Spring Stiffness Research Articles

Application of pole allocation to optimize passive viscous dampers represented by the Maxwell model

The Maxwell model is used to understand the realistic effectiveness of vibration reduction in the building structures. The model consists of a dashpot and spring in series. The dashpot and spring represent a viscous damper and joint between the damper and the structural frame, respectively. However, few studies have investigated the optimal damper capacity and the corresponding spring stiffness in the practical parameter ranges. Additionally, the optimal damper derived using the fixed-point theory has no relationships with that derived via pole allocation. Therefore, in this study, to examine the effects of the joint spring, the pole allocation method was used to design a structural system wherein the Maxwell model was incorporated into a single-degree-of-freedom damped model. We introduced a closed-form expression, which explicitly described the relationships between the target structural damping ratio, damper capacity, and joint spring. The Maxwell model constrained the control effectiveness using damper parameters and simultaneously suggested an optimal and realistic joint spring for the damper. Pole allocation was also applied to a multi-degree-of-freedom (M-DOF) damped structural system employing multiple Maxwell models. The newly proposed optimized joint spring could easily control the additional damping effect on the M-DOF system. The Maxwell model can be optimized almost manually. The scheme is more useful in the preliminary design stage than the previous numerical optimizations. This study extended the mathematical equation governing the building vibrations to the Maxwell model. The extended equation served as a unified description for considering structural passive control.

Dynamic analysis of high-rise residential structures through the cone method

In this article, to analyze the behavior of surface foundations based on reinforced soil, a simple physical method based on material resistance called (cone method) has been used. Which is used as an alternative to the exact solution methods that are based on the three-dimensional elastodynamic theory. The purpose of this research is to introduce a simple physical method and analytical tool to analyze the surface foundation based on geocell-reinforced soil. With the cone method, it is possible to examine the effects of layered soil. Obtaining the dynamic stiffness of the foundation based on reinforced soil using the cone method for the vertical degree of freedom and also investigating the effect of various parameters related to geocell reinforcement on the dynamic stiffness of the surface foundation are the goals of this research. In general, the longer the geocell cushion has, the smaller the holes, and the higher the modulus of elasticity, the better the results. By increasing the number of geocell layers, the amount of spring stiffness coefficient and damping coefficient and as a result the dynamic stiffness coefficient increases. On the other hand, the closer the geocell is placed to the soil surface, the higher the stiffness of the spring and the lower the damping coefficient. In general, among these two parameters related to dynamic stiffness, the role of the damping coefficient has been more prominent and decisive in the investigation of foundations based on reinforced soil. One of the most obvious characteristics of geosynthetic materials in the soil, regardless of the greater hardness it gives to the soil, is the damping percentage of its constituent materials.

Theoretical and experimental investigation of flexible air spring stiffness in a tuned liquid column gas damper for vertical vibration control

This study investigates the effectiveness of equivalent linear air spring stiffness in a Vertical Tuned Liquid Column Gas Damper (VTLCGD) for reducing vertical structural vibrations, particularly considering different sealing conditions. The VTLCGD system, designed with a single sealed vertical air column, allows flexible frequency tuning by adjusting the air spring stiffness. It aims to be used in low-frequency structures where conventional VTLCGD designs with two sealed ends are less efficient. The geometric configuration and working principle of the VTLCGD are first described. The equation of motion is derived using liquid dynamic equilibrium and the pressure-volume relationship, with expressions for natural frequency and control force validated through shaking table tests conducted with varying VTLCGD lengths. Experimental investigations are conducted to examine the parameters affecting damper performance using pressure data. The system's vibration reduction performance is then numerically evaluated on a cantilevered floor subjected to human walking. The numerical results, based on the equation of motion for liquid displacement and natural frequency, show good agreement with experimental data, which confirms the effectiveness of using linearized air spring stiffness for frequency tuning. The effects of total liquid length and height difference on control force are further investigated, and a polytropic index of 1.2 is determined for the VTLCGD frequency formula. The VTLCGD system achieves a maximum vibration reduction of 52.63 % in acceleration response on the cantilevered floor compared to the uncontrolled case. Moreover, the variation in stiffness due to changes in the sealing condition is presented to validate the damper's adaptability over a wide frequency range.

Improving mass lumping and stiffness parameters of bar and hinge model for accurate modal dynamics of origami structures.

The bar and hinge framework uses truss elements and rotational springs to efficiently model the structural behaviour of origami. The framework is especially useful to investigate origami metamaterials as they have repeating geometry, which makes conventional finite element simulations very expensive due to a large number of degrees of freedom. This work proposes improvements to the parameters of bar and hinge model within the context of structural dynamics, specifically modal analysis under small deformations, which has not been carried out previously in the literature. A range of low-frequency modes involving origami folding and panel bending deformations that can be accurately captured by the bar and hinge framework are identified. Within this range, bar and hinge parameters like the lumped masses and the rotational spring stiffness values are derived using conservation laws and finite element tests. The best among the proposed schemes is found to predict natural frequencies of the considered origami structures to within 10% maximum error, improving the accuracy by more than three times from existing schemes. In most cases, the errors in natural frequencies are less than 5%. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Novel approaches to assessing brake squeal propensity: comparative evaluation of two index formulations

The persistence of brake squeal as a significant NVH issue within automotive brake systems necessitates the development of robust methods for assessing squeal propensity. This study introduces two novel squeal index formulations for assessment purposes and evaluates their performance against the formulation defined by the SAE J2521 standard. Experimental data essential for formulating the squeal index are collected on a controlled mass-sliding belt experiment. Experiments are performed at three different spring stiffness levels, and squeal index formulations are derived based on time and frequency domain data. The first formulation places particular emphasis on the time domain, explicitly considering the duration of individual squeal events. Conversely, the second formulation is based on the frequency response characteristics of the system, which considers the number of super-harmonic peaks in frequency spectra. To evaluate the performance and discriminatory capabilities of the newly developed squeal index formulations, extensive comparisons are made against the Squeal Index based on the test procedure described in SAE J2521. In conclusion, through meticulous analysis and interpretation of the experimental results, it is discerned that the squeal index formulation defined by time domain exhibits superior efficacy in discerning the nuanced effects of varying design parameters on the occurrence of squeal-like behavior.

Deep subwavelength sound absorption by a buckling plate resonator

Conventional sound-absorbing materials face great difficulties in achieving deep subwavelength broadband sound absorption due to the causal constraint on the minimum thickness. A buckling plate resonator is proposed to alleviate the causal constraint on the thickness of sound-absorbing materials. The resonator consists of a back cavity sealed by an elastic circular thin plate. When the plate is subjected to a uniform in-plane compressive force up to its critical load, it buckles and behaves as a negative stiffness spring. The positive stiffness of the back cavity is counteracted by the negative stiffness of the buckling plate, resulting in resonance absorption at the deep subwavelength scale. The sound absorption performance of the buckling plate resonator under different in-plane loads was measured in an impedance tube. Quasi-perfect sound absorption was observed in a buckling plate resonator with a thickness less than 1% of the wavelength corresponding to the resonance frequency. This work provides a solution for the design of ultra-thin broadband sound absorbers.

Developing a virtual physical system for vortex-induced vibration studies of a bluff body

A virtual physical system (VPS) for VIV studies of a bluff body is developed to replace the actual physical systems. It can arbitrarily and accurately control and edit the physical parameters, including mass, damping ratio and spring stiffness, specifically for the mass-spring-damper system. The recursive Duhamel integral method (DIM) with unconditional stability was used for the VPS control system, addressing real-time noise filtering problem and simplifying the system as a single input and single output (SISO) one. Delay compensation and inertial force elimination methods were investigated and proposed to overcome the crucial unwanted damping effects. An experimental facility for VIV model tests by VPS was manufactured, and the bluff body model with a measurement system was specially designed to accurately sense the hydrodynamic force during VPS operation. Systematic verification experiments for parameter editing and control of an actual physical target system were conducted, showing that the VPS can reproduce the equivalent spring-damper-mass system in high fidelity with an accuracy error of less than 5%. VIV model tests for a bluff body at Reynolds numbers (Re= UD/υ, where U is the flow velocity, D represents the diameter of cylinder model, and υ is the kinematic viscosity coefficient) of 5.7E4 and 2.3E5 were performed using the VPS experimental facility, presenting well-repeated VIV responses at low Re and unexpected VIV response with a large amplitude of 2.4 D at high Re, which can cause severe fatigue damage for relevant structures. The present VPS will provide promising and powerful experimental tools for VIV studies of a bluff body to reveal the related sensitive parameter effects.

Design and analysis of a piezo-actuated hydraulic micro-displacement amplifier with high bandwidth and large amplification ratio

Micro-displacement amplifiers show great potential for enhancing the performance of piezoelectric ceramic stacks. However, designing a compact micro-displacement amplifier with a high amplification ratio and broad bandwidth remains challenging. This article introduces a hydraulic micro-displacement amplifier driven by a piezoelectric ceramic stack with a large amplification ratio, high bandwidth, and eliminating parasitic displacement. Theoretical and simulation analyses are performed for the hydraulic micro-displacement amplifier, optimizing parameters such as edge thin ring thickness and spring stiffness. A prototype was fabricated and tested in a series of experiments. The experimental results demonstrate that, under steady-state conditions, the hydraulic micro-displacement amplifier achieves an amplification ratio of 26.12 with a standard deviation of 2.28. The maximum output displacement is 172.4 μm at 150 V, and the resonant frequency is 445 Hz under a 20 N load. This study provides a novel approach to designing micro-displacement amplifiers.

Effects of distribution valve spring stiffness and opening pressure on the volumetric efficiency of micro high-pressure plunger pump

With the rapid development of material science and manufacturing capabilities, hydraulic technology is increasingly high-pressure, lightweight and miniaturizing. Micro plunger pump is widely used in the field of deep-sea hydraulic equipment and advanced intelligent hydraulic equipment, owing to its high-power density, high output pressure and many other advantages. Its broad application aims to examine the inlet and outlet distribution valve spring parameters change on the micro high-pressure plunger pump volumetric efficiency, by changing the inlet and outlet distribution valve. This paper is based on the simulation of AMESim engineering software to derive multiple sets of data. It compared and analysed the specific effect of different spring stiffness and opening pressure on the volumetric efficiency of the micro high-pressure plunger pump which was then verified through experiment. Results of this study have certain reference significance for the design of the spring of the inlet and outlet distribution valve of the micro high-pressure plunger pump, which facilitates the optimization and improvement of the dynamic performance of the micro high-pressure plunger pump.

Synchronization analysis of a four-motor excitation with torsion spring coupling vibrating system

In oil drilling processes, vibrating screen is a crucial solid control equipment for separating harmful solid-phase particles. However, the conventional utilization of either dual-motor or triple-motor is inadequate demanding the excitation force of large vibrating equipment. Moreover, self-synchronization in zero phase is difficult to achieve in multiple motors system. To avoid cancellation of the excitation force caused by the phase inconsistency of multiple motors, the adoption of a four-motor excitation with torsion spring coupling system is proposed this work. Initially, the mathematical model of the vibrating system is formulated through the Lagrange's equation. Subsequently, the steady-state solution is determined using Laplace transform. Then the conditions and characteristics in stable equilibrium state are elucidated employing the small parameter averaging method. Ultimately, the reliability and applicability of the theoretical results are substantiated through numerical simulations. The findings reveal that with the increase of the torsion spring stiffness, the phase difference between the motors connected by torsion springs (MCTSs) is exponentially decreased, and desired zero-phase synchronization in engineering is eventually achieved. Furthermore, the present study uncovers two distinct synchronization mechanisms in the torsion spring coupling system: mechanical controlled synchronization between MCTSs is inevitably achieved, while the self-synchronization between motors unconnected by torsion springs (MUTSs) is limited by the synchronization conditions. The study provides valuable insights for designing mechanical controlled synchronization and self-synchronization vibrating machines.

Vortex-induced vibration and energy harvesting of cylinder system with nonlinear springs

Vortex-induced vibration is a common fluid–structure coupling phenomenon in ocean engineering. As a nonlinear stiffness structure caused by anisotropic deformation and support gap, nonlinear spring is of great significance in the field of vortex-induced vibration. Based on the structural dynamics equations and the wake oscillator model, the vibration model of the 2-Degree of Freedom cylinder system with the nonlinear stiffness spring is established. The stiffness ratio, mass ratio, and damping ratio of the cylinder system are changed, and the amplitude and frequency response are analyzed to obtain the effect of nonlinear spring on vortex-induced vibration. The results show that the cylinder system with a nonlinear stiffness spring can maintain a high vibration amplitude in a large Reynolds number range, and the smaller the mass ratio, the more significant the effect is. When m*=1.5, the amplitude of the cylinder system in a cross flow is greater than 0.6 at Re = 180 000–480 000. The energy harvesting in the vortex-induced vibration of cylinder systems with nonlinear stiffness springs is also studied. The efficiency depends on the damping ratio and Reynolds number. When the damping ratio is constant, the energy harvesting efficiency increases first and then decreases with the Reynolds number increases. When m*=1.5, the energy harvesting efficiency of the cylinder system is 19.27% with Re= 35 000 and damping ratio ξ=0.155. As a kind of renewable energy, energy harvesting in vortex-induced vibration is a potential development direction in the future energy field.

Exact Second-Order Stiffness Matrix of Axial-Loaded Beam–Columns on Winkler Foundation

This paper establishes the exact second-order stiffness matrix of axial-loaded beam–columns on Winkler foundation. This Winkler foundation model has been widely used in engineering applications. The equilibrium analysis of an axial-loaded beam–column on Winkler foundation is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. Two dimensionless factors for the axial force and foundation spring stiffness are defined, respectively. The exact second-order element stiffness matrix of the axial-loaded beam–column on Winkler foundation is then formulated, showing the relationship between the element-end deformations (translation and rotation angle) and the corresponding forces (shear force and bending moment). Such an exact second-order stiffness matrix may be used for the buckling and second-order solutions with allowance for the use of one element per member for the exact solution. The classical buckling problem of axial-loaded beam–columns on Winkler foundation with typical element-end boundary conditions is then analyzed by the developed matrix stiffness method. By comparing with classical buckling criterion expressions of axial-loaded beam–columns on Winkler foundation with various boundary conditions, it is demonstrated that the derived exact second-order stiffness matrix can be used to obtain the exact buckling solutions systematically.

A novel method for shape pre-optimization of fir-tree couplings in blade-disk connections

This paper presents an innovative approach to the pre-optimization of the blade-disk fir-tree coupling, with the intent of advancing the design and analysis of such connections in turbomachinery. These couplings are subjected to extreme mechanical loads, primarily from centrifugal forces during turbine operation, compounded by aerodynamic and vibrational forces. These conditions pose substantial challenges to ensuring the structural integrity and long-term reliability of the connections. Failure of the blade-disk attachment can lead to catastrophic consequences, making the optimization of these joints crucial.The proposed pre-optimization method addresses this by optimizing the distribution of material between the blade-root, disk-post, and engaging teeth, ensuring uniform load distribution across all teeth. A novel one-dimensional radial spring model is introduced, which simplifies the representation of the joint. The method achieves accurate load-sharing without requiring knowledge of the spring stiffness values, instead relying on stiffness ratios, which in turn are approximated equal to ratios between the areas of throat cross-sections. This approach effectively balances tensile stresses, contact pressures, and, proportionally, local stress peaks, all while streamlining the design process.Validation against finite element models demonstrates the model’s accuracy across different joint configurations with varying numbers of teeth. This method serves as a foundation for more detailed analysis and future optimization efforts, including those targeting specific failure mechanisms such as fretting fatigue and peak stress mitigation. Overall, this work offers a practical and efficient approach to enhancing the performance and reliability of blade-disk connections in turbomachinery.

A three filament mechanistic model of musculotendon force and impedance

The force developed by actively lengthened muscle depends on different structures across different scales of lengthening. For small perturbations, the active response of muscle is well captured by a linear-time-invariant (LTI) system: a stiff spring in parallel with a light damper. The force response of muscle to longer stretches is better represented by a compliant spring that can fix its end when activated. Experimental work has shown that the stiffness and damping (impedance) of muscle in response to small perturbations is of fundamental importance to motor learning and mechanical stability, while the huge forces developed during long active stretches are critical for simulating and predicting injury. Outside of motor learning and injury, muscle is actively lengthened as a part of nearly all terrestrial locomotion. Despite the functional importance of impedance and active lengthening, no single muscle model has all these mechanical properties. In this work, we present the viscoelastic-crossbridge active-titin (VEXAT) model that can replicate the response of muscle to length changes great and small. To evaluate the VEXAT model, we compare its response to biological muscle by simulating experiments that measure the impedance of muscle, and the forces developed during long active stretches. In addition, we have also compared the responses of the VEXAT model to a popular Hill-type muscle model. The VEXAT model more accurately captures the impedance of biological muscle and its responses to long active stretches than a Hill-type model and can still reproduce the force-velocity and force-length relations of muscle. While the comparison between the VEXAT model and biological muscle is favorable, there are some phenomena that can be improved: the low frequency phase response of the model, and a mechanism to support passive force enhancement.

A three filament mechanistic model of musculotendon force and impedance.

The force developed by actively lengthened muscle depends on different structures across different scales of lengthening. For small perturbations, the active response of muscle is well captured by a linear-time-invariant (LTI) system: a stiff spring in parallel with a light damper. The force response of muscle to longer stretches is better represented by a compliant spring that can fix its end when activated. Experimental work has shown that the stiffness and damping (impedance) of muscle in response to small perturbations is of fundamental importance to motor learning and mechanical stability, while the huge forces developed during long active stretches are critical for simulating and predicting injury. Outside of motor learning and injury, muscle is actively lengthened as a part of nearly all terrestrial locomotion. Despite the functional importance of impedance and active lengthening, no single muscle model has all these mechanical properties. In this work, we present the viscoelastic-crossbridge active-titin (VEXAT) model that can replicate the response of muscle to length changes great and small. To evaluate the VEXAT model, we compare its response to biological muscle by simulating experiments that measure the impedance of muscle, and the forces developed during long active stretches. In addition, we have also compared the responses of the VEXAT model to a popular Hill-type muscle model. The VEXAT model more accurately captures the impedance of biological muscle and its responses to long active stretches than a Hill-type model and can still reproduce the force-velocity and force-length relations of muscle. While the comparison between the VEXAT model and biological muscle is favorable, there are some phenomena that can be improved: the low frequency phase response of the model, and a mechanism to support passive force enhancement.

Experimental study on the free and constrained vibration performance of composite wave springs

This article proposes a glass fiber reinforced plastic (GFRP) wave spring and conducts a study on its free and constrained vibration characteristics and compression performance. Initially, a stiffness prediction model for composite wave spring is established using laminated plate theory and Moore’s theorem. Additionally, the failure mode of the composite wave spring was predicted and experimentally confirmed using ABAQUS and the 3D-Hashin failure criterion. Finally, the vibration characteristics of composite wave spring and metal helical spring are investigated using the B&K testing system in both free and constrained states. The experimental stiffness and ultimate load of the composite wave spring are 20.67 N/mm and 1386.67 N. The composite wave spring exhibits good vibration reduction performance in both free and constrained states, with maximum vibration attenuation amplitudes of 122.71 and 144.93 dB. In comparison, the metal helical spring has maximum vibration attenuation amplitudes of 41.65 and 111.89 dB.

Current Loads on a Horizontal Floating Flexible Membrane in a 3D Channel

A 3D analytical model is formulated based on linearised small-amplitude wave theory to analyse the behaviour of a horizontal, flexible membrane subject to wave–current interaction. The membrane is connected to spring moorings for stability. Green’s function approach is used to obtain the dispersion relation and is utilised in the solution by applying the velocity decomposition method. On the other hand, a brief description of the experiment is presented. The accuracy level of the analytical results is checked by comparing the results of reflection and the transmission coefficients against experimental data sets. Several numerical results on the displacements of the membrane and the vertical forces are studied thoroughly to examine the impact of current loads, spring stiffness, membrane tension, modes of oscillations, and water depths. It is observed that as the value of the current speed (CS) rises, the deflection also increases, whereas it declines in deeper water. On the other hand, the spring stiffness has minimal effect on the vibrations of the flexible membrane. When vertical force is considered, higher oscillation modes increase the vertical loads on the membrane, and for a mid-range wavelength, the vertical wave loads on the membrane grow as the CS increases. Further, the influence of the phase and group velocities are presented. The influences of CS and comparisons between them in terms of water depth are presented and analysed. This analysis will inform the design of membrane-based wave energy converters and breakwaters by clarifying how current loads affect the dynamics of floating membranes at various water depths.

Free vibration of spinning stepped functionally graded circular cylindrical shells in a thermal environment

This paper investigates the free vibration of spinning stepped functionally graded material (FGM) circular cylindrical shells with general boundary conditions in a thermal environment. The artificial spring is implemented to simulate the general boundary constraints. Temperature-dependent material properties change constantly along the thickness direction. The Donnell-Mushtari shell theory is applied to derive the energy functions of thin stepped circular cylindrical shells, where the effect of centrifugal force and Coriolis force is taken into account. The modified Fourier-Ritz method is utilized to derive the frequency equations of the spinning stepped FGM circular cylindrical shells. The influences of spring stiffness, power-law index, and temperature on the traveling wave frequencies are investigated. The influence of configuration on the forward wave frequency and critical speed of three stepped shells, including the one-end thickened, the two-end thickened, and the middle-thickened stepped shells, is discussed. For each configuration, design strategies for the stepped shells are proposed.

Examination of the complementary energy principle in modified couple stress theory and its application for analysis of size effects in the internal force field of functionally graded microbeams

The precise quantification of the internal force field in MEMS piezoresistive sensor micro structure is crucial for ensuring accurate signal output. This paper proposes and proves the complementary energy principle associated with modified couple stress theory (MCST) based on the linear elasticity assumption. And this principle is used to discuss and analyze whether there is a size effect in the internal force field of functionally graded (FG) micro beams for the MEMS piezoresistive sensor. The results show that the size effects exist in the internal forces and the stationary point coordinate x of the total moment for the micro beam. The size effects are related to the dimensionless material scale parameter l/h, the supported spring stiffness, the temperature and the support rotational movements. Intuitive explanations of the size effects are given for the internal force fields and its stationary point coordinates. Behaviors of the solutions agree with the published data when the MCST degenerates into the classical theory (CT).

Complex Dispersion Analysis of Flexural Wave Propagation in Periodic Track Structure

The periodic characteristics of the track structure have filtering characteristics for the propagation of vibration waves within it, and the dispersion analysis is of great significance to understand the vibration transmission characteristics of the track structure. The accuracy of the traditional virtual spring method depends on the value of spring stiffness, which leads to poor stability. Therefore, this paper adopts a null space method with both stability, accuracy and efficiency, and takes the CRTSIII ballastless track structure as the research object. Based on the null space method and energy functional variational principle, the damping factor and the material frequency variation effect are considered in detail, and the complex frequency dispersion of the CRTSIII ballastless track structure is analyzed. The accuracy of the proposed method is verified by comparing it with the results of the existing literature, and the influence of fastener stiffness frequency variation, fastener damping and concrete support layer damping on the complex dispersion characteristics of vibration waves of track structures is analyzed by using this method. The results show that the stiffness frequency effect, fastener damping and concrete support layer damping have a great influence on the attenuation domain and attenuation rate of vibration, which must be considered in the vibration transmission analysis of track structures.