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Article: Wagg, D. orcid.org/0000-0002-7266-2105 and Pei, J.-S. (2020) Modeling a helical fluid inerter system with time-invariant mem-models. Structural Control and Health Monitoring. ISSN 1545-2255 https://doi.org/10.1002/stc.2579
This is the peer reviewed version of the following article: Wagg, DJ, Pei, J‐S. Modeling a helical fluid inerter system with time‐invariant mem‐models. Struct Control Health Monit. 2020;e2579, which has been published in final form at https://doi.org/10.1002/stc.2579. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
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Modeling a Helical Fluid Inerter System with Time-Invariant Mem-Models
Journal: Structural Control and Health Monitoring
Manuscript ID STC-20-0045.R1
Wiley - Manuscript type: Research Article
Date Submitted by Forthe Peer Review 13-Feb-2020 Author:
Complete List of Authors: Wagg, David; University of Sheffield, Department of Mechanical Engineering Pei, Jin-Song; The University of Oklahoma, School of CEES
hysteresis, inerter, mem-models, dual input-output pairs, higher-order Keywords: element, Masing model, mem-inerter
Structural Control and Health Monitoring Page 1 of 23 http://mc.manuscriptcentral.com/stc Received: Added at production Revised: Added at production Accepted: Added at production
DOI: xxx/xxxx 1 2 3 RESEARCH ARTICLE 4 5 Modeling a Helical Fluid Inerter System with Time-Invariant 6 7 Mem-Models 8 9 10 David J. Wagg1 | Jin-Song Pei2 11 12 13 1Department of Mechanical Engineering, 14 Sheffield University, Sheffield, UK Abstract 2School of Civil Engineering and 15 In this paper, experimental data from tests of a helical fluid inerter are used to model 16 Environmental Science, University of the observed hysteretic behavior. The novel idea is to test the feasibility of employing 17 Oklahoma, Oklahoma, USA mem-models, which are time-invariant herein, to capture the observed phenomena 18 Correspondence For Peer Review 19 *Jin-Song Pei, School of Civil Engineering by using physically meaningful state variables. Firstly we use a Masing model con- 20 and Environmental Science, University of cept, identified with a multilayer feedforward neural network to capture the physical 21 Oklahoma, Norman, OK 73019. USA Email: characteristics of the hysteresis functions. Following this, a more refined problem 22 [email protected] 23 formulation based on the concept of a multi-element model including a mem-inerter Present Address 24 Present address is developed. This is compared with previous definitions in the literature and shown 25 to be a more general model. Through-out this paper, numerical simulations are used 26 to demonstrate the type of dynamic responses anticipated using the proposed time- 27 28 invariant mem-models. Corresponding experimental measurements are processed to 29 demonstrate and justify the new mem-modeling concepts. Focusing on identifying 30 the unknown function forms in the proposed problem formulations, the results show 31 that it is possible to formulate a unified model constructed using both the damper and 32 33 inerter from the mem-model family. This model captures many of the more subtle 34 features of the underlying physics, not captured by other forms of existing model. 35 36 KEYWORDS: 37 hysteresis; inerter; mem-models; dual input-output pairs; higher-order element; Masing model; mem- 38 inerter 39 40 41 42 43 1 INTRODUCTION 44 45 The inerter is a novel dynamic device that has been the subject of substantial research interest in both academia and industry. 46 In particular, systems with inerters have been developed to improve passive control of dynamic systems, such as automotive 47 suspensions, and more recently applications in civil engineering structures. This paper describes the development of a series of 48 hysteresis models for an inerter by using new concepts recently introduced from other disciplines. The results are compared to 49 experimental results from a fluid inerter system based on a fluid filled cylinder that induces flow in a helical pipe system. 50 The term inerter was first introduced by Smith (1) using the force-current analogy between mechanical and electrical networks. 51 In this context, the inerter is considered to represent the equivalent of the capacitor. As a result it has the property that the force 52 generated is proportional to the relative acceleration between its end points (or nodes). The constant of proportionality for the 53 inerter is called inertance and is measured in kilograms. 54 55 56 57 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 2 of 23
2 Wagg and Pei 1 2 Of course, in mechanical engineering the flywheel concept had been used for many centuries previously, but typically without 3 relative accelerations. However in earthquake engineering, particularly in Japan since the late 1990s, several researchers have 4 used inertial components to enhance the effectiveness of damping devices – see summary by (2) and references therein. There 5 are several types of inerters: the rack and pinion inerter (1), the ball screw inerter (3), the fluid inerter (4, 5, 6, 7), and the 6 electromagnetic inerter (8, 9). There are three main application areas, vehicle suspensions systems (3, 10, 11, 12, 13, 14, 15), train 7 suspension systems (16, 17), and civil engineering systems (18, 2, 19, 20, 21, 22, 23, 24). The optimal performance of inerter- 8 based vibration isolation systems has been considered by several authors, see for example the discussions in 25, 26, 27, 28. In 9 general inertance is assumed to be fixed, but some recent devices have investigated the idea of variable inertance (29, 8, 5). The 10 modeling in this study elucidates the natural variations in inertance that can occur in this type of fluid inerter. 11 Fluid inerters have the advantage of minimal moving parts, with the inertia effect being generated by the motion of the fluid. 12 13 This inertial effect has two main ways of being realized. It is either due to the fluid driving a mechanical flywheel, which is 14 usually called the hydraulic inerter (4), or the mass of the fluid itself moving in a helical pipe, which is defined as the helical fluid 15 inerter, or just the helical inerter (5). In this paper we will describe the behavior of a helical inerter using memristive models, 16 otherwise known as mem-models. 17 The models developed in this study, are based on two strategies: The first is based on the Masing model for hysteresis, which 18 uses the experimental data to build the hysteresis curves. This model is cast into the memristive model framework defined in (30), 19 specifically by using dual input-outputFor pairs of Peer variables. The Review second modeling approach assumes the system is modeled by a 20 combination of mem-elements. Specifically in this case a mem-inerter and memristor. In this case the experimental data can be 21 used to (i) identify the functional form of the mem-inerter function, and (ii) compare the performance of the mem-inerter model 22 to an alternative nonlinear inerter model. 23 The rest of this paper is structured as follows. An overview of mem-modeling including the basic concepts and a suite 24 of carefully designed numerical demonstrations named “idealized responses” will be given in Section 2. An overview of all 25 26 experimental data sets, preprocessing, and the extraction of typical one-cycle loops of the steady-state responses are given in 27 Appendix A. The Masing modeling approach and results are presented in Section 3, while the multi-element mem-modeling 28 approach and results are given in Section 4. We conclude our study in Section 5. 29 30 31 2 CONCEPTS AND NUMERICAL DEMONSTRATIONS 32 33 2.1 Definitions for Mem-Models 34 35 Mem-models for dampers and springs are briefly introduced here, following the derivations of (31). The general mem-dashpot 36 model, also called memristive system model, is defined as follows for a flow-controlled setting: 37 state equations: ẏ (t) = g y (t), ̇x(t), t (1) 38 d d d 39 input-output equation: rd (t) = D yd (t), ̇x(t), t ̇x(t) (2) 40 where y (t) is the state vector, x is relative displacement, and an overdot represents differentiation with respect to time t. Here in 41 d this constitutive relation, the input is ̇x while the output is the force r . D represents a generalized damping function of the states 42 d and velocity. When t is dropped from the right-hand side of both Eqs. (1) and (2), the definition becomes that of a time-invariant 43 mem-dashpot. The exact functional forms for both g and are to be determined, which is a key challenge in applications and 44 d D 45 will be addressed in this study. 46 The general mem-spring model, also called memcapacitive system model, is defined as follows for a flow-controlled setting: 47 state equations: ẏ (t) = g y (t), x(t), t (3) 48 s s s ( ) = ( ) ( ) ( ) 49 input-output equation: rs t S ys t , x t , t x t (4) 50 where ys(t) is the state vector. Here in this constitutive relation, the input is x while the output is the force rs. S represents a 51 generalized stiffness function of the states and the displacement. When t is dropped from the right-hand side of both Eqs. (3) 52 and (4), the definition becomes that of a time-invariant mem-spring. 53 For hysteretic systems, the state variables displacement and velocity can be augmented with additional states to account for 54 memory effects. For example, by including absement and generalized momentum , which are defined as the time integral 55 a p 56 of displacement x, and the time integral of the restoring force r, respectively. Note that r will be taken as either rd , rs, the 57 inerter restoring force ri (to be introduced later), or a combined element restoring force depending on the specific context. 58 59 Structural Control and Health Monitoring 60 Page 3 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 3 1 2 The corresponding p will be pd , ps, and pi, respectively. In general, absement and generalized momentum are given by the 3 relationships 4 t t 5 a(t) = x( )d , p(t) = r( )d (5) Ê Ê 6 −∞ −∞ 7 An important property of mem-models is the so-called “zero-crossing” or “origin-crossing” property, which means that for 8 mem-dashpots ̇x and r become zero simultaneously, and for mem-springs x and r become zero simultaneously. Mem-models 9 d s have been developed to model nonlinear phenomena (32, 33, 34). Specifically, in engineering mechanics applications, mem- 10 11 models are for modeling memory effects, i.e., hysteresis. For example, the use of a and p as state variables are introduced 12 in 31, 30, 35 to capture history-/path-dependency. 13 We also would like to highlight the adjective “relative” for displacement to pave the road for the introduction of the mem- 14 inerter. In fact, all kinematic quantities involved here including absement, displacement, velocity (and later, acceleration) are 15 relative quantities being the difference of the two nodes that define the element. Mem-dampers and mem-springs are generalized 16 dampers and springs, respectively. Relative velocity and relative displacement are specified for dampers and springs, e.g., in (36). 17 Consider the following mem-dashpot and mem-spring example defined using D and S, respectively, each as a simple time- 18 invariant system model (31, 30, 35): 19 For Peer Review 20 mem-dashpot: D = −sgn( ̇x) sin x + 2 (6) 2 21 mem-spring: S = sgn(x) cos a + 2 (7) 22 2 23 Subjecting the mem-dashpot and mem-spring to the defined sinusoidal relative displacement input x(t) gives 24 25 ̈x(t) = −A!2 sin(!t) (8) 26 ̇x(t) = A! cos(!t) (9) 27 28 x(t) = A sin(!t) (10) 29 A A a(t) = − cos(!t) (11) 30 ! ! 31 where A = 1 and ! = 1 in this example. The results are shown in Figs. 1 and 2, respectively. Throughout this study, numerical 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 FIGURE 1 A particular mem-dashpot subject to the prescribed sinusoidal displacement input. For comparison using a linear 47 dashpot, would change the behavior to: from the left to right columns, there would be ellipses, straight lines with a positive 48 slope, ellipses, and constants of the value of the slope for D 49 50 51 differentiation with respect to time is carried out by using the central difference method. The MATLAB (37) code adopted here 52 is central_diff.m (38) to ensure forward and backward differences at the left and right ends, respectively, and with the same 53 second-order of accuracy as the central difference for the mid-portion. Numerical integration with respect to time is carried out 54 by using the trapezoidal rule to obtain the integral counterpart for a specified input or solved output. Four colors, red, orange, 55 56 green, and blue, highlight the responses to the four quarters of the sinusoidal input and are used in the figures to indicate the 57 behavior of the hysteresis loops. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 4 of 23
4 Wagg and Pei 1 2 3 4 5 6 7 8 9 10 11 12 13 FIGURE 2 A particular mem-spring subject to the prescribed sinusoidal displacement input. For comparison a linear spring, 14 would change the behavior to: from the left to right columns, there would be straight lines with a positive slope, ellipses, straight 15 lines with the negative slope, and constants of the value of the positive slope for S 16 17 18 19 2.2 Inerter Models For Peer Review 20 There are three sub-models of inerters. A linear inerter is defined as follows: 21 22 pi = B ̇x (12) 23 where B stands for inertance, with the unit of mass. The origin-crossing behavior is trivial, meaning that ̇x and p become zero 24 simultaneously. This definition is consistent with the other one in (39), where a linear relationship between ̈x and r is used. 25 As mentioned previously, it should be emphasized that is a relative motion. Referring to Fig. 2 for a linear inerter subject to 26 x 27 the defined sinusoidal input, the response would be from the left to right columns: straight lines with a negative slope, ellipses, 28 straight lines with the positive slope, and constants of the value of the positive slope for B. 29 A nonlinear inerter is defined as ̂ 30 ̂pi = B ( ̇x) (13) 31 where the origin-crossing behavior is assumed because it is physically meaningful. Notice that this definition differs from that 32 in (39), where the linear relationship between and is generalized. 33 ̈x r 34 The mem-inerter, which is assumed to be a displacement-dependent inerter function, is defined as 35 p = B(x) ̇x (14) 36 i 37 Or, equivalently, there is a one-to-one mapping as follows 38 = H(x) (15) 39 40 where the integrated generalized-momentum is denoted as: 41 t 42 (t) = p ( )d (16) 43 Ê i 44 −∞ 45 and the subscript i can be dropped for a more general definition of . The time derivative of H(x) is B(x) ̇x after applying the 46 chain rule. 47 The general mem-inerter model may be defined as follows for a flow-controlled setting by exercising mathematical parallelism: 48 49 state equations: ẏ i(t) = gi yi(t), ̇x(t), t (17) 50 input-output equation: pi(t) = B yi(t), ̇x(t), t ̇x(t) (18) 51 52 where yi(t) is the state vector. Here in this constitutive relation, the input is ̇x while the output is pi. B is a function that represents 53 the generalized inertance, i.e. a function of the states and the velocity. When t is dropped from the right-hand side of both 54 Eqs. (17) and (18), the definition becomes that of a time-invariant mem-inerter. For example, if the state variable is chosen as 55 x, we could have a simple time-invariant mem-inerter model as follows: 56 57 pi = B(x, ̇x) ̇x (19) 58 59 Structural Control and Health Monitoring 60 Page 5 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 5 1 2 Equations (17), (18), and (19) are the results of extending the mathematical parallelism in mem-models. In this study, we will 3 demonstrate that these formulas are useful in modeling certain types of inerter, such as the helical inerter. However, as will be 4 seen, the behavior of a physical inerter is complex, and it therefore unlikely that it could be modeled as a mem-inerter alone. It 5 should also be noted that the signals recorded from an inerter test (discussed in detail in Section 3.1) could be interpreted using 6 other nonlinear models. What we aim to show here is that the mem-model approach provides a useful framework for capturing 7 some of the more subtle behaviors exhibited by these systems. To do this, we will first provide a set of numerical examples to 8 demonstrate the modeling capability of Eq. (19) alone. Fig. 3 presents three gradually varying time-invariant mem-inerters in 9 terms of Eq. (19) and subject to the defined sinusoidal input. These mem-inerters are defined using B as follows: 10 11 sgn ( ̇x) + 1 0.2 sgn ( ̇x) − 1 −0.2 12 mem-inerter: B = + (20) 1 2 −10x 2 10x 13 1 + e 1 + e sgn ( ̇x) + 1 0.2 sgn ( ̇x) − 1 −0.2 14 B = + (21) 2 −10x−3 10x+3 15 2 1 + e 2 1 + e 16 sgn ( ̇x) + 1 0.2 sgn ( ̇x) − 1 −0.2 B3 = + − 0.1 (22) 17 2 1 + e−10x−3 2 01 + e10x+3 1 18 19 Later, we will seek to reveal the inspirationFor ofPeer Eq. (19) and Review the adopted function forms in Eqs. (20) to (22) by using the 20 experimental data of the helical inerter. 21 22 23 24 0.2 0.2 0.1 25 0 0 0 -0.1 26 -0.2 27 -0.2 -0.2 28 -1 0 1 -1 0 1 -1 0 1 29 30 0.2 0.2 0.1 31 0 0 0 -0.1 32 -0.2 33 -0.2 -0.2 34 -1 0 1 -1 0 1 -1 0 1 35 36 0.2 0.2 0.1 37 0 38 0 0 -0.1 -0.2 39 -0.2 -0.2 40 0 1 2 0 1 2 0 1 2 41 42 0 0 43 0 -0.1 44 -0.1 -0.2 -0.2 45 -0.2 -0.4 46 -1 0 1 -1 0 1 -1 0 1 47 48 49 0.2 0.2 0.2 50 0.1 0.1 51 0.1 0 0 0 52 -1 0 1 -1 0 1 -1 0 1 53 54 55 FIGURE 3 Three particular mem-inerters (following subscripts 1, 2, and 3) with each subject to the prescribed sinusoidal 56 displacement input 57 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 6 of 23
6 Wagg and Pei 1 2 The origin-crossing property for the mem-dashpot, mem-spring and mem-inerter is prominent in Figs. 1 to 3 showing r vs 3 ̇x, r vs x, and p vs ̇x,respectively. The “figure of eight" structure of the response can be seen from the corresponding plots. 4 Furthermore, the orientation of a loop can be observed following (31). For dashpots, r vs x rotates clockwise — it is symmetrical 5 for a linear dashpot, while anti-symmetrical for a mem-dashpot. Similarly for springs, r vs ̇x rotates counter-clockwise, and 6 likewise it is symmetrical for a linear spring, while anti-symmetrical for a mem-spring. Note also that the pairs of p vs x, and 7 p vs a rotate clockwise for the dashpots, while p vs x rotates counter-clockwise for the springs. For p vs a, a mem-dashpot and 8 mem-spring differ in terms of the shearing direction. 9 Column-wise in Fig. 3, three variations of a time-invariant mem-inerter model in the formulation of Eq. (19) are studied. 10 While their different origin-crossing behaviors can be observed in the panels of p vs ̇x in the first row, the piecewise defined B 11 vs x curves are plotted in last row with the equations given in Eqs. (20) to (22). Other rows in the middle reveal more inner- 12 = 0 13 workings of the models. It can be seen that when B1 vs x switches between the two curves mirroring about x , p1 vs ̇x 14 is anti-symmetrical in the first and third quadrants. When the two curves on B2 vs x do not mirror about x = 0, p2 vs ̇x does 15 not have the same loop in the first and third quadrants. When the two curves on B3 vs x become even more complex, p3 vs ̇x 16 displays different loops in the first and third quadrants. Eq. (19) and the functional forms in Eqs. (20) to (22) demonstrate how 17 the data-based models can be used to capture the complex and subtle physical behaviors exhibited by the helical fluid inerter. 18 This observation appears to be robust to the fact that the quality of the available data does not always lead to good fitting for all 19 data in this study. For Peer Review 20 Reference 40 proposes the concept of a mem-inerter and contrasts it with definitions of a linear and nonlinear inerter. The 21 physical motivation seems to compare well with this study. However, the definition for the mem-inerter in Reference 40 is not 22 as comprehensive as those for memristive and memcapacitive system models, such as those defined here. 23 24 25 2.3 Dual Input-Output Pairs, Masing Model, and Higher-Order Elements 26 27 There are a wide ranging set of models for hysteresis — for recent treatments of the subject see for example (41, 42). Commonly 28 used hysteresis models include Ramberg-Osgood (43), Bouc-Wen (44, 45, 41), bilinear hysteresis (46, 47), and classical Preisach 29 models (48, 49). In this paper mem-models will be used based on the connection between mem- and classical models, as 30 described by Reference 30. In the definition of the Masing model, as illustrated in Fig. 4, a virgin loading curve is critical 31 because any other unloading and reloading curves are the scaled and shifted version of the virgin loading curve. However, in 32 this specific application of the Masing model, we look into steady-state responses where we do not encounter the virgin loading 33 curve anymore. Instead, we see only unloading and reloading curves, as will be described in the next Section. 34 35 36 r 37 38 39 40 41 42 43 x 44 45
46 virgin loading unloading 47 reloading tangent to virgin loading 48 tangent to unloading tangent to reloading 49 50 FIGURE 4 A sketch of Masing model highlighting the identical initial slopes as in Fig. 8 and later utilized in Fig. B7 for Groups 51 1 and 2 modeling, respectively 52 53 54 The key idea in (30) is that the commonly seen hysteresis models are treated in the plane of vs where there are no origin- 55 r x 56 crossing properties due to permanent deformations involved, however all these models can be treated on the plane of ̇r vs ̇x 57 instead, where there are origin-crossing properties given the piecewise-defined restoring forces. That is, both ̇x and ̇r become 58 59 Structural Control and Health Monitoring 60 Page 7 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 7 1 2 zero simultaneously. For these models, r vs x form an input-output pair called an integral pair, where ̇r vs ̇x form an input- 3 output pair called a differential pair. Apart from the origin-crossing properties, these two pairs are inherently connected, for 4 example, tangent quantities for an integral pair are secant quantities for a differential pair. Available models for integral pairs 5 can be transformed to models for differential pairs. 6 In this study, we will investigate an alternative set of dual input-output pairs: the integral pair of p vs x and differential pair of 7 r vs ̇x. Initially, we will create a model for the inerter based on the pair of p vs x, from which we will obtain the other quantities 8 needed to define the pair of r vs ̇x. The model for the p vs x is inspired by the Masing model that is typically used for modeling 9 hysteretic behavior in the r vs x domain. See (50, 51, 52) for the Masing model for r vs x or equivalently, stress vs strain. This 10 modeling approach is new, and we show that it is successful in using one model (i.e., the Masing model on p vs x plane) to 11 capture some dominant features for multiple test data sets. The effectiveness of this simplified modeling strategies, one of the 12 13 two main approaches in this study, will be demonstrated in Section 3. The results using this strategy appear to be more accurate 14 and insightful than the linear modeling approach in (53). 15 This initial success does not come as a surprise if we think broadly in terms of higher-order element (HOE) in the multi- 16 physics domain. Among other papers on this subject, (54) introduced the concept of ( , ) elements, or higher-order elements for 17 two-terminal electronic devices. Also, (55) further generalized the concept to other multiple physical domains. For mechanics, 18 HOE promotes examining the derivatives and integrals of kinematic-kinetic pairs that we normally examine in order to define 19 constitutive relations. Mem-modelsFor and inerters Peer are just special Review cases of these type of HOEs. 20 21 22 23 3 THE MASING MODEL 24 25 In this Section we will use experimental data from tests on the inerter system in order to create a Masing type model. 26 27 28 29 3.1 Experimental data overview 30 The experimental inerter test set-up has been previously described by (5), and for completeness a brief description is included in 31 Appendix A. The inerter was subject to a sinusoidal excitation with a prescribed frequency and amplitude. Only the displacement 32 (x) and restoring force (r) time histories were recorded from the experiment. All the other time histories to be shown were 33 obtained by post-processing the recorded results based on standard signal processing methods to obtain close approximations to 34 the quantities defined in Section 2. The integrated quantities, absement a, generalized-momentum p and its integral were seen 35 to have drift and/or offset. This is primarily caused by numerical drift that is well known to occur in integrated quantities (56, 57). 36 In order to mitigate these effects the MATLAB detrend function was applied to the recorded restoring force data to limit the 37 38 effects on generalized momentum p and its integral . The resulting signals of a specific sample dataset are shown and discussed 39 in Appendix A. 40 41 42 3.2 Integral Input-Output Pair using Masing Model 43 44 In seeking to create an effective model of the inerter, we utilize the concept of higher-order elements in conjunction with the 45 Masing model. The initial step is to form two groups using the majority of the available test data sets. See Table 1. Group 1 46 consists of the data sets which clearly exhibit memristive type behavior. Group 2 consists of the data sets that show a mixture 47 of memristive and other behavior. Within each group, the analysis is the same. It is important to emphasize that this division 48 criterion is arbitrary. However, it allowed us to carry out the subsequent analysis in a consistent manner. 49 Figures 5 and 6 contrast Groups 1 and 2 in terms of r vs ̇x origin-crossing behavior. Recalling the panel of r vs ̇x in Fig. 1, 50 it can be seen that Group 1 display features of memristive system models given both the origin crossing and the switching 51 behaviors from the unloading in the first quadrant (in red arrow) to the loading in the third quadrant (in orange arrow). Group 2, 52 on the other hand, does not so clearly display the origin-crossing behavior (although some are close) and thus we assume that 53 these would not necessarily be approximated well with memristive system models. 54 An important question is whether it is possible to approximate all test results within Group 1 using one model, and those within 55 56 Group 2 using another model. This calls for an examination of the features of all test results within each group in a collective 57 manner. To do so, we extract five consecutive loops from one test, and collect all test results within Group 1 that contains a 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 8 of 23
8 Wagg and Pei 1 2 TABLE 1 Two groups of typical results selected from test data to demonstrate the usefulness of integral input-output pair using 3 Masing model. Note that the data sets are labeled in terms of the input amplitude and frequency of the sinusoidal excitation 4 5 Group 1: memristive system Group 2: mixed behavior 6 20mm 2Hz 30mm 2Hz 7 15mm 2Hz 5mm 2Hz 8 10mm 2Hz 15mm 3Hz 9 10mm 3Hz 2.5mm 7Hz 10 5mm 4Hz 2.5mm 10Hz 11 3mm 7Hz 12 13 2mm 10Hz 14 15 16 400 400 400 500 500 200 17 200 200 0 18 0 0 0 0 r r r r 19 -200 For Peer Review -200 -200 20 -400 -600 -400 -400 -500 -500 21 -200 0 200 5 -200 0 200 5 -200 0 200 5 -200 0 200 5 -200 0 200 22 x˙ x˙ x˙ x˙ 23 24 FIGURE 5 Useful loops of a typical one-cycle steady-state inerter response showing the origin-crossing behavior on the r vs 25 ̇x plane in an approximate sense: Group 1 in Table 1. From left to right, they are: 20mm 2Hz, 15mm 2Hz, 10mm 2Hz, 10mm 26 3Hz, and 5mm 4Hz 27 28 29 1000 1000
30 500 500 31 0 0 32 r r 33 -500 -500 -1000 -1000 34 1 -500 0 500 1 -500 0 500 35 x˙ x˙ 36 1000 1000 1000 1000 37 500 500 500 38 0 0 0 0 r r r r
39 -500 -500 -500 40 -1000 -1000 -1000 -1000 41 5 -200 0 2005 -200 0 2005 -200 0 2005 -200 0 200 42 x˙ x˙ x˙ x˙ 43 44 FIGURE 6 Useful loops of a typical one-cycle steady-state inerter response not showing the origin-crossing on the r vs ̇x plane 45 in an approximate sense: Group 2 in Table 1. From top to bottom and left to right, they are: 30mm 2Hz, 5mm 2Hz, 15mm 3Hz, 46 2.5mm 7Hz, 2.5mm 10Hz, 3mm 7Hz, and 2mm 10Hz 47 48 49 total of 25 cycles. For Group 1, Fig. 7 helps examine possible underlying structure in terms of higher-order input-output pairs 50 including p vs ̇x, p vs x, p vs a, and vs x, respectively. 51 In order to make sense of these figures, we first focus on the p vs x plot in Fig. 7. Then we shift each loop in the p vs x so that 52 they are all positioned with the same upper right vertex. This is the so-called shifted p vs x plot in Fig. 8. It indicates that all the 53 data sets conform to one Masing model because each data set only corresponds to a different range of the same Masing model. 54 Note that the Masing model (50, 51, 52) has been used to model restoring force vs displacement , or stress vs strain 55 r x 56 " relations. Here we use the Masing model to model generalized momentum p vs displacement x relation instead. After the 57 approximation, we will simply perform time derivatives on both generalized momentum p and displacement x to obtain a model 58 59 Structural Control and Health Monitoring 60 Page 9 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 9 1 20 2 20 3 4 10 10 5
6 0 0 7 8 -10 9 -10 p 10
11 -20 -20 12 13 -30 -30 14 15 -40 -40 16 -400 -300 -200 -100 0 100 200 300 400 -20 -15 -10 -5 0 5 10 15 20 25 x 17 20 2 18 1 19 For Peer Review 10 20 0 21 22 0 -1
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-10 φ 24 p 25 -3 26 -20 -4 27
28 -5 29 -30 30 -6
31 -40 -7 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -20 -15 -10 -5 0 5 10 15 20 25 32 a x 33 34 FIGURE 7 p vs ̇x, p vs x, p vs a, and vs x plots of all data sets with a somewhat ambiguous underlying structure: Group 1 in 35 Table 1 36 37 38 for restoring force r vs velocity ̇x, which is our true interest for characterizing the inerter behavior. This modeling strategy 39 including leveraging and taking time derivatives follows the approach described in (30). 40 In summary, p vs x pairs from the steady-state in different tests form different loops, the so-called hysteresis loops as in Fig. 7. 41 These loops can be shifted and form a set in Fig. 8, which can be considered from the same Masing model in terms of vs 42 p x 43 but with different turning points. Notice that shifting in an integral pair p vs x will not affect the differential pair r vs ̇x based 44 on calculus. 45 Looking at Fig. 8, we decide to assume that all five tests are following the same unloading curve. Since the 20mm 2Hz test 46 covers the largest range, we fit its unloading curve with a multilayer feedforward neural network (FFNN), a universal approxi- 47 mator. Five hidden nodes were chosen to produce the best approximation result. An alternative would be to collect the unloading 48 curves of all tests for fitting, say, using an FFNN. Figure 9(a) presents the experimental unloading curve and trained unloading 49 curve. In terms of the reloading curve, the Masing rule was used given the approximated unloading curve. The corresponding 50 time histories are consistent with those in Fig. A4 if following the colors in the order of orange, green, blue and red. 51 To predict the r vs ̇x plots for all other tests in Group 1, we will have to predict their corresponding p vs x plots first. We used 52 linear interpolation applied to a given time history x with respect to that of 20mm 2Hz to obtain a portion of the 20mm 2Hz’s 53 time history p. This is not a perfect choice because not all time histories of x fall 100% to the path of that for 20mm 2Hz, and so 54 approximation errors are thus anticipated. We then numerically differentiate the given time history and predicted time history 55 x 56 p to obtained time histories of ̇x and r, respectively, before combining them. Fig. 10 presents all predictions for the rest of the 57 tests in Group 1. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 10 of 23
10 Wagg and Pei 1 0 2 3 4 -10 5 6 -20 7
8 p 9 -30
10 shifted 11 -40 12 13 14 -50 15 16 -60 17 -40 -35 -30 -25 -20 -15 -10 -5 0 shifted x 18 19 FIGURE 8 p vs x plots with shiftedForx and p of Peer all data sets indicating Review a clearer structure hinting an application of the Masing 20 model: Group 1 in Table 1 21 22
23 (a) 20mm-2hz training (b) 20mm-2hz differentiation 10 500 all data 24 extended data data for NN training and differentiation 25 400 26 0 27 300
-10 28 200 29 100 30 -20
31 r p 0 32 -30 33 -100 34 experimental -200 predicted 35 -40 36 -300 37 -50 38 -400
39 -60 -500 -50 -40 -30 -20 -10 0 -400 -200 0 200 400 40 x x˙ 41 FIGURE 9 The experimental and approximated dual input-output pairs for the 20mm 2Hz test in terms of (a) p vs x, and (b) r vs 42 43 ̇x plots. In detail, the FFNN was used to approximate the lower branch of p vs x, while the upper branch was placed by following 44 the Masing’s rule. The r vs ̇x was simply obtained by differentiating the time histories of p and x before combining them 45 46 47 The capability of the proposed modeling method has thus been demonstrated. Even though some of the more subtle behaviors 48 are not entirely captured, the overall features and maximum r values are represented to a reasonable degree using a unified 49 model, especially when comparing with linear modeling where there isn’t a unified model. Recall that we only need one data 50 set for modeling, where the excitation frequency is relatively low (thus the range of x is relatively high), to predict all of the rest 51 of the responses given known displacement inputs. This is similar to some other techniques, when we intend to predict dynamic 52 responses (with high frequencies) by using static or pseudo-static tests (with low frequencies). 53 The same analysis and data processing are carried out to the Group 2 test results. See Figs. B6 to B9 in Appendix B for Group 54 2. The performance is not as good as that for Group 1, although there are some similar features. 55 56 It deserves serious thoughts on why the p vs x pair could be selected for fitting the Masing model by entirely following the 57 experimental data. p vs x is an integral input-output pair for a mem-dashpot. A non-one-to-one but a looping p vs x as in a Masing 58 59 Structural Control and Health Monitoring 60 Page 11 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 11 1 (a) 15mm-2hz interpolation (b) 15mm-2hz differentiation (a) 10mm-3hz interpolation (b) 10mm-3hz differentiation 10 400 5 500 2 all data all data extended data extended data data for interpolation and differentiation data for interpolation and differentiation 0 400 3 300 0 4 -5 300 5 200 -10 200 6 -10 100 7 -15 100 r r 8 p -20 0 p -20 0
experimental 9 predicted -25 -100 -100
-30 experimental 10 -30 -200 predicted 11 -200 -35 -300 12 -40 -300 13 -40 -400
-50 -400 -45 -500 14 -35 -30 -25 -20 -15 -10 -5 0 -200 -100 0 100 200 -25 -20 -15 -10 -5 0 -200 -100 0 100 200 x x˙ x x˙ (a) 10mm-2hz interpolation (b) 10mm-2hz differentiation (a) 5mm-4hz interpolation (b) 5mm-4hz differentiation 15 5 400 5 500 all data experimental all data extended data predicted extended data 16 data for interpolation and differentiation data for interpolation and differentiation 0 400 300 17 0 -5 300
18 200 -5 19 -10 For Peer Review 200
100 20 -15 100 -10 21 r r p -20 0 p 0
22 -15 experimental -25 -100 predicted 23 -100 -30 -200 24 -20 -200 25 -35 -300 -25 -300 26 -40 -400 27 -45 -400 -30 -500 -25 -20 -15 -10 -5 0 -150 -100 -50 0 50 100 150 -12 -10 -8 -6 -4 -2 0 -150 -100 -50 0 50 100 150 28 x x˙ x x˙ 29 30 FIGURE 10 The experimental and predicted dual input-output pairs for the 15mm 2Hz, 10mm 3Hz, 10mm 2Hz, and 5mm 4Hz 31 test in terms of (a) p vs x, and (b) r vs ̇x plots. In detail, linear interpolation was used to obtain the lower branch of p vs x, while 32 the upper branch was placed by following the Masing’s rule. The r vs ̇x was simply obtained by differentiating the time histories 33 of p and x before combining them 34 35 36 37 38 39 model corresponds to a memristive system model as was studied in (30). This means that we chose the first approximation of the 40 inerter by using a memristive system model (see both Eqs. (1) and (2) for the most comprehensive mathematical definition for 41 a flow-controlled memristive system model). We do not claim that this modeling is completely accurate, however the feasibility 42 should not come as a surprise. At least, both a helical fluid inerter and a mem-dashpot are variable energy-dissipative devices. 43 Finally we observe that the mem-models fit best with input that are low frequency and large amplitude. This applies to most 44 45 data sets in Group 1 and few in Group 2. This occurs because the time and amplitude range for the history-dependent effects to 46 develop were more significant, and as a result we believe this is why the mem-modeling approach works better for this situation. 47 48 49 50 51 4 MEM-INERTER MODELING 52 53 We now assume the inerter to be modeled is a parallel combination of mem-elements, dominated by the mem-inerter and a 54 memristor. Initially we pose the problem by assuming that we are unsure of the exact type of elements in the model, and want 55 56 to use the experimental data to identify this. We also investigate whether the definition for the mem-inerter in (40) is adequate 57 in this context. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 12 of 23
12 Wagg and Pei 1 2 With these uncertainties regarding the sub-models in mind, we start with the following overall equations that are general and 3 flexible enough to capture the points mentioned above. Thus we define our possible models as 4 5 pi + pd = B(x) ̇x + G(x) , or (23) «¯¬ «¯¬ 6 ⎧ mem-inerter memristor 7 p = ⎪ ̂pi + pd = ̂p( ̇x) + G(x) (24) 8 ⎪ «¯¬ «¯¬ 9 ⎨ nonlinear inerter memristor ⎪ 10 ⎪ 11 where the first terms on the righthand-side (RHS)⎩ account for the contribution from the inerter, while the second terms on RHS 12 is from a damping element, assumed in the first instance to be a memristor, the simplest form of a mem-dashpot. Capturing 13 both variable inertance and variable damping follows well with the physics observed in the experiments. If the definition for 14 the mem-inerter in Reference (40) is inadequate, then it would show up in the term of B(x). We treat this as a model selection 15 problem: Based on our identification, we choose the model that fits the data better.
16 First, when ̇x = 0 in either Eqs. (23) or (24), we can extract all points to model pd = G(x) for the memristor alone. If the 17 identified function form is linear (without memory), then it means that there is only a linear dashpot. If the identified function 18 is a one-to-one nonlinear function, then it means that there is a mem-dashpot. Two sets of these points are presented in Figs. 11 19 for Groups 1 and 2. An FFNN withFor three hidden Peer nodes was used Review to fit each sets of the points; the fitted results are presented in 20 the same figures. We highlight that when x = 0, p = 0, and the first and third quadrant behaviors are not symmetrical. 21 22 23
24 20 25 10 26 0
27 0 28 29 -20 30 -10
31 -40
32 -20 33 -60 34 35 -30
-80 36 experimental experimental fitted fitted 37 -40 38 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 25 39
40 FIGURE 11 The experimental and approximated pd = G(x) for the mem-dashpot component of all experiments in Table 1 41 using a FFNN with three hidden nodes: Left plot Group 1, right plot Group 2. 42 43 44 45 To appreciate what the fitted pd = G(x) curves mean, Fig. 12 shows the r vs ̇x plots belonging to the memristor portions for 46 all Group 1 tests. Since the fitted pd = G(x) functions are the integral pairs of the memristors, we used the experimental input 47 x as the domain of each pd = G(x), and obtained pd by simulating the trained FFNN. Then we numerically differentiated the 48 time histories of both x and pd before combining them for a rd vs ̇x loop. The origin-crossing is anticipated for rd vs ̇x with 49 r = D(x) ̇x, where D = dpd . These loops do not have a switching feature because they are for memristors not memristive system d dx 50 models. 51 The generalized momentum for the memristor portion, p , as just obtained was removed from the total generalized momentum 52 d to obtain the generalized momentum for the mem-inerter portion, p . Fig. 13 presents all plots belonging to the mem-inerter 53 i portions for Group 1 tests. These plots include vs x, p vs ̇x, and B vs x, where both numerical differentiation and numerical 54 i integration were used, and B = pi . 55 ̇x 56 Four out of five pi vs ̇x plots in Fig. 13 for Group 1 display the origin-crossing behavior, which is the signature of an inerter. 57 Nonetheless, negative B values on B vs x are not meaningful. It is not difficult to match B < 0 portion to those branches in the 58 59 Structural Control and Health Monitoring 60 Page 13 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 13 1 1500 2 3
4 1000 5 6 7 500 8 9 0 10 11 12 -500 13 14 15 -1000 16
17 -1500 18 19 For Peer Review 20 -2000 -400 -300 -200 -100 0 100 200 300 400 21 22 23 FIGURE 12 The approximated rd = D(x) ̇x for the mem-dashpot component of all experiments in Group 1 in Table 1 using a 24 FFNN with three hidden nodes 25 26 27 second and fourth quadrants on pi vs ̇x, very similar to disallowing D < 0 for the mem-dasphot in terms of rd vs ̇x. We can trim 28 those physically meaningless portions. 29 Four out of five B vs x plots in Fig. 13 show some consistency, indicating the possibility of using one model for all tests in 30 Group 1. It seems that there is no one-to-one mapping between x and B. Instead, it seems that B switches between two functions 31 32 of x according to ̇x. With this said, we can redefine the functional form of the model to be 33 p = pi + pd = B(x, ̇x) ̇x + G(x) (25) 34 «¯¬ «¯¬ 35 mem-inerter memristor 36 37 meaning that indeed, the behavior is better modeled by a mem-inerter, not a nonlinear inerter. In addition, this mem-inerter 38 is more complicated than the definition in (40). This mem-inerter is a system-level model as proposed previously in Eq. (19). 39 Similar to the system-level mem-models studied in (31), ̇x switches between two functions of B(x). 40 Extracting physically meaningful portion of the data from (the four out of five cases in) Fig. 13 and overlaying them to identify 41 a unified function form, e.g., in terms of B vs. x, would not lead to a good result given the quality of the data. Nonetheless we 42 can qualitatively compare these five cases with those three numerical examples given previously in Fig. 3. In addition to finding 43 similarities in terms of p vs ̇x and B vs x, vs x is now qualitatively comparable. In fact, these three numerical examples were 44 inspired by the experimental results discussed here. As a result, we believe it is possible that Eqs. (20) to (22) could thus inspire 45 more refined models using future experimental data. 46 In terms of mem-inerter modeling, when x = 0 in either Eqs. (23) or (24) and when we also assume G(0) = 0, we can extract 47 all points belonging to the inerter component meaning the B and ̇x terms, and the corresponding difference between the p and G 48 terms. This is done but not presented here due to space considerations. It is uncertain how these points should be fitted making 49 50 an origin-crossing curve. The differential pair is always more challenging to fit than its integral pair counterpart. 51 Finally, differentiating Eqs. (23) and (24), we have 52 ̇ 2 ri + rd = B(x) ̇x + B(x) ̈x + D(x) ̇x , or (26) 53 «¯¬ «¯¬ 54 r = ⎧ mem-inerter memristor 55 ⎪ ̇ ̂ri + rd = ̂p( ̇x) ̇x + ̂p( ̇x) ̈x + D(x) ̇x (27) 56 ⎪ «¯¬ «¯¬ ⎨ nonlinear inerter memristor 57 ⎪ 58 ⎪ 59 Structural⎩ Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 14 of 23
14 Wagg and Pei 1 2 0 0 0 0 0 3 -1 -1 -1 -1 -1 4 φ φ φ φ φ 5 -2 -2 -2 -2 -2 6 7 -3 -3 -3 -3 -3 8 -20 0 20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 9 x x x x x 10 20 20 20 20 20 11 10 10 10 10 10
12 0 0 0 0 0 i i i i i p p p p p 13 -10 -10 -10 -10 -10 14 -20 -20 -20 -20 -20 15 -30 -30 -30 -30 -30 16 -200 0 200 -200 0 200 -200 0 200 -200 0 200 -200 0 200 17 x˙ x˙ x˙ x˙ x˙ 18 0.2 0.2 0.2 0.2 0.2 19 For Peer Review 0.1 0.1 0.1 0.1 0.1 20 0 0 0 0 0
21 B B B B B
22 -0.1 -0.1 -0.1 -0.1 -0.1 23 -0.2 -0.2 -0.2 -0.2 -0.2 24 -20 0 20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 25 x x x x x 26 27 FIGURE 13 The approximated pi = B(x) ̇x for the mem-inerter component for a typical one cycle: Group 1 in Table 1. From 28 left to right, they are: 20mm 2Hz, 15mm 2Hz, 10mm 2Hz, 10mm 3Hz, and 5mm 4Hz. Notice: There are physically meaningless 29 portions 30 31 32 When ̇x = 0, we can extract all corresponding ̈x so as to verify the existence of B(x). If this does not go well for a specific set of 33 data, it indicates that perhaps the inerter is a nonlinear inerter rather than a mem-inerter. Note that Eq. (26) differs from Eq. (11) 34 in Reference (40) because the latter is differentiated from Eq. (14), a less general version than Eq. (23). 35 36 37 5 CONCLUSION 38 39 The helical fluid inerter is a new device with important properties for passive structural control. Specifically it can provide both 40 inertance and viscous damping simultaneously in a device with minimal moving components. However, its inherent hysteretic 41 characteristics are quite challenging in terms of creating an effective model for the device. Our exploration in this study shows that 42 43 some of the physics exhibited by the helical fluid inerter can be accurately captured using a newly developed theory called mem- 44 models for hysteresis. More specifically, the functional relationships beyond the commonly-seen combinations of displacement, 45 velocity, and restoring force have been demonstrated. These relationships depend on the integral quantities including absement, 46 generalized momentum, and the integral of generalized momentum, which represent the natural memory effects that occur in 47 the device. 48 By following the origin-crossing formulation of mem-modeling, which comes from the underlying instantaneous switching of 49 the governing quantities, we have been able to demonstrate the effectiveness of the mem-modeling approach for the helical fluid 50 inerter. In particular the results reported in this paper included both the Masing model (see Figs. 9 and B8 for training results 51 and Figs. 10 and B9 for validation) and a mem-elements model that revealed the functional form of the mem elements (see both 52 Figs. 11 and 13). These models indicate the potential usefulness that mem-modeling can offer. In future work, the current data 53 could be potentially augmented by using different excitation. For example, creep or relaxation-like tests as suggested in (35) or 54 quasi-static force tests to estimate = ( ) for mem-models. 55 pd G x 56 We have considered how a combination of mem-elements, such as mem-inerter and memristor may be used to capture more 57 fully the physical behavior of the helical fluid inerter device. This also included a discussion of the subtle difference between the 58 59 Structural Control and Health Monitoring 60 Page 15 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 15 1 2 mem-inerter and a nonlinear inerter model – an important distinction when modeling practical devices. The results demonstrate 3 that mem-modeling can enable engineers to build models of this type of device that more closely match the experimental data, 4 and therefore give better validation results. Ultimately, we believe this will enable engineers to design helical fluid inerters with 5 increased accuracy and therefore increased design confidence. 6 7 8 How to cite this article: D. J. Wagg, and J. S. Pei (2019), Modeling a Helical Fluid Inerter System with Time-Invariant Mem- 9 Models, Structural Control and Health Monitoring 10 11 12 13 APPENDIX 14 15 A EXPERIMENTAL SETUP 16 17 Details of the experimental testing can be found in (5). For completeness a brief description of the system is included here. A 18 schematic diagram of the fluid inerter system considered in this paper is shown in Fig. A1. The inerter system is designed with a central fluid filled cylinder, radiusFor, which isPeer attached to a helicalReview coil on the outside of the cylinder. The helical coil has an 19 r2 20 internal radius of r3, and the helix radius is r4. The fluid is driven through the cylinder using a plunger of radius r1. 21 22 23 Cross Section 24 25 d 26 27 28 29 30 31 32 33 34 h 35 36 FIGURE A1 Schematic diagram of the inerter design showing the longitudinal cross section, and the top view of the system 37 38 39 Having a larger helix radius will lead to a higher inertance being generated by the fluid inerter system. In this paper the radius 40 was set at its maximum possible value during testing of 120mm. 41 For the purpose of testing the inerter was installed in a test rig as shown in Fig. A2. The test rig consists of a hydraulic test 42 43 machine that can be used to give a dynamic displacement signal to the upper input of the inerter. The lower input of the inerter 44 is constrained not to move, and connected to a force transducer. The upper input is attached to a hydraulic actuator, the position 45 of which is measured with a linear variable differential transformer (LVDT). The actuator is controlled using bespoke computer 46 software associated with the test machine. Finally, a laser thermometer was used to monitor the inerter temperature to ensure a 47 consistent operating temperature. A close up photograph of the inerter system is shown in Fig. A2, where the helical pipe system 48 can be clearly seen. The helical pipe was made of flexible tubing held in place by pipe clips. The central chamber was made of 49 steel. 50 Details of the specific set-up parameters for each test phase are given in Table A1. 51 In Fig. A3, the restoring force r has been detrended before any other numerical manipulation. Figure A3 also shows approx- 52 r r p imations to the quantities D = , S = , and B = . It is important to clarify these computations are only meaningful for 53 ̇x x ̇x a mem-dashpot, mem-spring, and mem-inerter, respectively, and it is therefore to be expected that not all values in these time 54 histories are physically meaningful, e.g., there are values of 0, 0, and 0. This supports the assertion that this type 55 D < S < B < 56 of inerter cannot be modeled as a single mem-element alone. The analysis of the recorded responses will enable us to develop 57 both a simplified model and a more sophisticated model. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 16 of 23
16 Wagg and Pei 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 For Peer Review 20 21 FIGURE A2 Photograph of the inerter mounted vertically in the test machine showing the external helix piping 22 23 24 25 TABLE A1 System parameters 26 27 Property Value Units 28 Helix pitch, ℎ 30 mm 29 30 Wall thickness, d 5 mm 31 r1 14 mm 32 r2 25 mm 33 r3 6 mm 34 r4 120 mm 35 Stroke of cylinder 150 mm 36 Kinematic viscosity at 40◦ 2.1 cSt 37 Oil density 802 kg/m3 38 39 40 41 In order to understand the physical behavior of the inerter system in more detail, the data from Fig. A3, was processed into 42 a representative set of data for a single period of oscillation. The results are shown in Fig. A4. Here absement and generalized- 43 momentum are numerically integrated from zero for each cycle. 44 In addition to time series representations, the data was plotted in the form of cyclic loops for a single steady-state oscillation, 45 and the results are shown in Fig. A5. Four colored arrows, red, orange, green, and blue, again, highlight the responses to the 46 four quarters of the sinusoidal input. The top row contains r vs ̈x, r vs ̇x, r vs x, and ̇x vs x plots, which are commonly used in 47 analysis of cyclic signals. The bottom row contains p vs ̇x, p vs x, p vs a, and vs x plots, which will be examined closely in 48 the rest of the paper following the concept of HOE. The key idea is that it may be challenging to create a model by using the 49 50 commonly seen (top row) plots, however we may be able to seek insights and even solutions from those less commonly seen 51 (bottom row) plots. 52 53 54 55 B RESULTS OF GROUP 2 IN SIMPLIFIED MODELING 56 57 Figures B6 to B9 are results of Group 2 in Section 3. 58 59 Structural Control and Health Monitoring 60 Page 17 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 For Peer Review 20 21 22 23 24 25 26 27 FIGURE A3 Detrended time histories of inerter response from the onset of a test: 20mm 2Hz 28 29 30 C ACKNOWLEDGEMENTS 31 32 The Faculty Investment Program awarded by the Research Council at the University of Oklahoma is acknowledged for launching 33 this collaboration. Professor Keith Worden is acknowledged for facilitating this collaboration. A discussion on the Masing model 34 offered by Professor Jim Beck but not utilized in this study is appreciated. 35 36 37 38 REFERENCES 39 40 1. Smith MC. Synthesis of Mechanical Networks: The Inerter. IEEE Transactions on Automatic Control 2002; 47: 1648-1662. 41 42 2. Ikago K, Kenji S, Norio I. Seismic control of single-degree-of-freedom structure using tuned viscous mass damper. 43 Earthquake Engineering & Structural Dynamics 2012; 41(3): 453-474. 44 3. Chen MZQ, Papageorgiou C, Scheibe F, Wang FC, Smith MC. The missing mechanical circuit. IEEE Circuits and Systems 45 Magazine 2009; 1531-636X: 10-26. 46 47 4. Wang FC, Hong MF, Lin TC. Designing and testing a hydraulic inerter. In: Proceedings of the Institution of Mechanical 48 Engineers, Part C: Journal of mechanical Engineering Science. 225. ; 2010: 66-72. 49 50 5. Smith NDJ, Wagg DJ. A fluid inerter with variable inertance properties. In: Proceedings of the 6th European Conference 51 on Structural Control; 2016; Sheffield: 1-8. Paper 192. 52 53 6. Liu X, Jiang JZ, Titurus B, Harrison A. Model identification methodology for fluid-based inerters. Mechanical Systems and 54 Signal Processing 2018; 106: 479–494. 55 56 7. De Domenico D, Deastra P, Ricciardi G, Sims ND, Wagg DJ. Novel fluid inerter based tuned mass dampers for optimised 57 structural control of base-isolated buildings. Journal of the Franklin Institute 2019; 356: 7626–7649. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 18 of 23
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4 -5 -200 5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 6 20 3 7 10 2 8 0 -10 1 9 0 10 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 11 500 12 0 0 13 -20 14 -500 -40 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 15
16 200 0 17 -2 100 18 -4 -6 For Peer Review0 19 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 20 21 0 10000 22 5000 -100 23 0 -200 24 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 25 26 FIGURE A4 Time histories of a typical one-cycle steady-state inerter response: 20mm 2Hz. Four colored arrows, red, orange, 27 green, and blue, highlight the responses to the four quarters of the sinusoidal input 28 29 30 400 400 400 31 200 200 200 200 32 0 0 0 33 0 -200 -200 -200 34 -400 -400 -400 35 -200 -600 -600 -600 36 -5 0 5 -200 0 200 -10 0 10 20 -10 0 10 20 37 104 38 20 20 20 2 0 39 0 0 0 40 -2 -4 41 -20 -20 -20 42 -6 -40 -40 -40 -8 43 -200 0 200 -10 0 10 20 0 1 2 3 -10 0 10 20 44 45 46 FIGURE A5 Useful loops of a typical one-cycle steady-state inerter response: 20mm 2Hz 47 48 49 8. Gonzalez-Buelga A, Clare LR, Neild SA, Jiang JZ, Inman DJ. An electromagnetic inerter-based vibration suppression 50 device. Smart Materials and Structures 2015; 24(5): 055015. 51 52 9. Sugiura K, Watanabe Y, Asai T, Araki Y, Ikago K. Experimental characterization and performance improvement evaluation 53 of an electromagnetic transducer utilizing a tuned inerter. Journal of Vibration and Control 2019: 1077546319876396. 54 55 56 10. Papageorgiou C, Smith MC. Laboratory experimental testing of inerters. In: 44th IEEE Conference on Decision and Control 57 and the European Control Conference; 2005; Seville, Spain: 3351-3356. 58 59 Structural Control and Health Monitoring 60 Page 19 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 19 1 40 2 40 3
4 20 20 5 6 7 0 0 8 9 10 -20 -20
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12 -40 -40 13 14 15 -60 -60 16 17 18 -80 -80 19 For Peer Review
20 -100 -100 -300 -200 -100 0 100 200 300 -30 -20 -10 0 10 20 30 21 x 22 40 5 23 24 20 25 0 26
27 0 28
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35 -60 36 37 -15 38 -80 39 40 -100 -20 41 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -30 -20 -10 0 10 20 30 42 a x 43 FIGURE B6 p vs ̇x, p vs x, p vs a, and vs x plots of all data sets without a strong indication of underlying structure: Group 2 44 in Table 1 45 46 47 11. Papageorgiou C, Houghton NE, Smith MC. Experimental Testing and Analysis of Inerter Devices. Journal of Dynamic 48 Systems, Measurement and Control, ASME 2009; 131: 011001-1 - 011001-11. 49 50 12. Shen Y, Chen L, Liu Y, Zhang X. Influence of fluid inerter nonlinearities on vehicle suspension performance. Advances in 51 Mechanical Engineering 2017; 9(11): 1687814017737257. 52 53 13. Kuznetsov A, Mammadov M, Sultan I, Hajilarov E. Optimization of improved suspension system with inerter device of the 54 quarter-car model in vibration analysis. Arch. Applied Mechanics 2011; 81: 1427-1437. 55 56 14. Wang FC, Chan HA. Vehicle suspensions with a mechatronic network strut.. International Journal of Vehicle Mechanics 57 and Mobility 2011; 49(5): 811-830. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 20 of 23
20 Wagg and Pei 1 2 0 3 4 5 -20 6 7 8 -40 9
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12 shifted 13 14 -80 15 16
17 -100 18 19 For Peer Review
20 -120 -60 -50 -40 -30 -20 -10 0 21 shifted x 22 23 FIGURE B7 p vs x plots with shifted x and p of all data sets indicating a clearer structure hinting an application of the Masing 24 model: Group 2 in Table 1 25 26 27 15. Evangelou S, Limebeer DJN, Sharp RS, Smith MC. Mechanical steering compensators for high-performance motorcycles. 28 Transactions of the ASME 2007; 74: 332-346. 29 30 16. Wang FC, Liao MK, Liao BH, Su WJ, Chan HA. The performance improvements of train suspension systems with 31 mechanical networks employing inerters. International Journal of Vehicle Mechanics and Mobility 2009; 47: 805-830. 32 17. Lewis T, Jiang J, Neild S, Gong C, Iwnicki S. Using an inerter-based suspension to improve both passenger comfort and 33 track wear in railway vehicles. Vehicle System Dynamics 2019: 1–22. 34 35 18. Wang FC, Hong MF, Chen CW. Building suspensions with inerters. Proceedings of the Institution of Mechanical Engineers, 36 Part C: Journal of Mechanical Engineering Science 2010; 224(8): 1605–1616. 37 38 19. Lazar IF, Neild SA, Wagg DJ. Using an inerter-based device for structural vibration suppression. Earthquake Engineering 39 & Structural Dynamics 2014; 43(8): 1129-1147. 40 41 20. Giaralis A, Taflanidis AA. Reliability-based design of tuned mass-damper-inerter (tmdi) equipped multi-storey frame build- 42 ings under seismic excitation. In: Proc. of 12th International Conference on Applications of Statistics and Probability in 43 Civil Engineering, ICASP12; 2015. 44 21. Lazar IF, Neild SA, Wagg DJ. Vibration suppression of cables using tuned inerter dampers. Engineering Structures 2016; 45 46 122: 62-71. 47 22. Pan C, Zhang R, Luo H, Li C, Shen H. Demand-based optimal design of oscillator with parallel-layout viscous inerter 48 damper. Structural Control and Health Monitoring 2017. 49 50 23. Gonzalez-Buelga A, Lazar IF, Jiang JZ, Neild SA, Inman DJ. Assessing the effect of nonlinearities on the performance of 51 a tuned inerter damper. Structural Control and Health Monitoring 2017; 24(3). 52 53 24. Giaralis A, Taflanidis A. Optimal tuned mass-damper-inerter (TMDI) design for seismically excited MDOF structures with 54 model uncertainties based on reliability criteria. Structural Control and Health Monitoring 2018; 25(2): e2082. 55 56 25. Scheibe F, Smith MC. Analytical solutions for optimal ride comfort and tyre grip for passive vehicle suspensions. Journal 57 of Vehicle System Dynamics 2009; 47: 1229-1252. 58 59 Structural Control and Health Monitoring 60 Page 21 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 21 1 (a) 30mm-2hz training (b) 30mm-2hz differentiation 2 20 1500 all data experimental 3 extended data predicted data for NN training and differentiation 4 0 5 1000 6
7 -20
8 500 9 10 -40 11 r p 0 12 13 -60 14 -500
15 -80 16 17 -1000 18 -100 19 For Peer Review 20 -120 -1500 21 -60 -50 -40 -30 -20 -10 0 -300 -200 -100 0 100 200 300 ˙ 22 x x 23 FIGURE B8 The experimental and approximated dual input-output pairs for the 30mm 2Hz test in terms of (a) p vs x, and (b) 24 r vs ̇x plots. In detail, the FFNN was used to approximate the lower branch of p vs x, while the upper branch was placed by 25 following the Masing’s rule. The r vs ̇x was simply obtained by differentiating the time histories of p and x before combining 26 them 27 28 29 26. Smith MC, Wang FC. Performance benefits in passive vehicle suspensions employing inerters. Journal of Vehicle System 30 Dynamics 2004; 42: 235-257. 31 32 27. Wang FC, Su WJ. Impact of inerter nonlinearities on vehicle suspension control. International Journal of Vehicle Mechanics 33 and Mobility 2008; 46: 575-595. 34 35 28. Krenk S, Høgsberg J. Tuned resonant mass or inerter-based absorbers: unified calibration with quasi-dynamic flexibility 36 and inertia correction. Proc. R. Soc. A 2016; 472: 20150718. 37 38 29. Brzeski P, Kapitaniak T, Perlikowski P. Novel type of tuned mass damper with inerter which enables changes of inertance. 39 Journal of Sound and Vibration 2015; 349: 56-66. 40 41 30. Pei JS, Gay-Balmaz F, Wright JP, Todd MD, Masri SF. Dual input-output pairs for modeling hysteresis inspired by mem- 42 models. Nonlinear Dynamics 2017; 88(4): 2435-2455. 43 44 31. Pei JS, Wright JP, Todd MD, Masri SF, Gay-Balmaz F. Understanding memristors and memcapacitors for engineering 45 mechanical applications. Nonlinear Dynamics 2015; 80(1): 457-489. 46 32. Chua LO. Memrister - The Missing Circuit Element. IEEE Transactions on Circuit Theory 1971; CT-18(5): 507-519. 47 48 33. Chua LO, Kang SM. Memristive Devices and Systems. In: Proceedings of the IEEE. 64. ; 1976: 209-223. 49 50 34. Di Ventra M, Pershin YV, Chua LO. Circuit Elements with Memory: Memristors, Memcapacitors, and Meminductors. In: 51 Proceedings of IEEE. 97. ; 2009: 1717-1724. 52 53 35. Pei JS. Mem-spring models combined with hybrid dynamical system approach to represent material behavior. ASCE Journal 54 of Engineering Mechanics 2018. 55 56 36. Karnopp DC, Margolis DL, Rosenberg RC. System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. 57 John Wiley & Sons, Inc. 5 ed. 2012. 58 59 Structural Control and Health Monitoring 60 http://mc.manuscriptcentral.com/stc Page 22 of 23
22 Wagg and Pei 1 (a) 5mm-2hz interpolation (b) 5mm-2hz differentiation (a) 15mm-3hz interpolation (b) 15mm-3hz differentiation 10 500 20 1500 2 all data all data extended data extended data 3 data for interpolation and differentiation 400 data for interpolation and differentiation 4 0 0 1000 5 300
6 -10 200 7 -20 500 8 100 -20 9 r p 0 r 10 p -40 0 -30 11 -100
12 experimental -60 -500 experimental -200 predicted predicted 13 -40
14 -300 -80 -1000 15 -50 16 -400
17 -60 -500 -100 -1500 -12 -10 -8 -6 -4 -2 0 -100 -50 0 50 100 -30 -25 -20 -15 -10 -5 0 -300 -200 -100 0 100 200 300 18 x x˙ x x˙ (a) 2.5mm-7hz interpolation (b) 2.5mm-7hz differentiation (a) 2.5mm-10hz interpolation (b) 2.5mm-10hz differentiation 5 1000For Peer Review5 1500 19 all data all data extended data extended data data for interpolation and differentiation data for interpolation and differentiation 20 800 0 0 21 1000 600 22 -5 -5
23 400 24 -10 500 -10 25 200 26 -15 r r p 0 p -15 0
27 -20
-200 28 -20 29 -25 -500 experimental -400 predicted
30 -25 experimental -30 predicted 31 -600 -1000 32 -30 -35 33 -800
34 -40 -1000 -35 -1500 -5 -4 -3 -2 -1 0 -150 -100 -50 0 50 100 150 -5 -4 -3 -2 -1 0 -200 -100 0 100 200 x x˙ x x˙ 35 (a) 3mm-7hz interpolation (b) 3mm-7hz differentiation (a) 2mm-10hz interpolation (b) 2mm-10hz differentiation 5 1500 5 1500 36 all data experimental all data experimental extended data predicted extended data predicted data for interpolation and differentiation data for interpolation and differentiation 37 0 0 38 1000 1000 39 -5 40 -5 41 -10 500 500
42 -10 -15 r r 43 p 0 p 0
-20 44 -15 45 46 -25 -500 -500 -20 47 -30
48 -1000 -1000 -25 49 -35 50
-40 -1500 -30 -1500 51 -6 -5 -4 -3 -2 -1 0 -150 -100 -50 0 50 100 150 -4 -3 -2 -1 0 -150 -100 -50 0 50 100 150 52 x x˙ x x˙ 53 FIGURE B9 The experimental and predicted dual input-output pairs for the 5mm 2Hz, 15mm 3Hz, 2.5mm 7Hz, 2.5mm 10Hz, 54 3mm 7Hz, and 2mm 10Hz test in terms of (a) p vs x, and (b) r vs ̇x plots. In detail, linear interpolation was used to obtain the 55 lower branch of p vs x, while the upper branch was placed by following the Masing’s rule. The r vs ̇x was simply obtained by 56 57 differentiating the time histories of p and x before combining them 58 59 Structural Control and Health Monitoring 60 Page 23 of 23 http://mc.manuscriptcentral.com/stc
Wagg and Pei 23 1 2 37. The Mathworks, Inc.MATLAB. 2010. http://www.mathworks.com/. 3 38. Canfield R. central_diff.m. http://www.mathworks.com/matlabcentral/fileexchange/12-central-diff-m/content/central_diff. 4 5 m; 2015. 6 39. Biolek D, Biolek Z, Biolkova V, Kokla Z. Nonlinear Inerter in the Light of Chua’s Table of Higher-Order Electrical 7 Elements. In: 2016 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS); 2016; Jeju, South Korea: 617 - 620. 8 9 40. Zhang XL, Gao Q, Nie J. The mem-inerter: A new mechanical element with memory. Advances in Mechanical Engineering 10 2018; 10(6): 1-13. 11 12 41. Ikhouane F, Rodellar J. Systems with hysteresis: analysis, identification and control using the Bouc-Wen model. John Wiley 13 & Sons . 2007. 14 15 42. Visintin A. Differential models of hysteresis. 111. Springer Science & Business Media . 2013. 16 17 43. Jennings PC. Periodic Response of a General Yielding Structure. Journal of the Engineering Mechanics Division, 18 Proceedings of the American Society of Civil Engineers 1964; 90(EM2): 131-166. 19 For Peer Review 44. Bouc R. Modèle mathématique d’hystérésis. Acustica 1971; 21: 16-25. 20 21 45. Wen YK. Equivalent Linearization for Hysteretic Systems Under Random Excitation. ASME Journal of Applied Mechanics 22 1980; 47: 150-154. 23 24 46. Caughey TK. Sinusoidal Excitation of a System with Bilinear Hysteresis. Journal of Applied Mechanics 1960; 27: 640-643. 25 26 47. Caughey TK. Random Excitation of a System with Bilinear Hysteresis. Journal of Applied Mechanics 1960; 27: 649-652. 27 48. Krasnosel’skii MA, Pokrovskii AV. Systems with Hysteresis. Springer-Verlag . 1989. 28 29 49. Mayergoyz I. Mathematical Models of Hysteresis. Springer-Verlag . 1991. 30 31 50. Jayakumar P. Modeling and Identification in Structural Dynamics. PhD thesis. California Institute of Technology, Pasadena, 32 CA; 1987. 33 34 51. Jayakumar P, Beck JL. System Identification Using Nonlinear Structural Models. In: Nake HG, Yao JTP., eds. Structural 35 Safety Evaluation Based on System Identification Approaches. Proceedings of the Workshop at Lambrecht/Pfalz of Vieweg 36 International Scientific Book Series. Friedr. Vieweg & Sohn Braunschweig/Wiesbaden; 1988: 82-102. 37 38 52. Beck JL, Jayakumar P. Class of Masing Models for Plastic Hysteresis in Structures. In: Proceedings 14th ASCE Structures 39 Congress; 1996; Chicago, IL. 40 53. Wagg DJ. Modelling the hysteretic behaviour of a fluid inerter. In: X International Conference on Structural Dynamics, 41 42 EURODYN 2017; 2017. 43 54. Chua L. Nonlinear circuit foundations for nanodevices. I. The four-element torus. In: Proceedings of the IEEE. 91. ; 2003. 44 45 55. Biolek D, Biolek Z, Biolkova V. Memristors and other higher-order elements in generalized through-across domain. In: 46 Proceedings of the 2016 IEEE International Conference on Electronics, Circuits and Systems (ICECS); 2016; Monte Caro, 47 Monaco. 48 49 56. Worden K. Data Processing and Experiment Design for the Restoring Force Surface Method, Part I: Integration and 50 Differentiation of Measured Time Data. Mechanical Systems and Signal Processing 1990; 4(4): 295-319. 51 52 57. Worden K. Data Processing and Experiment Design for the Restoring Force Surface Method, Part II: Choice of Excitation 53 Signal. Mechanical Systems and Signal Processing 1990; 4(4): 321-344. 54 55 The DOI for the data is https://doi.org/10.15131/shef.data.11843202.v1. 56 57 58 59 Structural Control and Health Monitoring 60