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Abstract

The high-static-low-dynamic-stiffness (HSLDS) isolator is sensitive to mass deviation, which causes a shift in static equilibrium and results in softening of the dynamic response. This study introduces sliding mass inertia (SMI) to an HSLDS to mitigate the softening and enhance the dynamic performance of the system under mass deviation. The harmonic balance method (HBM) is employed to obtain the dynamic response of the vibrating system with the proposed nonlinear isolator. Key characteristics, such as the backbone curve, amplitude-frequency response, resonance peak, and displacement transmissibility are analyzed in detail through parametric analysis. The results indicate that mass deviation worsens the softening behavior of the HSLDS isolator. A higher mass deviation ratio results in an increase in both the amplitude and the resonance frequency of the system. Fortunately, however, the sliding mass eliminates the softening behavior, reduces the peak response, and improves the overall dynamic performance.

Keywords

sliding mass inertia high-static-low-dynamic-stiffness mass deviation anti-resonance frequency harmonic balance method passive vibration isolation

Introduction

Nonlinear vibration isolators have garnered significant attention due to their superior isolation performance compared to linear isolators. They excel in providing very low isolation frequencies while maintaining excellent load-bearing capabilities. Ibrahim provided a comprehensive review on nonlinear vibration isolators, highlighting numerous mechanisms employing negative stiffness that result in ultra–low-frequency isolation.¹ Carrela et al. performed an extensive examination on the static characteristics of a quasi-zero-stiffness (QZS) isolator, comprising three oblique springs.² They introduced the notion of high-static-low-dynamic-stiffness (HSLDS) isolator³ and optimized the minimal stiffness range.

Recently, different methods have been employed to achieve HSLDS, such as scissor-like structures,⁴ bio-inspired structures,⁵ cam-roller-spring mechanisms,⁶ origami-inspired structures,⁷ cross-ring structures,⁸ horizontal springs,⁹ Euler buckled beams,¹⁰ and magnetic elements.¹¹ Advances in nonlinear vibration isolation have increasingly focused on HSLDS and QZS systems due to their ability to extend the isolation range to lower frequencies. Yan et al. reviewed recent progress in nonlinear vibration isolation, highlighting HSLDS and QZS mechanisms for low-frequency applications.¹² Ma et al. discussed QZS isolators, emphasizing negative stiffness mechanisms and the performance of passive and active systems.¹³ Zhu et al. explored compact QZS isolators, addressing design approaches and trade-offs between miniaturization and performance.¹⁴ Liu et al. summarized advancements in QZS isolators, demonstrating their advantages over linear systems in low-frequency isolation.¹⁵ Hamzehei et al. reviewed the design of QZS mechanical metamaterials, examining deformation mechanisms such as stiffness elements and origami-inspired structures for passive vibration isolation.¹⁶

Recent studies on nonlinear systems provide powerful tools for analyzing stiffness asymmetry and mass variation effects in vibration isolation systems. Baleanu et al. used fractional variational principles to model asymmetric oscillators with quadratic nonlinearity and memory effects.¹⁷ Florea et al. applied the multi-step differential transformation method (Ms-DTM) to accurately solve stiff pendulum–spring systems, demonstrating strong handling of nonlinearities.¹⁸ Al-Khader et al. developed the Optimal and Modified Homotopy Perturbation Method (OM-HPM) for oscillators with time-dependent mass, improving efficiency and accuracy.¹⁸ Roy et al. extended OM-HPM to pendulums with vibrating supports, capturing parametric excitation and resonance phenomena,¹⁹ and further applied it to highly nonlinear oscillators such as Duffing and Van der Pol systems, achieving superior convergence and precision.²⁰

The HSLDS and QZS vibration isolators are effective in providing low-frequency vibration isolation, but they may demonstrate hardening stiffness in specific scenarios, which can reduce isolation effectiveness. One approach to mitigate this problem is the addition of inertia to an HSLDS isolator.^21,22

In the above-mentioned literature, researchers primarily concentrated on examining isolation systems under rated mass where the minimum stiffness aligns with the static equilibrium position. Nevertheless, achieving perfect balance at the desired equilibrium point is challenging in real-world scenarios due to mass variation, namely, the deviation of the actual mass of a system from its intended or rated value. Overload or underload conditions shift the system to a new static equilibrium position. This shift alters the dynamic stiffness, making it asymmetric and resulting in an overall asymmetric dynamic system. Several researchers tried to address this issue by examining the dynamic behaviors of certain HSLDS isolators for a mistuned mass. These studies revealed that the isolation performance deteriorated, but remains superior to that of a linear isolator when subjected to base or force excitations.^23–26 Through these studies, the effect of mass deviation on the dynamic response of the isolators is explored, and many useful conclusions are drawn.

To reduce or eliminate the performance degradation of an HSLDS isolator under mass deviation, the present research investigates the potential of a novel nonlinear isolator. The proposed isolator, developed by the authors and referred to as the HSLDS-SMI isolator,²¹ combines an HSLDS isolator with sliding mass inertia (SMI). The HSLDS characteristic is obtained using a negative stiffness structure, whereas the inertia is obtained through the sliding motion of the secondary masses. The dynamic equation of motion of the HSLDS-SMI isolation system under mass deviation is derived using the Lagrange equation. The harmonic balance method (HBM) is then utilized to obtain the dynamic response of the vibration system under harmonic excitation, and the backbone curve of the system is examined. The isolation performance in terms of displacement transmissibility is obtained and studied for different system parameters.

The remainder of this paper is organized as follows. In Section 2, the design and construction of the proposed isolator are introduced, and the dynamic equation of a vibrating system with the HSLDS-SMI isolator and mass deviation is derived. In Section 3, the dynamic response is obtained using the harmonic balance method (HBM), and the backbone curve is examined in detail. The amplitude-frequency response and peak response under overload and underload conditions are analyzed in Sections 4 and 5, respectively. In Section 6, the isolation performance of the proposed isolator is studied in terms of displacement transmissibility under both overload and underload conditions. Section 7 provides the conclusions.

The HSLDS-SMI isolator with mass deviation and its modeling

This section presents the design of the HSLDS-SMI isolator and establishes the dynamic model for a system with a nonlinear isolator and mass deviation.

Vibration system with an HSLDS-SMI isolator

The HSLDS-SMI isolator considered in this study is constructed by adding two sliding masses to an HSLDS isolator, as illustrated in Figure 1. The HSLDS-SMI isolator comprises top and bottom plates, connecting rods, two sliding masses, two vertical springs, and two horizontal springs with the same stiffness and pre-compression. The combination of pre-compressed horizontal springs and connecting rods generates negative stiffness in the vertical direction. Two sliding masses are symmetrically incorporated into the top plate using inclined connecting rods. The mass and installation angle of these sliding masses can be adjusted to optimize vibration isolation performance. This study assumes the sliding masses are equal and symmetrically positioned to maintain balanced inertial forces, avoid the generation of net moments, and ensure a stable and predictable response. Asymmetric or unequal configurations are undesirable, as they can introduce rotational dynamics, modeling complexities, and destabilizing effects that compromise the isolation performance.

Figure 1.

The HSLDS-SMI isolator at static equilibrium position with rated mass.

To focus on the dominant dynamic behavior governed by the primary and sliding masses, it is assumed that the sliding and rotational joints are frictionless, whereas the connecting rods and the top and bottom plates are rigid and massless. In practical applications, the mass of these components is usually much smaller than that of the primary mass. Furthermore, the mass of the top plate is usually included in the primary mass, as it moves together with the supported object. The same treatment applies to the bottom plate, since it is often rigidly integrated with the base supporting structure. The mass of the connecting rods is very small, less than one percent of the primary mass, and has a negligible effect on the vibration isolation performance. Therefore, no separate mass is defined in the model for these components.

Under the application of a rated mass, the HSLDS-SMI isolator attains static equilibrium, with the lower connecting rods becoming horizontal, as depicted in Figure 1. The positive stiffness of the vertical springs is offset by the negative stiffness, producing the desired characteristic behavior of the HSLDS. When there is a deviation in the supported mass from the rated mass intended for HSLDS-SMI, the isolator can encounter overload or underload scenarios. Consequently, the static equilibrium position of the isolator shifts away from the ideal static equilibrium location, as shown in Figure 2. This results in the minimum stiffness of the isolator, which is no longer aligning with the static equilibrium location, leading to an asymmetric stiffness/force-displacement curve around the new static equilibrium location.

Figure 2.

The HSLDS-SMI isolator at (a) overload and (b) underload conditions.

Under the shifted equilibrium condition caused by mass deviation from the rated mass, the stiffness of the isolator becomes asymmetric with a net positive value. This asymmetry in the stiffness/force-displacement curve is heavily influenced by the load condition, as shown in Figure 3, and further discussed in our previous work.²⁷ Consequently, subsequent analysis is conducted in this work to thoroughly examine how this mass deviation impacts the dynamic response of the HSLDS-SMI isolator.

Figure 3.

Dimensionless (a) stiffness and (b) force-displacement curve: “ $•$ ” under ideal condition, “ $*$ ” under overload, and “o” under underload condition.

System modeling

When the isolation mass deviates from the intended mass, the mass is denoted as $m_{d}$ and the displacement $y$ remains relative to the static equilibrium point of the HSLDS-SMI at its rated mass, acting as the generalized coordinate, as depicted in Figure 2. To derive the equation of motion for this system, the Lagrange principle can be applied, where the total kinetic energy T, potential energy U, and dissipation force D of the isolation system are formulated as

T = \frac{1}{2} m_{d} {\dot{y}}^{2} + m_{2} ({\dot{x}}^{2} + {\dot{\hat{y}}}^{2})

(1)

U = \frac{1}{2} K_{v} {\hat{y}}^{2} + K_{h} δ_{h}^{2} + m_{f} g \hat{y}

(2)

D = c \dot{\hat{y}}

(3)

In the equations,

m_{2}

denotes the sliding mass, while

K_{v}

and

K_{h}

represent the vertical and horizontal spring stiffness, respectively. The lengths of the connecting rods of HSLDS and SMI are

L

and

l

, respectively. The compression of the horizontal spring is denoted as

δ_{h}

, the damping coefficient of the vertical damper is represented as

c

and the initial compression in the horizontal spring is designated as

δ_{h 0}

. Additionally,

m_{f}

stands for the mass deviation within the isolation system, defined as the difference between the deviated and rated masses.

Using the Lagrange equation

\frac{d}{d t} (\frac{\partial Q}{\partial \dot{\hat{y}}}) - \frac{\partial Q}{\partial \hat{y}} = - D, Q = T - U

(4)

The dynamic equation of motion can be obtained as

m_{D} \ddot{\hat{y}} + (K_{v} - 2 K_{h} (1 - \frac{L_{0} - δ_{h 0}}{\sqrt{L^{2} - {\hat{y}}^{2}}})) \hat{y} + c \dot{\hat{y}} = (m_{D} - m_{T}) g - m_{d} \ddot{z}

(5)

where

m_{D} = m_{d} + \frac{2 m_{2} l^{2}}{l^{2} - {(l \sin (θ) + \hat{y})}^{2}}

m_{T} = M_{1} + \frac{2 m_{2} l^{2}}{l^{2} - {(l \sin (θ) + \hat{y})}^{2}}

The dimensionless dynamic equation of motion for the isolation system can be rewritten as

\frac{m_{D}}{m_{T}} {\hat{Y}}^{″} + (1 - 2 β (1 - \frac{l_{0} - δ}{\sqrt{1 - {\hat{Y}}^{2}}})) \hat{Y} + 2 ξ {\hat{Y}}^{'} = (\frac{m_{D}}{m_{T}} - 1) Δ \hat{Y} - \frac{μ M_{1}}{m_{T}} {\hat{Z}}^{″}

(6)

where the dimensionless parameters are defined as follows.

μ

represents the mass deviation ratio. More details about the nondimensionalization process and other dimensionless parameters can be found in our previous work.²¹

\hat{Y} = \frac{\hat{y}}{L}, β = \frac{K_{h}}{K_{v}}, δ = \frac{δ_{h 0}}{L}, l_{0} = \frac{L_{0}}{L}, ξ = \frac{c}{2 m_{T} ω_{n}}, ω_{n} = \sqrt{\frac{K_{v}}{m_{T}}}, Ω = \frac{ω}{ω_{n}}, τ = ω_{n} t, \hat{Z} = \frac{z}{L}, μ = \frac{m_{d}}{M_{1}}, r = \sin (θ), Δ \hat{Y} = \frac{Δ \hat{y}}{L}, l = 2 L, Δ \hat{Y} = \sqrt{1 - l_{0}^{2}}, m_{D} = m_{d} + \frac{8 m_{2}}{4 - {(2 r + \hat{Y})}^{2}}, m_{T} = M_{1} + \frac{8 m_{2}}{4 - {(2 r + \hat{Y})}^{2}}

Using polynomial forms to represent the fractional and radical expressions in the equation of motion can greatly facilitate subsequent dynamic analysis. This simplification is accomplished through the application of the Taylor series expansion, resulting in the following reduced equation of motion

μ_{e} {\hat{Y}}^{″} + γ_{1} \hat{Y} + γ_{2} {\hat{Y}}^{3} + 2 ξ {\hat{Y}}^{'} = d - \frac{μ M_{1}}{M_{T}} {\hat{Z}}^{″}

(7)

where

μ_{e} = \frac{M_{D}}{M_{T}}

M_{D} = m_{d} + \frac{2 m_{2}}{1 - r^{2}}

M_{T} = M_{1} + \frac{2 m_{2}}{1 - r^{2}}

d = (μ_{e} - 1) Δ \hat{Y}

γ_{1} = 1 + (2 l_{0} - 2 δ - 2) β

γ_{2} = β (l_{0} - δ)

Such a treatment has been proven to be appropriate in previous work.²⁷

Steady-state harmonic response

Solution by the harmonic balance method

The equation of motion can be solved using the harmonic balance method (HBM). Assuming the solution has the form

\hat{Y} = a_{0} + a_{1} \cos (Ω τ + ϕ)

(8)

The base excitation has the form

\hat{Z} = z_{0} \cos (Ω τ)

(9)

In equation (8),

a_{0}

is a dimensionless bias or constant amplitude term,

a_{1}

represents the amplitude of the harmonic term, and

ϕ

is the initial phase angle.

By substituting equations (8) and (9) into equation (7) and setting the constant terms and coefficients of the $\sin (Ω τ)$ and $\cos (Ω τ)$ terms equal on both sides, the following equations can be obtained

γ_{2} a_{0}^{3} + γ_{1} a_{0} + \frac{3}{2} γ_{2} a_{0} a_{1}^{2} = d

(10)

- μ_{e} a_{1} Ω^{2} + γ_{1} a_{1} + 3 γ_{2} a_{0}^{2} a_{1} + \frac{3}{4} γ_{2} a_{1}^{3} = \frac{μ M_{1}}{M_{T}} z_{0} Ω^{2} \cos (ϕ)

(11)

- 2 ξ Ω a_{1} = \frac{μ M_{1}}{M_{T}} z_{0} Ω^{2} \sin (ϕ)

(12)

By manipulating equations (10)–(12), implicit equations can be formulated for the amplitude-frequency response in terms of

a_{1}

and

a_{0}

, respectively, and given as

\begin{array}{l} \frac{μ^{2} M_{1}^{2}}{M_{T}^{2}} z_{0}^{2} Ω^{4} - 4 ξ^{2} a_{1}^{2} Ω^{2} \\ - (γ_{1} a_{1} + \frac{3}{4} γ_{2} a_{1}^{3} + 3 γ_{2} a_{1} ({(\frac{d}{2 γ_{2}} + \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}} \\ {{+ {(\frac{d}{2 γ_{2}} - \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}})}^{2} - μ_{e} a_{1} Ω^{2})}^{2} = 0 \end{array}

(13)

\begin{array}{l} 2 μ_{e}^{2} (2 d a_{0}^{2} - 2 γ_{1} a_{0}^{3} - 2 γ_{2} a_{0}^{5} - \frac{3 μ^{2} M_{1}^{2} γ_{2} a_{0}^{3} z_{0}^{2}}{M_{T}^{2} μ_{e}^{2}}) Ω^{4} \\ + 2 a_{0} (d - γ_{1} a_{0} - γ_{2} a_{0}^{3}) (8 ξ^{2} a_{0} - 2 μ_{e} (γ_{1} a_{0} + 5 γ_{2} a_{0}^{3} + d)) Ω^{2} \\ + (d - γ_{1} a_{0} - γ_{2} a_{0}^{3}) {(γ_{1} a_{0} + 5 γ_{2} a_{0}^{3} + d)}^{2} = 0 \end{array}

(14)

The trend and the peak amplitude values of the amplitude-frequency curve, along with the corresponding resonance frequencies, can be predicted using the backbone curve.

The equations for the backbone curve, expressed in terms of the harmonic amplitude term $a_{1}$ and the amplitude bias term $a_{0}$ , can be derived from equations (10) and (11). The expressions for $a_{1}$ and $a_{0}$ are given as follows:

\begin{array}{l} Ω_{1}^{2} = \frac{1}{μ_{e}} (γ_{1} a_{1} + \frac{3}{4} γ_{2} a_{1}^{3} + 3 γ_{2} a_{1} ({(\frac{d}{2 γ_{2}} + \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}} \\ {+ {(\frac{d}{2 γ_{2}} - \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}})}^{2} \end{array}

(15)

Ω_{0}^{2} = \frac{1}{2 μ_{e}} (γ_{1} + 5 γ_{2} a_{0}^{2} + \frac{d}{a_{0}})

(16)

Backbone curve

Figures 4 and 5 depict the backbone curve of the HSLDS-SMI vibration isolator under $μ$ and $m_{2}$ , respectively.

Figure 4.

Backbone curve for different $μ$ for (a) $a_{1}$ and (b) $a_{0}$ .

Figure 5.

Backbone curve for varying $m_{2}$ for (a) $a_{1}$ and (b) $a_{0}$ .

The backbone curves of the harmonic amplitude $a_{1}$ exhibit different changing trends with varying values of $μ$ , as shown in Figure 4(a) when $β = 0.7$ , $δ = 0.7$ , $l_{0} = 0.9$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ . At $μ = 1$ and $μ = 1.01$ , the curves bend distinctly to the right, indicating classical hardening behavior. As $μ$ increases, the backbone curves gradually shift to the right in the frequency domain, reflecting an increase in resonance frequency. At the same time, the curvature of the response changes, and the curves begin to bend leftward for higher values of $μ$ , indicating a transition toward softening behavior.

Regarding the constant amplitude term $a_{0}$ , all the backbone curves follow the same changing trend across different $μ$ values as depicted in Figure 4(b). Initially, the backbone curve exhibits softening behavior, which decreases with increasing $μ$ as the amplitude grows. Moreover, an increase in $μ$ results in a higher resonance frequency response.

The backbone curves of the harmonic amplitude $a_{1}$ and constant amplitude term $a_{0}$ , for different values of $m_{2}$ , exhibit a reverse effect compared to $μ$ , as shown in Figure 5 when $μ = 1.03$ , $β = 0.7$ , $δ = 0.7$ , $l_{0} = 0.9$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ and $z_{0} = 0.05$ . As $m_{2}$ increases, the backbone curve shifts to the left, resulting in a reduced resonance frequency. Furthermore, increasing $m_{2}$ significantly reduces the softening behavior in the backbone curve.

Analysis on the amplitude-frequency response

Under overload conditions

Figures 6 and 7 illustrate the amplitude-frequency responses of $a_{1}$ and $a_{0}$ for different $μ$ and $m_{2}$ , respectively.

Figure 6.

Amplitude-frequency response under overload for different $μ$ for (a) $a_{1}$ and (b) $a_{0}$ .

Figure 7.

Amplitude-frequency response under overload for varying $m_{2}$ for (a) $a_{1}$ and (b) $a_{0}$ .

Under the condition of rated mass, the harmonic term behaves as expected under normal conditions when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ as shown in Figure 6(a). However, the constant term $a_{0}$ exhibits a very small response, with no apparent peak as depicted in Figure 6(b). When $μ = 1.01$ , the amplitude-frequency response of both terms reveals a notable change, with the harmonic and the constant terms reaching maximum and minimum values, respectively, at the resonance frequency. Furthermore, both amplitude-frequency curves exhibit a leftward bend, indicating a gradual softening of the stiffness characteristics of the isolation system. As $μ$ increases, the amplitudes of both terms increase, causing the amplitude-frequency curve to shift towards higher frequencies.

The amplitude-frequency curves for varying sliding mass $m_{2}$ exhibit a reverse effect compared to $μ$ when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ , $μ = 1.05$ and $z_{0} = 0.05$ , as depicted in Figure 7. When $m_{2} = 0$ , corresponding to the HSLDS isolator, the system demonstrates a softening behavior. However, when there is a sliding mass, $m_{2} = 0.5$ , denoting the HSLDS-SMI system, the amplitude of both terms decreases. As $m_{2}$ further increases, the softening behavior of the system decreases. Therefore, increasing the sliding mass $m_{2}$ can mitigate the softening stiffness induced by mass deviation while dampening the response amplitude within the resonance zone. Moreover, at higher frequencies, there is a significant improvement in the dynamic response.

Under underload conditions

The amplitude-frequency curve of the HSLDS-SMI under the underload condition, compared to the overload condition, is presented in Figure 8 when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ .

Figure 8.

Amplitude-frequency response under underload and overload for different $μ$ for (a) $a_{1}$ and (b) $a_{0}$ .

When $μ < 1$ , representing the underload condition, both the harmonic and constant terms exhibit softening behavior, with the constant term $a_{0}$ having a negative response, as depicted by the solid black and red lines. When there is an equal mass deviation between underload and overload responses, the comparison shows that the amplitude-frequency curve of the harmonic term $a_{1}$ exhibits a similar trend with minimal change, while the amplitude-frequency curves of the constant term $a_{0}$ in both cases are nearly symmetrical about the horizontal axis.

The curves depicting the variation in other parameters closely resemble those observed during overload. Consequently, this section does not delve into the amplitude-frequency response under underload conditions.

Peak responses to harmonic excitation

Under overload conditions

The backbone curve, which is the locus of the amplitude-frequency curve, is associated with the resonance frequency and peak of the amplitudes of the harmonic terms $a_{1 p}$ of the system. Substituting equation (15) into equation (13) results in the equation for the peak amplitudes of the harmonic terms $a_{1 p}$ as

\begin{array}{l} {\{{(\frac{d}{2 γ_{2}} + \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1 p}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}} + {(\frac{d}{2 γ_{2}} - \sqrt{\frac{d^{2}}{4 γ_{2}^{2}} + \frac{{(2 γ_{1} + 3 γ_{2} a_{1 p}^{2})}^{3}}{216 γ_{2}^{3}}})}^{\frac{1}{3}}\}}^{2} \\ + \frac{γ_{1}}{3 γ_{2}} + \frac{1}{4} a_{1 p}^{2} - \frac{4 μ_{e} M_{T}^{2} ξ^{2} a_{1 p}^{2}}{3 μ^{2} M_{1}^{2} γ_{2} z_{0}^{2}} = 0 \end{array}

(17)

Similarly, the backbone curve of the peak amplitudes of the constant term

a_{0}

can be obtained by substituting equation (16) into equation (14) as follows:

\begin{array}{l} μ_{e} (γ_{1} a_{0 p} + 5 γ_{2} a_{0 p}^{3} + d) \\ ((2 d - 2 γ_{1} a_{0 p} - 2 γ_{2} a_{0 p}^{3} - \frac{3 μ^{2} M_{1}^{2} γ_{2} a_{0 p} z_{0}^{2}}{M_{T}^{2} μ_{e}^{2}}) + 2 (d - γ_{1} a_{0 p} - γ_{2} a_{0 p}^{3})) \\ + 2 (d - γ_{1} a_{0 p} - γ_{2} a_{0 p}^{3}) (8 ξ^{2} a_{0 p} - 2 μ_{e} (γ_{1} a_{0 p} + 5 γ_{2} a_{0 p}^{3} + d)) = 0 \end{array}

(18)

Figure 9(a) and (b) depict the peak value of the harmonic term $a_{1 p}$ , the minimum value of the constant term $a_{0 p}$ , and the resonance frequency $Ω_{p}$ of the HSLDS-SMI vibration isolator under overload conditions for different $μ$ and $m_{2}$ , respectively.

Figure 9.

The curves of the peak $a_{1 p}$ , minimum $a_{0 p}$ and $Ω_{p}$ for different (a) $μ$ and (b) $m_{2}$ under overload.

Figure 9(a) illustrates variations in the peak value of $a_{1 p}$ , the minimum value of $a_{0 p}$ , and the resonance frequency $Ω_{p}$ relative to changes in the mass deviation ratio $μ$ when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ . The results indicate that as $μ$ increases, there is a corresponding increase in the peak value of $a_{1 p}$ , the minimum value of $a_{0 p}$ and the resonance frequency $Ω_{p}$ . This suggests that a higher $μ$ is associated with an amplified dynamic response of the HSLDS-SMI.

The peak value of $a_{1 p}$ , the minimum value of $a_{0 p}$ and the resonance frequency $Ω_{p}$ , for different values of $m_{2}$ under overload conditions, show a reverse effect compared to $μ$ , as shown in Figure 9(b) when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ , $μ = 1.05$ and $z_{0} = 0.05$ . It can be observed that an increase in the sliding mass $m_{2}$ leads to a decrease in the peak value of the harmonic term $a_{1 p}$ and the resonance frequency $Ω_{p}$ . Additionally, the minimum value of the constant term $a_{0 p}$ first increases and then decreases.

Under underload conditions

Figure 10(a) and (b) show the peak value of the harmonic term $a_{1 p}$ , the minimum value of the constant term $a_{0 p}$ , and the resonance frequency $Ω_{p}$ of the HSLDS-SMI vibration isolator under underload for varying $μ$ and $m_{2}$ , respectively.

Figure 10.

The curves of the peak $a_{1 p}$ , minimum $a_{0 p}$ and $Ω_{p}$ for varying (a) $μ$ and (b) $m_{2}$ under underload.

Figure 10(a) depicts the variations in the peak value of $a_{1 p}$ , the minimum value of $a_{0 p}$ , and the resonance frequency $Ω_{p}$ relative to changes in the mass deviation ratio $μ$ when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ . The results demonstrate that a decrease in $μ$ leads to an increase in the peak of the harmonic term, the minimum value of the constant term, and the resonance frequency.

Figure 10(b) shows the changes in the peak value $a_{1 p}$ , the minimum value of $a_{0 p}$ , and the resonance frequency $Ω_{p}$ relative to variation in the sliding mass $m_{2}$ under underload when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ , $μ = 1.05$ , and $z_{0} = 0.05$ . The results reveal that an increase in $m_{2}$ leads to a decrease in the peak of the harmonic term and the resonance frequency, while the minimum value of the constant term increases.

Displacement transmissibility analysis

When the HSLDS-SMI system is displaced from the ideal static equilibrium position due to mass deviation by a deviation distance $\partial$ due to loading effects, the static equilibrium equation for this loaded configuration can be written as

γ_{1} \partial + γ_{2} \partial^{3} = d

(19)

Taking into account the deviation distance

\partial

arising from mass deviation, the harmonic term

a_{1}

and the constant term

a_{0}

, the absolute displacement transmissibility of the HSLDS-SMI under mass deviation can be derived as follows:

T_{d} = 20 \lg (| \frac{a_{0} - \partial}{z_{0}} | + \frac{\sqrt{a_{1}^{2} + z_{0}^{2} + 2 a_{1} z_{0} \cos (ϕ)}}{z_{0}})

(20)

Under overload conditions

Figure 11(a) and (b) depict displacement transmissibility $T_{d}$ of the HSLDS-SMI vibration isolator under overload conditions for different $μ$ and $m_{2}$ , respectively.

Figure 11.

Displacement transmissibility for different (a) $μ$ and (b) $m_{2}$ under overload.

Figure 11(a) shows the displacement transmissibility $T_{d}$ for various mass deviation ratios $μ$ when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ . Under the condition of rated mass, the displacement transmissibility $T_{d}$ exhibits expected behavior. However, when $μ > 1$ , the transmissibility begins to increase. As $μ$ increases, the peak transmissibility, resonance frequency, initial isolation frequency, and anti-resonance frequency of the HSLDS-SMI also increase. Moreover, larger values of $μ$ tend to shift the peak transmissibility towards the higher left region, thereby softening the dynamic response. At higher frequencies, the transmissibility becomes relatively large. Consequently, a larger $μ$ leads to degraded isolation performance of the HSLDS-SMI.

Table 1 shows the effect of varying the mass deviation ratio

μ

on the performance of the HSLDS-SMI isolator. As

μ

increases from 1.0 to 1.05, the peak amplitude increases by 78%, from 8.87 to 15.80, the resonance frequency increases by 86%, from 0.145 to 0.269, and the isolation frequency increases by 108%, from 0.177 to 0.368. These changes indicate that higher mass deviation ratios deteriorate the dynamic response of the HSLDS-SMI isolator, leading to higher peak amplitudes, resonance frequencies, and isolation frequencies.

Table 1.

Influence of $μ$ on peak amplitude, resonance, and isolation frequency.

Parameter value	Peak amplitude	Resonance frequency	Isolation frequency
$μ = 1.0$	8.87	0.145	0.177
$μ = 1.01$	11.160	0.168	0.216
$μ = 1.03$	13.955	0.230	0.306
$μ = 1.05$	15.80	0.269	0.368

The displacement transmissibility $T_{d}$ , for different values of $m_{2}$ , exhibits a reverse effect compared to $μ$ , as shown in Figure 11(b) when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ , $μ = 1.05$ , and $z_{0} = 0.05$ . It can be observed that when $m_{2} = 0$ , representing an HSLDS isolator, the isolation system exhibits softening behavior. However, with the presence of $m_{2}$ , the softening behavior is reduced, distinguishing the SMI from traditional tuned mass dampers (TMD). While TMDs use resonance to absorb energy at specific frequencies, the SMI uses the sliding mass or inertia element to counteract the nonlinear stiffness term. The sliding mass $m_{2}$ influences the natural frequency, as it affects the equivalent mass $M_{T}$ . Increasing $m_{2}$ results in a larger $M_{T}$ , which significantly reduces the natural frequency and introduces an anti-resonance in the system, creating a minimum in oscillation amplitude at certain frequencies and enhancing vibration attenuation. This reduction lowers the transmissibility peak and decreases the amplitude response, thereby improving vibration isolation and system stability. Further increase in $m_{2}$ eliminates the stiffening softening effect, suppresses the peak transmissibility, and reduces the resonant frequency in the resonance zone, but it increases in response at higher frequencies.

Table 2 shows the effect of varying the

m_{2}

on the performance of the HSLDS-SMI isolator. As

m_{2}

increases from 0 to 3.0, the peak amplitude decreases by 51%, from 20.24 to 9.932, the resonance frequency decreases by 19%, from 0.290 to 0.234, and the isolation frequency decreases by 44%, from 0.50 to 0.277. These results demonstrate that

m_{2}

has the reverse effect of

μ

on the isolation performance, enhancing the HSLDS-SMI isolator by reducing peak amplitude, resonance frequency, and isolation frequency.

Table 2.

Influence of $m_{2}$ on peak amplitude, resonance, and isolation frequency.

Parameter value	Peak amplitude	Resonance frequency	Isolation frequency
$m_{2} = 0$	20.24	0.290	0.50
$m_{2} = 0.5$	17.73	0.279	0.418
$m_{2} = 1.5$	13.77	0.260	0.335
$m_{2} = 3.0$	9.932	0.234	0.277

Under underload conditions

Figure 12(a) and (b) show the displacement transmissibility $T_{d}$ under underload for varying $μ$ and $m_{2}$ , respectively.

Figure 12.

Displacement transmissibility for different (a) $μ$ and (b) $m_{2}$ under underload.

When operating under the rated mass, the displacement transmissibility $T_{d}$ behaves as expected when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $m_{2} = 1$ , $r = 0.707$ , $ξ = 0.02$ , and $z_{0} = 0.05$ , as illustrated in Figure 12(a). However, for $μ < 1$ , the $T_{d}$ increases. As $μ$ decreases, there is a noticeable increase in the peak $T_{d}$ , resonance frequency, initial isolation frequency, and anti-resonance frequency of the isolator. Additionally, smaller values of $μ$ shift the peak transmissibility towards the higher left region, effectively softening the dynamic response. At higher frequencies, the transmissibility becomes relatively high, leading to diminished isolation performance as $μ$ decreases.

The displacement transmissibility $T_{d}$ displays an opposite trend for varying values of $m_{2}$ compared to $μ$ , as illustrated in Figure 12(b) when $β = 0.7$ , $δ = 0.7$ , $M_{1} = 8$ , $r = 0.707$ , $ξ = 0.02$ , $μ = 1.05$ , and $z_{0} = 0.05$ . When $m_{2} = 0$ , the system exhibits softening behavior. The presence of $m_{2}$ , however, mitigates this softening effect. Increasing $m_{2}$ further suppresses the softening behavior, reduces the peak transmissibility, and lowers the resonance frequency.

Conclusions

This study explores the applicability of a newly proposed nonlinear isolator, consisting of a high-static-low-dynamic-stiffness (HSLDS) and sliding mass inertia (SMI), to vibration isolation with primary mass deviation. The harmonic balance method (HBM) is used to obtain the steady-state response of the vibrating system. The backbone curve, resonance peak, displacement transmissibility, and other typical performance indices are determined by means of detailed parametric analysis.

The findings reveal that for the system with the HSLDS isolator, the mass deviation causes the backbone curve to bend leftward, renders the amplitude-frequency response of the isolator to demonstrate a softening stiffness characteristic, and results in an increase in the displacement transmissibility in the whole frequency region. A higher mass deviation amplifies these performance degradations. For the studied cases, the mass deviation ratio $μ$ increases from 1.0 to 1.05, the peak amplitude increases by 78%, the resonance frequency by 86%, and the isolation frequency by 108%

To such performance degradations, the sliding mass has the opposite effect by mitigating the leftward bend and reducing the resonance frequency. The inclusion of SMI in the isolation system suppresses the softening exhibited by the HSLDS, and an increase in the sliding mass eliminates the softening behavior in the amplitude-frequency response and mitigates peak transmissibility. As the sliding mass $m_{2}$ increases from 0 to 3.0, the peak amplitude decreases by 51%, the resonance frequency by 19%, and the isolation frequency by 44%.

Footnotes

ORCID iD

Muhammad Umair

Funding

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

Declaration of conflicting interests

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

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Dynamic performance of an HSLDS-SMI isolator with primary mass variation

Abstract

Keywords

Introduction

The HSLDS-SMI isolator with mass deviation and its modeling

Vibration system with an HSLDS-SMI isolator

System modeling

Steady-state harmonic response

Solution by the harmonic balance method

Backbone curve

Analysis on the amplitude-frequency response

Under overload conditions

Under underload conditions

Peak responses to harmonic excitation

Under overload conditions

Under underload conditions

Displacement transmissibility analysis

Under overload conditions

Under underload conditions

Conclusions

Footnotes

ORCID iD

Funding

Declaration of conflicting interests

References