Simple Mass Spring Research Articles

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.

Application of Laplace-based variational iteration method to analyze generalized nonlinear oscillations in physical systems

In this paper, the variational iterative method (VIM) with the Laplace transform is utilized to solve the nonlinear problems of a simple pendulum and mass spring oscillator, which corresponds to the Duffing equation. Finding the Lagrange multiplier (LM) is a significant phase in the VIM, and variational theory is frequently employed for this purpose. This paper demonstrates how the Laplace transform can be utilized to locate the LM in a more efficient manner. The frequency obtained by Laplace-based VIM is the same as that defined in the already existing methods in the literature in order to ensure the clarity of the results. Numerous analytical techniques can be used to solve the Duffing equation, but we are the first to do it using a Laplace-based VIM and a distinctive LM. The fundamental results of my paper are that LM is also the same in the Elzaki transformation. In the vast majority of instances, Laplace-based VIM only requires one iteration to arrive at an answer with high precision and linearization, discretization or intensive computational work is required for this purpose. Comparing analytical results of VIM by Laplace transform to the built-in Simulink command in MATLAB which gives us the surety about the method’s applicability for solving nonlinear problems. Future work on the basic pendulum may examine the effects of nonlinearities and damping on its motion and the application of advanced control mechanisms to regulate its behavior. Future research on mass spring oscillators could examine the system’s response to random or harmonic input. The mass spring oscillator could also be used in vibration isolation to minimize vibrations from one building to another.

Solutions of the Nonlinear Evolution Problems and their Applications

Abstract In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

A new approach to model a system with both friction and geometric nonlinearity

This paper presents a new reduced-order model that can capture the nonlinear dynamics of a structure containing both friction and geometric nonlinearity, and proposes a procedure to derive the parameters of the model from quasi-static simulations. The Single-mode Implicit Condensation and Expansion (SICE) method was recently shown to be capable of capturing the resonant behavior of a geometrically nonlinear structure, and the modal Iwan model has proven effective at capturing the hysteretic behavior of a mode that exhibits nonlinearity due to friction. This paper combines the two into a new model, referred to here as the Iwan model with Geometric Nonlinearity (or the IGNL model). It consists of an Iwan model with an additional spring-slider unit, where the slider has infinite strength and the spring consists of polynomial terms that define the SICE-ROM. A procedure is also proposed to use Quasi-Static Modal Analysis (QSMA) to estimate the parameters of the IGNL model from a set of nonlinear force–displacement curves. Existing literature shows how QSMA can be used to characterize the force–displacement behavior of a nonlinear mode where the nonlinearity is due to friction or bending–stretching coupling, but not both simultaneously. This paper proposes certain adjustments to the QSMA procedure so that the two forms of nonlinearity can be isolated. Two case studies are presented to test the efficacy of the IGNL model, one consisting of a simple spring–mass model and the second, a finite element model containing both contact and geometric nonlinearity. The proposed method is shown to be significantly faster than performing dynamic simulations to estimate the amplitude-dependent frequency and damping behavior.

Bifurcation, stability, and critical slowing down in a simple mass–spring system

We analyse a simple mass–spring system as an accessible context for showcasing how continuous changes to system parameters can lead to critical transitions (‘tipping points’). Two kinds of transition are explored in particular: saddle–node bifurcations, due to changes in a mass forcing parameter a; and pitchfork bifurcations, due to changes in a spring separation parameter X. Both types of bifurcation arise as features of a cusp catastrophe characterised in X−a parameter space by the critical curve X2/3+a2/3=1, leading to hysteresis cycles, as described by C. Ong (2021), and non-reversible pitchfork catastrophes, which are discussed here for the first time. In each case we demonstrate critical slowing down of the oscillation period τ→∞ as the system approaches bifurcation.

Modelling and Validation of Joint Properties for NVH Simulation Model of an Electrified Passenger Vehicle Drivetrain


 
 
 The constantly changing automotive market, which is characterized by the progressing electrification of vehicles, presents challenges with respect to the acoustic behaviour of powertrains. As indicated frequently, the characteristic noise of electric cars differs from that of conventional drives. In particular, energy dissipation of the assembled housing parts from material and joint damping; is an important aspect. In modelling the NVH behavior of electric vehicles or drivetrains in general, the breakdown of global material and local joint damping is not yet state of the art. Instead, measured global modal damping values from similar constructions are frequently used. Often, a constant degree of damping is defined for all modes. However, moderate changes in the geometry of components or in the load level can lead to severe fluctuations in the damping. In order to improve predictability and to be able to estimate the effects of constructive changes in natural frequency, mode shape, and damping, the goal will be to take into account the global material and local joint damping separately. For that, a local model which is able to take the joint properties into account has to be developed. The analytical based Iwan model for the contact interface was expanded upon to facilitate model updating against forced response measurements on a simple mass spring structure in one dimension and to extend the model from one-dimensional to three-dimensional.
 
 

Estimation of Structural Parameters Using Transfer Matrices and State Vectors

A new system identification (SI) method based on Transfer Matrix (TM) concept is proposed here to identify structural stiffness. Transfer matrices based on lumped mass and the more accurate consistent mass models are used here, based on displacement measurements in the time domain. The consistent mass based TM is derived from the dynamic stiffness matrix of a beam element. The state vector at a location is the sum of the internal and external contributions of displacements, forces and moments at that point - when multiplied with the (TM), we obtain the adjacent state vectors. The method of identification proposed here involves predicting the displacements at certain locations using the TM, and comparing them with the measured displacements at those locations. The mean square deviations between the measured and predicted responses at all locations are minimized using an optimization algorithm, and the optimization variables are the unknown stiffness parameters in the TM. A non-classical heuristic Particle Swarm Optimization algorithm (PSO) is used, since it is especially suited for global search. Different strategies for calculating the initial state vector, as well as two identification processes viz., simultaneous and successive methods, are discussed. Numerical simulations are carried out on four examples ranging from a simple spring mass system to a nine member framed structure. The speed and accuracy of identification using this method are good. One main advantage of this method is that it can be applied at any portion of the structure to identify the local parameters in that zone without the need to model the entire global structure.

Response of a linear viscoelastic splitter plate attached to a cylinder in laminar flow

The flow-induced deformation of a viscoelastic thin plate attached to the lee side of a circular cylinder subjected to laminar flow is numerically investigated. An in-house fluid–structure interaction solver couples the sharp-interface immersed boundary method for fluid dynamics with a finite-element method for structural dynamics. Extending published results on elastic materials, the Standard Linear Solid (SLS) model is used to represent viscoelasticity of the plate, governed by the following two parameters: (a) the ratio of the non-equilibrium to equilibrium Young’s modulus (R), and (b) the ratio of dimensionless material damping to the dimensionless non-equilibrium Young’s modulus (τ). The focus of the present study is to examine the dynamic response of the viscoelastic plate at Reynolds number of Re=100. The amplitude and the time to achieve a dynamic steady state of a plate with 0.1<R<5 and 0.1<τ<10 have been investigated. The plate attains a self-sustained time-periodic oscillation with a plateau amplitude for the range of R=[0.1–5] for all values of τ. The dimensionless tip displacement amplitude (AY,tip) is found to be a non-monotonic function of τ. When the forcing frequency (vortex shedding frequency) is lesser (greater) than the natural frequency (considering the equilibrium Young modulus) of the plate, AY,tip decreases (increases) asymptotically. The theoretical analysis of a simple spring–mass–dashpot model of SLS is undertaken to understand the effect of sinusoidal forcing on the plate dynamics. Analytical predictions show general agreement with the non-monotonic behaviour of the numerically computed AY,tip. The results suggest that careful tuning of the damping may be effectively employed to enhance power output for energy extraction applications or to suppress flow-induced vibration when it is detrimental to the structure.

The Hanging Column as a Dynamic Vibration Absorber

A simple mass–spring system with an attached hanging column is investigated. The problem is formulated and the frequencies obtained with an efficient initial value method. Under forced vibration, the amplitude of the mass may be greatly reduced by adding a hanging column. The possibility of using such a hanging column as a dynamic vibration absorber is shown for the first time.

Parametric Study of Single Bolted Composite Bolted Joint Subjected to Static Tensile Loading

The use of composites is increasing in the engineering applications in order to reduce the weight, building energy efficient systems, designing a suitable material according to the requirements of the application. But at the same time, building a structure is possible only by bonding or bolting or combination of them. There are limitations for the bonding methods and problems with the bolting such as stress concentration near the neighborhood of the bolt hole, tensile or shear failure, delamination etc. Hence the design of a composite bolted structure needs a special attention. This paper focuses on the performance of the composite bolted joint under static tensile loading and the effect of variation in the parameters such as the bolt pitch, plate width, thickness, bolt tightening torque, composite material, coefficient of friction between the bolt and plate etc. A simple spring mass model is used to study the single bolted composite bolted joint.The influencing parameters are identified through the developed model and compared with the results from the literature. The best geometric parameters for the applied load are identified for the composite bolted joints.

Adaptation in Variable Parallel Compliance: Towards Energy Efficiency in Cyclic Tasks

We present a compliance adaptation method for online natural dynamics modification of multijoint robots performing cyclic tasks. In this method, parameters of multibasis nonlinear compliances acting in parallel with actuators are adapted to minimize actuation forces that results in joint-by-joint energy consumption reduction. Stability, convergence, and optimality of this method are proved analytically for a general compliance structure. We do not impose any specific constraint on the controller structure and tracking performance, yet stable tracking of cyclic motions is necessary for the convergence to the optimal solution. Extensive simulations on a set of systems ranging from simple mass–spring system to robotic manipulator (with linear and nonlinear compliances), along with the experimental results on a 1-degree of freedom compliant revolute joint with two basis functions in the compliance profile, demonstrate the efficiency of our method in terms of stability, convergence, and optimality; i.e., actuation force and energy consumption reduction.

Effect of the mandible on mouthguard measurements of head kinematics

Wearable sensors are becoming increasingly popular for measuring head motions and detecting head impacts. Many sensors are worn on the skin or in headgear and can suffer from motion artifacts introduced by the compliance of soft tissue or decoupling of headgear from the skull. The instrumented mouthguard is designed to couple directly to the upper dentition, which is made of hard enamel and anchored in a bony socket by stiff ligaments. This gives the mouthguard superior coupling to the skull compared with other systems. However, multiple validation studies have yielded conflicting results with respect to the mouthguard׳s head kinematics measurement accuracy. Here, we demonstrate that imposing different constraints on the mandible (lower jaw) can alter mouthguard kinematic accuracy in dummy headform testing. In addition, post mortem human surrogate tests utilizing the worst-case unconstrained mandible condition yield 40% and 80% normalized root mean square error in angular velocity and angular acceleration respectively. These errors can be modeled using a simple spring–mass system in which the soft mouthguard material near the sensors acts as a spring and the mandible as a mass. However, the mouthguard can be designed to mitigate these disturbances by isolating sensors from mandible loads, improving accuracy to below 15% normalized root mean square error in all kinematic measures. Thus, while current mouthguards would suffer from measurement errors in the worst-case unconstrained mandible condition, future mouthguards should be designed to account for these disturbances and future validation testing should include unconstrained mandibles to ensure proper accuracy.

On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction

After previous works related to the equilibrium states, this paper goes deeper into the study of the effect of coupling between smooth and non-smooth non-linearities on the qualitative behavior of low dimensional dynamical systems. The non-smooth non-linearity is due to non-regularized unilateral contact and Coulomb friction while the smooth one is due to large strains of a simple mass spring system, which lead to a nonlinear restoring force. The main qualitative differences with the case of a linear restoring force are due to the shape of the set of equilibrium states.

COLLISION DETECTION FOR CLOTH SIMULATION USING BOUNDING SPHERE HIERARCHY

For the past 2-decades, the challenges of collision detection on cloth simulation have attracted numerous researchers. Simple mass spring model is used to model the cloth where the movement of the particles within the cloth was controlled by applying the Newton’s second law. After the modeling stage, implementation of the collision detection algorithm took place on cloth has been done. The collision detection technique used is bounding sphere hierarchy. Then, quad tree is being used to partitioning the bounding sphere and the collision search was based on the top-down approach. A prototype of the collision detection system is developed on cloth simulation and several experiments were conducted. Time taken for this system to be executed is around 235.258 milliseconds. Then the frame rate is at the average of 22 frames per second which is close to the real time system. Times taken for the collision detection system travels from root to nodes were 23 seconds. As a conclusion, the computational cost for bounding sphere hierarchy is much higher because the bounding sphere required more vertices for generation process, however the execution time for bounding sphere hierarchy is faster than the AABB hierarchy.

Experimental Validation of a Feed-Forward Predictor for the Spring-Loaded Inverted Pendulum Template

Widely accepted utility of simple spring–mass models for running behaviors as descriptive tools, as well as literal control targets, motivates accurate analytical approximations to their dynamics. Despite the availability of a number of such analytical predictors in the literature, their validation has mostly been done in simulation, and it is yet unclear how well they perform when applied to physical platforms. In this paper, we extend on one of the most recent approximations in the literature to ensure its accuracy and applicability to a physical monopedal platform. To this end, we present systematic experiments on a well-instrumented planar monopod robot, first to perform careful identification of system parameters and subsequently to assess predictor performance. Our results show that the approximate solutions to the spring-loaded inverted pendulum dynamics are capable of predicting physical robot position and velocity trajectories with average prediction errors of 2% and 7%, respectively. This predictive performance together with the simple analytic nature of the approximations shows their suitability as a basis for both state estimators and locomotion controllers.

Numerical Computation of a Lined Duct Instability Using the Linearized Euler Equations

The two-dimensional linearized Euler equations in the time domain are computed to obtain the sound field in a channel with a rigid upper wall and an acoustic liner on the bottom wall in the presence of a shear flow. The liner is a simple mass–spring–damper system. The shear flow has a rather thin boundary layer, and the mesh has to be refined close to the walls. The objective is to assess whether an instability can be computed and whether a computed instability is physical. The computations are backed by a linear stability analysis of the flow that is performed by solving a matrix eigenvalue problem. By comparing the results of the simulations of the linearized Euler equations with the results of the stability analysis, it is found that the computed instabilities are physical and that their characteristics can be predicted. The effect of using selective filtering on the wave-number spectrum is also discussed.

Elastic moduli of simple mass spring models

Mass spring models (MSMs) are a popular choice for representation of soft bodies in computer graphics and virtual reality applications. In this paper, we investigate physical properties of the simplest MSMs composed of mass points and linear springs. The nodes are either placed on a cubic lattice or positioned randomly within the system. We calculate the elastic moduli for such models and relate the results to other studies. We show that there is a well-defined relationship between the geometric characteristics of the MSM systems and physical properties of the modeled materials. It is also demonstrated that these models exhibit a proper convergence to a unique solution upon mesh refinement and thus can represent elastic materials with a high precision.

Basilar membrane and tectorial membrane stiffness in the CBA/CaJ mouse.

The mouse has become an important animal model in understanding cochlear function. Structures, such as the tectorial membrane or hair cells, have been changed by gene manipulation, and the resulting effect on cochlear function has been studied. To contrast those findings, physical properties of the basilar membrane (BM) and tectorial membrane (TM) in mice without gene mutation are of great importance. Using the hemicochlea of CBA/CaJ mice, we have demonstrated that tectorial membrane (TM) and basilar membrane (BM) revealed a stiffness gradient along the cochlea. While a simple spring mass resonator predicts the change in the characteristic frequency of the BM, the spring mass model does not predict the frequency change along the TM. Plateau stiffness values of the TM were 0.6 ± 0.5, 0.2 ± 0.1, and 0.09 ± 0.09 N/m for the basal, middle, and upper turns, respectively. The BM plateau stiffness values were 3.7 ± 2.2, 1.2 ± 1.2, and 0.5 ± 0.5 N/m for the basal, middle, and upper turns, respectively. Estimations of the TM Young's modulus (in kPa) revealed 24.3 ± 25.2 for the basal turns, 5.1 ± 4.5 for the middle turns, and 1.9 ± 1.6 for the apical turns. Young's modulus determined at the BM pectinate zone was 76.8 ± 72, 23.9 ± 30.6, and 9.4 ± 6.2 kPa for the basal, middle, and apical turns, respectively. The reported stiffness values of the CBA/CaJ mouse TM and BM provide basic data for the physical properties of its organ of Corti.

Theoretical modeling and experimental realization of dynamically magnified thermoacoustic-piezoelectric energy harvesters

Abstract Conventional thermoacoustic-piezoelectric (TAP) harvesters convert thermal energy, such as solar or waste heat energy, directly into electrical energy without the need for any moving components. The input thermal energy generates a steep temperature gradient along a porous medium. At a critical threshold of the temperature gradient, self-sustained acoustic waves are developed inside an acoustic resonator. The associated pressure fluctuations impinge on a piezoelectric diaphragm, placed at the end of the resonator. In this study, the TAP harvester is coupled with an auxiliary elastic structure in the form of a simple spring–mass system to amplify the strain experienced by the piezoelectric element. The auxiliary structure is referred to as a dynamic magnifier and has been shown in different areas to significantly amplify the deflection of vibrating structures. A comprehensive model of the dynamically magnified thermoacoustic-piezoelectric (DMTAP) harvester has been developed that includes equations of motions of the system׳s mechanical components, the harvested voltage, the mechanical impedance of the coupled structure at the resonator end and the equations necessary to compute the self-excited frequencies of oscillations inside the acoustic resonator. Theoretical results confirmed that significant amplification of the harvested power is feasible if the magnifier׳s parameters are properly chosen. The performance characteristics of experimental prototypes of a thermoacoustic-piezoelectric resonator with and without the magnifier are examined. The obtained experimental findings are validated against the theoretical results. Dynamic magnifiers serve as a novel approach to enhance the effectiveness of thermoacoustic energy harvested from waste heat by increasing the efficiency of their harvesting components.

Phonon dispersion of graphene revisited

The phonon dispersion of graphene is derived by using a simple mass spring model and considering up to the first, second, third, and fourth nearest-neighbor interactions. The results obtained from different nearest-neighbor interactions are compared and it is shown that the k2 dependence for the out-of-plane transverse acoustic mode obtained in other sophisticated methods as well as experiment occurs only after including the fourth nearest-neighbor interaction.