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The performance benefits of passive vibration suppression with network configurations consisting of stiffness, damping and inertance elements have been demonstrated for a wide range of mechanical systems. Considering physical implementations of these beneficial network configurations, hydraulic realisations have the advantages of durability and simplicity for integration with existing hydraulic dampers. Such designs are exemplified by fluid inerters and fluid-inerter-damper devices. However, in contrast to the convenience of realising inertance and damping elements, realising ‘embedded’ stiffness is very challenging. We use ‘embedded’ to refer to a network element, which is not purely in series or in parallel with the remainder of the network but instead lies within the network layout. In this work, a setup using a rubber membrane to realise such embedded stiffness is proposed, together with a procedure for hydraulic implementations of any stiffness-damping-inertance configurations. The nonlinear properties of the embedded stiffness due to rubber membrane properties are then investigated both theoretically and experimentally. In addition, the effectiveness of both the membrane setup and the design procedure are demonstrated via a case study of suspension design for passenger vehicle ride comfort enhancement.

In the field of structural dynamics, system identification usually refers to building mathematical models from an experimentally-obtained data set. To build reliable models using the measurement data, the mathematical model must be representative of the structure. In this work, attention is given to robust identification of nonlinear structures. We draw inspiration from reduced order modelling to determine a suitable model for the system identification. There are large similarities between reduced order modelling and system identification fields, i.e. both are used to replicate the dynamics of a system using a mathematical model with low complexity. Reduced Order Models (ROMs) can accurately capture the physics of a system with a low number of degrees of freedom; thus, in system identification, a model based on the form of a ROM is potentially more robust. Nonlinear system identification of a structure is presented, where inspiration is taken from a novel ROM to form the model. A finite-element model of the structure is built to simulate an experiment and the identification is performed. It is shown how the ROM-inspired model in the system identification improves the accuracy of the predicted response, in comparison to a standard nonlinear model. As the data is gathered from simulations, system identification is first demonstrated on the high fidelity data, then the fidelity of data is reduced to represent a more realistic experiment. A good response agreement is achieved when using the ROM-inspired model, which accounts for the kinetic energy of unmodelled modes. The estimated parameters of this model are also demonstrated to be more robust and rely on the underlying physics of the system.

Oshore wind turbines have the potential to capture the high-quality wind resource. However the signicant wind and wave excitations may result in excessive vibrations and decreased reliability. To reduce vibrations, passive structural control devices, such as the tuned mass damper (TMD), have been used. To further enhance the vibration suppression capability, inerter-based absorbers (IBAs) have been studied using the structure-based approach, i.e. proposing specic stiness-damping-inertance elements layouts for investigation. Such an approach has a critical limitation of being only able to cover specic IBA layouts, leaving numerous benecial congurations not identied. This paper adopts the newly introduced structure-immittance approach, which is able to cover all network lay out possibilities with a predetermined number of elements. Linear monopile and spar-buoy turbine models are rst established for optimisation. Results show that the performance improvements can be up to 6.5% and 7.3% with 4 and 6 elements, respectively, compared to the TMD. Moreover, a complete set of benecial IBA layouts with explicit element types and numbers have been obtained, which is essential for next step real-life applications. In order to verify the eectiveness of the identied absorbers with OpenFAST, an approach has been established to integrate any IBA transfer functions. It has been shown that the performance benets preserve under both the Fatigue Limit State (FLS) and the Ultimate Limit State (ULS). Furthermore, results show that the mass component of the optimum IBAs can be reduced by up to 25.1% (7486kg) to achieve the same performance as the TMD.

The inerter completes the force-current analogy between mechanical and electrical components, providing the mechanical equivalent to the capacitor. As such, it is a two-terminal passive element that, when implemented ideally, is normally said to generate a force proportional to the relative acceleration between its two terminals. However, this is applicable only if the inerter does not rotate, so the only relative motion between the device’s terminals is axial. In many applications, this restriction is acceptable, such as in car suspension systems. However, in this paper, it is shown that the relationship between the terminal accelerations and the generated force is more complex if the inerter is used in a 2-dimensional (2D) or 3-dimensional (3D) environment, such as within a multi-bar mechanism (e.g., robotic arms or railway pantographs). Specifically, the inerter force is not given by simply the relative acceleration between the two terminals. The centripetal acceleration, resulting from the rotation of the inerter, needs to be accounted for to find the second derivative of the inerter length, which defines the generated force. Two case studies are presented to demonstrate the effects of this normally neglected centripetal acceleration term. It is shown that when an inerter is operating in a 2D or 3D environment, significant errors may occur in evaluating the inerter force and also the system response if the centripetal acceleration term is neglected. Equations are provided for both modelling the inerter in different coordinate systems and for incorporating the inerter in 2D and 3D multibody systems.

Performance of a passive vehicle suspension can be improved with the help of an active actuator, however, with potentially problematic control requirements, such as high energy consumption and large actuator forces. To maximize performance benefits without requiring significant control efforts, the passive and active parts need to be designed and work synergistically. In this paper, a novel combined passive and active vibration suppression approach of which the passive part is enhanced by an inerter is proposed for improving the trade-off between dynamic performance and control requirements. Via this approach, the optimal passive configuration consisting of inerter(s), spring(s) and damper(s) with pre-determined numbers and the optimal active control parameter can be identified. The approach is demonstrated using a case study where the combined suspension is designed considering a quarter-car model and a typical active controller (i.e., the skyhook control). It will be shown that, compared with a conventional passive part of a spring-damper, adding an inerter in parallel can significantly improve the pareto optimality between the ride comfort and power (or force) requirements. The improvement is further enhanced by systematically exploring all passive network possibilities with a pre-determined complexity via the structure-immittance technique. This approach is also applicable to the vibration suppression of other engineering structures.

This paper presents a novel approach to identify inerter-based suspension struts, which can provide significant performance enhancement for passenger vehicles. The inerter has been used on Formula 1 racing cars, and several beneficial devices incorporating inerters have also been identified for ride comfort enhancement. However, previous investigations either were limited to simple network configurations with moderate performance improvement, or resulted in complicated configurations with a large number of elements which are impractical for real-life applications. In addition, some important practical performance constraints have not been taken into consideration, such as high-frequency dynamic stiffness which influences the NVH performance, and frequency content consideration of the sprung mass acceleration which more directly relates to passenger perception. In this paper, a quarter-car model including top mount is studied, with the performance of a conventional suspension strut presented as baseline. The structure-immittance approach, which can cover all networks with pre-determined numbers of each element type, is adopted for the identification of the optimal suspension configurations. Several configurations with up to a 14.7% performance improvement are identified with all other practical performance indices to be no worse than the baseline. The suspension devices proposed in previous works are also considered for a sake of comparison, demonstrating significant advantages of the structure-immittance approach. Subsequently, a sensitivity analysis against the sprung and unsprung mass changes is carried out, which represents cargo and tyre weight variations, respectively. Time domain response and other reality checks are then conducted for the out-performing configurations, which reconfirm the ride comfort enhancement and ensure no unexpected behaviour occurs.

This paper presents a novel method for gust loads alleviation in a truss-braced wing in which an inerter-based device located in the truss-structure is used to reduce peak-loads during a discrete “1-cosine” gust. Three candidate layouts are considered, and the device parameters are optimized to target the response of the first three structural modes. It is demonstrated that either a single damper or a combination of inerter-based devices can be used to achieve a reduction of approximately 4% for spanwise locations inboard of the strut attachment point and that this reduction is consistent across the full range of gust gradients. Furthermore, it is noted that the inerter-based device has a significantly smaller damping coefficient than the case where just a damper is used and that the device parameter values are viable within the scope of an aerospace application.

Improving the track friendliness of a railway vehicle can make a significant contribution to improving the overall cost effectiveness of the rail industry. Rail surface damage in curves can be reduced by using vehicles with a lower Primary Yaw Stiffness (PYS); however, a lower PYS can reduce high-speed stability and have a negative impact on ride comfort. Previous studies have shown that this trade-off between track friendliness and passenger comfort can be successfully combated by using an inerter in the primary suspension; however, these previous studies used simplified vehicle models, contact models, and track inputs. Considering a realistic four-axle passenger vehicle model, this paper investigates the extent to which the vehicle's PYS can be reduced with inertance-integrated primary lateral suspensions without increasing Root Mean Square (RMS) lateral accelerations when running over a 5km example track. The vehicle model, with inertance-integrated primary lateral suspensions, has been created in VAMPIRE, and the vehicle dynamics are captured over a range of vehicle velocities and wheel-rail equivalent conicities. Based on systematic optimisations using network-synthesis theory, several beneficial inertance-integrated configurations are identified. It is found that with such beneficial configurations, the PYS can be reduced by up to 47% compared to a base case vehicle, without increasing lateral RMS accelerations. This could result in a potential Network Rail Variable Usage Charge saving of 26%. With the beneficial inertance-integrated suspensions, further simulations are carried out to investigate the vehicle's performance in curve transitions and when subject to one-off peak lateral track irregularities.

This paper investigates the potential for improving the performance of hydraulic engine mounts through fluid passageway designs. In previous studies, a few simple inertia track designs have been investigated with moderate improvements obtained. However, there are countless alternative design possibilities existing; while analyzing each one of them in turn is impracticable. To this end, this paper introduces a systematic methodology to optimize fluid passageway designs in a hydraulic engine mount. First, beneficial fluid passageway configurations are systematically identified using a linearized low-frequency model that captures the relative displacement transmissibility. A nonlinear model is then used to fine-tune the fluid passageway designs for the low-frequency transmissibility improvement, and also for the assessment of high-frequency dynamic stiffness performance. The obtained beneficial designs present performance advantages over a wide frequency range. The design approach introduced in this study is directly applicable to other engine mount models and performance criteria.

In this paper, a nonlinear dynamic landing gear model considering the influence of the coupling of the shock absorber stroke variation and the landing gear longitudinal motion with an anti-skid PID braking control system that captures gear walk is established. This gear walk model is verified by comparing with the response from a virtual prototype model. Then a parameter sensitivity analysis is carried out to find out the parameters with greater effects on gear walk and braking performance. The short time Fourier transform is employed to study the transient gear walk amplitude-frequency response, whose results are used to define the optimization constraints. A feedforward controller is proposed as part of the braking control law. Single-objective optimizations are then carried out to improve the gear walk performance while maintaining the braking efficiency. It is shown that the feedforward control, together with the PID feedback controller, can provide 25.68% reduction of the maximum gear walk angle while satisfying other constraints. The stability and robustness of the optimized braking law is verified under different working conditions. Multi-objective optimization is then used to highlight the trade-off between the gear walk vibration and the braking efficiency.

Vibration absorbers with a combination of stiffness, damping and inertance have been shown to be effective through numerous theoretical studies. One way to realise inertance is by using the fluid-based inerter, with the advantages of durability, structural simplicity and the similarity with existing damper constructions. Previous studies focused on accurate modelling of a specific fluid-based inerter device, while there has been no investigation on whether the dynamic models are still valid when its design parameters change. A model is termed here as being generalisable when it is able to sufficiently accurately characterise the terminal behaviour while allowing its design parameters to vary within their pre-defined ranges. In this paper, a generalisable model is developed for an example fluid-inerter integrated damping (FID) design. The methodology to develop such model uses the hydraulic and the corresponding mechanical networks representing the device, and tailored experimental testing to characterise each network element. Such approach is applicable to other design parameter settings and also other designs of FID devices.

For modern electrical rail systems, the pantograph-catenary dynamic performance is one of the most critical challenges. Too much fluctuation in contact forces leads to either accelerated wear of the contacting components or losses of contact and, consequently, arcing. In this work, inertance-integrated pantograph damping systems are investigated with the objective of reducing the contact force standard deviation. Firstly, a multibody pantograph model is developed with its accuracy compared with experimental data. The model is improved through the calibration of the pantograph head suspension parameters and the introduction of both non-ideal joint and flexibility effects. Using the calibrated model, beneficial inertance-integrated damping systems are identified for the pantograph suspension. The results show that the configuration with one inerter provides the best performance among other candidate layouts and contends a 40% reduction of the maximum standard deviation of the contact force over the whole operating speed range in the numerical modelling scenario analysed. Considering the identified configuration, time-domain analysis and modal analysis are investigated. It has been shown that the achieved improvement is due to the fact that with the beneficial inertance-integrated damping system, the first resonance frequency of the pantograph system coincides with the natural frequency of the catenary system.

This paper proposes a new tool for the selection of optimal elementary shunts connected to piezoelectric patches, which are bonded on thin structures to control the multi-resonant flexural vibration produced by broadband stochastic excitations. More specifically, the study introduces a procedure for the identification of simple shunt architectures composed by a small-number of low-value RLC elements, which yet produce significant multi-resonant vibration control effects. The challenge to identify beneficial elementary shunts lies in the fact that numerous candidate circuits exist, and this number grows exponentially with the increase of allowed electrical components. To this end, a design optimisation procedure is proposed, which enables the characterisation of all candidate shunt networks with a pre-defined number of elements having a given range of values. The paper refers to a practical model-problem, which encompasses a thin plate equipped with five piezoelectric patches connected to shunts. A trial design study is illustrated, where the tool is used to find suitable simple shunts to control the resonant responses of 6 flexural modes of the plate in a frequency band comprised between 20 and 150 Hz. The full set of shunt circuits composed by 4 elements is systematically searched and two simple shunts are identified. The physics and control performance of these two shunts are contrasted with those of classical single-resonant and multi-resonant shunts. The study shows that these elementary shunts generate 6 to 16 dB reductions of the target resonant responses, which are comparable to those that would be produced by a classical six-branches multi-resonant shunt encompassing 18 elements. Moreover, it shows that the two shunts produce combined absorption and damping effects such that they can provide beneficial vibration control in presence of significant variations of the flexural response of the structure too.

As a two-terminal mechanical element, the inerter has been successfully deployed in various mechanical systems, such as automotives, multi-story buildings, and motorcycles. The introduction of the inerter allows the use of network synthesis to design a passive mechanical network, and can potentially facilitate the identification of practical and high performance mechatronic vibration absorbers. This paper provides an approach for optimal design of both the mechanical and the electrical parts for an inerter-based mechatronic device in vehicle suspension. The system considered includes a mechanical ball-screw inerter alongside an electric motor that is not driven but instead used passively with an electrical load applied across the terminals. The trade-offs in designing the ball-screw inerter and the permanent magnet electric machinery is discussed in detail. Two factors, namely, the coil resistance and the inductor resistance, are taken into account in the performance evaluation. Results show that the improvements in the road holding performance can reach 9.24% for the ideal suspension system with no diverse effect on the ride comfort and suspension travel performance, while a 5.77% improvement can be obtained when the effects of the coil resistance and the inductor are included.

This paper investigates the use of two two-terminal vibration suppression devices in a building and assesses the performance benefits over those achieved using a single device. The inerter-combined configurations for a multi-storey building structure are considered. The inerter is a two-terminal device, with the property that the applied force is proportional to the relative acceleration across its terminals. In this paper, a fivestorey building model with two absorbers of the same kind subjected to base excitation is studied, where one is located between ground and the first floor and the other is between the first and second floors of the building. Three passive suppression layouts, two dampers, two tuned inerter dampers and two tuned viscous mass dampers, are considered. The optimal configurations for minimising the maximum interstorey drifts of the building are obtained with respect to the inerter’s size and the damping boundary. The corresponding parameter values are also presented. For the sake of comparison, the single device mounted between the ground and first floor is also considered. Finally, with specific inertance and damping values, the frequency response is provided to show the potential advantage of the proposed optimal configurations. It is demonstrated that the optimal configurations with a pair of devices is more effective than the optimal single device with equal total inertance and the same total damping boundary. The approach demonstrated in this paper is applicable to the investigation of using more than two devices for multistorey buildings.

Road damage caused by heavy vehicles is a serious problem experienced worldwide. This paper investigates the potential for reduction in road damage by incorporating the inerter element into truck suspension systems. Initially, quarter-car, pitch-plane and roll-plane models with two low-complexity inerter-based linear suspension layouts are investigated in the frequency domain. Reductions of the J95 road damage index for each model are identified against conventional parallel spring–damper truck suspension layouts. It is also shown that the proposed suspensions are capable of enhancing the roll stability while keeping the road damage at a given level. Subsequently, the nonlinear relationship between force and displacement as manifested by leaf springs is incorporated into the pitch-plane and roll-plane time-domain models. These confirm the potential advantage of inerter-based suspension layouts for road damage reduction.

Improving the track friendliness of a railway vehicle is highly beneficial to the rail industry, as it substantially increases its cost effectiveness. Rail surface damage under curving conditions can be reduced by using vehicles with a reduced Primary Yaw Stiffness (PYS); however, a lower PYS often leads to a reduction in high-speed stability and can negatively impact ride comfort. Previous studies have suggested that this trade-off, between track friendliness and passenger comfort, can be successfully improved by using an inerter in the primary suspension; however, these studies used simplified two-axle vehicles and simplified contact models, and track inputs. Considering a more realistic four-axle passenger vehicle model, this paper investigates the extent to which the PYS can be reduced using inertance-integrated primary lateral suspensions without increasing Root Mean Square (RMS) lateral carbody accelerations when running over a 5 km example track (with a number of vertical, lateral and longitudinal irregularities, and gauge variations). The vehicle, with inertance-integrated primary lateral suspensions, has been modelled in \(\mathrm{VAMPIRE}^{\textregistered }\), and the vehicle dynamics are captured over a range of different velocities and wheel-rail equivalent conicities. Several inertance-integrated suspensions are optimised, leading to permissible PYS reductions of up to 47% compared to the original vehicle, whilst lateral carbody accelerations remain at acceptable levels. This level of PYS reduction would result in a potential Network Rail Variable Usage Charge saving of 26%.

Passive mechatronic vibration absorbers have demonstrated great performance potential in previous studies. For such devices, optimal design of the electrical circuits therein is critical but challenging since existing techniques have evident limitations: those investigating a few specific circuits leave huge possibilities unexplored; those optimising circuit impedances potentially lead to circuits which cannot be physically implemented. Another challenge lies in the need for considering device parasitic effects (e.g., transducer 1 resistance) to guarantee the predicted performance accuracy—this can be extremely time-consuming, especially when exploring numerous design possibilities (e.g., circuits, transducers). To address these two challenges, this paper proposes a novel design methodology, which (1) allows the optimal and practically implementable circuit to be identified among all layouts with predefined complexity; (2) considers the device parasitic effects where necessary, to efficiently explore various possibilities. The validity of this methodology is demonstrated via an automotive suspension design case study, where the obtained significant performance improvement is successfully verified via experiments. This methodology is directly applicable to vibration suppression of other engineering structures and can also be adopted for other mechatronic absorber types.

Inerter is the mechanical dual of the capacitor via the force-current analogy. It has the property that the force across the terminals is proportional to their relative acceleration. Compared with flywheel-based inerters, fluid-based forms have advantages of improved durability, inherent damping and simplicity of design. In order to improve the understanding of the physical behaviour of this fluid-based device, especially caused by the hydraulic resistance and inertial effects in the external tube, this work proposes a comprehensive model identification methodology. Firstly, a modelling procedure is established, which allows the topological arrangement of the mechanical networks to be obtained by mapping the damping, inertance and stiffness effects directly to their respective hydraulic counterparts. Secondly, an experimental sequence is followed, which separates the identification of friction, stiffness and various damping effects. Furthermore, an experimental set-up is introduced, where two pressure gauges are used to accurately measure the pressure drop across the external tube. The theoretical models with improved confidence are obtained using the proposed methodology for a helical-tube fluid inerter prototype. The sources of remaining discrepancies are further analysed.

This paper presents the possibility of suppressing the train-induced vibration on bridges using a linear passive vibration suppression device incorporating inerter. The inerter is a two-terminal mechanical element with the property that the applied force is proportional to the relative acceleration across its terminals. The inerter has been applied to various engineering structures where performance benefits have been identified. However, currently these studies are either limited to simple network configurations with moderate performance improvement, or resulted in complicated configurations with a large number of elements which are impractical for real-life applications. In addition, the potential of using inerter-based absorbers for suppressing the train-induced bridge vibrations has not been taken into consideration. In this paper, a simplified bridge model equipped with a vibration suppression device subjected to series of moving forces is presented. The structure-immittance approach, which can cover all networks with pre-determined numbers of each element type, is adopted for the identification of the optimal absorber configurations. Numerical studies for two bridges, the THSR and the Kum-gang, under three types of railway excitations, the French T.G.V., the German I.C.E. and the Japanese S.K.S. are conducted. It is demonstrated that the overall performance of the bridge can be improved by using the proposed inerter-based configuration. It is also checked that the absorber stroke is much smaller than the inner space of bridge box girder, indicating the practical installation of the absorber. Furthermore, the robustness of the proposed absorber is verified by sensitivity analysis subjected to the change of bridge parameters and the device component values.

This paper considers several passive shimmy-suppression devices for a dual-wheel main landing gear (MLG) and proposes a method of selecting the device parameter values for which no shimmy occurs. Two of these devices include an inerter, a novel mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals. A nonlinear mathematical model is developed to represent the MLG dynamics. A bifurcation study is then carried out to investigate the effects of the shimmy-suppression devices on the gear steady-state response. The aircraft forward speed and the device damping are chosen as the continuation parameters. A range of device parameter values that ensure the aircraft is free from shimmy instability for any forward speed within its operating region are identified. It is shown that the use of a proposed spring-damper configuration can result in a more robust device in terms of the device damping over that of a conventional shimmy damper. Two inerter-based shimmy-suppression devices are then considered and yield further benefits on expanding the zero-shimmy regions in the two-parameter bifurcation diagrams.

Summary This paper provides a complete realisation of a special class of positive-real bicubic impedances. The problem is motivated by the concept of the inerter, which is the mechanical dual of a capacitor. This device allows mechanical network synthesis, by completing the electrical mechanical analogy. With mechanical synthesis, the emphasis is to minimise the number of elements required to allow feasible implementation. The definitions of simple-series-parallel networks and essential-regular positive-real functions are introduced. The simple-series-parallel minimum-reactive networks that can realise all essential-regular bicubics are identified and grouped into six network quartets. One of the advantages of these networks is that they contain the minimal number of reactive elements. The necessary and sufficient realisability conditions for all these networks, as well as corresponding element values, are then derived. Finally, numerical examples are provided to illustrate the validity of the theoretical results. In the course of the argument, interesting conclusions regarding essential-regular bilinear and biquadratic functions have also been presented. Copyright © 2017 John Wiley & Sons, Ltd.

This paper presents the possibility of using inerter-based shock strut in a landing gear to improve aircraft touch-down performance. The inerter is a mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals. The baseline performance of a traditional oleo-pneumatic shock strut is established using a simplified landing gear touch-down model. Several simple layouts and general transfer functions are used to represent the shock struts and time-domain optimisations are carried out to minimise the maximum strut load transmitted to the fuselage during touch-down. The performance benefits of several inerter-based shock strut configurations with the corresponding parameter values have been identified.

This paper investigates a nonlinear inertance mechanism (NIM) for vibration mitigation and evaluates the performance of nonlinear vibration isolators employing such mechanism. The NIM comprises a pair of oblique inerters with one common hinged terminal and the other terminals fixed. The addition of the NIM to a linear spring-damper isolator and to nonlinear quasi-zero-stiffness (QZS) isolators is considered. The harmonic balance method is used to derive the steady-state frequency response relationship and force transmissibility of the isolators subjected to harmonic force excitations. Different performance indices associated with the dynamic displacement response and force transmissibility are employed to evaluate the performance of the resulting isolators. It is found that the frequency response curve of the inerter-based nonlinear isolation system with the NIM and a linear stiffness bends towards the low-frequency range, similar to the characteristics of the Duffing oscillator with softening stiffness. It is shown that the addition of NIM to a QZS isolator enhances vibration isolation performance by providing a wider frequency band of low amplitude response and force transmissibility. These findings provide a better understanding of the functionality of the NIM and assist in better designs of nonlinear passive vibration mitigation systems with inerters.

Cables are widely used in cable-stayed bridges and other civil engineering structures, but they often experience large amplitude vibrations due to their low inherent damping. Recent studies have shown that inerter-based vibration absorbers with two or three elements can provide significant performance improvements with very large inertance. Such large inertance leads to difficulties in physical implementation. Meanwhile, alternative inerter-based layouts with more elements could potentially provide better performance with significantly smaller inertance. However, studying these configurations one by one is impractical because the number of possible absorber layouts increases exponentially with the number of elements. This paper, using two types of fixed-sized-inerter (FSI) layouts, presents an efficient and systematic optimum configuration identification methodology. A simplification procedure is also adopted to then simplify the obtained configurations while not compromising the performance gains. Using this approach, it is shown that when the number of elements is increased from three to four, significant enhancement can be obtained even with small inertance values. The proposed approach can also be applied to vibration problems of other mechanical structures and with other performance criteria.

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.

Vibration suppression capabilities of linear passive vibration absorbers, such as traditional tuned mass damper (TMD), and recently proposed inerter-based vibration absorbers, have been studied for multiple mechanical systems. In particular, significant performance advantages have been obtained with a specific device making use of both inerter and mass elements, namely the tuned mass damper inerter (TMDI). However, there are still countless mass-included inerter-based configurations that have not been studied, which can potentially provide more preferred dynamic properties. In this paper, an immittance-function-layout (IFL) is introduced, which can cover a large range of topological connection possibilities with both mass and inerter elements. With the recently proposed structural immittance format, a systematic approach is established to identify the most beneficial IFL type mass-included inerter-based configurations with pre-determined number of each element type. Vibration suppression performance with single-IFL type device and two parallel-connected IFLs (i.e. dual-IFL) type devices are investigated in this paper. Three optimal configurations are identified for mitigating the maximum inter-storey drift of an example 3-storey building model subjected to base excitation. With this 3-storey building model, results show that, for the optimum single-IFL configuration, the performance improvement is 7.3% compared with the optimum TMDI, and with identified beneficial dual-IFL configurations, up to 34.9% performance advantages are obtained. Furthermore, consistent performance gains are shown under real-life earthquake inputs and with a 10-storey building model using identified absorber configurations.

There is an increasing desire in the railway industry to improve the longevity of wheels and rails without reducing performance in other ways (e.g. worsening passenger comfort). One way of reducing track and wheel wear is to reduce the primary yaw stiffness, significantly diminishing the costs associated with maintenance and emergency repairs, resulting however in reduced passenger comfort and high-speed stability. This paper, using a two-axle railway vehicle case study, demonstrates the potential of using passive, inerter-based suspensions to concurrently improve ride comfort and reduce track wear. The industrial parameter T Gamma is used to quantify the frictional energy lost at the contact patch under curving conditions, and the lateral RMS carbody acceleration is used to quantify passenger comfort under straight running conditions, with lateral track disturbances taken from real track data. Optimisation results conclude that, with the default yaw stiffness value, compared with the default spring- damper configuration in the primary lateral suspension, employing beneficial inerter-based configurations can improve passenger comfort by up to 43%. If the yaw stiffness is reduced such that the track wear is improved, similar improvements in passenger comfort can still be achieved with lateral inerter-based suspensions; for example, an improvement of 33% can still be achieved with a 50% reduction in yaw stiffness. Furthermore, when an inerter-based lateral suspension is used together with a Hall-Bush longitudinal suspension, the passenger comfort rises to 40%, which relates to a 25% improvement when compared with a non-inerter lateral plus Hall-Bush longitudinal setup.

Stay cables are prone to vibrations due to their low inherent damping. This paper presents an approach for systematic identification of beneficial passive vibration absorber layouts consisting of a damper, a spring, or an inerter. With the proposed method, optimal configurations among all layouts with three elements or fewer are identified. The inerter is a two-port mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals. In this work, a finite-element taut cable model, with a generic vibration absorber represented by its admittance function, is first established. Two performance measures, depending on the length of the cable and the forcing conditions, are introduced to assess the effect of candidate absorbers. Potential advantages of low-complexity inerter-based absorber layouts are then systematically investigated, with corresponding element values in these layouts identified. Building on this, the effect of series compliance is also examined for beneficial absorber layouts. It is shown that, up to a certain inertance, which depends on the stiffness of the series compliance, the performance advantages over a viscous damper can be maintained, or even increased in some cases, if the element values are properly tuned.

A key part of the main landing gear (MLG) of a civil aircraft is its locking mechanism that holds the gear in the deployed or down-locked state. The locking is driven by a spring mechanism and its release by the unlock actuator. This paper considers this mechanism in terms of its stability and the locking and unlocking forces required for down-locking. To study this an analytical model was developed. The equations, consisting of geometric constraints and force/moment equilibriums, were derived using the coordinate transformation method. Using numerical continuation to solve these equations, the effect ofthe unlock force on the MLG retraction cycle was analyzed. The variation of a fold bifurcation point, which indicates the transition between the locked state and the unlocked state, gives further insight into the required unlock force that governs the sizing of the unlock actuator. Moreover, some important information, such as the critical position for the lock-links’ stops, the unlock position and the unlock force, are discussed using the bifurcation diagrams for the MLG retraction/extension cycle. Then, the effect of three key geometry parameters of the locking spring (the spring stiffness, unstrained spring length and spring attachment point) on the critical over-center angle and the unlock force are investigated. Finally, an optimization of the critical unlock force is carried out with a constraint on the initial over-center angle. The results show that the spring parameters have significant effects on the MLG’s retraction performance. A 37% reduction of the required unlock force is obtained through optimizing for the gear considered here.

This paper investigates the possibility of improving aircraft landing gear touch-down performance by adding an inerter alongside a linear passive shock strut. The inerter is a novel mechanical element with the property that the applied force is proportional to the relative acceleration between its terminals. A simplified landing gear model is presented and the baseline performance of a conventional oleo-pneumatic shock absorber is established. Candidate layouts with linear mechanical components including inerters are considered using three objective functions: the strut efficiency, the maximum strut load and the maximum stroke. It is demonstrated that improved touch-down performance can be achieved with a linear inerter-based configuration. However it is also observed that the potential energy stored in the gear at the end of the first compression stroke exceeds that of the baseline nonlinear system. This suggests a poorer elongation stage might be observed. To address this, an additional constraint on energy dissipation is then considered. To achieve a reduced potential energy, a double-stage compression spring is introduced. With this, inerter-based configurations that provide improvements for the performance indices of interest are identified and presented.

Nonlinear stiffness isolation mounts, which offer a high static stiffness alongside a low dynamic stiffness or even quasi-zero-stiffness (QZS) over a displacement range have been proposed. These vibration isolators offer a higher isolation frequency band of low transmissibility than conventional linear devices. Here, three kinds of nonlinear two degree-of-freedom (DOF) vibration isolators with QZS characteristic are analysed in order to further improve the isolation performance. The dynamic response is obtained using the Harmonic Balance Method (HBM) and the peak dynamic displacement is obtained using backbone curve analysis and energy balancing method. The optimum isolation performance of the nonlinear 2DOF vibration isolators are evaluated for four performance indexes and compared with three baseline vibration isolators. These are a linear and a QZS 1DOF vibration isolator as well as a linear 2DOF vibration isolator. To ensure a fair comparison, the static displacement of each vibration isolator is kept constant. Thecomparison demonstrates that a nonlinear 2DOF vibration isolator can be tuned to achieve a better isolation performance in the higher isolation frequency band than the baseline vibration isolators, while retaining a moderate peak dynamic displacement and peak transmissibility. In addition, the best vibration isolator is identified for each of the four performance indexes.

The work reported in this paper concentrates on the possibility of suppressing shimmy oscillations of landing gear more effectively using a linear passive suppression device incorporating inerter. The inerter is a one-port mechanical device with the property that the applied force is proportional to the relative acceleration between its terminals. A linear model of a Fokker 100 aircraft main landing gear equipped with a shimmy-suppression device is presented. Time-domain optimizations of the shimmy-suppression device are carried out using cost functions of the maximum amplitude and the settling time of torsional-yaw motion. Applying two types of excitations that trigger the shimmy oscillations, the performance advantages of inerter-based configurations for suppressing shimmy of main landing gear, together with corresponding parameter values, are identified.

This paper proposes a new method, the structure-immittance approach, for designing the passive vibration absorbers consisting of inerters, dampers and springs. When considering possible configurations for these elements broadly, one of two exist approaches may be taken, either a structure-based or an immittance-based approach. Both methods have their advantages and disadvantages. In this paper, the new approach combines the advantages of the existing ones. It can both consider a full set of absorber layouts together (the advantage of immittance-based approach), and restrict the complexity, topology and element values of the candidate layouts (the advantage of structure-based approach). The structural immittances covering a full set of possible networks with one damper, one inerter and at most one spring are derived and applied to a civil engineering study. This demonstrates the advantages of the new methodology in being able to identify the optimum configurations for different element constraints.

The inerter is a dynamic physical dual of a capacitor via the force-current analogy, having the property that the force across the terminals is ideally proportional to their relative acceleration. Fluid-based forms of inerter have physical advantages of improved durability, inherent damping and simplicity of design in comparison to mechanical flywheel-based forms. Apart from the inertial effect, linear and nonlinear damping also occur in the helical-tube fluid inerter arrangement. In previous studies, discrepancies between experimental and theoretical results have been found. These are believed to arise from imperfect modelling of damping and pressure losses within the helical tube. To model these effects more accurately, this paper introduces a new experimental set-up. Pressure gauges are used to measure the pressure drop across the helical channel during constant velocity tests. This approach delivers improved agreement between experimental and theoretical results. The sources of minor remaining discrepancies are further analysed. Furthermore, a new fluid-based inerter design is first proposed with different damping characteristics, the theoretical damping comparison is also presented between these two designs.

A common approach to the mitigation of vibrations of structural cables, e.g. on cable-stayed bridges, is to install a viscous damper transverse to the cable axis quite close to one end. The damping coefficient can be optimized for suppressing vibrations in one cable mode but the damper is then sub-optimal for other modes. This paper proposes the use of a passive inerter-based vibration absorber for suppressing multiple unwanted cable vibration modes. The inerter has the property that the force between its two terminals is proportional to their relative acceleration. A previous study has shown that inerter-based vibration absorber configurations can provide greater modal damping ratios than a viscous damper alone for vibration modes around the first undamped natural frequency. In this study, a finite-element model of a taut cable, together with a generic absorber is built. The absorber is located close to one end of the cable and represented by a general positive-real impedance function. The absorber layouts for maximizing the modal damping ratios over lower-frequency modes while perverting deterioration those for higher-frequency modes are identified. It will be shown that, compared with traditional viscous dampers, the proposed inerter-based vibration absorbers can give enhanced damping performance over multiple modes.

Reichert’s theorem (1969), a fundamental theorem of network synthesis, completely characterises minimum reactive synthesis of positive-real biquadratic impedances. The crucial part of the original approach depends on a complicated topological argument. This paper provides an alternative proof using the recently introduced concept of regular positive-real functions.

Retrofitting of existing buildings with seismic protection devices is an important approach which is commonly adopted by the civil engineering community. This paper investigates the potential of inerter-integrated seismic vibration absorber for building structures. A real case study of a high-rise building in Colombia is considered herein to demonstrate the effectiveness of the proposed device. An inerter generates a force proportional to the relative acceleration between its two terminals and it fundamentally enlarges the range of realisable passive mechanical impedances. In this work, an interter-integrated device is modelled into the open-source structural engineering software OpenSees and implemented into a simplified two-dimensional (2D) version of the Colombian test-bed structure. The performances of the structure retrofitted with the proposed absorber are compared to those of the same structure retrofitted with a traditional damper device, a more classical retrofitting. This preliminary work demonstrates the effectiveness of the inerter-integrated retrofitting strategies considering the non-structural damage caused by a moderate earthquake. It has been found that with an optimal inerter-integrated device, the maximum inter-storey drift of the building will be reduced by 45.64% comparing with the structure without any retrofitting, which is 20.8% further reduction comparing with the structure retrofitted with an optimal damper.

The aim of this paper is to investigate the possibility of improving the ride quality of a two-axle railway vehicle with a single-stage suspension by means of passive suspensions employing an inerter device. The inerter is a mechanical one-port element that is analogous to a capacitor in electrical circuits. The goal is to improve the ride quality in both the vertical and lateral motions in response to track irregularities. Performance benefits for several simple passive suspension layouts are demonstrated and compared with the conventional scheme. The elastic effects of the damper and inerter device are then taken into consideration for practical purposes. The optimum parameter values of the damper, inerter and the parameters representing the elastic effects provide guidance for mechanical design purposes.

This paper reworks and amplifies Reichert’s proof of his theorem (1969) which asserts that any impedance function of a one-port electrical network which can be realised with two reactive elements and an arbitrary number of resistors can be realised with two reactive elements and three resistors.

This paper investigates the use of inerters for vibration suppression of a multistorey building structure. The inerter was proposed as a two-terminal replacement for the mass element, with the property that the applied force is proportional to the relative acceleration across its terminals. It completes the force-current mechanical-electrical network analogy, providing the mechanical equivalent to a capacitor. Thus allows all passive mechanical impedances to be synthesised. The inerter has been used in Formula 1 racing cars and applications to various systems such as vehicle suspension have been identified. Several devices that incoporate inerter(s), as well as spring(s) and damper(s), have also been identified for vibration suppression of building structures. These include the tuned inerter damper (TID) and the tuned viscous mass damper (TVMD). In this paper, a three-storey building model with an absorber located at the bottom subjected to base excitation is studied. Four simple absorber layouts, in terms of how spring, damper and inerter components should be arranged, have been studied. In order to minimise the maximum relative displacement of the building, the optimum parameter values for each of the layouts have been obtained with respect to the inerter's size.

The object of this paper is to give a complete treatment of the realizability of positive-real biquadratic impedance functions by six-element series-parallel networks comprising resistors, capacitors, and inductors. This question was studied but not fully resolved in the classical electrical circuit literature. Renewed interest in this question arises in the synthesis of passive mechanical impedances. Recent work by the authors has introduced the concept of a regular positive-real functions. It was shown that five-element networks are capable of realizing all regular and some (but not all) nonregular biquadratic positive-real functions. Accordingly, the focus of this paper is on the realizability of nonregular biquadratics. It will be shown that the only six-element series-parallel networks which are capable of realizing nonregular biquadratic impedances are those with three reactive elements or four reactive elements. We identify a set of networks that can realize all the nonregular biquadratic functions for each of the two cases. The realizability conditions for the networks are expressed in terms of a canonical form for biquadratics. The nonregular realizable region for each of the networks is explicitly characterized.

This paper investigates the possibility of improving the performance of railway vehicle suspensions by incorporating a newly developed mechanical device known as the inerter. A comparative study of several low-complexity passive suspension layouts is made. Improved performance for the lateral and vertical ride comfort, as well as lateral body movement when curving are demonstrated in comparison with the conventional suspension layout. The constraints imposed are to maintain the same level of other performance metrics. The calculations and optimisations are based on linearised plan-view and side-view high-speed train mathematical models.

This paper classifies the positive-real biquadratic functions which can be realized by five-element networks. The concept of regular positive-real functions is introduced to facilitate this classification. Networks are grouped into quartets which may sometimes reduce to two or one network(s). It is shown that a biquadratic can be realized by a series-parallel network with two reactive elements if and only if it is regular. Moreover, there are two such network quartets which can realize all regular biquadratics. It is shown that the only five-element networks which can realize nonregular biquadratics can be arranged into three network quartets. The quartets comprise: 1) two bridge networks with two reactive elements; 2) four series-parallel networks with three reactive elements; and 3) two bridge networks with three reactive elements. The necessary and sufficient realizability conditions are derived for each of these networks. The results are motivated by an approach to passive mechanical control which makes use of the inerter device.