Piecewise Linear Stiffness Research Articles

A Hybrid Continuation Framework for Analyzing Nonlinear Normal Modes of Systems With Contact Nonlinearity

Abstract The prediction of the modal properties of structural systems with contact clearances and prestress presents a computational challenge, as the nonlinearity induced by piecewise-linear stiffness eliminates the use of efficient linear modal analysis techniques. The most common approach to obtaining the nonlinear normal modes (NNMs) of these structural systems is a numerical framework that integrates numerical integration, the shooting method, and the pseudo-arc-length continuation scheme. This numerical continuation framework (NCF) computes NNMs through iterative numerical calculations; thus, the computational cost of the nonlinear modal analysis of complex nonlinear systems, particularly piecewise-linear systems, becomes prohibitively expensive as the model size increases. In this study, a hybrid continuation framework (HCF) combining analytic and numerical methods is proposed to enable efficient computations of NNMs for systems with contact boundaries. This new hybrid framework utilizes a semi-analytic method to conduct the iterative shooting procedure; thus, the computational burden of the numerical continuation can be significantly reduced. The proposed method is demonstrated on spring-mass oscillators with contact elements, and the NNMs obtained using the proposed method are validated by those computed using the traditional numerical continuation framework. The modal properties of the systems can be computed using the proposed framework with significant speed-up. Furthermore, the modal properties, including internal resonance and sharp turning in NNM curves, of the piecewise-linear systems are identified and discussed.

Equivalent continuous method based novel design of a nonlinear TMD with piecewise stiffness

Tuned mass Dampers (TMDs) are extensively used passive control devices for high-rise buildings to mitigate undesirable responses. In practice, limit devices are frequently applied due to the large displacement of TMD, resulting in the emergence of piecewise nonlinearity. In this study, a TMD with piecewise stiffness (PSTMD) is developed to investigate the piecewise nonlinearity, including piecewise linear stiffness (PLS) and piecewise nonlinear stiffness (PNS). Initially, a novel equivalent continuous method (ECM) is proposed based on the principles of the least squares method (LSM) to approximate the PSTMD using a nonlinear TMD with Duffing stiffness. The equivalent nonlinear TMD is proven to be accurate enough to approximate the PSTMD. Subsequently, frequency response analysis is conducted on the PSTMD designed by linear method, results indicate that the linear method fails to design PSTMD. Thirdly, the optimal parameters of PSTMD are determined based on the ECM and the improved design method for nonlinear TMD proposed by the author. Results indicate that this novel design method effectively enhances the performance of PSTMD and mitigates the effect of piecewise nonlinearity. Finally, the influence of parameters of PSTMD on the optimal tuning frequency is investigated. The influence of damping ratio of PSTMD is explored by numerical simulation, with results indicating that employing the damping ratio suggested by traditional linear design method as the optimal parameter is reasonable. In summary, the novel method proposed in this study holds significance for the engineering design of TMDs with piecewise characteristics.

A semi-analytical time-domain model with explicit fluid force expressions for fluidelastic vibration of a tube array in crossflow

It is widely acknowledged that fluidelastic instability (FEI), among other mechanisms, is of the greatest concern in the flow-induced vibration (FIV) of tube bundles in steam generators and heat exchangers. A range of theoretical models have been developed for FEI analysis, and, in addition to the earliest semi-empirical Connors' model, the unsteady model, the quasi-steady model and the semi-analytical model are believed to be three advanced models predominant in the literature. The unsteady and the quasi-static models share the merits of having explicit fluid force expressions and ease of being implemented but require more experimental inputs, whereas the semi-analytical model requires fewer parameters due to its analytical nature but is hard, if not prohibitive, to derive explicit fluid force expressions. Since the fluid force formulations set in the core of development of FEI models, the understanding and in particular the implementation of the semi-analytical model has been impaired by the nonexistence of explicit fluid force expression. This issue becomes more profound in time-domain analysis whereby the simple harmonic assumption is discarded. Here we report a new semi-analytical time-domain (SATD) FEI model with explicit fluid force expressions. The new model allows a consistent frequency-domain stability analysis and more importantly a truly time-domain response analysis. The theory was validated by calculating linear stability thresholds of two typical tube array patterns and comparing against reported experimental data. We then present a nonlinear time-domain analysis of a single loosely-supported tube with piece-wise linear stiffness. The nonlinear and nonsmooth dynamics was probed in details by utilizing various techniques, playing an emphasis on characterizing and distinguishing the chaotic vibration. We found that the system follows a quasi-periodic route to chaos. Such an in-depth study of the nonlinear dynamics of tubes in crossflow has never been reported in the context of SATD model. Our results enrich the theory and provide a different approach for linear and nonlinear dynamics of tube bundles, which are essential for the subsequent fretting wear analysis.

Vibration characteristics of a bolted joint rotor-bearing system with rubbing and pedestal looseness coupled fault: Simulation and experiment

Rotor-stator rub and pedestal looseness are common faults in rotating machinery, such as large gas turbines that typically use bolted joint structures to fasten adjacent disks together. It is of great significance to investigate the vibration behaviors of the bolted joint rotor-bearing system with pedestal looseness and rubbing coupled fault, which can provide proofs for the fault diagnosis. In the present work, a dynamic model of the bolted joint rotor-bearing system with coupled fault is developed by using the finite element (FE) method, and then the vibration characteristic analysis is carried out. The rotor shafts that connected by the bolted joint element (BJE) are discretized as two sets of Timoshenko beam elements. The BJE is composed with four springs (two linear translational springs and two rotational springs with piecewise-linear stiffness). The interactions of rotor-stator contact are then modeled based on the Hertz contact theory and Coulomb's friction law, and the bearing forces are also calculated based on the Hertz contact theory. Moreover, the pedestal looseness fault is modeled through the piecewise-linear stiffness spring. The equations of motions of the bolted joint rotor-bearing system with coupled fault are then developed through FE theory and solved by the Newmark method. The dynamic responses are then calculated, and the frequency spectrums, bending stiffness of bolted joint, and vibration responses under different rotation speeds under the coupled fault are analyzed. The experimental study is finally conducted based on a bolted joint rotor test rig. Simulation results are in agreement with experiment results. The research results can provide a basis for understanding the rotor dynamic features of a bolted joint rotor system with coupled faults of rubbing and pedestal looseness.

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Research on Nonlinear Vibration of Dual Mass Flywheel Considering Piecewise Linear Stiffness and Damping

Nonlinear torsional vibration differential equation of the nested arc-shaped short spring dual mass flywheel (DMF) is established, considering the piecewise linear stiffness and damping of the spring. The first-order approximate analytical solution under sinusoidal excitation and the amplitude–frequency characteristic function are obtained by means of the average method which verified by the Runge–Kutta (R–K) method. The effects of the parameters of input excitation, inertia, and piecewise linear stiffness and damping of DMF on the resonant amplitude, resonant frequency band, and equivalent linear natural frequency of the system are analyzed. The results show that the amplitude–frequency characteristic curve bending and jumping with the changes of excitation frequency and the peak of resonant amplitude can be obviously reduced by increasing the inertia of the primary flywheel and decreasing the inertia of the secondary flywheel. The complex nonlinear dynamic phenomena such as Period 1, quasi-periodic, and chaos are obtained by analyzing the forced vibration response under the different excitation frequencies.

Rub-impact dynamic analysis of a dual-rotor system with bolted joint structure: Theoretical and experimental investigations

Large gas turbines often utilize dual-rotor structures to improve efficiency. To facilitate manufacture and maintenance, the large rotor systems are typically constructed by connecting components made from different materials together through the bolted joint. In this work, a dynamic model of a dual-rotor system with bolted joint included in the high-pressure (HP) rotor is established based upon the finite element theory of Timoshenko beam element, as well as taking into account the bearing forces and fixed-point rubbing fault. To evaluate the influence of rubbing faults on the dynamic behavior of a dual-rotor system, this study analyzed the response characteristics of the rotor under the cases of the LP rotor and the HP rotor subjected to the rubbing fault, respectively. In order to further reveal the effect of bolted joint structure on the system response while rubbing fault occurs, piecewise linear bending stiffness is also considered in this study. The nonlinear vibration responses of the bolted joint dual rotor-bearing system are studied through numerical simulation. Effect of rotor–stator contact stiffness and the occurrence of rubbing faults at the low-pressure (LP) rotor and the HP rotor are investigated through frequency-amplitude curves, waterfall diagrams, time-domain responses, and bending stiffness of the bolted joint. The results indicate that the presence of rubbing faults, whether occurring at the LP rotor or the HP rotor, leads to a decrease in the critical speed. Furthermore, under a rubbing fault, the bending stiffness of the bolted joint enters the stiffness softening region earlier and exhibits a wider range, which is exacerbated with increased contact stiffness. This phenomenon is considered one of the main causes of system response instability. Finally, the experimental studies are carried out on a bolted joint dual-rotor test rig to validate the numerical simulation results.

Energy Transfer and Dissipation in Combined-Stiffness Nonlinear Energy Sink Systems

Abstract Nonlinear energy sinks (NES) are highly efficient vibration energy absorption and dissipation devices, and play an important vibration-suppression role in many types of structures. In this study, the influence of parameters on the combined stiffness nonlinear energy sink system is revealed from the perspective of energy, in which combined-stiffness terms are composed of piecewise linear stiffness and cubic stiffness. First, the slow-varying derivative of the combined-stiffness nonlinear energy sink system is calculated based on the complexification-averaging and multiscale methods. Second, an approximate expression for the extreme points on the slow-invariant manifold (SIM) of the system is derived by polynomial approximation, and the energy dissipation equation of the combined-stiffness nonlinear energy sink system is derived. The impacts of the stiffness gap, piecewise linear stiffness coefficient, and cubic stiffness coefficient on the system are analyzed by studying the energy transfer efficiency equation. Additionally, the relationship between the damping ratio of the primary structure and the dissipation time is analyzed.

Analytical study on nonlinear frequency response and vibration isolation of multistage clutch damper in automotive powertrain

This paper studies the nonlinear frequency response and vibration isolation performance of the multistage clutch damper at resonance. First, a 2-DOF drivetrain model is established considering the piecewise-linear stiffness, piecewise-linear damping and asymmetric transition angles of the multistage clutch damper. Then, the averaging method is employed to deduce the analytical solution of the nonlinear motion equation, and the stability of the periodic solution is studied using Routh-Hurwitz criterion. The nonlinear behaviors of the oscillator are discussed based on the theoretical results, and the numerical simulation is carried out to verify the credibility of the proposed method. Finally, the sensitivity analysis is conducted to investigate the influence of the system parameters on the amplitude-frequency response of the multistage clutch damper. The results indicate that the isolation performance of the multistage clutch damper is sensitive to the first and second stages of damping and the first-stage transition angles. By increasing the first and second stages of viscous damping by 50% and simultaneously reducing the first-stage angle range by 3° with 1° asymmetry, the isolation performance of the clutch damper is effectively improved, reducing the vibration amplitude at resonance by 28.8%.

An inerter nonlinear energy sink for torsional vibration suppression of the rotor system

To reduce size and weight of the inertial mass required in traditional nonlinear energy sink (NES), an inerter nonlinear energy sink (INES) for suppressing the torsional vibration of the rotor system is developed in this paper. The INES is a combination of a NES with piecewise linear stiffness and an inerter whose vibration can be enlarged by levers. The structure of the INES is introduced, the inerter mechanism is analyzed and the dynamic model of the INES-rotor system is built. Vibration attenuation performance of INES on the rotor system is evaluated by the transient torsional vibration and steady-state torsional vibration, both of which are optimized by the genetic algorithm (GA). It is found that the inerter mass of INES can save about 14.2 times its own rotary inertia when suppressing the torsional vibration of the rotor system. By using the inerter, a significant damping effect also appears. In transient torsional vibration suppression, the percentage of accumulated energy dissipation of INES can reach 95.1%. In steady-state torsional vibration suppression, the peak vibration eliminations of INES for the rotor system can achieve 89.1% in simulation and 80.6% in test.

Investigation on Vibro-Impacts of Driveline System Based on a Nonlinear Clearance Element With Time-Varying Stiffness and Oil-Squeeze Damping

Abstract The transient vibro-impacts induced by clearance between the connected rotors in driveline system easily causes serious transient noise and vibration, especially between the gear teeth with backlash. To analyze the transient vibro-impacts of the driveline system excited by a step-down engine torque, a new piecewise nonlinear clearance element with time-varying stiffness and oil squeeze damping is proposed, and an eight degrees-of-freedom (DOF) lumped parameters model with the new piecewise nonlinear clearance elements is established. The transient vibro-impact phenomena of the vehicle driveline during fast disengagement of the clutch are numerically simulated. Colormaps of angular acceleration and vibro-impact force shows the difference of frequency components from transient impact to stable tooth-meshing. The phase plane reveals the phenomenon of multiple impacts and rebounds in each transient impact and shows the relationship between the relative contact displacement and velocity. The frequency responses of the angular velocity, angular acceleration, and vibro-impact forces with time-varying stiffness and linear stiffness are compared, respectively. Compared with the widely used clearance element with piecewise linear stiffness, the new nonlinear clearance element with the piecewise nonlinear time-varying stiffness can better reveal the transient vibro-impact responses between the driving and driven gears. Lastly, the transient vibro-impact results of driveline system are verified by the vehicle experiments.

Dynamic Performance of Time-Domain Piecewise Linear Stiffness System

AbstractThis paper studies the dynamic performance of two kinds of time-domain piecewise linear stiffness systems, a piecewise positive-stiffness system (PPS) and a piecewise negative-stiffness sys...

Investigation on clearance-induced vibro-impacts of torsional system based on Hertz contact nonlinearity

The clearance-induced vibro-impacts between the connected rotors in torsional system easily causes serious transient noise and vibration. To analyze the transient vibro-impacts of torsional system excited by a step-down torque, taking a rear-wheel-drive vehicle driveline system as an example, a 7 degree-of-freedom lumped parameters model with new piecewise nonlinear clearance elements is established based on nonlinear Hertz contact theory. The transient vibro-impact phenomena of the vehicle driveline system during fast disengagement of the clutch are numerically simulated. The phase plane reveals the phenomenon of multiple impacts and rebounds in each transient impact, and shows the relationship between the relative contact displacement and velocity. The vibro-impact force and time-frequency response of linear contact stiffness model and Hertz contact stiffness model are analyzed and compared. Compared with the widely used clearance element with piecewise linear stiffness, the clearance element with piecewise nonlinear stiffness can better capture the transient vibro-impact forces and time-varying frequency responses between the driving and driven rotors. The vibro-impact results are verified by the vehicle driveline experiments.

Nonlinear vibrations of beams with bilinear hysteresis at supports: interpretation of experimental results

Experimental vibration responses of beams with nonlinear boundary conditions were interpreted using a bilinear spring model. Two systems of beams were considered: one is a single rod and the second one is a cluster of parallel rods linked together. The boundary conditions at the beam ends, for both systems, were provided by spacer grids, which are specific supports used in nuclear industry. The experiments were conducted with the beams immersed in still water. The nonlinear boundary conditions imposed to the beams by the spacer grids were experimentally characterized. The boundary conditions changed from having a rotational linear stiffness at low vibration amplitudes to bilinear with hysteresis at higher vibration amplitudes. The bilinear stiffness was found to be of the softening type. A bilinear stiffness model with viscous damping was used to interpret the forced vibration response of the beams. The stiffness and damping ratio used in the model were identified at any excitation level by minimizing, in the frequency domain, the weighted distance between the set of experimental points and the corresponding numerical results. The equivalent viscous damping of the system grew with the vibration amplitude. Very good agreement between numerical and experimental results was obtained for both the beam systems studied in the frequency and force-displacement domains. The model was further refined by using a piecewise linear stiffness function with additional switching points and introducing nonlinear damping. The identified model was capable of reproducing the experimental results with accuracy and without the requirement to adjust the parameters at different excitation levels.

Experimental Modelling and Amplitude-Frequency Response Analysis of a Piecewise Linear Vibration System

The amplitude-frequency response of a nonlinear vibration system with the coexistence of stiffness and viscous damping piecewise linearities are analysed by means of analytical, numerical and experimental investigations. First, a mechanical model of the piecewise linear system under simple harmonic base excitation is established, and the amplitude-frequency response equation is obtained by the averaging method. Second, an experimental device is built to realize the piecewise linear system. The stiffness and damping coefficients are identified by the least square method. Third, case studies are conducted to illustrate the effect of the clearance and base excitation amplitude on the amplitude-frequency response. The experimental results show that the introduction of the piecewise linear stiffness and damping significantly decreases the response amplitude at the primary resonance. The piecewise linear stiffness, damping coefficients, primary resonance frequency and frequency range of the bi-stable state depend on the clearance and excitation amplitude. The experimental results are consistent with the theoretical predictions and numerical simulation results of the method of backward differentiation formulas. This research provides instructive ideas to the design of the nonlinear isolator in practical engineering.

Nonlinear vibrations of a fluid-filled, soft circular shell: experiments and system identification

Vibration experiments are carried out on a slightly corrugated circular cylindrical shell made of polyethylene terephthalate fabric. The shell is liquid-filled, it is pressurized by a liquid column that applies a pressure of 100 mmHg, and the two edges are clamped to fix supports. Forced vibrations of the shell are experimentally studied in the linear (small amplitude) and in the geometrically nonlinear (large amplitude) regime. The large-amplitude vibrations of the liquid-filled shell are characterized by a strong softening behavior that cannot be captured by any quadratic nonlinear stiffness. Since compressed fibers do not carry load, a piecewise linear stiffness with viscous damping is thus introduced in a reduced-order model, resulting in a very good agreement between experimental and simulated responses. The stiffness parameters and the damping ratios are identified from the experimental results. The damping ratio grows linearly with the excitation amplitude, indicating a predominant hydrodynamic damping. In particular, the damping ratio increases 2.75 times from the small-amplitude vibrations to a maximum amplitude of 1.26 mm. This is a very significant increase that highlights the necessity to introduce nonlinear damping to model shell structures.

Nonlinear vibration localisation in a symmetric system of two coupled beams

We report nonlinear vibration localisation in a system of two symmetric weakly coupled nonlinear oscillators. A two degree-of-freedom model with piecewise linear stiffness shows bifurcations to localised solutions. An experimental investigation employing two weakly coupled beams touching against stoppers for large vibration amplitudes confirms the nonlinear localisation.

Vibration mitigation and dynamics of a rotor-blade system with an attached nonlinear energy sink

Excessive vibration of rotor-blade systems has been a main reason for the failure of rotating machinery. Therefore, methods capable of simultaneously suppressing vibrations rotor and blade are urgently needed. Considering this, a nonlinear energy sink (NES) with piecewise linear stiffness is used to satisfy the requirements. Firstly, the structure and working mechanism of the NES are introduced. And then, the dynamic model of rotor-blade-NES system is established by Lagrangian method The vibration suppression ability of the NES is studied under transient and steady state excitations. Dynamic behavior of the rotor-blade system with a NES is also investigated through global bifurcation analyses in this work. Finally, a simple test of NES’s vibration suppression ability on rotor-blade system is completed. The results show that, under given parameters, the vibration suppression performances of the NES on both rotor and blade are superior both in steady state and transient state. In numerical simulation, a vibration suppression rate of 93% on the rotor is achieved and in test a rate of 88% is achieved.

On the Design of a Type of Mechanical Molecular Spring Isolator

The mechanical molecular spring isolator (MMSI) presents the characteristics of high-static-low-dynamic-stiffness to isolate the low-frequency vibration of a heavy payload. An MMSI consists of a piston-cylinder container full of the liquid and some hydraulic spring accumulators. This paper focuses on a particular type of the MMSI that all hydraulic spring accumulators applying the conventional coil spring are identical. In the beginning, the MMSI stiffness model is modelled and the influences of critical parameters on the stiffness property are studied. The MMSI presents a piecewise linear stiffness property. The averaging method is applied to determine the primary resonance response, and the energy transmissibility is adopted to estimate the vibration isolation performance. A jump phenomenon is inevitable to be occurred around the resonance region. The jump avoidance is a critical design part of the MMSI isolation system. On the other hand, based on the stiffness model, it reveals that any stiffness of the coil spring or any size of the accumulator in the MMSI will be able to achieve the same desired stiffness property. The most significant difference between all solutions depends on the coil spring lateral stability. The coil spring lateral stability magnitude is defined as the ratio of the deflection to the buckling deflection to evaluate the quality of each solution. Then, the fundamental design criterion on the MMSI based on jump avoidance and the coil spring lateral stability is established.

Experimental characterization and performance of dynamic vibration absorber with tunable piecewise-linear stiffness

Dynamic vibration absorber (DVA) is one of the ways to control the level of vibration. Among many mechanisms and control strategies of the DVA, passively tuned DVA emerges as one of the most effective absorbers due to its simple mechanism. However, the performance of the passive tuned DVA suffers from narrow suppression bandwidth. This paper introduces a nonlinear dynamic vibration absorber (NDVA) with an adjustable piecewise-linear stiffness mechanism which has a characteristic almost similar to hardening stiffness to broaden the vibration suppression bandwidth. The mechanism is made of a cantilever beam constrained by two vertically and horizontally adjustable limit blocks on either side of the beam’s equilibrium position. The static and dynamic properties of the proposed NDVA were first investigated for different positions of the limit blocks. Then the performance of the NDVA was compared with the linear DVA in terms of the vibration suppression bandwidth, separation of resonance frequencies and the ability to cope with mistuning. Overall, the NDVA seems to outperform the linear DVA, and it is more forgiving in the case of mistuning.