Proportional Damping Research Articles

A Foundational Study on Rational Optimization of Damping Ratio for Accurate Dynamic Simulation with Ultra Large Displacement

The integration of dynamic simulation analysis has become widespread in general-purpose software, providing enhanced capabilities. However, accurately tracking deformations based on complete equilibrium solutions remains a significant challenge in problems characterized by strong geometric nonlinearity. This study examines the accuracy of the combined Newmark β method and Tangent stiffness method in dynamic analysis with ultra large displacements and evaluates the utility of Rayleigh proportional damping in numerical simulations compared to experimental models. An experimental model of a slender steel plate undergoing free vibrations after being released from a deformed state was created. Video footage capturing the deformation histories was compared to computational simulations to verify accuracy. The study also examines the appropriate values of the damping ratio (ζ) and the Newmark β value in the simulations. The results indicate that adjusting various damping ratio and a β value of 1/2 yield more realistic simulations with longer conservation of mechanical energy. The findings suggest that incorporating numerical damping into actual damping settings can achieve a more realistic simulation of dynamic behaviour with ultra-large displacements. Furthermore, experiments and analyses were performed to correct natural frequency by changing Young’s modulus, which is a key factor influencing natural frequencies, with observed correlations to plate thickness, setting the stage for further research under varied conditions to develop a more rational methodology for structural analysis.

An Optimization Method for Stiffness and Damping Mistuning Identification From Blade Tip Timing Data

Abstract System identification of dynamic properties is of large interest in the turbomachinery industry to create more accurate computational models and more effective designs. Previous identification techniques were able to accurately capture the blade variability, known as mistuning, in the stiffness as well as the average damping of all the blades. Mistuning is vital to accurately identify and study because the symmetry of the system is broken and can lead to vibration localization and high amplitudes. In this work, a new method is proposed to not only capture the stiffness mistuning values, but also the blade-to-blade variability in damping. These damping mistuning values have been shown to have a significant effect on the dynamics of bladed disk systems. Incorporation of the damping mistuning into the computational model can be done utilizing an augmented component mode mistuning method with either structural or proportional damping. The mistuning values for this new identification method were compared to a well-established direct method and a previously studied optimization method. Blade responses were then found using a harmonic analysis and the newly identified mistuning values. These blade responses were then compared to experimental tip timing data from full scale rotating experiments. These comparisons show that the new model is able to better reproduce experimental data using computational models that incorporate both stiffness and damping mistuning values.

Flutter in functionally graded conical shell under follower force

Truncated conical shells are essential components of rocket booster nozzles and thrust vector control (TVC) systems for the propulsion of multi-stage launch vehicles. The TVCs assist the ascent of launch vehicles by directing the resultant thrust from the boosters, acting as a follower force that might trigger instability. Previous studies on the instability of aerospace structures have mostly focused on beams, plates, and cylindrical shells. This study analyses a truncated conical shell under a follower force of constant magnitude, considering thickness-wise gradation of material properties employed for thermal management. The governing equations are derived following Hamilton's principle, considering first-order shear deformation theory, and solved using the finite element method. Clamped and free boundaries are assumed at the small and large ends. The influence of mass and stiffness proportional damping is considered. Although strong flutter appears as the dominant instability mode, instances of erratic weak instabilities are also observed for undamped and lightly damped cases. Damping enhances stability, but its effect becomes saturated. The flutter loads are presented for varying non-dimensional parameters, characterizing the shell geometry and material properties.

Improving the Step Response of Flexible Systems in the Presence of Real Nonminimum Phase Zeros

Abstract It is well known that real nonminimum phase (RNMP) zeros impose a tradeoff between the settling time and undershoot in the step response of flexible systems. Existing methods to alleviate this tradeoff predominantly rely on various advanced control strategies without delving into a broader mechatronic approach that combines physical system and control system design. To address this gap, this article proposes a proportional viscous damping-based physical system design in combination with feedback control and prefilter design. First, the effect of proportional viscous damping on RNMP zeros of flexible systems is established to propose a damping strategy that pushes all the RNMP zeros further away from the imaginary axis. Then, a step-by-step mechatronic system design process is presented to apply this damping strategy along with a full-state feedback control strategy and prefilter to a multi-degree-of-freedom (DoF) flexible system. The application of this design process yields simultaneous improvement in the settling time and undershoot in the step response of this flexible system.

A comparative analysis of damping models for the vibratory properties of a bent beam

Much of Eric Unger's career was devoted to the study of dissipation mechanisms. This paper will explore the effect of various ways that damping effects have been incorporated in the solution of equations of motion. The analytical model of an L-shaped beam, cantilvered at one end and loaded at its free end by a force that is transverse to the plane of the bent beam, is the basis for the study. Equations of motion that account for coupling of flexure and torsion are created by merging the Ritz series method and the method of Lagrange multipliers. Energy is dissipated by a dashpot transversely mounted at the corner. An “exact” solution of the equations of motion using state-space modal analysis provides the reference for approximate methods including proportional damping, modal decoupling, and structural damping. Impulse response and complex frequency response at various levels of damping will be addressed.

Modal identification of non‐classically damped structures using generalized sparse component analysis

SummaryModal identification method based on blind source separation (BSS) technique has gained extensive attentions for civil structures. Developing the complex modes estimation method is important in practical applications because the assumption of proportional damping is not always satisfied. Sparse component analysis (SCA) performs well in underdetermined BSS problems. However, SCA is confined to the situation of proportional damping. In this study, a generalized SCA method is proposed to extend the original SCA method to both real and complex modes identification. First, the general formulation of complex modes is extended by the analytic form to eliminate the complex conjugate part in the BSS model. A new single‐source‐point detection method that is available to handle real and complex modes is proposed. Local outlier factor method is adopted to remove the outliers in single source points. Subsequently, complex‐valued modal matrix is calculated by the clustering technique. Then, modal responses are recovered using the complex version of smoothed zero norm method, where modal frequencies and damping ratios can be extracted. Finally, the effectiveness of the proposed method is demonstrated for identification of real and complex modes, close modes, and underdetermined problem. The application to a benchmark structure demonstrates the effectiveness for practical applications.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

Dynamics of a double pendulum with viscous friction in the joints. II. Dissipative oscillation modes and optimization of damping parameters

This paper is a continuation of the article "Dynamics of a double pendulum with viscous friction in the joints. I. Mathematical model of motion and construction of the regime diagram", in which a linear mathematical model of the motion of a double mathematical pendulum with identical parameters of links and end loads in the presence of viscous friction in both articulations was given, and a diagram of dissipative regimes of its motion was also constructed. The question of a particular variant of proportional damping is considered, in which the oscillation modes of a dissipative system are not distorted by friction forces, and basic formulas are given that describe the dynamics of the system in this situation. For the general case of damping, through a rational combination of analytical and numerical research methods, all key quantities characterizing the motion of the system on each of the dissipative oscillation modes are identified and determined. In addition, several problems of optimal damping of system oscillations are considered, and the best dissipative parameters are selected based on the criterion of the maximum degree of stability. The obtained results are accompanied by a series of graphical illustrations, which make it possible to establish their dependence on the damping coefficients and note their main qualitative and quantitative features. The solutions found can be useful in practice when designing two-link manipulators and studying their dynamic behavior.

Experimental evaluation of damping models for a nonlinear pendulum system

Popular damping models for a single pendulum system were experimentally evaluated in this study. The comparison between the numerical and experimental responses showed that neither linear viscous damping model nor squared velocity proportional viscous damping model was able to well capture the free vibration amplitudes of the pendulum in both large and small amplitude ranges. Specifically, the amplitudes of the numerical model with linear viscous damping decreased slower at the large amplitude vibration but faster at the small amplitude vibration than the experimental amplitudes did. The trend was reverse in the numerical model with the squared velocity proportional viscous damping model. Based on the experimental data and a numerical method combined with a fitting approach, a complete quadratic viscous damping model for the pendulum was derived. The model was able to confidently predict the free vibration amplitudes of the pendulum in both large and small amplitude ranges. This study provides analysts with the sense of the accuracy of some popular damping models for a nonlinear pendulum system. It also demonstrates the use of a fitting approach to retrieve physical properties of a system from experimental data. The contents of this paper are suitable for gifted high school students and undergraduate students.

Frequency Fluctuation Mitigation in a Single-Area Power System Using LQR-Based Proportional Damping Compensator

To maintain the stability of the power system, frequency fluctuations must be reduced in the shortest possible timeframe. Load frequency control (LFC) plays a critical role in achieving this objective by regulating the system frequency and the desired demand or output power in an interconnected network, thereby enabling the system to adapt the load disturbances. In order to effectively mitigate the frequency fluctuation caused by load variation in a single-area power system, a new control strategy integrating a linear quadratic regulator (LQR), a proportional controller, and a damping compensator is proposed in this paper. The proposed controller is named as the LQR-based proportional damping compensator which mitigates the frequency fluctuation of a single-area power system. MATLAB/Simulink simulation is conducted on a single-area power system to demonstrate the efficacy of the proposed control technique. The simulation results demonstrate that the proposed method successfully reduces frequency variations, maintains system frequency within reasonable limits, and substantially reduces the settling time as compared to other existing control techniques. Apart from the simulation analysis, to experimentally validate the performance of the proposed controller, a hybrid multiprocessor-based processor-in-loop (PIL) technique is also introduced in the paper. Both the simulation and experimental results prove the promising performance of the proposed controller for mitigating the frequency fluctuation of a single-area power system.

A Novel Adaptive Finite-Time Position Tracking Control Strategy for Teleoperation System with Varying Communication Delays

Based on the traditional control approach, the position-tracking performance of the teleoperation system with communication delay is generally asymptotically stable. In practical applications, the closed-loop system is expected to achieve stable and finite-time convergence performance. A novel finite-time bilateral control scheme for a telerobotics system with communication delay is presented in this paper. On the basis of the traditional proportional damping injection control, this paper proposes and designs a new finite-time control method by introducing the non-integer power to the position error, velocity, and the combined error with position error and velocity. In comparison to existing proportional damping injection and finite-time control structures, the proposed method not only achieves the finite-time convergence performance of position tracking, but it also has the advantages of a simple structure and fewer gain coefficients. The controller also incorporates the radial basis function (RBF) neural network and adaptive approach to compensate unknown dynamics and external forces, thus also avoiding the measurement of force signals. The Lyapunov–Krasovskii function is then defined, and it is demonstrated that the position tracking of closed-loop teleoperation system has bounded stability and finite-time control performance. The simulation experiment is also performed, and the results further illustrated the bounded stability of the system. Moreover, compared to the position tracking errors of other non-finite-time control methods, it is demonstrated that the proposed finite-time control scheme has a faster convergence rate and higher convergence precision.

FRF-based finite element model updating for non-viscous and non-proportionally damped systems

All structures exhibit some form of damping, but the characterization of damping is not well-understood, and there is no universal damping model for the dynamic systems. Recently, model updating methods have been used to update or identify the damping matrix in dynamic systems. Most of the finite element updating methods assume viscous and proportional damping models for updating or identifying of damping matrix. In this paper, a new finite element model updating method is proposed in which the damping model is assumed as non-viscous and non-proportional. A parametric exponential non-viscous damping model has been used to model the damping in the dynamic system. The proposed method is the frequency response function (FRF)-based updating model, which updates the non-viscous and non-proportional damping matrix in the dynamic system. The effectiveness of the proposed damped finite element updating method is demonstrated by a numerical example and actual laboratory experiments. First, a numerical study is performed on a cantilever beam structure with non-viscous and non-proportional damping. The numerical study is followed by cases involving actual measured data. Joints and boundary conditions are assumed as a major source of damping, therefore joints and boundary conditions are modelled using relaxation functions and damping coefficients. The updated results have shown that the proposed damped element model updating method can be used to derive accurate models for the non-viscous and non-proportional damped systems. This is illustrated by matching complex FRFs obtained from the updated model with from the experimental data.

DEVELOPMENT OF CAPPED VISCOUS DAMPING MODEL BASED ON RESTORING-FORCE AMPLITUDE

Stiffness proportional or Rayleigh damping are well-known typical viscous damping, but since it depends on the frequency, it becomes a problem when the damping coefficient is to be kept constant over a wide period band. As a countermeasure, mode damping is sometimes used, but it is a problem that it lacks practicality in large-scale analysis because the computer load is generally large. In this study, we focus on the capped damping model that gives an upper limit to the damping force of the initial stiffness proportional damping, which is expected to have a certain degree of frequency insensitiveness.

Influence of Viscous Damping Models on the Inelastic Response of Reinforced Concrete Columns and Bridges

The appropriate estimation of displacements is crucial in performance-based design. Among the options to assess deformation demands, Nonlinear Time History Analysis (NLTHA) is the most sophisticated. However, previous research has shown that NLTHA is sensitive to the viscous damping model definition, and there is substantial disagreement in the engineering community regarding damping model choices. Thus, the goal of this paper is to show the impact of viscous damping model assumptions on the nonlinear response of bridges. A displacement sensitivity study was conducted on several multi-span bridges using various viscous damping models and earthquake records. The results indicate that the mean displacement varies as a function of the displacement ductility level and damping model. In order of ascending displacement demand, the Wilson-Penzien model had the lowest demands followed by the Rayleigh-Initial stiffness, Mass proportional, Rayleigh-Tangent stiffness, Tangent stiffness proportional damping, and Zero-damping. Also, this paper proposes a new viscous damping model. We expect these findings to help practitioners understand the implications of the choice of the damping model and guide the analyst when selecting damping parameters.

Dynamics of hierarchical beam lattice structures by an exact reduced-order dynamic-stiffness model

An exact reduced-order dynamic-stiffness model is proposed to study the dynamics of two-dimensional hierarchical beam lattices. Upon modelling every member of the lattice by a single exact two-node beam element, the model retains a few primary nodes of the unit cell and involves an exact dynamic condensation of the nodal degrees of freedom of the lattice structure internal to the unit cell and connected to the primary nodes. It is shown that the modes of the reduced-order model exhibit two orthogonality conditions, with dynamic-stiffness terms pertinent to the condensed lattice structures. The orthogonality conditions are the basis to decouple the motion equations and derive the exact modal response to external loading, in both time and frequency domains, under the assumption of proportional damping. The modal response is obtained in simple and elegant analytical form for the whole lattice, including the condensed lattice structures internal to the unit cells. Moreover, it can be calculated very expeditiously thanks to an exact representation of the eigenfunctions based on frequency dependent shape functions, which applies regardless of geometry and complexity of the hierarchical lattice structure. Numerical results confirm the exactness of the proposed reduced-order dynamic-stiffness model by comparison with standard finite-element solutions in ABAQUS.

Leader-follower and leaderless pose consensus of robot networks with variable time-delays and without velocity measurements

This paper proposes a novel distributed controller that solves the leader-follower and the leaderless consensus problems in the task space for networks composed of robots that can be kinematically and dynamically different (heterogeneous). In the leader-follower scenario, the controller ensures that all the robots in the network asymptotically reach a given leader pose (position and orientation), provided that at least one follower robot has access to the leader pose. In the leaderless problem, the robots asymptotically reach an agreement pose. The proposed controller is robust to variable time-delays in the communication channel and does not rely on velocity measurements. The controller is dynamic, it cancels-out the gravity effects and it incorporates a proportional to the error term between the robot and the controller virtual position. The controller dynamics consists of a simple proportional scheme plus damping injection through a second-order (virtual) system. The proposed approach employs the singularity free unit-quaternions to represent the orientation of the end-effectors, and the network is represented by an undirected and connected interconnection graph. The application to the control of bilateral teleoperators is described as a special case of the leaderless consensus solution. The paper presents numerical simulations with a network composed of four 6-Degrees-of-Freedom (DoF) and one 7-DoF robot manipulators. Moreover, we also report some experiments with a 6-DoF industrial robot and two 3-DoF haptic devices.

Modal Analysis of Non-Conservative Combined Dynamic Systems

Abstract The emergence of the use of mechanical metamaterials for vibration suppression and the creation of frequency gaps in structures require an understanding of the fundament underlying dynamics partial differential equations coupled to ordinary differential equations. Essentially periodic structures consist of a distributed parameter structure connected (embedded) to a series of spring-mass-dampers. Such systems in the past have been studied as combined dynamical systems. This work deals with the modal analysis of non-conservative combined dynamic systems formed by assembling distributed parameter structures and linear, viscously damped, lumped parameter oscillators. The mathematical model of the forced response of such dynamic systems is presented via differential operators. The related non-linear eigenproblem is formulated next and a proper solution is provided. Furthermore, the orthogonality of the eigenfunctions is studied and the completeness of the generated solution space is verified. Coupled modal coordinate differential equations result through modal analysis, thus revealing the non-proportional damping configuration, while the proportional damping conditions are also derived and discussed. The theory is applied to non-conservative Euler–Bernoulli beams subject to different types of boundary conditions and coupled to linear, viscously damped oscillators. A numerical example yields interesting conclusions about the non-proportionality and the applicability of the associated methods to solving the coupled differential equations.

A study on small scale thermal dynamic instability of rotating GPL-reinforced microbeams under principal parametric resonance stimulation of axial and transversal modes regarding the proportional damping

A study on the principal parametric stimulation of rotating microbeams strengthened by means of graphene platelet is presented. The microbeam is exposed to a temperature gradient. On the basis of the assumptions of the hypothesis of the Timoshenko for beams alongside the modified couple stress hypothesis the nonlinear motion equations are achieved. It is supposed that the rotating speed of the microbeam varies harmonically about a mean quantity. The frequency of the harmonic term of the rotating velocity is presumed to be almost two times an axial or a transversal natural frequency. In such situation, the principal parametric agitation is stimulated. Assuming a proportional damping, a least square scheme is employed to define the Rayleigh’s coefficients. The impacts of the rotating speed, the weight fraction as well as the scattering pattern of the graphene platelets, and the damping coefficient on the presented results are examined. The results illuminate that as a consequence of the addition of the damping coefficient, the critical incitement amplitude constant increases more for reinforced microbeams rather than not-reinforced microbeams. Moreover, by means of enlarging the weight fraction of the graphene platelets, the critical incitement amplitude constant develops more for a rotating X microbeam rather than an O microbeam. Additionally, at moderate to high magnitudes of the damping coefficient, the instability area border aligned with the fundamental axial mode is impressed by the graphene platelet scattering pattern although it is invariant for small values of the damping coefficient. Moreover, by the development of the temperature the instability area border aligned with the fundamental transversal mode gets broader, while the critical incitement amplitude coefficient decreases; the both more for on O scattering pattern for the graphene platelet. Furthermore, the instability region boundary is more influenced by the graphene platelet weight fraction, while the critical incitement amplitude coefficient is more impressed by the damping coefficient design value.

Applicability of Proportional Damping for Seismic Response Evaluation Using Three-Dimensional Finite Element Method Model Consisting of Structures with Multiple Damping Properties

The purpose of this study is to investigate the applicability of proportional damping to reduce the computational load, which is a common issue for seamless structural integrity evaluation with a 3D-FEM model. When the proportional damping is applied, it is necessary to compare this method with current design method (strain energy proportional damping) regarding on the seismic response calculation precision. Therefore, the research target is to construct a modeling method of proportional damping that realizes the same calculation precision as current design method. The ratios of the maximum stresses in strain energy proportional damping and proportional damping were confirmed using the 3D-FEM model with multiple damping properties, and the applicability of the proportional coefficients setting method based on the magnitude of the effective mass ratio was examined. As a result, it was confirmed that the proportional damping method can calculate the same stress distribution as current design method by setting the proportional coefficients for vibration modes with large effective mass ratios and for the upper limit frequency of the target frequency range.

Multi-objective optimum design of nonlinear viscous dampers in steel structures based on life cycle cost

An algorithm for optimum design of nonlinear viscous dampers with the goal of achieving minimum total cost is presented. An appropriate cost model is developed to determine the initial cost of equipping structures with viscous dampers, and the expected costs of the studied structures related to the possible earthquakes over their life time are estimated using life cycle cost analysis (LCCA). The results of this analysis are used in a multi-objective optimization algorithm with the aim of achieving minimum initial and life cycle cost. To evaluate the seismic behavior of structures, the Endurance Time (ET) method is used as a dynamic response-history analysis method which requires much less computational effort in comparison with conventional time history methods. To this end, the optimum arrangement of nonlinear viscous dampers along with their damping exponent (Alpha) for two weak steel moment frames with 4 and 8 stories are acquired using multi-objective genetic algorithm. Results from the proposed method are compared with two classic methods; uniform and stiffness proportional damping methods. With the aim of proposing an optimal Alpha value, a detailed discussion is provided on its effects on the overall performance of studied structures in terms of life cycle cost.