Physical Damping Research Articles

Towards optimal use of the explicit [formula omitted]-Bathe time integration method for linear and nonlinear dynamics

In an earlier publication, we proposed a new explicit time integration scheme, the β1/β2-Bathe method, which is simple in its formulation and showed remarkable accuracy in the solution of problems [1]. A particular strength of the method is that it can directly be used as a first-order or second-order scheme by a change of the values of β1 and β2. While good results are obtained with reasonable values of β1 and β2, for excellent accuracy better values of the parameters need to be chosen. We propose in this paper values of β1 and β2 for the first-order scheme, best used in wave propagation analyses, and separate values for the second-order scheme, best used in analyses of structural vibrations. In each case, one set of values of (β1,β2) is given and to possibly improve the results only one of the parameters needs to be changed, that is, β1 for wave propagations and β2 for structural vibrations, making the scheme a one-parameter method. Another strength of the procedure is that physical damping can directly be included in the solution, the effect of which on the stability and accuracy of the solutions we analyze in the paper. The use of the solution scheme in nonlinear analysis is, as we show in the paper, a simple extension from linear analysis. Finally, we give various solutions using the explicit β1/β2-Bathe method in linear and nonlinear analyses to illustrate the performance of the method with the given recommendations for its use.

Investigation of a perturbation-based model updating approach for structural health and event monitoring

With the intention to detect structural events, for example, occurrence of damage and assessing their severity under service, structural health and event monitoring proved itself as a reliable evaluation tool. While many algorithmic approaches exist, model-based monitoring offers the prominent advantage to enable extensive analysis of damage states based on a structural digital twin. In order to accurately replicate the dynamical behaviour of a real-world structure, inherent physical properties, for example, mass, stiffness and structural damping, of the digital twin might be corrected from nominal values. This can be achieved by numerical model updating procedures. In this study, the application of a model updating approach is presented to eliminate modal discrepancies. Based on the perturbated modal dynamic residual, an updating equation is formulated. Stiffness and mass correction terms with preserved orthogonality and symmetry conditions are determined by the method of least squares. A case study using computationally generated modal data is conducted, evaluating the numerical updating performance in terms of correction accuracy and reproduction of modal parameters. Finally, the transient structural response to an impulse excitation is compared between updated and reference model. The gathered results prove the method’s suitability as accurate and robust updating procedure, fostering its application in model-based monitoring frameworks.

Design and control of a compliant robotic actuator with parallel spring-damping transmission

AbstractPhysically compliant actuator brings significant benefits to robots in terms of environmental adaptability, human–robot interaction, and energy efficiency as the introduction of the inherent compliance. However, this inherent compliance also limits the force and position control performance of the actuator system due to the induced oscillations and decreased mechanical bandwidth. To solve this problem, we first investigate the dynamic effects of implementing variable physical damping into a compliant actuator. Following this, we propose a structural scheme that integrates a variable damping element in parallel to a conventional series elastic actuator. A damping regulation algorithm is then developed for the parallel spring-damping actuator (PSDA) to tune the dynamic performance of the system while remaining sufficient compliance. Experimental results show that the PSDA offers better stability and dynamic capability in the force and position control by generating appropriate damping levels.

High-order composite implicit time integration schemes based on rational approximations for elastodynamics

A new approach for developing implicit composite time integration schemes is established starting with rational approximations of the matrix exponential in the solution of the equations of motion. The rational approximations are designed to have the same effective stiffness matrix in all sub-steps. An efficient algorithm is devised so that the implicit equation for a sub-step is in the same form as that of the trapezoidal method. The proposed M-schemes are Mth order accurate using M sub-steps. The amount of numerical dissipation is controlled by a user-specified parameter ρ∞. The (M+1)-schemes that are (M+1)th order accurate using M sub-steps can also be constructed, but the amount of numerical dissipation is built in the schemes and not adjustable. The order of accuracy of all the schemes is not affected by external forces and physical damping. Numerical examples demonstrate that the proposed high-order composite schemes are effective for introducing numerical dissipation and allow the use of large time step sizes in wave propagation problems. The source code written in MATLAB is available for download at: https://github.com/ChongminSong/CompsiteTimeIntegration.

Variable Damping Actuator Using an Electromagnetic Brake for Impedance Modulation in Physical Human–Robot Interaction

Compliance actuation systems are efficient and safe, drawing attention to their development. However, compliance has caused bandwidth loss, instability, and mechanical vibration in robotic systems. Variable physical damping was introduced to address these issues. This paper presents a technique for obtaining variable damping properties using an electromagnetic brake. The relationship mapping of the voltage and the braking torque is studied and applied to the variable damping concept. A new model is proposed to demonstrate the actuation system performance gained by introducing physical damping. The experimental setup comprises an electromagnetic brake and a motor with an integrated controller for speed control and torque feedback. The motor provides the motion, while the electromagnetic brake replicates the damping through a friction mechanism. The variable damping concept was evaluated experimentally using a 1-degree-of-freedom rotational system. Experimental results show that the proposed concept can generate the desired mechanical damping with a high degree of fidelity.

An explicit integration method with third-order accuracy for linear and nonlinear dynamic systems

Computational efficiency and accuracy are two of the most important properties of integration methods. In this study, an explicit single-step integration method with third-order accuracy is proposed to improve computational efficiency and accuracy. The proposed method can achieve fourth-order accuracy in terms of displacement, velocity, and acceleration responses without physical damping. The algorithmic dissipation property is also improved to filter out high-frequency spurious responses on the premise of sufficient accuracy in the low-frequency response domain. Moreover, since the proposed method can advance the calculation step-by-step via the known displacement and velocity items, the time-consuming factorization of an effective stiffness matrix is not required in the calculation for the structure with a lumped mass matrix. In other words, high computational efficiency is ensured in the proposed method. The properties of the proposed method (i.e., the accuracy, convergence, dissipation, efficiency, and effectiveness of the nonlinearity) are demonstrated by using four representative examples. Specifically, a single-degree-of-freedom (SDOF) dynamic system with a theoretical solution is adopted to comparatively evaluate the accuracy and convergence of the proposed method, a typical nonlinear dynamic system is used to investigate the effectiveness of the nonlinear calculation; a linear Howe truss model is employed to explore the effectiveness and efficiency in the calculation of a multi-DOF structure, and a nonlinear wellbore modelis used to demonstrate the potential to solve the large-scale nonlinear system. Results show that the proposed method can obtain more accurate results in the nonlinear dynamic calculations; and its time consumption is around 65% of that of the Yuan method, 38% of that of the Kim method, and 30% of that of the Zhai method.

Physical, thermal, mechanical, sound absorption and vibration damping characteristics of natural fiber reinforced composites and hybrid fiber reinforced composites: A review

Physical, thermal, mechanical, sound absorption and vibration damping characteristics of natural fiber reinforced composites and hybrid fiber reinforced composites: A review

A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

AbstractAn important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high‐frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first‐order overshooting components enter into several well‐known time integration algorithms which were previously thought to exhibit zero‐order overshooting in displacement. The Newmark method, Wilson‐ method, Bazzi‐ method, HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method are analyzed, as well as the generalized single‐step single‐solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.

A massless boundary component mode synthesis method for elastodynamic contact problems

We propose to combine the ideas of mass redistribution and component mode synthesis. More specifically, we employ the MacNeal method, which readily leads to a singular mass matrix, and an accordingly modified version of the Craig-Bampton method. Besides obtaining a massless boundary, we achieve a drastic reduction of the mathematical model order in this way compared to the parent finite element model. Contact is modeled using set-valued laws and time stepping is carried out with a semi-explit scheme. We assess the method’s computational performance by a series of benchmarks, including both frictionless and frictional contact. The results indicate that the proposed method achieves excellent energy conservation properties and superior convergence behavior. It reduces the spurious oscillations and decreases the computational effort by about 1–2 orders of magnitude compared to the current state of the art (mass-carrying component mode synthesis method). We believe that the computational performance and favorable energy conservation properties will be valuable for the prediction of vibro-impact processes and physical damping.

Coaxial damping series elastic actuator for improved performance of transmissibility

There are many flexible joint designs that are delivering torque and absorbing shocks simultaneously. One of the most active areas of the research is the Series Elastic Actuator (SEA), which connects the motor and load by lowering the stiffness. The problem, however, is that the non-rigid stiffness causes the system to have a lower bandwidth, thereby slowing down the system’s response. On the other hand, the high stiffness of SEA adversely affects energy efficiency and performance of impact relief. In other words, concerning the application of conventional SEAs, it is important to determine the joint stiffness. Therefore, other researchers have developed Variable Stiffness Actuators (VSAs) that can control the stiffness in the system. To improve further the controllability and transmissibility of SEAs, Variable Physical Damping Actuators (VPDAs) that can change the damping element of the system has been developed. Although previous studies have provided valuable findings, VSAs and VPDAs make the system complex so that complicated control theories are required. In this paper, a new Coaxial Damping SEA (CODSEA) that can efficiently adjust the stiffness and dynamic damping is designed. The biggest advantage of CODSEA is that it can change the stiffness and damping efficiently by converting the geometry of the driving shaft and particle materials inside the driving shaft without additional systems. In other words, only the driving shaft becomes the SEA and physical damping system.

A novel predictor–corrector explicit integration scheme for structural dynamics

A novel predictor–corrector explicit scheme is presented to solve structural dynamic problems. It is a three sub-steps method, in which the previous two sub-steps are set as predictors and the last sub-step is regarded as correctors. The explicit scheme is third-order accuracy and can achieve forth-order accuracy in the absent of physical damping. The stability limit of the proposed scheme is much larger than the existing methods. Also, the numerical dissipation and dispersion of the explicit scheme can be controlled through the different algorithm parameters. The explicit scheme not only possesses adequate numerical dissipation and dispersion in high-frequency responses, but also gets small numerical errors in the whole frequency domain for dynamic system. These performances of the proposed explicit scheme are further highlighted in comparison with other typical explicit schemes.

A‐stable linear two‐step time integration methods with consistent starting and their equivalent single‐step methods in structural dynamics analysis

AbstractA spectral consistent starting procedure is proposed for the first‐order‐type A‐stable linear two‐step (LTS) time integration methods in structural dynamics analysis. The accuracy analysis for the LTS methods in structural dynamics is presented based on the first‐order model, which enables the algorithms in first‐order transient systems to be extended to structural dynamics under the umbrella of a consistent framework. It is indicated that the approximation of the loads plays an essential role in the equivalence between the LTS methods and some single‐step methods. An algorithmic equivalence feature, which is stricter than the spectral equivalence, is revealed between the two‐leg generalized‐α methods in first‐order systems and the A‐stable LTS methods. The measures of the error constant in the LTS methods and the ultimate spectral radius are extended to optimize a class of single‐step methods of a given convergence order. The spectral consistent starting procedure for the A‐stable LTS methods is developed by utilizing the algorithmic equivalence of the single‐step methods, which results in an optimal A‐stable LTS (OALTS) method possessing a consistent spectral radius with controllable numerical dissipation in the starting step and the steps thereafter. Comparing with the equivalent single‐step methods, the present OALTS method does not need the auxiliary variables, and its displacement, velocity and acceleration can achieve second‐order accuracy simultaneously. The performance of the present OALTS method is verified by numerical examples including physical damping, external loads, and/or nonlinearity.

Effective Viscous Damping Enables Morphological Computation in Legged Locomotion.

Muscle models and animal observations suggest that physical damping is beneficial for stabilization. Still, only a few implementations of physical damping exist in compliant robotic legged locomotion. It remains unclear how physical damping can be exploited for locomotion tasks, while its advantages as sensor-free, adaptive force- and negative work-producing actuators are promising. In a simplified numerical leg model, we studied the energy dissipation from viscous and Coulomb damping during vertical drops with ground-level perturbations. A parallel spring- damper is engaged between touch-down and mid-stance, and its damper auto-decouples from mid-stance to takeoff. Our simulations indicate that an adjustable and viscous damper is desired. In hardware we explored effective viscous damping and adjustability, and quantified the dissipated energy. We tested two mechanical, leg-mounted damping mechanisms: a commercial hydraulic damper, and a custom-made pneumatic damper. The pneumatic damper exploits a rolling diaphragm with an adjustable orifice, minimizing Coulomb damping effects while permitting adjustable resistance. Experimental results show that the leg-mounted, hydraulic damper exhibits the most effective viscous damping. Adjusting the orifice setting did not result in substantial changes of dissipated energy per drop, unlike adjusting the damping parameters in the numerical model. Consequently, we also emphasize the importance of characterizing physical dampers during real legged impacts to evaluate their effectiveness for compliant legged locomotion.

Necessary and Sufficient Conditions for the Passivity of Impedance Rendering With Velocity-Sourced Series Elastic Actuation

Series elastic actuation (SEA) has become prevalent in applications involving physical human–robot interaction, as it provides considerable advantages over traditional stiff actuators in terms of stability robustness and force control fidelity. Several impedance control architectures have been proposed for SEA. Among these alternatives, the cascaded controller with an innermost velocity loop, an intermediate torque loop, and an outermost impedance loop is particularly favored for its simplicity, robustness, and performance. In this article, we derive the necessary and sufficient conditions for passively rendering null impedance and virtual springs with this cascade-controller architecture. Based on the newly established conditions, we provide nonconservative passivity design guidelines to haptically display these two impedance models, which serve as the basic building blocks of various virtual environments, while ensuring the safety of interaction. We also demonstrate the importance of including physical damping in the actuator model for deriving the passivity conditions, when integrators are utilized. In particular, we prove the unintuitive adversary effect of physical damping on the passivity of the system by noting that the damping term reduces the system $Z$ -width, as well as introducing an extra passivity constraint. Finally, we experimentally validate our theoretical results using an SEA brake pedal.

A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics

SummaryA comprehensive study of A‐stable linear two‐step time integration methods for structural dynamics analysis is presented in this paper. An optimal A‐stable linear two‐step (OALTS) time integration method is revealed with desirable performance on low‐frequency accuracy and high‐frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second‐order equations of structural dynamics; is implicit, A‐stable, and second‐order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.

Identification of physical nonlinearities of a hybrid aeroelastic\u2013pressure balance

This study has presented an improved method for determining physical nonlinearities of weakly nonlinear spring-suspension system and successfully applied to a novel hybrid aeroelastic–pressure balance (HAPB) system used in wind tunnel, which can be used for simultaneously obtaining the unsteady wind pressure and aeroelastic response of a test model. A nonlinear identification method of equivalent linearization approximation was firstly developed on the basis of the averaging method of Krylov–Bogoliubov to model the physical nonlinearity of a weakly nonlinear system. Subsequently, the nonlinear physical frequency and damping were identified using a modified Morlet wavelet transform method and a constant variant method. Using these methods, the physical nonlinear frequency and damping of the HAPB system with a vertical test model were determined and were validated by a time domain method and the Newmark-beta method. Finally, the nonlinear mechanical frequency and damping of the HAPB system with inclined test models were determined in a similar way. This study has not only provided an identification method for determining physical nonlinearities of weakly nonlinear system, but presented the detail for developing a hybrid aeroelastic–pressure balance used in wind tunnel.

Time-varying system identification by enhanced Empirical Wavelet Transform based on Synchroextracting Transform

In this paper, an enhanced Empirical Wavelet Transform (EWT) approach based on Synchroextracting Transform (SET) is proposed for time-varying system identification. When a structure of time-varying physical properties, i.e. mass, stiffness or damping, is under external excitations, structural dynamic responses are usually non-stationary because the system has time-varying dynamic vibration characteristics. Under this circumstance, it would be difficult to determine the number of Intrinsic Mode Functions (IMFs) included in structural dynamic responses by using Fourier spectrum. Considering that the filtering boundaries of traditional EWT method are defined based on the segmental Fourier Spectrum of a processed signal, directly using it for non-stationary signal decomposition may not be effective and accurate. To apply the EWT method for time-varying system identification, in this study, time-frequency analysis based on SET is first performed to determine the frequency components of a non-stationary vibration signal instead of using Fourier spectrum. The filtering boundaries for EWT analysis are determined based on the time-frequency representation. Then, the IMFs are extracted from the non-stationary vibration signals by using EWT with the above defined filtering boundaries. When the IMFs are accurately obtained, the instantaneous frequencies of IMFs are identified by using Hilbert Transform (HT). In numerical simulations, a simulated signal with a high level noise is analyzed to verify the feasibility of using SET to define the filtering boundaries. Then the proposed approach is used to identify the instantaneous frequencies of a time-varying two-storey shear type building under earthquake and Gaussian white noise excitations, respectively. Experimental investigations on a time-varying bridge-vehicle system are conducted to verify the effectiveness of the proposed approach. The results in both numerical simulations and experimental validations demonstrate that the enhanced EWT approach can effectively and reliably identify the instantaneous frequencies of time-varying systems.

Two novel explicit time integration methods based on displacement-velocity relations for structural dynamics

Two novel explicit time integration methods are proposed based on displacement-velocity relations in this paper for structural dynamics. They avoid the factorization of damping and stiffness matrices, and are truly self-starting due to the exclusion of acceleration vectors. The first method employs the motion equation of expanded form and a linear relation of the displacement and velocity vectors. The recommended parameters, derived from linear analysis, enable the method to possess first-order accuracy mostly and second-order accuracy in the absence of numerical and physical damping. The second method adopts the idea of composite methods, and employs two motion equations of different expanded forms per step. Theoretical analysis indicates that this method can achieve a maximum stability limit of 4, and provides a single-parameter optimal scheme, controlled by the degree of numerical dissipation, for this method. The resulting scheme is second-order accurate with the stability limit ranged from 3.5708 to 4 for the undamped case, and some numerical experiments show that it has better numerical performance compared with some up-to-date explicit methods.

Physical Parametric Model of an Automotive Electrohydraulic Semiactive Damper

A full mathematical model is developed to formulate the nonlinear characteristics of an automotive electrohydraulic semiactive damper. This multidisciplinary physical parametric model considers the force equilibrium, flow continuity with flow losses due to the volumetric change and pressure difference, total flow through the valve system under different control modes, and dynamics of the floating piston which are balanced by the pressures of the fluid and gas. Both computer simulation and bench testing of the damper demonstrate that the damper has four levels of damping characteristics: “hard mode”, “hard rebound and medium-soft compression mode”, “medium-hard rebound and soft compression mode” and “soft mode”. The considerable consistency between the simulation and experimental results verifies that the mathematical model accurately captures the physical damping characteristics of the damper. Thus, the developed physical damper model is experimentally validated. The obtained full parametric damper model with mode controls is valuable for accurate and reliable damper development and crucial for the next step in vehicle system dynamics simulation and suspension control system synthesis.

Analysis of high-frequency ANCF modes: Navier–Stokes physical damping and implicit numerical integration

This paper is concerned with the study of the high-frequency modes resulting from the use of the finite element absolute nodal coordinate formulation (ANCF) in multibody system (MBS) applications. The coupling between the cross-sectional deformations and bending and extension of ANCF beam and plate elements produces high-frequency modes which negatively impact the computational efficiency. In this paper, two new and fundamentally different approaches are proposed to efficiently solve stiff systems of differential/algebraic equations by filtering and/or damping out ANCF high-frequency modes. A new objective large rotation and large deformation viscoelastic constitutive model defined by the Navier–Stokes equations, widely used for fluids, is proposed for ANCF solids. The proposed Navier–Stokes viscoelastic constitutive model is formulated in terms of a diagonal damping matrix, allows damping out insignificant high-frequency modes, and leads to zero energy dissipation in the case of rigid body motion. The second approach, however, is numerical and is based on enhancing the two-loop implicit sparse matrix numerical integration (TLISMNI) method by introducing a new stiffness detection error control criterion. The new criterion avoids unnecessary reductions in the time step and minimizes the number of TLISMNI outer loop iterations required to achieve convergence. The TLISMNI method ensures that the MBS algebraic constraint equations are satisfied at the position, velocity, and acceleration levels, efficiently exploits sparse matrix techniques, and avoids numerical force differentiation. The performance of the TLISMNI/Adams algorithm using the proposed error criterion is evaluated by comparison with the TLISMNI/HHT method and the explicit predictor–corrector, variable-order, and variable step-size Adams methods. Several numerical examples are used to evaluate the accuracy, efficiency, and damping characteristics of the new nonlinear viscoelastic constitutive model and the TLISMNI procedure.