x_{1}) d x_{1} = 0

(7a)

and

\int_{0}^{L_{2}} {\begin{cases} ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} + C_{B 2} \frac{\partial u_{2}}{\partial t} + E_{2} I_{2} \frac{\partial^{4} u_{2}}{\partial {x_{2}}^{4}} + δ (x_{2} - x_{S 2}) k_{S 2} u_{2} \\ + δ (x_{2}) k_{E} [u_{2} - u_{1} (L_{1}, t)] + δ (x_{2} - x_{F}) F_{0} \sin (ω t) \\ + δ (x_{2} - x_{N 2}) [k n_{N 2} {(u_{2} - u_{N 2})}^{3} + C_{N 2} (\frac{\partial u_{2}}{\partial t} - \frac{\partial u_{N 2}}{\partial t})] \end{cases}} ψ_{2 n} (x_{2}) d x_{2} = 0

(7b)

where ψ_1m (x₁) and ψ_2n (x₂) are the m^th and n^th weight functions of beams. It should be noted that equations (7a) and (7b) are called the residual equations, where the corresponding solution procedure is verified and discussed in Ref. 44. By employing the above process, one can derive the governing equations of the primary system with NESs. By changing the accelerator terms of equations (5a) and (5b) into one side, equations (5a) and (5b) can be rederived as

\begin{array}{l} \int_{0}^{L_{1}} {ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} + δ (x_{1}) m_{A} \frac{\partial^{2} u_{1}}{\partial t^{2}}} ψ_{1 m} (x_{1}) d x_{1} \\ = - \int_{0}^{L_{1}} {\begin{cases} C_{B 1} \frac{\partial u_{1}}{\partial t} + δ (x_{1} - x_{N 1}) [k n_{N 1} {(u_{1} - u_{N 1})}^{3} + C_{N 1} (\frac{\partial u_{1}}{\partial t} - \frac{\partial u_{N 1}}{\partial t})] \\ + E_{1} I_{1} \frac{\partial^{4} u_{1}}{\partial {x_{1}}^{4}} + δ (x_{1} - L_{1}) k_{E} [u_{1} - u_{2} (0, t)] + δ (x_{1} - x_{S 1}) k_{S 1} u_{1} \end{cases}} ψ_{1 m} (x_{1}) d x_{1} \end{array}

(8a)

and

\int_{0}^{L_{2}} ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} ψ_{2 n} (x_{2}) d x_{2} = - {\int_{0}^{L_{2}}}^{\int_{2 n} (x_{2}) d x_{2}} {\begin{cases} C_{B 2} \frac{\partial u_{2}}{\partial t} + E_{2} I_{2} \frac{\partial^{4} u_{2}}{\partial {x_{2}}^{4}} + δ (x_{2} - x_{S 2}) k_{S 2} u_{2} \\ + {k [u_{2} (0, t) - u_{1} (L_{1}, t)] {(x_{2} - x_{F 2})}_{0} \sin (ω t)}_{E} \\ + δ (x_{2} - x_{N 2}) [k n_{N 2} {(u_{2} - u_{N 2})}^{3} + C_{N 2} (\frac{\partial u_{2}}{\partial t} - \frac{\partial u_{N 2}}{\partial t})] \end{cases}}

(8b)

The trail and weight functions utilized within the Galerkin discretization method should satisfy the boundary conditions of the primary system with NESs. Importantly, to ensure that the number of generalized coordinate equations is equal to the number of residual equations. Namely, the max value of m is equal to i, while the max of n is equal to j. Considering the boundary conditions of the primary system with NESs are linear, suggests that the mode functions of the beams with two linear springs fortunately satisfy the boundary conditions of the primary system with NESs. One can get the mode functions of the beams with two linear springs by employing the boundary-smoothed Fourier series combined with the Rayleigh–Ritz method.¹⁰

In addition, equations (4a) and (4b) can be rewritten as the following by using the equation operation

\frac{d^{2} u_{N 1}}{d t^{2}} = - \frac{1}{m_{N 1}} {C_{N 1} [\frac{d u_{N 1}}{d t} - \frac{d u_{1} (x_{N 1})}{d t}] + k n_{N 1} {[u_{N 1} - u_{1} (x_{N 1}, t)]}^{3}}

(9a)

and

\frac{d^{2} u_{N 2}}{d t^{2} \frac{1}{m_{N 2} {C_{N 2} [\frac{d u_{N 2}}{d t \frac{d u_{2} (x_{N 2} ())}{d t}} + k n_{N 2} [u_{N 2} - u_{2} (x_{N 2} ()) {[]}^{3}] {}]}}}

(9b)

By numerically solving equations (8a), (8b), (9a), and (9b), transverse vibration responses of the primary system with NESs can be obtained. The Runge–Kutta method is employed for this study to solve the above equations. The Runge–Kutta method can obtain a series of stable numerical results of the primary system with NESs. The numerical solving processes are programmed in the MATLAB software. By using the order “ODE45,” one can get the periodic solutions of the primary system with NESs.

Numerical results and discussion

Considering the prosperous development of the material industry, the beams in this work are made of alloy aluminum. Parameters of beams, boundary-supporting springs, internal supporting springs, and external excitation are shown in Table 1.

The reliability of the GTM

The reliability of the GTM in predicting forced transverse vibration of the primary system with NESs is studied. The forced transverse vibration calculated by the GTM is verified by the harmonic balance method (HBM) and Lagrange method (LM). In HBM, the solution terms are set as the fundamental harmonic. Then, the stability of the GTM is also studied. Additionally, Table 2 presents the parameters of NESs.

Firstly, Figure 2 presents forced transverse vibration of the primary system with NESs under 2-term, 4-term, 6-term, and 8-term truncation numbers, where the truncation numbers correspond to the number of trail and weight functions. The amplitude of the primary system is selected as the y-coordinate, which may be more intuitive for engineers to understand the system’s vibration response. It is worth mentioning that the structural parameters of Beam 1 are close to those of Beam 2. Thus, the truncation numbers of Beam 1 and Beam 2 are set to the same. According to Figure 2, two primary resonance regions appear when the truncation number is 2-term, while five resonance regions appear when the truncation number is 4-term, 6-term, and 8-term. The forced transverse vibration of the primary system can be accurately obtained as the truncation number reaches six terms. Therefore, the truncation number of the GTM is set to six terms in the subsequent study. Additionally, it can be found that the max amplitude of the vibration displacement of the primary system is less than 10⁻³ m. The above max amplitude of the vibration displacement of the primary system conforms to the general engineering situation.

Figure 2.

Forced transverse vibration of the primary system with NESs under different truncation numbers.

Figure 3 presents the forced transverse vibration responses of the primary system with NESs obtained by different methods (GTM, HBM, and LM). It should be noted that the forced transverse vibration of the primary system predicted by the HBM is calculated from the frequency domain, while that predicted by the GTM is calculated from the time domain. It should be noted that the calculation time domain of GTM and LM is chosen as 500 T_E, where T_E is the single-acting period of the external force. To ensure the transient responses of the primary system with NESs die away, calculating results in [401 T_E, 500 T_E] are chosen as the stable results. It should be noted that the amplitude of this work is the local maximum steady-state displacement of the primary system. One can obtain such amplitudes by selecting the local maximum steady-state displacement of the time domain results under the stable calculating regions, namely, [401 T_E, 500 T_E]. Based on the stable time-domain calculating results, frequency response curves of the primary system with NESs can be graphed. In plotting frequency response curves, the amplitudes under the time-domain calculating results are chosen as the y-axis coordinate. Excitation frequencies are selected as the x-axis coordinate. Then, the frequency responses of the primary system with NESs can be obtained. In calculating the frequency responses of the primary system with NESs using the HBM, the coefficients of the solution terms are introduced into the assumed vibration displacements of the primary system with NESs. Then, one can get the amplitude of the primary system. The corresponding amplitudes are set as the y-axis coordinate. From Figure 3, the forced transverse vibration of the primary system with NESs obtained by GTM and HBM match each other well. The max relative errors are less than 1%, indicating the correctness of the GTM in calculating forced transverse vibration of the primary system with NESs established in Section 2.

Figure 3.

Forced transverse vibration of the primary system with NESs obtained by the GTM and HBM.

Additionally, NESs exert a nonlinear restoring force on the primary system. Generally, the existence of nonlinear restoring forces can motivate the nonlinear vibration responses of the primary system. Since the nonlinear restoring force is generated only by NESs, it is positively correlated with the nonlinear stiffness and the relative displacement between NESs and beams. In Figures 2 and 3, the nonlinear stiffness of NESs is under a relatively low value, making the nonlinear restoring force acting on the primary system also under a low value. Thus, nonlinear phenomena do not occur in the amplitude-frequency response curves in Figures 2 and 3.

The influence of NESs on frequency responses

This section focuses on the influence of NESs on the forced transverse vibration of the primary system with NESs, where the excitation frequency range is [1 Hz, 200 Hz]. To improve the engineering acceptance of NESs, the parameters of NES 1 and NES 2 are set as the same (m_N1 = m_N2 = m_N, C_N1 = C_N2 = C_N, kn_N1 = kn_N2 = kn_N). Considering the limitations of installation space and ways, it is difficult to attach NESs directly to beams. Fortunately, supporting structures can provide an installation platform for NESs. For beams in engineering, changing the position of supporting structures may cause some unwanted problems. Therefore, the influence of x_N1 and x_N2 on transverse forced vibration responses will not be investigated in the subsequent study.

Firstly, Figure 4 presents the forced transverse vibration of the primary system with NESs under different kn_N, where kn_N = 10⁸ N/m³, kn_N = 10⁹ N/m³, and kn_N = 10¹⁰ N/m³. In this part, C_N is set as 6 Ns/m and other parameters of NESs are the same as those in Table 2. From Figure 4, the existence of NESs beneficially influences the vibration suppression at each primary resonance region. With the increase of kn_N, some unusual phenomena appear in the response curve of forced transverse vibration. Namely, multiple vibration amplitudes exist in the response curve for certain single frequencies. Under the above phenomena, the vibration response of the primary system is more complex than the normal response. Therefore, the above phenomena are classified as complex transverse vibration responses. In addition, the complex transverse vibration responses of the primary system with NESs are strengthened with the increase of kn_N. For this study, complex transverse vibration responses of the primary system appear under kn_N = 10⁹ N/m³ and kn_N = 10¹⁰ N/m³. It should be noted that NESs can effectively absorb the vibration energy of the primary system, limiting the vibration level of the primary system in a small range. Considering the nonlinear phenomenon of the primary system is mainly influenced by the nonlinear restoring force, the nonlinear restoring force motivated by NESs is always limited to a value. Therefore, the jump phenomenon cannot be obviously observed in this work. To study the above complex transverse dynamic responses, phase diagrams are also graphed in Figure 4, in which Poincaré points are also graphed. Poincaré points tend to compose a closed curve, and the phase path remains stable, which indicates that the vibration state of the complex transverse vibration responses in Figure 4 is quasi-periodic.

Figure 4.

Forced transverse vibration of the primary system with NESs under different kn_N.

Secondly, Figure 5 presents forced transverse vibration of the primary system with NESs under different C_N, where C_N = 2 Ns/m, C_N = 5 Ns/m, and C_N = 10 Ns/m. Other parameters of NESs are the same as those in Table 2. According to Figure 5, complex transverse vibration responses at the boundary of the primary system appear under C_N = 2 Ns/m. With the increase of C_N, the complex transverse vibration responses of the primary system with NESs gradually vanish. Furthermore, the increase of C_N beneficially influences the vibration suppression at each resonance region. Phase diagrams and Poincaré points of the complex transverse vibration responses in Figure 5 are plotted to study their vibration states. According to the subfigures in Figure 5, there are finite Poincare points and the phase path remains stable, suggesting the vibration state of the complex transverse vibration responses in Figure 5 is multiple-periodic.

Figure 5.

Forced transverse vibration of the primary system with NESs under different C_N.

Thirdly, Figure 6 shows the forced transverse vibration of the primary system with NESs under different m_N, where m_N = 0.025 kg, m_N = 0.050 kg, and m_N = 0.075 kg. Other parameters of NESs are the same as those in Table 2. According to Figure 6, complex transverse vibration responses at the boundary of the primary system appear under m_N = 0.050 kg. With the increase or decrease of m_N, the complex transverse vibration responses of the primary system with NESs gradually vanish. For the parameters studied in this section, the changing of m_N has a slight influence on the vibration suppression at each primary resonance region. To study the complex transverse vibration responses in Figure 6, phase diagrams and Poincaré points are also graphed. It can be seen from each phase diagram that there are finite Poincaré points, and the phase path remains stable, which indicates that the complex transverse vibration responses in Figure 6 are in the multiple-periodic vibration state.

Figure 6.

Forced transverse vibration of the primary system with NESs under different m_N.

To further investigate the complex transverse vibration responses shown in Figures 4–6, Figure 7 presents the kinetic energy of the beams and NESs. In this work, the kinetic energy of beams and NESs is defined as

E_{S} = \int_{0}^{L_{1}} ρ_{1} S_{1} \frac{\partial^{2} u_{1}}{\partial t^{2}} d x_{1} + δ (x_{1}) {m \frac{\partial^{2} u_{1}}{\partial t^{2}} \int_{0}^{L_{2}} ρ_{2} S_{2} \frac{\partial^{2} u_{2}}{\partial t^{2}} d x_{2}}_{A}

(10a)

and

E_{NES} = \frac{1}{2} m_{N 1} {(\frac{d u_{N 1}}{d t})}^{2} + \frac{1}{2} m_{N 2} {(\frac{d u_{N 2}}{d t})}^{2}

(10b)

Figure 7.

Kinetic energy of the primary system with NESs under 29 Hz.

Parameters of the NESs are set as kn_N = 2×10⁸ N/m³, C_N = 2 Ns/m, and m_N = 0.050 kg. The excitation frequency is set as 29 Hz. From Figures 7(a) and 7(b), the kinetic energy of beams and NESs presents the characteristics of periodic oscillation. Figure 7(c) is the comparison of the kinetic energy of beams and NESs. According to Figure 7(c), the kinetic energy of the beams targeted transfers to NESs during the time interval T_S. The above phenomenon suggests that the targeted energy transfer phenomenon appears between the primary system and NESs under some appropriate parameters of NESs.

The influence of NESs on single-frequency vibration responses

In engineering, power equipment works in its rated condition for a long time to improve its lifetime. Such an engineering phenomenon suggests that the external excitation introduced by the power equipment is typically in a single frequency. Considering the engineering practice, this section studies the influence of NESs on the primary system under a single-frequency excitation. In Section 3.3, the excitation frequency is selected.

Firstly, Figure 8 shows amplitude-kn_N responses of the primary system with NESs (30 Hz), where kn_N varies from 10⁶ N/m³ to 10¹⁰ N/m³, m_N = 0.05 kg, and C_N = 4 Ns/m. From Figure 8, when kn_N stays in regions [10⁶ N/m³, 10^8.8 N/m³] and [10^9.4 N/m³, 10¹⁰ N/m³], the vibration state of beams is single-periodic. When kn_N stays in the region [10^8.8 N/m³, 10^9.4 N/m³], the complex transverse vibration responses of beams appear in single-frequency responses. 10^8.8 N/m³ and 10^9.4 N/m³ are defined as the vibration state converted value of kn_N. As kn_N gets close to its converted value, the change in kn_N has a significant influence on the vibration state of beams. The vibration state of the region [10^8.8 N/m³, 10^9.4 N/m³] is studied by plotting phase diagrams and Poincaré points. From each phase diagram, there are multiple Poincaré points, and the phase path remains stable. It should be noticed that Poincaré points tend to form a closed curve, suggesting that the vibration state of the region [10^8.8 N/m³, 10^9.4 N/m³] is quasi-periodic.

Figure 8.

Amplitude-kn_N responses of the primary system with NESs (30 Hz).

Secondly, Figure 9 shows amplitude-C_N responses of the primary system with NESs (30 Hz), where C_N varies from 1 Ns/m to 100 Ns/m, m_N = 0.05 kg, and kn_N = 3×10⁸ N/m³. From Figure 9, when C_N stays in the region [2.5 Ns/m, 100 Ns/m], the vibration state of the beams is single-periodic. When C_N stays in the region [1 Ns/m, 2.5 Ns/m], the complex transverse vibration responses of beams appear in single-frequency responses. 2.5 Ns/m is defined as the vibration state converted value of C_N. When C_N ranges from 2.5 Ns/m to 20 Ns/m, the increase of C_N has a beneficial influence on the vibration suppression of the primary system. After C_N exceeds 20 Ns/m, the changing of C_N slightly influences the forced transverse vibration of the primary system with NESs. In addition, the vibration state of the region [1 Ns/m, 2.5 Ns/m] is studied by plotting phase diagrams and Poincaré points. From subfigures, the phase path remains stable, and Poincaré points present finite, suggesting the region [2 Ns/m, 2.5 Ns/m] is in the multiple-periodic vibration state.

Figure 9.

Amplitude-C_S responses of the primary system with NESs (30 Hz).

Thirdly, Figure 10 is amplitude-m_N responses of the primary system with NESs (30 Hz), where m_N varies from 0.01 kg to 0.1 kg, C_N = 2 Ns/m, and kn_N = 2×10⁸ N/m³. From Figure 10, when m_N stays in regions [0.01 kg, 0.025 kg] and [0.034 kg, 0.1 kg], the vibration state of beams is single-periodic. When m_N stays in the region [0.025 kg, 0.034 kg], the complex transverse vibration responses of beams appear in single-frequency responses. 0.025 kg and 0.034 kg are defined as the vibration state converted value of m_N. For parameters employed in this section, the change of m_N significantly influences the vibration state of beams with NESs when m_N approaches its converted value. To further study the vibration state of the region [0.025 kg, 0.034 kg], its phase diagrams are plotted. From each phase diagram, Poincare points form a closed curve and the phase path remains stable, indicating that the vibration state of the region [0.025 kg, 0.034 kg] is quasi-periodic.

Figure 10.

Amplitude-m_N responses of the primary system with NESs (30 Hz).

According to analysis in Figures 8–10, the effect of C_N on the forced transverse vibration of beams with NESs is monotonous while the influence of m_N and kn_N on transverse forced vibration responses of beams with NESs is nonmonotonic. For the viscous damping of NESs, a great value of C_N can effectively suppress the vibration level of beams. For the nonlinear stiffness and motion mass of NESs, it is of great significance to study the influence of simultaneous changes in nonlinear stiffness and motion mass of NESs on the max amplitude of the primary system under the single-frequency excitation. Figure 11 presents the max amplitudes of the primary system with NESs with the variation of kn_N and m_N, where the single-frequency excitation is 30 Hz. In plotting Figure 11, m_N varies from 0.01 kg to 0.15 kg, kn_N varies from 10⁸ N/m³ to 10¹⁰ N/m³, and C_N = 5 Ns/m. From Figure 11, simultaneous changes in nonlinear stiffness and motion mass of NESs significantly impacts the max amplitude of the primary system with NESs. An amplitude sensitive zone exists in Figure 11, where the max amplitude of the primary system with NESs is more sensitive to the variation of nonlinear stiffness and motion mass belonging to NESs. Importantly, a suitable combination of kn_N and m_N beneficially influences vibration suppression of the primary system. For the forced transverse vibration at x₁ = 0, x₁ = L₁, and x₂ = 0, the suitable combination of kn_N and m_N is m_N = 0.09 kg and kn_N = 10¹⁰ N/m³. For the forced transverse vibration at x₂ = L₂, the suitable combination of kn_N and m_N is m_N = 0.025 kg and kn_N = 10¹⁰ N/m³. Furthermore, when kn_N is within the interval [10⁹ N/m³, 10¹⁰ N/m³], maximum values of the forced transverse vibration of beams with NESs are more susceptible to the change of kn_N.

Figure 11.

Max amplitudes of the primary system with NESs with the variation of kn_N and m_N (30 Hz).

Conclusions

This work establishes the vibration analysis model of a primary system with NESs, where the primary system consists of two beams, internal supporting springs, boundary-supporting springs, and an elastic coupling element. The GTM is utilized to calculate the forced transverse vibration. After studying the reliability of the GTM, the effect of NESs on the forced transverse vibration of the primary system is investigated. Some conclusions are drawn as

(1) Transverse forced vibration responses of beams with NESs can be accurately predicted by the GTM. For the structural parameters employed in this work, a 6-term truncation number of the GTM guarantees its stability.

(2) The forced transverse vibration of the system is significantly influenced by the parameters of NESs, including their nonlinear stiffness, viscous damping, and mass. Under certain parameters of NESs, beams’ complex transverse vibration responses appear in both frequency and single-frequency responses. The targeted energy transfer phenomenon appears in the primary system under some appropriate parameters of NESs.

(3) The influence of viscous damping of NESs on single-frequency responses of the primary system is monotonous, while the influence of motion mass and nonlinear stiffness of NESs on single-frequency responses of the primary system is nonmonotonic. A suitable combination of nonlinear stiffness and motion mass of NESs beneficially influences vibration suppression of the primary system.

Footnotes

ORCID iD

Haijian Cui

Funding

The author(s) 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.

Data Availability Statement

The datasets generated during and/or analyzed during the current study are available from the first author and the corresponding author upon reasonable request.*

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