Sage Journals: Discover world-class research

Abstract

A new two-terminal mechanical element named the mem-inerter described by a relation between integrated momentum and displacement is introduced as the memory counterpart of an inerter. It exhibits an individual “fingermark” featured by a pinched hysteresis loop located within the momentum-velocity plane. The mem-inerter is attached to a simple mass-spring-damper system. The system equipped with a mem-inerter is mathematically modeled, and its nonlinear vibration equation is derived. To ensure a fair performance comparison between the systems equipped with the mem-inerter and the inerter, the nonlinear mem-inerter with an appropriate helix pitch can be proved to be equivalent to the linear inerter with a fixed inertance by the fact that the systems have the same displacement transmissibility for forced response. Under such a premise, it is found that the system with the mem-inerter having positive initial displacement has better performance for free response than the system with the inerter. Furthermore, the application scenario that both systems are arranged on an inclined plane is taken as an example of the positive initial displacement. The example demonstrates that the system with the mem-inerter has significantly better transient performance than the system with the inerter.

Keywords

Mechanical memory element displacement-dependent inerter pinched hysteresis loop nonlinear differential equation vibration analysis

Introduction

In 2002, a new ideal mechanical element named the inerter was proposed by Smith.¹ With two independently movable terminals, the inerter has the nature that the acting force on the terminals is directly proportional to the relative acceleration between them. Unlike the terminals of the inerter, those of the mass are its centroid and a fixed point in a frame of reference. Therefore, in the force-current analogy, the mass is corresponding to a grounded capacitor. Since the inerter has two independently movable terminals without the need for a frame of reference, it is analogous to an ungrounded capacitor. And thus, the damper, spring, and inerter (instead of the mass) completely correspond to the resistor, inductor, and capacitor, respectively. By applying these analogs, electrical circuits can be “translated into” mechanical systems in an unambiguous manner. As a new mechanical element, the inerter brings performance improvements for various mechanical systems.^2–5

The two types of inerter devices, a ball-screw inerter and a rack-and-pinion inerter, were developed in the Engineering Department at Cambridge University.⁶ Such devices have been successfully used in Formula One cars.² Recently, this report describes a new inerter implementation^7,8 which uses the moving mass of a fluid inside a helical path to provide the inertance. Swift et al.⁹ proved that durability and simplicity were the main advantages of this new implementation. These devices show that inerters can be mechanically implemented in different forms.

In 2014, Chen et al.¹⁰ introduced the concept of semi-active inerter. Hu et al.¹¹ provided the physical embodiment of semi-active inerter with a controllable-inertia flywheel and proved that the semi-active inerter could be realized. These research results indicate that the inertance-variable inerter can be mechanically realized, and furthermore, its inertance can be adjusted online.

In 1971, Chua¹² presented a new elementary circuit element, named the memristor, described by the relation between flux linkage and charge. Although Chua postulated the existence of the memristor, a physical passive two-terminal memristive prototype could not be constructed until 2008, when researchers at HP Labs announced that they found its engineering realization in nature.¹³ In 1976, the concept of memristor was generalized to a nonlinear dynamical system by Chua and Kang;¹⁴ as a result, a mathematical theory involving memristive systems was developed. Recently, Di Ventra et al.¹⁵ and Yin et al.¹⁶ extended this theory to memcapacitive systems and meminductive systems, thereby creating a whole class of mem-models. According to mechanical-electrical analogies, the corresponding mem-models of the mechanical counterparts of memristors, memcapacitors, and meminductors were created.¹⁷ For example, a tapered dashpot is a two-terminal mechanical resistor with its resistance having a dependence on the relative displacement.¹⁸ Another example is a rotational mechanical memcapacitor which is the subsystem comprised of the wounded part of the cable and the reel.¹⁹ From a mechanical perspective, a tapered dashpot is the memory counterpart of a damper. Likewise, a cable-reel subsystem is the memory counterpart of an angular mass. But what is the memory counterpart of an inerter with two independently movable terminals?

In this study, a new ideal mechanical element named the mem-inerter is proposed as the memory counterpart of an inerter, and then, a scheme for the mem-inerter device is presented. The proposed scheme uses the moving mass of a fluid inside a helical path to provide the inertance, and furthermore, the helical path length changes with the relative displacement between its terminals. As a result, the inertance provided by the mass is displacement-dependent, in other words, the device is a displacement-dependent inerter. Moreover, some simulation results show that the device presents pinched hysteresis loops which have been accepted as the fingermarks of memory circuit elements in the electrical domain.^12,15 The novelty of this work is that the displacement-dependent inerter is proposed to be a physical embodiment of the mem-inerter. Furthermore, in this article, the nonlinear vibration equation of a mass-spring-damper (MSD) system equipped with a mem-inerter is derived. The free and forced responses of the system are achieved by the fourth-order and fifth-order Runge–Kutta algorithm.

Displacement-dependent inertance

An internal-helix fluid inerter in Figure 1 consists of a moving piston with a helical channel surrounding its outer surface and a cylinder filled with fluid. A helical path is formed between the piston and the cylinder when the piston is inserted into the cylinder. Movement of the piston causes the fluid to flow through the helical path which generates an inertial force due to the moving mass of the fluid inside the cylinder.

Figure 1.

Schematic of an internal-helix fluid inerter.

Let $F$ be the opposite and equal force acting on the terminals which are the cylinder and the piston as shown in Figure 1. Let $A_{1}$ and $A_{2}$ be the effective cross-sectional areas of the piston and the helical channel, respectively. According to Swift et al.,⁹ the device depicted in Figure 1 can be modeled as an ideal linear inerter element; as a result, the relative acceleration between the terminals and the force applied to them are related linearly as follows

F = B \overset{\cdot\cdot}{x}

(1)

where $\overset{\cdot\cdot}{x}$ is the relative acceleration between the cylinder and the piston, and $B$ is the inertance in kg.

For a linear inerter, equation (1) can, via integration, be transformed into the following equivalent relation

p = B \overset{\cdot}{x}

(2)

where $\overset{\cdot}{x}$ is the relative velocity between the cylinder and the piston, and $p$ is the momentum deduced from $\overset{\cdot}{x}$ . Due to the fact that the capacitor provides a constitutive relation between charge $q$ and voltage $u$ , the constitutive relation of the inerter should be described as velocity-momentum relationship by applying force–current analogies.

Let $l$ be the channel length. According to Swift et al.,⁹ the inertance $B$ can be expressed as

B = ρ l \frac{A_{1}^{2}}{A_{2}}

(3)

Let $m_{F}$ be the mass of the fluid in the channel, then equation (3) can be represented in the form

B = m_{F} {(\frac{A_{1}}{A_{2}})}^{2}

(4)

It indicates that the inertance is essentially proportional to the mass of the fluid in the channel, and square of the area ratio between the piston and the channel.

Let $R$ be the radius of the piston, $r$ be the radius of the piston rod, $r_{h}$ be the radius of the helical channel, $P_{h}$ be the pitch of the helix, $w$ be the piston width, and $ρ$ be the fluid density. Equation (3) will be rewritten as follows

B = b_{0} w

(5)

where $b_{0} = (2 π ρ (R^{2} - r^{2})^{2} \sqrt{P_{h}^{2} + {(2 π R)}^{2}}) / (P_{h} r_{h}^{2})$ .

For a set of selected parameters $R$ , $r$ , $P_{h}$ , $r_{h}$ , and $w$ , the inertance is a constant value. The idea of making a constant inertance into a variable inertance can be used to design an inerter with displacement-dependent inertance. To this end, a modified cylinder with enlarged radius of the internal surface in the right half part is considered as shown in Figure 2, so that the length of the helical path is changed during the motion of the piston, and the mass of the fluid in the helical path is successively varied. According to equations (3) or (4), the inertance is consequently varied. Therefore, the linear inerter whose inertance is a constant value is transformed into the displacement-dependent inerter whose inertance is a variable value.

Figure 2.

Schematic of the displacement-dependent inerter.

For the designed internal-helix fluid inerter device with displacement-dependent inertance shown in Figure 2, the working width of the piston, which is linearly related to the length of the helical path, is equal to $(w / 2) - x$ , where $x$ is the relative displacement between the cylinder and the piston, the origin of $x$ axis is fixed on the center of the cylinder. Obviously, the piston must run between $- w / 2$ and $w / 2$ , that is, $x \in [- w / 2, w / 2]$ .

Therefore, it is needed to substitute $(w / 2) - x$ for $w$ into equation (5) as follows

B (x) = b_{0} (\frac{w}{2} - x)

(6)

It indicates that the inertance at a given time is a function of the relative displacement between the terminals.

Note that the nonlinear capacitor was defined by Chua,²⁰ as $q = \hat{q} (u)$ . Based on the force–current analogies, the nonlinear inerter should be defined by the following equation

p = \hat{p} (v)

(7)

Obviously, according to equation (7), the displacement-dependent inerter device shown in Figure 2 cannot be modeled as the nonlinear inerter.

It is worth noticing that according to Biolek’s definition for the nonlinear inerter;²¹ namely, $F = \hat{F} (a)$ or $a = \hat{a} (F)$ , the displacement-dependent inerter device cannot be modeled as the nonlinear inerter, either. It is not difficult to find that the two definitions of the nonlinear inerter mentioned above are different. The reason is that the constitutive relation of the inerter is identified as velocity–momentum relationship in this article, whereas as acceleration–force relationship by Biolek. Anyway, the displacement-dependent inerter cannot be defined as a nonlinear inerter.

The mem-inerter and pinched hysteresis loops

As can be seen in the triangular periodic table of elementary circuit elements²², the resistor, capacitor, and inductor, together with their counterparts with memory, the memristor, memcapacitor, and meminductor, are really particular cases of a whole group of higher-order elements. By applying force–current analogies, the triangular periodic table may be transferred from electrical to mechanical domain to obtain a triangular periodic table of elementary mechanical elements as depicted in Figure 3. Such a table can be used to predict new mechanical elements just as Mendeleev’s periodic table can be used to find new chemical elements. In the table, the mem-inerter is proposed and arranged in a basic element class linking integrated momentum $\int p$ and displacement $x$ . The mem-inerter, as a memory counterpart of the inerter, has two independently movable terminals and is analogous to a memcapacitor. It is worth noticing that the mem-inerter and the inerter cover the mem-mass and the mass in concept, respectively. Moreover, such a relationship can be expanded inward and outward to the apexes’ higher-order or lower-order elements in the class linking integrated momentum $\int p$ and displacement $x$ in the table.

Figure 3.

Triangular periodic table of elementary mechanical elements (a mechanical analog of the triangular periodic table of elementary circuit elements by Wang²²).

As pointed out by Jeltsema and Dòria-Cerezo,¹⁹ the mechanical analog of a memristor can be described by the relation between momentum $p$ and displacement $x$ , which are the time integrals of force $F$ and velocity $v$ , respectively. Consider a constitutive relation between displacement $x$ and integrated momentum $δ (\int p)$ , as shown in Figure 3, that is

δ = \hat{δ} (x), with δ : = \int_{- \infty}^{t} p (τ) d τ

(8)

The derivative of the latter with respect to time is in the following form

p = \frac{d \hat{δ} (x)}{d x} v

(9)

By defining $B (x) = d \hat{δ} (x) / d x$ , the displacement-dependent inerter shown in Figure 2 can be represented by the following equation

p = B (x) v

(10)

Suppose one actuates the device with dimensions from Table 1, using a sinusoidal displacement $x = A \sin (ω t + (π / 2))$ and compare the displacement-dependent inerter for different values of $P_{h}$ . Considering the maximum working stroke of the device, it is assumed that $A = 0.05 m$ and $ω = 2 π$ .

Table 1.

Displacement-dependent inerter details.

Description	Value
Piston radius $R$	50 mm
Piston rod radius $r$	6 mm
Helical channel radius $r_{h}$	8 mm
Helix pitch $P_{h}$	20 mm
Piston width $w$	100 mm
Working stroke $L$	100 mm
Fluid density $ρ$	1000 kg m⁻³

Figure 4(a) manifests that the inertance of the device is a function of the relative displacement, as mentioned previously. In addition, it is worthwhile to note that the function is monotonically decreasing. The mapping relation between the momentum and the velocity is multivalued except at the origin as shown in Figure 4(b). This means one is unable to define a one-to-one correspondence relationship between $p$ and $v$ . Such hardship is evaded by modeling the displacement-dependent inerter as a nonlinear mem-inerter, as achieved in equation (8), making use of the one-to-one correspondence relationship between integrated momentum and relative displacement demonstrated in Figure 4(c). Pinched hysteresis loops as seen in Figure 4(b) have been accepted as the fingermarks of memory circuit elements in the electrical domain.^12,15 To summarize, it is necessary and appropriate to define the displacement-dependent inerter shown in Figure 2 as a nonlinear mem-inerter.

Figure 4.

Characteristic curves of the displacement-dependent inerter: (a) the curves of inertance versus displacement; (b) the curves of momentum versus velocity; (c) the curves of integrated momentum versus displacement. —— helix pitch $P_{h} = 0.02 m$ ; ---- helix pitch $P_{h} = 0.04 m$ ; ······· helix pitch $P_{h} = 0.12 m$ , respectively.

The concept of mem-inerter helps to differentiate the differences between the mem-inerter and the ordinary displacement-dependent inerter. Notice that the mem-inerter proposed in this research is an inerter with displacement-dependent inertance. However, not all inerters with displacement-dependent inertance are mem-inerters. Only those which exhibit pinched hysteresis loops within the momentum–velocity plane (see Figure 4) are mem-inerters. Most importantly, the mem-inerter, as a two-terminal mechanical element with memory, is completely analogous to a memcapacitor, which enriches mechanical and electrical analogies. This analog allows electrical circuits with the memcapacitor to be translated over to mechanical systems with the mem-inerter in an unambiguous manner, as a result, the mathematical theory involving memcapacitive systems can be translated directly into the mechanical context. Thus, the mathematical theory involving mem-inerter systems will be better developed in the future.

Now, review and concentrate on the six two-terminal ideal elements: inductor, capacitor, resistor, spring, damper, and inerter. Each of these elements is used to ideally and approximately describe behavior of physical devices. It is the same for the mem-inerter. As pointed out by Swift et al.,⁹ in the process of modeling the fluid inerter device as an ideal element, suppose that the fluid is inelastic and dissipative effects are caused by viscosity of the fluid, the inertia generated by the fluid in the cylinder chamber and the inertia generated by the piston may be neglected. Only the inertia generated by the fluid moving in the path is considered. Such simplifications are also made in the process of modeling the device shown in Figure 2 as an ideal mem-inerter element. Besides, suppose that the mass of the fluid in the helical path is gained or lost at zero-velocity with respect to the cylinder. In this case, the momentum equation, $\overset{\cdot}{p} = F$ , that is

F = B (x) \overset{\cdot}{v} + \overset{\cdot}{B} (x) v

(11)

holds, without reference to the reactive force proportional to the velocity of the fluid which is being accreted to or expelled from the helical path.²³

MSD system equipped with a mem-inerter

Figure 5 depicts a MSD system equipped with a mem-inerter. A cart with mass $m$ is linked to the wall by a mem-inerter with inertance $B (x)$ , a linear damper with damping coefficient $c$ , and a linear spring with stiffness coefficient $k$ .

Figure 5.

MSD system equipped with a mem-inerter without any external force.

Indeed, in Figure 5, suppose that the system is in equilibrium position where the piston of the mem-inerter stays at central position, then the governing differential equation of the system without any external force can be written as

m \overset{\cdot\cdot}{x} + B (x) \overset{\cdot\cdot}{x} + \overset{\cdot}{B} (x) \overset{\cdot}{x} + c \overset{\cdot}{x} + kx = 0

(12)

Substituting equation (6) into equation (12) leads to

[m + b_{0} (\frac{w}{2} - x)] \overset{\cdot\cdot}{x} - b_{0} {\overset{\cdot}{x}}^{2} + c \overset{\cdot}{x} + kx = 0

(13)

Obviously, equation (13) is a nonlinear differential equation. If the inertance is constant, that is $B (x) = b$ , a mem-inerter becomes an ordinary inerter, then equation (12) can be expressed as

(m + b) \overset{\cdot\cdot}{x} + c \overset{\cdot}{x} + kx = 0

(14)

Note that it is a linear differential equation corresponding to a MSD system with a linear inerter without any external force.

Consider a MSD system equipped with a mem-inerter under displacement excitation, as shown in Figure 6.

Figure 6.

MSD system equipped with a mem-inerter under a displacement excitation.

The differential equation of the system is as follows

\begin{matrix} m {\overset{\cdot\cdot}{x}}_{2} + B (x) ({\overset{\cdot\cdot}{x}}_{2} - {\overset{\cdot\cdot}{x}}_{1}) + \overset{\cdot}{B} (x) ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) \\ + c ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + k (x_{2} - x_{1}) = 0 \end{matrix}

(15)

where $x_{1}$ and $x_{2}$ denote absolute displacements of the terminals of the mem-inerter, respectively. Substituting equation (6) into equation (15) leads to

\begin{matrix} [m + b_{0} (\frac{w}{2} - x_{2} + x_{1})] {\overset{\cdot\cdot}{x}}_{2} - b_{0} {\overset{\cdot}{x}}_{2}^{2} + (2 b_{0} {\overset{\cdot}{x}}_{1} + c) {\overset{\cdot}{x}}_{2} \\ + (b_{0} {\overset{\cdot\cdot}{x}}_{1} + k) x_{2} = b_{0} (\frac{w}{2} + x_{1}) {\overset{\cdot\cdot}{x}}_{1} + b_{0} {\overset{\cdot}{x}}_{1}^{2} + c {\overset{\cdot}{x}}_{1} + k x_{1} \end{matrix}

(16)

Obviously, equation (16) is a nonlinear differential equation. If the inertance is constant, that is, $B (x) = b$ , then equation (15) can be written as

(m + b) {\overset{\cdot\cdot}{x}}_{2} - b {\overset{\cdot\cdot}{x}}_{1} + c {\overset{\cdot}{x}}_{2} + k x_{2} = c {\overset{\cdot}{x}}_{1} + k x_{1}

(17)

Note that it is a linear differential equation corresponding to a MSD system with a linear inerter under displacement excitation.

Numerical examples

The system parameters, $m$ , $k$ , and $c$ , have the same values for the system equipped with the mem-inerter and the system equipped with the inerter, that is, $m = 317.5 kg$ , $k = 22, 000 N m^{- 1}$ , and $c = 1500 N s m^{- 1}$ . For the system with the mem-inerter, the inertance $B (x)$ of the mem-inerter, as one of the system parameters, is affected by the helix pitch $P_{h}$ . The values of all structural parameters of the mem-inerter including the helix pitch are given in Table 1. For the system with the inerter, the inertance is constant. Table 2 exhibits the selected values for the numerical examples in this article.

Table 2.

The selected values for the numerical examples.

	Inerter	Mem-inerter
Case	$b$ (kg)	$P_{h}$ (mm)
1	472	20
2	237	40
3	158	60

As a numerical example of forced response, the systems have zero initial values with the sinusoidal displacement input $x_{1} = A \sin (2 π ft)$ , where $f$ varies from 0.5 to 3 Hz. Considering the maximum working stroke of the mem-inerter device, here choose $A = 0.03 m$ as maximum amplitude of the displacement input, at the same time, take $A = 0.01 and 0.02 m$ for comparison. As a numerical example of free response, the systems have the initial values, $x = \pm 0.05 m$ and $\overset{\cdot}{x} = 0 m s^{- 1}$ , without any external force. In addition, the initial values, $x = \pm 0.025 m$ and $\overset{\cdot}{x} = 0 m s^{- 1}$ , are chosen for comparison.

In order to achieve the numerical solutions of the systems, MATLAB^TM is used for all computations. In view of the accuracy of the solutions, the fourth-order and fifth-order Runge–Kutta method is applied to solve these differential equations. Ode45, as a MATLAB ODE solver based on the fourth-order and fifth-order Runge–Kutta method, is used with RelTol = 10⁻⁶, AbsTol = 10⁻³, and MaxSep = 10⁻³.

Results and discussion

Forced response

For the purpose of comparing the dynamic behavior of the mem-inerter with the inerter, the forced responses of the systems for the case (1) are demonstrated in Figure 7, under the sinusoidal displacement excitation $x_{1} = A \sin (2 π ft)$ with $A = 0.03 m$ and $f = 0.8 Hz$ . The responses include the inertial force, relative displacement, relative velocity, and power related to the mem-inerter and the inerter. As can be seen in Figure 7, all curves for the inerter are sinusoidal in shape due to its linearity, while those for the mem-inerter are non-sinusoidal. Moreover, in a period of cycle, the first half-period is unequal to the second half-period. Especially, in the second half-period, the inertial force curve for the mem-inerter is seriously distorted in shape, as depicted in Figure 7(a). In addition, according to the power curves shown in Figure 7(d), and due to the fact that the area under power-time curve represents the work done, the mem-inerter absorbs or releases more energy than the inerter in the second half-period, while the situation is exactly reversed in the first half-period.

Figure 7.

Comparison between the dynamic behavior of the mem-inerter and that of the inerter for the case (1) under the sinusoidal excitation $x_{1} = A \sin (2 π ft)$ with $A = 0.03 m$ and $f = 0.8 Hz$ : (a) inertial force curves; (b) relative displacement curves; (c) relative velocity curves; and (d) power curves. —— the mem-inerter; ······· the inerter.

Figure 8 reveals the relation between the pinched hysteresis loop and the distorted wave presented in Figure 7(a). Obviously, the pinched hysteresis loop, as the fingerprint for an element with memory, does bring about changes in dynamic behavior for the mem-inerter as mentioned above. In Figure 8, the energy expelled from or accreted to the mem-inerter $\int v (p) d p$ equals the area between the $p$ and the curve axis. The areas of regions $E_{1}$ and $E_{2}$ present the amount of gained or lost energy due to the inflow or outflow of inertance of the mem-inerter in each half-period. From another point of view, the area of $E_{1}$ equals the difference between the areas inside the power curve of the mem-inerter and the $t$ axis in the fourth and the first quarter period in Figure 7(d). Similarly, the area of $E_{2}$ also equals the difference between the areas inside the power curve and the $t$ axis in the third- and the second-quarter period in Figure 7(d). And furthermore, this explains why the mem-inerter absorbs or releases more energy than the inerter in the second half-period and why the situation is just reversed in the first half-period.

Figure 8.

Relationship between the pinched hysteresis loop and the dynamic behavior of the mem-inerter for the case (1) under the sinusoidal excitation $x_{1} = A \sin (2 π ft)$ with $A = 0.03 m$ and $f = 0.8 Hz$ .

The comparison of displacement transmissibility of the systems under the sinusoidal excitation for different amplitudes is performed for all cases presented in Table 2 and is illustrated in Figure 9(a)–(c). As is shown in Figure 9, the natural frequencies of both systems decrease with increasing inertance of the inerter and decreasing pitch of the mem-inerter, respectively, and so are the resonant peaks, which means improvement in the level of vibration isolation. Most importantly, Figure 9 reveals the displacement transmissibility curves of the systems for different amplitudes for all cases nearly identical, indicating that a nonlinear mem-inerter with an appropriate helix pitch is equivalent to a linear inerter with a fixed inertance. Such an equivalent relationship is necessary to ensure a fair performance comparison between the systems equipped with the mem-inerter and the inerter for free response.

Figure 9.

Comparison between displacement transmissibility of the system equipped with the mem-inerter and that of the system equipped with the inerter for the case (1) under the sinusoidal displacement excitation of different amplitudes: (a) $A = 0.01 m$ ; (b) $A = 0.02 m$ ; and (c) $A = 0.03 m$ . —— MSD system equipped with the mem-inerter for helix pitch P_h = 0.02 m; MSD system equipped with the mem-inerter for helix pitch P_h = 0.04 m; MSD system equipped with the mem-inerter for helix pitch P_h = 0.06 m; ---- MSD system equipped with the inerter for inertance b = 472 kg; MSD system equipped with the inerter for inertance b = 237 kg; MSD system equipped with the inerter for inertance b = 158 kg.

Free response

For the purpose of investigating the difference between the dynamic behavior of the mem-inerter and that of the inerter for free response, the comparison between the responses of the system with the mem-inerter and those of the system with the inerter for the case (1) is carried out with different initial displacement values and is illustrated in Figure 10. Unlike the constant inertance of the inerter, the inertance of the mem-inerter defined by equation (9) depends on its relative displacement as graphically presented by the inertance-displacement curves in Figure 10(a). As a result, the momentum–velocity curves for the mem-inerter and the inerter are pinched hysteresis loops and lines, respectively, as shown in Figure 10(b). For these pinched hysteresis loops, the size of the petals (similar to the area of shaded regions $E_{1}$ or $E_{2}$ in Figure 8) is relevant to the initial displacement value, moreover, the larger the value is, the bigger the size is. This implies that the amount of gained or lost energy due to the inflow or outflow of inertance of the mem-inerter, increases with the initial displacement value increasing.

Figure 10.

Characteristic comparison of the mem-inerter and inerter in the systems with various initial displacement values (from left to right: $x = - 0.05, - 0.025, 0.025, and 0.05 m$ ) for free response: (a) the curves of inertance versus displacement; (b) the curves of momentum versus velocity. —— MSD system equipped with the mem-inerter for the case (1); ---- MSD system equipped with the inerter for the case (1).

The curves of displacement versus time for all cases are presented in Figure 11. As is shown in Figure 11, for free response, the system with the mem-inerter has smaller (larger) vibration amplitude than the system with the inerter when the initial displacement is positive (negative), though both systems have the same displacement transmissibility for forced response (see Figure 9). For the positive initial values, the amplitude of the system with mem-inerter compared to the system with the inerter is reduced in all cycles. In the first cycle, the amounts of vibration amplitude reduction corresponding to initial values $x = 0.025 and 0.05 m$ , are 16.3% and 32.1% for the case (1); 12.0% and 23.8% for the case (2); and 9.4% and 18.9% for the case (3), respectively. For a linear system like the system with the inerter, its displacement curves corresponding to positive or negative initial value should be symmetric about the time axis. This point can be also be seen from Figure 11, and thus, it is easy to show that the two curves corresponding to positive or negative initial value have the same amplitude. In other words, regardless of whether initial value is positive or negative, the system with the inerter has always larger vibration amplitude than the system with the mem-inerter with positive initial value. Therefore, the system with the mem-inerter with positive initial value could be used to reduce vibration amplitude. Note that the mem-inerter could be rearranged in the system with its terminals exchanged, in this case the system with negative initial value is available to reduce vibration amplitude.

Figure 11.

Comparison between free responses of the system equipped with the mem-inerter and those of the system equipped with the inerter for all cases with various initial displacement values ( $x = - 0.05, - 0.025, 0.025, and 0.05 m$ ): (a) displacement curves for the case (1); (b) displacement curves for the case (2); and (c) displacement curves for the case (3). —— MSD system equipped with the mem-inerter; ---- MSD system equipped with the inerter.

An example for benefits of the mem-inerter

Vibration of a MSD system arranged on an inclined plane is a frequent topic of discussion. Especially, when the plane is perpendicular to horizontal, it becomes suspension vibration. Generally, suppose that the MSD system, equipped with a mem-inerter in Figure 6, is placed on an inclined plane with an inclination angle $α$ from horizontal, as is shown in Figure 12.

Figure 12.

MSD system equipped with a mem-inerter and arranged on an inclined plane.

The component of the gravity acting on the cart along the inclined plane can be calculated as $mg \sin α$ , where $g$ is the acceleration due to gravity. Owing to this component force, the spring in Figure 12 is compressed. Suppose that the $x$ axis is parallel to the inclined plane and that the origin of $x$ axis is located on the position where the cart stays when the spring is uncompressed. Then, when $α = 10 \circ$ , the cart stays there where $x_{2} = - 0.025 m$ due to deformation of the spring. In this case, one can always manage to ensure that the mem-inerter has a positive initial displacement, that is, $x = 0.025 m$ , by means of exchanging its terminals from each other.

To indicate advantages of the mem-inerter, the comparison of displacement responses of the systems under the sinusoidal excitation of different amplitudes is carried out for the case (1) given in Table 2 and is demonstrated in Figure 13. The peak-to-peak $(PTP)$ values of displacement are exhibited in Table 3. The $PTP$ values of displacement are calculated as $PTP = max (x_{2} (t)) - min (x_{2} (t))$ .

Figure 13.

Comparison between $PTP$ values of displacement of the system equipped with the mem-inerter and those of the system equipped with the inerter for the case (1) under the sinusoidal displacement excitation of different amplitudes and frequencies: (a) $A = 0.01 m$ , $f = 2 Hz$ ; (b) $A = 0.02 m$ , $f = 2 Hz$ ; (c) $A = 0.03 m$ , $f = 2 Hz$ ; (d) $A = 0.01 m$ , $f = 3 Hz$ ; (e) $A = 0.02 m$ , $f = 3 Hz$ ; and (f) $A = 0.03 m$ , $f = 3 Hz$ . —— MSD system equipped with the mem-inerter for helix pitch P_h = 0.02 m; ---- MSD system equipped with the inerter for inertance b = 472 kg.

Table 3.

The $PTP$ values of displacement of the systems under the sinusoidal excitation for the case (1).

	$f = 2 Hz$			$f = 3 Hz$
	$A = 0.01 m$	$A = 0.02 m$	$A = 0.03 m$	$A = 0.01 m$	$A = 0.02 m$	$A = 0.03 m$
$PTP$ value of displacement of the system with the mem-inerter (m)	0.027	0.041	0.056	0.032	0.048	0.064
$PTP$ value of displacement of the system with the inerter (m)	0.034	0.054	0.074	0.038	0.062	0.094
Percentage of reduction (%)	20.6	24.1	24.3	15.8	22.6	31.9

PTP: peak-to-peak.

As can be seen in Figure 13 and Table 3, under the sinusoidal excitation $x_{1} = A \sin (2 π ft)$ with $f = 2 and 3 Hz$ , the $PTP$ values of displacement of the system with the mem-inerter compared to the system with the inerter decrease for all amplitudes. Moreover, the amount of reduction in the $PTP$ values increases as the sinusoidal excitation amplitude increases. Table 3 shows that the percentages of reduction corresponding to the excitation amplitudes $A = 0.01, 0.02, and 0.03 m$ are 20.6%, 24.1%, and 24.3% for excitation frequency of 2 Hz, 15.8%, 22.6%, and 31.9% for excitation frequency of 3 Hz. Therefore, for a system arranged on an inclined plane, the system with the mem-inerter has better transient performance than the system with the inerter, though both systems have the same steady state performance.

Conclusion

A new ideal mechanical element called the mem-inerter was proposed as the memory counterpart of an inerter and arranged in an element class linking integrated momentum $\int p$ and displacement $x$ at the triangular periodic table of elementary mechanical elements. A displacement-dependent inerter was proposed to be a physical embodiment of the mem-inerter. Based on some numerical modeling and simulations, the displacement-dependent inerter was shown to exhibit an individual “fingermark” featured by a pinched hysteresis loop. The nonlinear vibration equation describing a MSD system equipped with the mem-inerter was derived. Moreover, the free and forced responses of the system were obtained by the fourth-order and fifth-order Runge–Kutta algorithm.

For forced response, it is found that due to the pinched hysteresis loop, unlike the inerter, the mem-inerter has peculiar behaviors: half-period inequality and wave distortion in the time domain. However, the mem-inerter, on the other hand, exhibits some behaviors similar to that exhibited by the inerter; for instance, the systems equipped with the mem-inerter and the inerter have similar displacement transmissibility in the low-frequency regime, so that the nonlinear mem-inerter with a displacement-dependent inertance can be equivalent to the linear inerter with a fixed inertance. Such an equivalent relationship serves as a good basis to ensure a fair performance comparison between the systems equipped with the mem-inerter and the inerter for free response.

For free response, the MSD system with the mem-inerter compared to the MSD system with the inerter can reduce vibration amplitude under conditions of positive initial values. For negative initial values, the mem-inerter has to be rearranged in the system with its terminals exchanged.

The presented example for benefits of the mem-inerter exhibits that the MSD system equipped with the mem-inerter has significantly better transient performance than the MSD system with the inerter, when the systems are arranged on an inclined plane. Moreover, this example indicates that the mem-inerter with positive initial displacement can be fully used to improve transient performance which is beyond the reach of the inerter.

As a next step, the influence of the adhered nonlinearities such as friction and inherent damping should be considered and analyzed in the mem-inerter device for the purpose of practical application. In addition, further studies focusing on experiments and simulations based on more near-practical models should also be carried out.

Footnotes

Handling Editor: Jan Torgersen

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. 51405202), the Natural Science Foundation of Jiangsu Province of China (grant no. BK20130521), the China Postdoctoral Science Foundation (grant no. 2015M570408), the Six Talent Peaks Program of Jiangsu Province of China (2013-JNHB-001), and the Startup Foundation for Advanced Professional Talents of Jiangsu University (grant no. 13JDG033).

ORCID iD

Xiao-liang Zhang

References

Smith

. Synthesis of mechanical networks: the inerter. IEEE T Automat Contr 2002; 47: 1648–1662.

Chen

MZQ

Papageorgiou

Scheibe

et al . The missing mechanical circuit element. IEEE Circ Syst Mag 2009; 9: 31738.

Chen

MZQ

Huang

et al . Influence of inerter on natural frequencies of vibration systems. J Sound Vib 2014; 333: 1874–1887.

Chen

Wang

. Modeling and analyses of a connected multi-car train system employing the inerter. Adv Mech Eng 2017; 9:01703.

Chen

MZQ

Shu

et al . Analysis and optimisation for inerter-based isolators via fixed-point theory and algebraic solution. J Sound Vib 2015; 346: 17–36.

Papageorgiou

Houghton

Smith

. Experimental testing and analysis of inerter devices. J Dyn Syst: T ASME 2009; 131: 011001.

Gartner

Smith

. Damping and inertial hydraulic device. US20150167773A1 Patent, 2011.

Smith

Houghton

Long

PJG

et al . Force-controlling hydraulic device. US8881876B2 Patent, 2012.

Swift

Smith

Glover

et al . Design and modelling of a fluid inerter. Int J Control 2013; 86: 2035–2051.

10.

Chen

MZQ

et al . Semi-active suspension with semi-active inerter and semi-active damper. IFAC Proc Vol 2014; 47: 11225–11230.

11.

Chen

MZQ

et al . Semiactive inerter and its application in adaptive tuned vibration absorbers. IEEE T Contr Syst T 2017; 25: 294–300.

12.

Chua

. Memristor-the missing circuit element. IEEE T Circuits Syst 1971; 18: 507–519.

13.

Strukov

Snider

Stewart

et al . The missing memristor found. Nature 2008; 453: 80.

14.

Chua

Kang

. Memristive devices and systems. Proc IEEE 1976; 64: 209–223.

15.

Di Ventra

Pershin

Chua

. Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc IEEE 2009; 97: 1717–1724.

16.

Yin

Tian

Chen

et al . What are memristor, memcapacitor, and meminductor?IEEE T Circuits: II 2015; 62: 402–406.

17.

Pei

Wright

Todd

et al . Understanding memristors and memcapacitors in engineering mechanics applications. Nonlinear Dynam 2015; 80: 457–489.

18.

Oster

Auslander

. The memristor: a new bond graph element. J Dyn Syst: T ASME 1972; 94: 249–252.

19.

Jeltsema

Dòria-Cerezo

. Mechanical memory elements: modeling of systems with position-dependent mass revisited. In: Proceedings of the 2010 49th IEEE conference on decision and control (CDC), Atlanta, GA, 15–17 December 2010, pp.3511–3516. New York: IEEE.

20.

Chua

. Nonlinear circuit foundations for nanodevices. I. The four-element torus. Proc IEEE 2003; 91: 1830–1859.

21.

Biolek

Biolkova

et al . Nonlinear inerter in the light of Chua’s table of higher-order electrical elements. In: Proceedings of the 2016 IEEE Asia pacific conference on circuits and systems (APCCAS), Jeju, 25–28 October 2016, pp.617–620. New York: IEEE.

22.

Wang

. A triangular periodic table of elementary circuit elements. IEEE T Circuits: I 2013; 60: 616–623.

23.

Pesce

Casetta

. Systems with mass explicitly dependent on position. In: Irschik

Belyaev

(eds) Dynamics of mechanical systems with variable mass. New York: Springer, 2014, pp.51–106.

The mem-inerter: A new mechanical element with memory

Abstract

Keywords

Introduction

Displacement-dependent inertance

The mem-inerter and pinched hysteresis loops

MSD system equipped with a mem-inerter

Numerical examples

Results and discussion

Forced response

Free response

An example for benefits of the mem-inerter

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References