Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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A methodology for identifying optimum vibration absorbers with a reaction mass

Sara Ying Zhang

Sara Ying Zhang

Institute of Urban Smart Transportation and Safety Maintenance, Shenzhen University, Shenzhen 518060, People's Republic of China

Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

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Yi-Yuan Li

Yi-Yuan Li

Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

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Jason Zheng Jiang

Jason Zheng Jiang

Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

[email protected]

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Simon A. Neild

Simon A. Neild

Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

Department of Civil Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

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John H. G. Macdonald

John H. G. Macdonald

Department of Civil Engineering, University of Bristol, Queen's Building, University Walk, Bristol, UK

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    Abstract

    Tuned mass dampers (TMDs), in which a reaction mass is attached to a structural system via a spring–parallel–damper connection, are commonly used in a wide range of applications to suppress deleterious vibrations. Recently, a mass-included absorber layout with an inerter element, termed the tuned mass damper inerter (TMDI), was introduced, showing significant performance benefits on vibration suppression. However, there are countless mass-included absorber layouts with springs, dampers and inerters, which could potentially provide more preferred dynamic properties. Currently, because there is no systematic methodology for accessing them, only an extremely limited number of mass-included absorber layouts have been investigated. This paper proposes an approach to identify optimum vibration absorbers with a reaction mass. Using this approach, a full class of absorber layouts with a reaction mass and a pre-determined number of inerters, dampers and springs connected in series and parallel, can be systematically investigated using generic Immittance-Function-Networks. The advan- tages of the proposed approach are demonstrated via a 3 d.f. structure example.

    1. Introduction

    The concept of attaching an additional reaction mass to a dynamically excited structural system for the suppression of its oscillatory motion is among the first passive vibration control strategies in the area of structural dynamics. In this context, Frahm [1] introduced the use of a linear spring–mass–damper device, the tuned mass damper (TMD) in 1909, to suppress excessive vibration in a variety of mechanical engineering applications [24]. Den-Hartog proposed an effective tuning approach [5], which has been extended by many researchers, e.g. Warburton [6]. An alternative connection arrangement was consequently proposed, termed the non-traditional TMD [7]. In contrast to the traditional TMD, which has one point attached to the primary structure, such a device has two attachment points, respectively, connected to the reaction mass via a spring and a damper. More effective seismic-vibration mitigation performance has been demonstrated using this type of device [8].

    More recently, by adding an inerter into the TMD and employing two structural attachment points, Marian & Giaralis [9] proposed a new vibration suppression device, the tuned mass damper inerter (TMDI). The inerter is a relatively new passive mechanical element, introduced by Smith [10], in which the generated force is proportional to the relative acceleration across its two terminals. Inerter-based absorbers have been shown to be effective in various applications, such as automotive [11], buildings [1215] and railway vehicles [1618]. For the TMDI, it has been shown in [9] that enhanced performance can be achieved compared with the TMD. It has also been shown in [19] that under some conditions, the TMDI outperforms the tuned inerter damper (TID [12]), which is equivalent to a TMDI with zero mass. The significant benefits of the TMDI demonstrate the potential advantages of mass-included vibration absorbers, with all three types of two-terminal mechanical elements (springs, dampers and inerters) included.

    The existing mass-included dynamic absorbers have all been proposed using an ad hoc approach. However, when considering mass-included absorbers with springs, dampers and inerters, there are countless topological connection possibilities, some of which can potentially provide much more advantageous performance. An approach, to systematically cover a wide range of these connection possibilities, is yet to be developed. This paper establishes an approach to systematically characterize a full set of passive vibration absorbers with series–parallel1 connections of one reaction mass and a pre-determined number of springs, dampers and inerters. Classical network synthesis (e.g. [21,22]) and graph theory (e.g. [23,24]) are used to facilitate the enumeration of topological connection possibilities. Using the proposed approach, the optimum reaction-mass absorber can be obtained for given vibration suppression problems.

    This paper is structured as follows. In §2, vibration suppression of an example structure is studied using existing absorbers. In addition, the problem of identifying the optimum vibration absorber with one reaction mass including one or two attachment points is formulated. Section 3 enumerates all the possible mass-included absorber layouts with one attachment point, with which the optimum vibration absorber and the corresponding performance benefits are identified for the example structure. Section 4 proposes an approach to construct generic Immittance-Function-Networks—this captures the full set of absorber possibilities with two attachment points. In §5, the proposed approach for absorbers with two attachment points is applied to the example structure, illustrating the performance advantages that can be achieved. Conclusions are drawn in §6.

    2. Existing approach and problem formulation

    (a) Previous-studied absorbers applied to an example structure

    Several passive absorbers, consisting of masses, springs and dampers, have been proposed for mitigating unwanted vibrations, the best known is the TMD [3], see figure 1a. In figure 1, 1 and 2 indicate two attachment points to the structure (or terminals in network synthesis terminology). As the TMD has a single attachment point, 2 is not connected. An alternative spring–damper–mass device, the non-traditional TMD [7] with two attachment points, has also been proposed, as shown in figure 1b. Popular examples of two terminal devices, which use an inerter rather than a mass, include the TID [12] and the L2 [25] shown in figure 1c,d. A third class of devices is one which uses both mass and inerter elements—an example is the TMDI [9], figure 1e.

    Figure 1.

    Figure 1. Schematic plot of example passive vibration absorbers (a) the TMD, (b) the non-traditional TMD, (c) the TID, (d) the L2 and (e) the TMDI, and a 3 d.f. structure model (f ).

    In order to demonstrate the vibration suppression abilities of these and other devices, a 3 d.f. structure model subjected to earthquake excitation, as shown in figure 1f , is introduced, with floor masses ms and inter-storey stiffness ks. The structural damping is taken to be zero as it is typically negligible compared with that introduced by absorbers. Note that the TID is more effective in vibration suppression when placed at the bottom of the building, while the TMD and the TMDI are more effective at the top. However, since the focus of this work is on the optimum mass-included vibration absorber identification methodology, we only consider the absorbers mounted at a single location, at the top of the building as shown in figure 1f . Here, it exerts forces −fu and −fl on the upper and lower floors, respectively. The forces are defined as positive to the right on the absorber. Vibration suppressors attached to two floors are usually installed via a brace that spans between storey levels. Inevitably this brace has some compliance, so a brace stiffness kb in series with the absorber and connecting to the lower floor is included. In this study, the structure parameters are adopted as ms = 10 000 kg, ks = 15 000 kNm−1, and the brace stiffness is taken as kb = 0.2ks in line with [26]. The equation of motion for the model of figure 1f , in the matrix form, in the Laplace domain, can be derived as

    ms000ms000mss2X+2ksks0ks2ksks0ksksX=ksX0FlFu,2.1
    where in the Laplace domain, X(s) = [XI, XII, XIII]T represents the floor displacement vector, X0(s) is the ground displacement (note xi(t)LXi(s),x0(t)LX0(s) and fu(t)LFu(s),fl(t)LFl(s)). Owing to the brace stiffness kb, an additional degree of freedom with displacement xb is introduced, where Fl = kb(XII − Xb) is in the Laplace domain. In this example, the inter-storey drift displacement, which is a measure of the potential seismic damage of a structure, is taken as the performance measure. The frequency response function of the ith floor inter-storey drift displacement, denoted as Xdi, can be obtained from (2.1) with Xdi = Xi − Xi−1, where i = I, II, III. The objective function, related to the inter-storey drift displacements, is then defined as
    Jd=Max(overi)Max(overω)|Ts2X0Xdi|,(i=I,II,III),2.2
    where Ts2X0 → Xdi denotes the transfer function from the earthquake acceleration to the inter-storey drift displacements and Max|Ts2X0 → Xdi| is the maximum magnitude of Ts2X0 → Xdi.

    Consider the four example vibration suppressors shown in figure 1ae. The value of the reaction mass in the vibration absorbers is selected as m = 1000 kg for the TMD and the TMDI, approximately 3.6% of the first modal mass. The optimization is carried out to minimize the value of the objective function Jd. By selecting the component coefficients of each absorber layout, the optimum value of Jd is obtained and summarized in table 1. Note that Matlab is used for the optimization throughout this work, using the optimization function patternsearch to obtain approximate optimum solutions, with the genetic algorithm ‘GPS basis 2N’ [27]. The solutions from the patternsearch algorithm are then used as initial estimates for the gradient-based function fminsearch for fine-tuning. The convergence criterion for both patternsearch and fminsearch is a certain pre-determined tolerance on the change in the value of the cost function over the iteration. In this study, the relative tolerance is set to be 1 × 10−4. Furthermore, to identify the global minimum, multiple starting points have been used for patternsearch optimization. From table 1, it can be seen that the TMDI provides a 42% performance improvement over the TMD and also outperforms the TID, the L2 and the non-traditional TMD with 26%, 43% and 74% smaller values of Jd for this example structure and reaction mass. The frequency responses of the inter-storey drift displacements with the three devices, the TMD, TID and TMDI are shown in figure 2, with short horizontal lines indicating the value of Jd. It can be noticed from this figure that the TMDI device results in the smallest drift displacements in the vicinity of all three structural modes of the 3 d.f. structure. Note that the responses of the non-traditional TMD and the L2 absorbers are not shown in figure 2 due to their large Jd values.

    Figure 2.

    Figure 2. Frequency response in inter-storey drifts for the TMD (dashed), TID (dash-dotted) and the TMDI (solid). (Online version in colour.)

    Table 1. Optimization results of the previous-studied layouts' example shown in figure 1 with m = 1000 kg where applicable.

    configurations performance Jd ( × 10−3s2) optimal parameter values (kg, kNsm−1, kNm−1)
    TMD 25.0 (–) c = 11.6, k = 2.75 × 102
    non-traditional TMD 55.0 (−120%) c = 22.0, k = 1.81 × 102
    TID 19.6 (21.6%) b = 7.94 × 103, c = 7.50 × 102, k = 4.35 × 103
    L2 25.5 (−2.00%) b = 4.79 × 103, c = 32.6, k = 3.53 × 103
    TMDI 14.5 (42.0%) b = 1.12 × 103, c = 9.50, k = 6.22 × 102

    The significant performance benefit of the TMDI over the studied TMD and the inerter-based TID demonstrates the potential advantages of mass-included inerter-based absorbers where all four mechanical elements, inerters, dampers, springs and masses are used. The possible topological connections with these four types of elements are numerous. Hence, it is extremely challenging to systematically identify the most beneficial configurations among them. To this end, a systematic approach will be developed in this work.

    (b) Problem formulation

    Series–parallel mechanical networks with one reaction mass and any pre-determined number of inerters, dampers and springs are considered in this paper. For the two-terminal networks consisting only of non-mass elements, the network synthesis theory (e.g. [21,22]), developed in the electrical domain, has been adopted to facilitate a systematic analysis, making use of the force–current analogy [28]. Compared with electrical realizations, vibration absorbers have strict weight and space constraints for real-life implementations. Hence, it is crucial to minimize the required element number in the mechanical networks. This observation led to the structure-immittance approach [29] being proposed for devices consisting of inerters, dampers and springs. This approach, which employed network synthesis theory, made use of the fact that all three of these element types have two terminals. When a reaction mass is included into the networks, a systematic approach becomes much more challenging, as the reaction mass is a one-terminal element (its centre of mass, figure 3a1).

    Figure 3.

    Figure 3. Absorber schematic plots and their network representations: (a1) a reaction mass with its network representation (a2), and (a3) an inerter with one terminal connected to ground (G); (b1) the TMD with its network representation (b2); (c1) the TMDI with its network representation (c2).

    In order for network synthesis to be directly applicable to systematic enumeration of vibration absorbers with a reaction mass, it is necessary to treat the mass as a special two-terminal element, with one terminal notionally connected to the ground, denoted as a notional-ground (NG). Note it is not actually connected to ground, in contrast to an electrically grounded capacitor which is its equivalent in the force–current analogy [10]. The network representation of the mass is shown in figure 3a2, with the property that F(s) = m(sVm − 0), where Vm is the velocity of the mass with its value defined as positive to the right. Accordingly, in this work, terminals connected to physical attachments are denoted as physical-terminals (PTs) when considering their network representations. Note that mathematically a mass element is equivalent to an inerter with one of the two terminals physically connected to ground (figure 3a3). However, the mass element has its unique merit since attaching absorbers' terminals physically to the ground is unrealistic for a lot of applications. With the proposed network representation of the mass (figure 3a2), the two mass-included suppression devices, the TMD and the TMDI (figure 3b1,c1), can be depicted as networks, as shown in figures 3b2,c2. By denoting the absorber attachment points 1 (2) as PT1 (PT2) in the network representation, the TMD with one attachment point becomes a two-terminal network with one PT and one NG, termed a ‘1PT1NG network’. It can also be noted that for the TMDI device with two attachment points, its corresponding network, shown in figure 3c2, is no longer two-terminal, but a three-terminal network with two PTs and one NG, denoted as a ‘2PT1NG network’. Similarly, the non-mass absorbers with two attachment points, e.g. the TID (figure 1c) are termed ‘2PT networks’. Note that the spring, damper and inerter elements can be regarded as special cases of 2PT networks, and are termed ‘2PT elements’.

    Considering the fact that most vibration suppression devices have no more than two attachment points, our investigation focuses on 1PT1NG and 2PT1NG networks. The 1PT1NG network, represented by its force–velocity transfer function H(s) = F1(s)/V1(s), is shown in figure 4a, where at the PT1, the force f1 (F1(s) in the Laplace domain) is applied and results in a velocity v1 (V1(s)). Figure 4b shows a 2PT1NG network, with forces f1, f2 and velocities v1, v2 at two PTs. Note that because of the reaction mass, in contrast to the 2PT network (whose immittance function is Y (s) = F(s)/(V1(s) − V2(s)), see figure 4c), the forces f1, f2 of the 2PT1NG network are not equal and opposite. To describe the relations between the velocities and the forces in the Laplace domain, an Immittance-Function-Matrix (IF-Matrix), denoted as L(s), is required. The derivation of the IF-Matrix for a given 2PT1NG network is detailed in appendix A.

    Figure 4.

    Figure 4. (a) 1PT1NG network, (b) 2PT1NG network and (c) 2PT network.

    The rest of the paper addresses the following two questions:

    (1)

    Given one reaction mass and any pre-determined number of inerters, dampers and springs, how to enumerate all possible series–parallel ‘1PT1NG’ and ‘2PT1NG’ network layouts?

    (2)

    Based on (1), how to systematically identify the optimum absorber configuration for a given vibration suppression problem?

    In order to address Question (1), procedures to construct 1PT1NG and 2PT1NG Immittance-Function-Networks (IF-Networks) need to be introduced. Here, an IF-Network refers to a network layout with its 2PT sub-networks represented by Immittance-Function-Blocks (IF-Block, e.g. figure 4c). Generic IF-Networks which capture all IF-Network possibilities for given conditions will be identified. Different distribution cases of the pre-determined numbers of inerters, dampers and springs in the IF-Blocks of the generic IF-Networks will then be discussed, to obtain all possible series–parallel 1PT1NG and 2PT1NG network layouts. In the subsequent sections, the 1PT1NG networks will be firstly discussed, followed by the 2PT1NG cases. It turns out that the 2PT1NG case is much more complicated, but can provide significantly enhanced performance.

    3. 1PT1NG network layout enumeration and case demonstration

    This section considers the 1PT1NG network layouts with a reaction mass. The series and parallel connections between a 2PT network (represented by an IF-Block) and a 1PT1NG network are first described using concepts defined in graph theory [23]. A generic IF-network is then formulated, from which all possible 1PT1NG networks with a pre-determined number of 2PT elements can be enumerated. Subsequently, a case study is demonstrated where the 2PT elements are specified as one inerter, one damper and one spring.

    (a) Connection between 2PT and 1PT1NG networks

    While connecting 2PT and 1PT1NG networks is relatively straightforward, we choose to introduce concepts defined in graph theory [23] to describe the processes as they are necessary for the more complicated 2PT1NG network constructions later. In the electrical domain [30], a graph is used as a general representation of topological connections, consisting of a finite number of vertexes and branches. Here, the correspondence between graphs and mechanical networks is introduced.

    For a two-terminal mechanical network with specific connection topology, by depicting each element of the network as a branch, the two terminals and internal connection points as vertexes, the network can be represented as a graph, with a set of branches interconnected at their vertexes. For example, consider a network with only one mechanical element, such as the spring shown in figure 5a1, this can be represented as a graph with one branch, figure 5b1, where two vertexes shown as solid circles correspond to the two PTs, termed the terminal-vertexes. The graph of an example 2PT network, the TID of figure 5a2, is provided in figure 5b2. In this example, an intersection vertex exists, shown as a hollow circle, representing the inter-connected node of the TID. For a 1PT1NG network example, consider the TMD (figure 5a3); its graph is shown in figure 5b3 with one terminal-vertex corresponding to an NG in the mechanical domain. In this way, all 2PT and 1PT1NG networks can be represented as a graph with two terminal-vertexes.

    Figure 5.

    Figure 5. Example of two-terminal mechanical networks and their corresponding graphs: (a1) a spring, (a2) the TID, (a3) the TMD and (b1), (b2), (b3) the corresponding graphs.

    In [23], the series and parallel connections between graphs with two terminal-vertexes are defined. It states that the series connection is to coalesce one terminal-vertex of each graph into an intersection-vertex of the resulting two-terminal graph, of which the two terminal-vertexes are the remaining ones of the two connected graphs. Alternatively, a parallel connection is where two terminal-vertexes of each graph are connected together to formulate the two terminal-vertexes of the resulting two-terminal graph. Note that both these connection types can only be applied to terminal-vertexes. Based on this definition, we consider joining a 2PT network with a 1PT1NG network.

    Taking the graphs of figure 6a1,a2 as an example, we can note that only the series connection is possible, as a parallel connection would necessitate an NG being connected with a PT. The series connection between these two graphs results in the graph shown in figure 6a3. As a graph is a general representation, each branch can represent different elements. Also there can be more branches between two vertexes and there can be more intersection vertexes in a graph. The three graphs, figure 6a1–a3, can correspond to the three general network representations shown in figure 6b1–b3, where Y (s) is an IF-Block representing any possible series–parallel 2PT network and H(s) is a 1PT1NG network. As an example by depicting the IF-Block Y (s) as a spring (figure 6c1) and the 1PT1NG network H(s) as a TMD of figure 6c2, the series connection between them formulates the network shown in figure 6c3.

    Figure 6.

    Figure 6. Example series connection between a 2PT network and a 1PT1NG network, represented as connections between (a) graphs, (b) IF-Networks and (c) example network layouts. (Online version in colour.)

    (b) Network layout enumeration using the generic 1PT1NG IF-Network

    In order to formulate series–parallel 1PT1NG IF-Networks, a collection of a reaction mass and a finite number of IF-Blocks is now considered. A non-unique connection sequence is proposed, with which all possible 1PT1NG IF-Networks can be obtained. In the procedure, after each step, any obvious network simplification will be carried out. For example, if two IF-Blocks are connected in series or in parallel, they will be reduced to a single IF-Block. Start with a single IF-Block, it can be connected in series or in parallel with other IF-Blocks; however, these always reduce to a single IF-Block. At a certain step, the resulting IF-Block is connected to the mass, based on figure 6, only a series connection is possible, resulting in a new 1PT1NG network. Further addition of IF-Blocks can only be connected in series with this 1PT1NG network, which can be reduced to a single IF-Block. Hence, all the IF-Networks can be represented by the generic IF-Network shown in figure 7 with a single IF-Block Y (s).

    Figure 7.

    Figure 7. The generic 1PT1NG IF-Network.

    Consider the 1PT1NG network layouts with one reaction mass and a pre-determined number, N, of 2PT elements. All the network possibilities can be obtained using the generic IF-Network of figure 7, by enumerating the full class of 2PT network possibilities consisting of N elements in the IF-Block Y (s). To this end, the structure-immittance approach [29], developed to systematically express all possible series–parallel networks with pre-determined number of 2PT elements, can be directly applied. The obtained structural immittance Y (s) is then used to express the transfer function of the generic IF-Network, see figure 7, as

    H(s)=Y(s)ms(ms+Y(s)).3.1
    With this transfer function, the optimum 1PT1NG vibration absorber with a reaction mass can be identified for a given vibration suppression problem.

    (c) 1PT1NG network case demonstration

    To illustrate the enumeration and the systematic identification approach, a 1PT1NG case with one reaction mass, one damper, one spring and one inerter is analysed. From the generic IF-Network of figure 7, in total eight network layouts can be enumerated by allocating all possible series–parallel 2PT networks consisting of one each of the three 2PT element types (see table 2 in [29]), into the IF-Block. The immittance function Y (s) is then formulated using the structure-immittance approach to cover all 2PT network possibilities, given by [29]

    Y(s)=bcs2+b(k4+k6)s+c(k2+k6)bc(1/k3)s3+bs2+cs+k2+k4,3.2
    or
    Y(s)=bc(1/k1+1/k2)s3+bs2+cs+k3b(1/k1+1/k5)s3+c(1/k2+1/k5)s2+s,3.3
    with the condition that b≥0, c≥0, ki≥0, and for the function (3.2), at least three of the parameters k2, 1/k3, k4, k6 must equal zero (covering four 2PT networks), and for (3.3), at least three of the parameters 1/k1, 1/k2, k3, 1/k5 must equal zero (covering the other four 2PT networks). The transfer function of the generic IF-Network, H(s) of (3.1), with Y (s) expressed as (3.2) or (3.3) can now be used for the optimum absorber identification.

    The 3 d.f. structure shown in figure 1f is taken as an application example. The 1PT1NG network layouts consisting of one reaction mass (1000 kg as before), one inerter, one damper and one spring, connected to the top storey of the structure, are considered as candidate absorber layouts. By optimizing the objective function (2.2) with the transfer function H(s) (3.1), the optimum absorber configuration, C1, is shown in figure 8a1, with the corresponding parameter values obtained as b = 1.05 × 103kg, c = 30.5 kNsm−1, k = 4.92 × 102kNm−1. With this configuration, the optimum result of the objective function is Jd = 0.016, providing a 35% and a 17% performance improvement compared with the TMD and the TID, respectively. However, when compared with the 2PT1NG network, the TMDI, it can be seen from table 1 that the optimal 1PT1NG configuration, C1, is unable to match its vibration suppression performance, resulting in a 11% larger value of Jd. In addition to the TMDI, there are numerous other 2PT1NG networks that use the same numbers of each type of element, for which even higher performance benefit might be achievable. For example, the mass-included absorber (termed C2) shown in figure 8a2 is a 2PT1NG network, with which the objective function Jd can be reduced to 0.014 with b = 1.21 × 104kg, c = 13.9 kNsm−1 and k = 2.98 × 102kNm−1. It can be seen that the C2 outperforms the TMDI by 6.7%. The inter-storey drift frequency responses of the structure model with all five proposed absorbers are shown in figure 9. These results motivate the following investigation into the full class of 2PT1NG networks made up of one reaction mass and a pre-determined number of inerters, dampers and springs.

    Figure 8.

    Figure 8. (a1) The optimum 1PT1NG configuration C1 and (a2) a 2PT1NG network example C2.

    Figure 9.

    Figure 9. Frequency response in inter-storey drifts with the TMD (dashed), TID (dash-dotted), the TMDI (solid), the C1 (dotted) and the C2 (thick dashed). (Online version in colour.)

    4. 2PT1NG network layout enumeration

    In this section, 2PT1NG network layouts with a reaction mass are considered. The series and parallel connections between a 2PT and a 2PT1NG network are firstly described, after which a procedure for formulating 2PT1NG IF-Networks is introduced. The generic 2PT1NG IF-Networks, covering all the IF-Network possibilities with a given number of IF-Blocks, are formulated. Using these generic IF-Networks, the enumeration of all possible network layouts is then discussed, together with the IF-Matrix derived for systematic optimization.

    (a) Connection between 2PT and 2PT1NG networks

    Using the same correspondence between mechanical networks and graphs, as used in §3a, any 2PT1NG network can be represented by a three-terminal graph with one of the terminal-vertexes corresponding to the NG. Taken the TMDI (figure 10a1) as an example, its graph representation can be depicted as figure 10a2, which consists of three terminal-vertexes and one intersection-vertex.

    Figure 10.

    Figure 10. An example 2PT1NG network, the TMDI, (a1) and its graph representation (a2).

    To formulate the series–parallel 2PT1NG networks, the series and parallel connections between a 2PT and a 2PT1NG network also need to be introduced based on the definitions for three-terminal graphs [24]. In [24], the series connection concept is similar to that for the connection between two two-terminal graphs, described in the previous section. For a parallel connection, both terminal-vertexes of a two-terminal graph and two of the three terminal-vertexes of a three-terminal graph are connected together. The resulting graph has three terminal-vertexes of the original three-terminal graph but now two of these are shared with the original two-terminal graph. Based on these observations, considering the connection of an example two-terminal graph corresponding to any 2PT network (figure 11a1) and a three-terminal graph shown in figure 11a2 representing a 2PT1NG network, the series connection results in the two possibilities, shown in figure 11a3,a4, respectively. By coalescing one terminal-vertex of figure 11a1 with the left terminal-vertex of figure 11a2, figure 11a3 is obtained, while figure 11a4 is formulated by connecting figure 11a1 with the right terminal of figure 11a2. Figure 11a5 shows the graph obtained by the parallel connection; because of the existence of NG, the parallel connection can only result in this possibility. The general network examples obtained from the graphs are shown in figure 11b1–b5. By depicting the general network representations Y (s) and L(s) as specific layout examples, the series and parallel connections between a 2PT network (spring) and a 2PT1NG network (TMDI) are shown in figure 11c1–c5.

    Figure 11.

    Figure 11. Example series and parallel connection between a 2PT network and a 2PT1NG network, represented as connections between (a) graphs, (b) IF-Networks and (c) example network layouts. (Online version in colour.)

    (b) Formulation of generic 2PT1NG Immittance-Function-Network

    In order to formulate 2PT1NG IF-Networks, similar to 1PT1NG case, a collection of a reaction mass and a finite number of IF-Blocks is now considered. A sequence of steps is introduced based on the work of Nishizeki & Saito [24], shown in figure 12. At each step, a sub-network (either a 1PT1NG network or an IF-Block) is connected in series or in parallel with the network formed in the previous step. While such a procedure is note unique, using this procedure, any series–parallel 2PT1NG IF-Network which can be formed from the original set can be obtained. In the procedure, after each formulation step, we will carry out any obvious network simplification, same as the simplifications we carried out for the 1PT1NG case, for example if two IF-Blocks are connected in series or in parallel, they will be reduced again to a single IF-Block. Similar to [24], in which the construction of a three-terminal series–parallel graph begins with an empty graph, figure 12–Step 1 is taken as the start. We first consider joining the two terminals, PT1 and NG. Using the generic 1PT1NG IF-Network obtained in §3b (figure 7), a new network shown in figure 12–Step 2 is obtained. The next step is to add a single IF-Block Y2(s) to the network, resulting in figure 12–Step 3 (via a parallel connection between the terminals PT1 and PT2). Consider adding the next IF-Block, Y3(s), resulting in the new IF-Network shown in figure 12–Step 4 using the Series1 connection in figure 11. Note that all the other connection possibilities between figure 12–Step 3 and the IF-Block Y3(s) can all be simplified to figure 12–Step 3. At this point, only a parallel IF-Block can be added, with the resulting IF-Network shown in figure 12–Step 5, since series IF-Blocks can be reduced to the network of Step 4. Following this parallel addition, only series additions modify the network. Both Series1 and Series2 connections in figure 11 need to be considered, and we define connecting to PT2 as Step 6, resulting in the network of figure 12–Step 6. An additional IF-Block is then added in series at PT1—the resultant network is shown in figure 12–Step 7. Consequent steps will be adding IF-Blocks in parallel then in series by repeating Steps 5–7, until all IF-Blocks in the original collection are used. Following this procedure, all possible IF-Networks can be obtained.

    Figure 12.

    Figure 12. A procedure to form all possible series–parallel 2PT1NG IF-Networks with a reaction mass and a pre-determined number of IF-Blocks.

    In order to formulate generic IF-Networks, whether a specific IF-Block exists need to be discussed. To this end, the terminologies removed and present are introduced here. An IF-Block is defined to be removed as its immittance function takes the value of 0 or ∞—the value is chosen to ensure that the included components (i.e. the IF-Blocks and the reaction mass) or any two of the terminals, PT1, PT2 and NG, are not locked rigid and that none of the terminals is disconnected. An IF-Block is regarded as present if it is not removed. Consider the IF-Block Y1(s) shown in figure 12; if it is removed, its immittance function must take the value of ∞; otherwise the NG will be disconnected. For Y2(s), its removal must correspond to Y2(s) = ∞; otherwise the PT2 will be disconnected. Also for Y3(s), we must set Y3(s) = ∞; otherwise the PT1 in figure 12–Step 4 will be disconnected. Consider the parallel added IF-Block Y4(s): if it is removed, it must take the value of 0 (otherwise PT1 and PT2 in figure 12–Step 5 will be locked rigid). For the series added IF-Blocks Y5(s) and Y6(s): we must set them as ∞ to ensure that the terminals of the resulting networks (see figure 12–Step 6 and Step 7) are not disconnected. Following the same argument, the additional IF-Blocks added in parallel (series) connection must take the value of 0 (∞) when they are removed.

    The rest part of this subsection focuses on generating generic IF-Networks. Different from 1PT1NG networks, where one generic IF-Network is sufficient (figure 7), for 2PT1NG networks, different generic IF-Networks are needed depending on the number of IF-Blocks which are present. Now consider the possible 2PT1NG IF-Networks with a pre-determined number, R, of IF-Blocks. Suppose that the generic IF-Network, representing R − 1 IF-Blocks, has been formulated, denoted as LR−1(s), all the IF-Networks with R IF-Blocks can be subsequently obtained by adding an extra IF-Block to it. The generic IF-Network, LR(s), covering all the obtained IF-Network possibilities, can then be formulated, satisfying the condition that R IF-Blocks are present. Following this argument, the generic IF-Network for any R value can be formulated. Note that the generic IF-Network has more IF-Blocks than is allowed, hence requires the condition that R IF-Blocks are present. We will now discuss the R = 1 case, then move on to R = 2 and beyond.

    For R = 1: We first consider the case where in Step 1 of figure 12, Y1(s) is present. The resulting network is figure 12–Step 2, which is a 1PT1NG network (with PT2 disconnected). Hence, this case is not valid and Y1(s) must be removed. When Y2(s) is present, a 2PT1NG IF-Network is obtained, shown in appendix B, figure 15a. If Y2(s) is not present, the connection of Y3(s) results in a new 2PT1NG IF-Network, see figure 15b, appendix B. If Y3(s) is removed, PT1 and PT2 are rigidly connected and any follow on steps in figure 12 only result in a 1PT1NG network. Hence, two network possibilities (figure 15, appendix B) are obtained for the R = 1 case, with which the generic IF-Network can be obtained as L1(s) in table 2, satisfying the condition that only one IF-Block is present.

    For R = 2: The IF-Networks can be formulated by adding one additional IF-Block to the generic IF-Network L1(s) in table 2. Firstly, we set both Y2(s) and Y3(s) in L1(s) as present, this results in an IF-Network shown in figure 16a, appendix B. We then consider the case where Y1(s) is present. This means either Y2(s) or Y3(s) can be present, resulting in two new IF-Networks shown in appendix B, figure 16b,c. At last, an extra IF-Block Y4(s) is added to L1(s) in parallel connection, and its presence will result in the removal of Y2(s) or Y3(s); however, the resulting networks can be simplified to the network with one IF-Block. Hence, the network L2(s) shown in table 2 is the generic IF-Network for R = 2 case, satisfying the condition that two IF-Blocks are present.

    For R = 3: An extra IF-Block is added to the generic IF-Network L2(s) in series or in parallel, to formulate all the IF-Networks with three IF-Blocks. We first set three IF-Blocks in L2(s) as present, an IF-Network shown in Appendix B, figure 17a is hence obtained. By adding an additional IF-Block Y4(s) in parallel to L2(s), its presence will result in the removal of Y1(s); otherwise the resulting network will be reduced to a network with two IF-Blocks. This results in an IF-Network of figure 17b. Hence, the generic IF-Network for R = 3 can be formulated as L3(s) in table 2, subject to the condition that three IF-Blocks are present.

    For R = 4: An IF-Network shown in figure 18a, appendix B can be firstly obtained by setting all four IF-Blocks in L3(s) as present. An extra IF-Block, Y5(s) is then added in series to the right of L3(s) and if it is present, only the removal of Y1(s) can result in a new IF-Block, see appendix B, figure 18b. Consider the case that Y6(s) is added in series to the left of L3(s). An IF-Network of figure 18c can be obtained. Hence, the generic IF-Network for R = 4 can be formulated as L4(s) in table 2, with the condition that four IF-Blocks are present.

    Generalization: Using the similar argument, the generic IF-Network representing more numbers of IF-Blocks can be obtained, as shown in table 2. For the odd R (with R≥5), the generic IF-Network is formulated by adding an IF-Block Y(3R−1/2)(s) in parallel to LR−1(s). The subscript (3R − 1/2) of the IF-Block represents the total number of IF-Blocks required in the generic IF-Network, LR(s). Also for the even R (with R≥6), the generic IF-Network can be formulated by series connecting two IF-Blocks Y(3R/2)−1(s) and Y(3R/2)(s) with LR−1(s), where in total (3R/2) numbers of IF-Blocks are required.

    Table 2. Generic 2PT1NG IF-Networks, LR(s), with the condition of R IF-Blocks present.

    Inline Graphic

    The generic IF-Network results of the above are summarized in table 2, which can then be used for enumerating all possible network layouts.

    (c) Network layout enumeration and Immittance-Function-Matrix derivation

    For the enumeration of 2PT1NG network layouts with a reaction mass element and N 2PT elements (springs, dampers and inerters), the obtained generic IF-Networks with R = 1, 2, …, N in table 2 can be used. We first distribute the N 2PT elements into R present IF-Blocks (other IF-Blocks are removed), and secondly allocating element types into each present IF-Block. After this, the element types and numbers will be determined for each IF-Block in the generic IF-Networks. The structure-immittance approach [29] is then adopted to derive the corresponding immittance functions of the blocks, which is able to cover the full set of possible 2PT network connections.

    With the obtained immittance functions of the included IF-Blocks, the IF-Matrix for each generic IF-Network is derived based on the procedure shown in appendix A, which can then be used for optimization studies. We denote that the IF-Matrix for a 2PT1NG network can be derived by deriving its equations of motion. However, there are a great number of possible network layouts, a more straightforward and simpler approach for deriving the IF-Matrix is needed—this is provided and explained in detail in appendix A.

    5. 2PT1NG network case demonstration

    In this section, the 2PT1NG network case with one damper, one spring and one inerter is analysed. All the possible 2PT1NG network layouts are enumerated using the generic IF-Networks in table 2. The IF-Matrix for each generic IF-Network is derived and applied to the example structure. Significant performance advantages with 2PT1NG network layouts will be demonstrated at the end of this section.

    (a) Network layouts enumeration

    For the 2PT1NG networks with one damper, one spring and one inerter, we have N = 3. Hence, three generic IF-Networks with R = 1, R = 2 and R = 3, shown as L1(s), L2(s) and L3(s) in table 2, will be used.

    Consider the generic IF-Network for R = 1, L1(s) of table 2, all the three 2PT elements should be distributed in one IF-Block and the other IF-Block is removed. As a result, in total 16 2PT1NG network layouts are enumerated and to systematically analyse all of them, the structure-immittance approach [29] is adopted to express the immittance function of the present IF-Block. The immittance function is the structural immittances for one inerter, one damper and one spring case, shown in (3.2) and (3.3). By expressing the force–velocity transfer function matrix of the generic IF-Network shown in table 2 and making use of the condition that one IF-Block is removed, we can obtain the IF-Matrix L1(s) as (A 3) in appendix A, with the condition that one of the immittance functions, Y1(s), Y2(s), takes the expression as structural immittances of (3.2) and (3.3), and the other one must equal ∞.

    For the generic IF-Network with R = 2, shown as L2(s) of table 2, two of the three IF-Blocks are present, between which the three 2PT elements are distributed in. For these two present IF-Blocks, one should include one 2PT element and the other one consists of the remaining two elements. For example, when Y1(s) and Y2(s) in L2(s) (table 2) are present and Y1(s) includes one element, such as a spring, the other two elements (the damper and inerter) are contained in Y2(s), resulting in two network possibilities by connecting damper in series or in parallel with the inerter. Consider that there are three different element types and switching Y1(s) with Y2(s) will result in different network possibilities, the generic IF-Network with one IF-Block removed covers 36 network layouts. Finally, the IF-Matrix for this generic IF-Network, L2(s) can be derived as (A 4) in appendix A, where one of the immittance functions Y1(s), Y2(s), Y3(s) equals ∞, one is the force–velocity transfer function of a single 2PT element, and the third requests structural immittance for the remaining two 2PT elements. Consider the case that Y1(s) represents a spring k and Y2(s) includes one damper c and one inerter b, we should make Y3(s) = ∞, Y1(s) = k/s and Y2(s) takes the expression of a structural immittance obtained for one damper and one inerter case, given as

    Y2(s)=c2(bs+c1)(bs+c1+c2),5.1
    with the condition that one of the parameters c1 or 1/c2 is positive and the other one equals zero.

    Considering the generic IF-Network obtained for R = 3 case, shown as L3(s) of table 2, three of the four IF-Blocks must be present and each of them contains one 2PT element. By removing one IF-Block and distributing three different element types into the present IF-Blocks, all the 2PT1NG network layouts for this case can be enumerated. Note that when Y2(s) or Y3(s) is removed with its immittance function as ∞, the resulting IF-Network will be reduced to that with two IF-Blocks, hence are omitted for the network layouts enumeration. From the generic IF-Network, in total 12 network layouts can finally be enumerated. These network layouts can be represented by the IF-Matrix L3(s), derived based on (A 4) and (A 6) in appendix A, given as

    L3(s)=L2(s)+Y4(s)1111,5.2
    where either Y1(s) = ∞ or Y4(s) = 0 and the remaining three immittance functions are the force–velocity transfer functions of three different 2PT element types.

    These obtained IF-Matrices can then be used for given vibration suppression problems. Importantly, using the method proposed here, all possible series–parallel 2PT1NG network layouts, with one reaction mass, one spring, one damper and one inerter are covered. In addition, by making use of the structure-immittance approach, all the possible layouts can be analysed in a systematic way and by optimizing these using the objective functions considered, the optimal configuration can be obtained across all the network possibilities. In the following subsection, the results obtained in this part will be applied to the 3 d.f. structure example to illustrate the benefits of the proposed design approach. For the one reaction mass, one inerter, one damper and one spring case considered, in total 64 2PT1NG network layouts can be enumerated and covered by three generic IF-Networks, among which the optimum configuration can be obtained by three optimizations.

    (b) Numerical application on the example structure

    Consider the 3 d.f. structure shown in figure 1f . The suppression system is taken to be the 2PT1NG network including one reaction mass, one inerter, one damper and one spring. The value of the reaction mass is taken as 1000 kg, as the same as that used in previous two application examples (figures 2 and 9).

    For the 2PT1NG network with each of the four element types, three generic IF-Networks, L1(s), L2(s) and L3(s), are formulated in §5a, covering all the 2PT1NG network possibilities. These obtained generic IF-Networks, together with their corresponding IF-Matrices (A 3), (A 4), (5.2)), are then used to minimize the objective function Jd. For example, for the 2PT1NG networks covered by the generic IF-Network, L1(s), the IF-Matrix (A 3) is adopted as the transfer function matrix of the vibration suppression device, with which the objective function Jd can be obtained based on (2.1), (2.2). In (A 3), the present immittance function Y2(s) or Y3(s) formulated using the structure-immittance approach is then optimized to identify the optimal 2PT network configuration out of all the eight possible layouts made up of one inerter, one damper and one spring (see table 2 in [29]). Together with this obtained 2PT network, the optimal 2PT1NG network configuration can then be obtained from the corresponding generic IF-Network, L1(s), of table 2. The optimal structures for three generic IF-Network cases with R = 1, 2, 3 have been obtained as figure 13a1,a2 and a3, respectively. The corresponding optimal results in value of the objective function Jd (2.2) are summarized in table 3, together with the optimum parameter values. The previously proposed layouts, such as the TMD, the TID, the TMDI, the C1 identified for the 1PT1NG networks (figure 8a1) and the C2 (figure 8a2) are also provided in table 3 for a sake of comparison.

    Table 3. Optimization results for the one reaction mass, one inerter, one damper and one spring case.

    configurations performance Jd ( × 10−3 s2) optimal parameter values (kg, kNs m−1, kN m−1)
    TMD (no b) 25.0 (–) c = 11.7, k = 2.75 × 102
    TID (no m) 19.6 (21.6%) b = 7.94 × 103, c = 7.50 × 102, k = 4.35 × 103
    C1 (a 1PT1NG device) 16.3 (34.8%) b = 1.05 × 103, c = 30.5, k = 4.92 × 102
    TMDI (a R = 2 case device) 14.5 (42.0%) b = 1.12 × 103, c = 9.50, k = 6.22 × 102
    C2 (a R = 3 case device) 13.8 (44.8%) b = 1.21 × 104, c = 13.9, k = 2.98 × 102
    C(1) 19.5 (22.0%) b = 8.95 × 103, c = 7.65 × 102, k = 16.8
    C(2) 10.2 (59.2%) b = 9.91 × 103, c = 4.28 × 102, k = 2.26 × 102
    C(3) 8.60 (65.6%) b = 8.10 × 103, c = 3.26 × 102, k = 2.13 × 102
    Figure 13.

    Figure 13. Optimal 2PT1NG configurations for the one inerter, one damper and one spring case for (a1) R = 1, (a2) R = 2 and (a3) R = 3 case.

    For the generic IF-Network, L1(s) of table 2, out of the 16 possible layouts, the optimization indicates that configuration C(1) is optimal. The corresponding optimal value of the objective function Jd is 0.0195 shown in table 3, which is slightly smaller than that of the TID. Considering generic IF-Network representing two IF-Blocks, we note that the TMDI is covered by this network (L2(s) of table 2), with Y1(s) = ∞, Y2(s) = bs and Y3(s) = k/s + c. However, the TMDI is not the optimum configuration for this case, and the resulting optimum configuration is shown in figure 13a2, with which the value of Jd is obtained to be 0.0102. It can be seen that compared with the TMDI, C(2) can provide almost 30% performance improvement when b = 9.91 × 103kg, c = 4.28 × 102kNsm−1 and k = 2.26 × 102kNm−1. Figure 13a3 shows the optimum absorber configuration C(3) for the R = 3 case, which gives the value of Jd as 0.0086, approximately 41% smaller than that of the TMDI and almost 66% better than the TMD. Also note that C2, figure 8a2 is a network layout included in the R = 3 generic IF-Network; however, the C(3) outperforms it, achieving a 38% smaller value of Jd. It can also be seen that C(3) is the most effective vibration suppression device, among all the 64 series–parallel 2PT1NG networks consisting of one reaction mass and each of the three 2PT element types. From table 3, we note that configuration C(3) outperforms C1 with 47% performance improvement in Jd. This suggests that the 2PT1NG network, i.e. one which has connection points on the second and third storeys, can provide better seismic performance than the 1PT1NG one (which is connected to the third storey only) with the same numbers of each element type.

    Figure 14 shows the frequency responses of the three inter-storey drifts of the structure model with the identified beneficial absorber configurations, subjected to the earthquake excitation. It can be seen that rather than splitting the first fundamental frequency into two separate frequencies, both the C(2) and the C(3) split it into three frequencies, hence resulting in smaller values of the drift displacements compared with the other configurations, such as the TMDI. This is because both C(2) and C(3) contain additional degree of freedom compared with the other devices. Also note that configuration C(3) results in the smallest inter-storey drift values of all the three floors in the vicinity of not only the first fundamental frequency, but also the second and third fundamental frequencies.

    Figure 14.

    Figure 14. Frequency response in inter-storey drifts with the TMD (dashed), the TMDI (solid), the C1 (dotted), the C(2) (thin solid) and the C(3) (thick solid). (Online version in colour.)

    6. Conclusion

    This paper presents a systematic characterization and analysis of passive vibration suppression devices with a reaction mass. Devices with one and two structural attachment points were considered, for which a full set of network layouts with pre-determined numbers of inerters, dampers and springs were captured and enumerated. This is achieved using generic IF-Networks to represent the topological connection possibilities of the mass and the IF-Blocks, and the structural immittances to describe the 2PT networks in each IF-Block. By using the force–velocity Immittance-Function-Matrices, the dynamics of vibration suppression devices with two attachment points can be described and used for vibration suppression problems. A 3 d.f. structure model under earthquake excitation is considered, and the number of inerter, damper and spring was restricted to be one for vibration suppression devices. Optimal configurations were obtained out of 16 and 64 candidate network layouts, respectively, for the one and two attachment point device cases. For one attachment point devices, compared with the TMD, up to 34.8% performance improvement can be obtained. When considering the devices with two attachment points, the most beneficial configuration can provide almost 66% better performance than the TMD and also outperforms the TMDI with 41% performance improvement.

    Data accessibility

    The source code for the results in tables 1 and 3 has been provided as electronic supplementary material. Other datasets supporting this article are already in the article.

    Authors' contributions

    S.Y.Z. led the development of the method with assistance from Y.-Y.L., J.Z.J., S.A.N. and J.H.G.M. All five contributed to the preparation of the manuscript.

    Competing interests

    We declare we have no competing interests.

    Funding

    This work was supported by the EPSRC, the University of Bristol: S.A.N. was supported by an EPSRC fellowship EP/K005375/1 and is currently supported by Programme grant no. EP/R006768/1, J.Z.J. and S.Y.Z. are supported by an EPSRC first grant no. EP/P013546/1.

    Acknowledgements

    We gratefully acknowledge the support of our funders.

    Appendix A. Derivation of Immittance-Function-Matrix

    Consider a 2PT1NG network shown in figure 4b with velocities v1, v2 and forces f1, f2 at the two physical-terminals PT1 and PT2. Its Immittance-Function-Matrix L(s), relating the velocities with the forces in Laplace domain, is defined as

    F1F2=L(s)V1V2,A 1
    where L(s) is a 2 × 2 matrix, made up of four immittance functions, denoted as L(1, 1), L(1, 2), L(2, 1) and L(2, 2).

    For the generic IF-Network, L1(s) shown in table 2, the equation of motion, in Laplace domain, can be derived as

    F1=sY3(s)(V1Vm),F2=sY2(s)(V2Vm)andmsVm=F1+F2,A 2
    where Vm is the velocity of the reaction mass with its value defined as positive to the right. By expressing Vm as a function of V1, V2, the IF-Matrix of the generic IF-Network, L1(s), can be written as
    L1(s)=Y2(s)Y3(s)Y2(s)+Y3(s)+ms(Y2(s)+ms)Y2(s)11(Y3(s)+ms)Y3(s).A 3
    The IF-Matrix of the generic IF-Network for the R = 2 case, shown in table 2, can then be obtained by replacing ms in (A 3) with Ym(s), to give
    L2(s)=Y2(s)Y3(s)Y2(s)+Y3(s)+Ym(s)(Y2(s)+Ym(s))Y2(s)11(Y3(s)+Ym(s))Y3(s),A 4
    where
    Ym(s)=Y1(s)ms(Y1(s)+ms)A 5
    represents a mass connected in series with the IF-Block Y1(s). Built on (A 4), the IF-Matrix of the generic IF-Networks, LR(s), for the even R case (R≥3, see the left-hand side of table 2), can be obtained by a parallel connection between LR−1(s) and Y(3R−1/2)(s), given by
    LR(s)=LR1(s)+Y(3R1/2)(s)1111.A 6
    For the series connection between LR−1(s) and two IF-Blocks Y(3R/2)−1(s), Y(3R/2)(s) (see the right-hand side of table 2), the IF-Matrix can be formulated by first deriving LR(s), using
    LR(s)=1Y(3R/2)1(s)+LR1(2,2)Y(3R/2)1(s)LR1(s)+|LR1(s)|000,A 7
    and then substituting this into
    LR(s)=1Y(3R/2)(s)+LR(1,1)Y(3R/2)(s)LR(s)+000|LR(s)|.A 8

    Appendix B. 2PT1NG IF-network possibilities with R = 1, 2, 3, 4

    Figure 15.

    Figure 15. The two IF-Network cases for R = 1.

    Figure 16.

    Figure 16. The three IF-Network cases for R = 2.

    Figure 17.

    Figure 17. The two IF-Network cases for R = 3.

    Figure 18.

    Figure 18. The three IF-Network cases for R = 4.

    Footnotes

    1 We use series, rather than serial, in this work as the more common usage when describing serial connections is series, e.g. [20].

    Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4582514.

    Published by the Royal Society. All rights reserved.

    References