Stick State Research Articles

Dynamic Analysis of a Wiper Blade in Consideration of Attack Angle and Clarification of the Jumping Phenomenon

Automobile windshields are typically curved, creating an oblique angle of attack between the wiper blade and the windshield. This attack angle means that the wiper may jump off the windshield while wiping, causing a chattering noise and preventing the rainwater from being fully wiped off the windshield. Thus, it is important to examine the dynamics of the wiper blade under friction. In this study, the relationship between the attack angle and the jumping phenomenon is clarified through dynamic analysis. We introduce an analytical two-link model corresponding to an actual wiper blade that considers the exchange of dynamic and static friction between the windshield and the blade. The dynamic friction is assumed to be negatively correlated with the relative velocity, and the static friction is described by a set-valued function. As the motion transitions from the stick state to the slip state, the equation to be solved changes. Hence, the initial condition after a transition must agree with the final condition before the transition. Because the governing equations are nonlinear and the solution is highly dependent on the initial condition, the transition time and corresponding state variables are vital. The slack variable method is used to obtain the exact transition time and initial conditions. The sign of the normal force acting on the blade from the windshield determines the occurrence of the jump phenomenon. A larger attack angle makes the jump phenomenon more likely. However, the jump phenomenon does not occur when the motion of the blade reverses. Experimental observations support the theoretical description of the wiper blade.

Coupled dynamic behaviours of the brake system considering wheel–rail interactions

ABSTRACT During vehicle operation, the brake system is influenced by wheel–rail interactions. To investigate the dynamic behaviours of the brake system, we propose a novel dynamic model of the brake system by considering the coupling effects between the disc–pad and the wheel–rail subsystems. Using the established model, the nonlinear coupled dynamic behaviours including wheel–rail adhesion characteristics, wheel–disc torsional vibrations and disc–pad stick–slip vibrations were comprehensively explored. The results showed that braking conditions remarkably influenced the longitudinal slip rate. The wheelset and brake units affect each other, and the wheels on one wheelset influence each other when they are asynchronous. The brake system is stable in the wheel–rail stick state, but becomes unstable in the wheel–rail stick–slip state or slip state. The limit cycle analysis showed that the brake pad stick–slip vibrations were easier to occur at a large slip rate.

Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving

Research on mathematics teachers’ knowledge has generally focused more on mathematics concepts than mathematical processes. This paper addresses the latter with a focus on mathematical problem solving (PS). It reports on a study that investigated the pedagogical knowledge for PS of prospective elementary school teachers of mathematics (PTs). Participants were 149 PTs at a university in Spain. They were at the end of their teacher education program. Data sources consisted of a questionnaire on knowledge of learning PS and a questionnaire on knowledge of teaching PS. Findings indicated that the PTs held combination of different levels of knowledge of PS learning and teaching. Many of them demonstrated appropriate knowledge of many characteristics for (1) PS learning consisting of student as a problem-solver, PS as a worthwhile task, non-cognitive factor related to PS, and (2) PS teaching consisting of PS teaching approaches, discourse in PS, intervention during stuck state in PS, PS assessment, and PS resources. However, there were also contradictions and limitations to their knowledge with implications for teacher education. These combination of appropriate and inappropriate knowledge resulted in some conflicts that are related to teaching actions and would limit student’ learning of PS.

Ultrasonic monitoring of friction contacts during shear vibration cycles

Complex high-value jointed structures such as aero-engines are carefully designed and optimized to prevent failure and maximise their life. In the design process, physically-based numerical models are employed to predict the nonlinear dynamic response of the structure. However, the reliability of these models is limited due to the lack of accurate validation data from metallic contact interfaces subjected to high-frequency vibration cycles. In this study, ultrasonic shear waves are used to characterise metallic contact interfaces during vibration cycles, hence providing new validation data for an understanding of the state of the friction contact. Supported by numerical simulations of wave propagation within the material, a novel experimental method is developed to simultaneously acquire ultrasonic measurements and friction hysteresis loops within the same test on a high-frequency friction rig. Large variability in the ultrasound reflection/transmission is observed within each hysteresis loop and is associated with stick/slip transitions. The measurement results reveal that the ultrasound technique can be used to detect stick and slip states in contact interfaces subjected to high-frequency shear vibration. This is the first observation of this type and paves the way towards real-time monitoring of vibrating contact interfaces in jointed structures, leading to a new physical understanding of the contact states and new validation data needed for improved nonlinear dynamic analyses.

Analysis of complex modal instability of a minimal friction self-excited vibration system from multiscale fractal surface topography

In order to reveal the stability and nonlinear characteristics of the friction self-excited vibration system from the perspective of microscopic multiscale fractal surface topography, a minimal two-degree-of-freedom mathematical model of the disc brake system is established in this paper. Considering the fractal characteristics of rough surface topography, the fractal contact stiffness of contact surfaces is introduced into the system model from the perspective of microscopic contact, and the effects of two important surface fractal parameters, fractal dimension D and fractal roughness G, on the stability and nonlinearity of the system are analyzed. Further finite element analysis and brake noise test improve the analysis of system stability and noise intensity by different surface topographies. The results show that the system stability can be improved with the increase of fractal dimension within a certain range, and the system will be in a stick state with the further increase of fractal dimension. With the increase of fractal roughness, the unstable modal coupling region of the system is increasing, which can reduce the stability of the system. The unstable region of the system is dependent on fractal dimension and fractal roughness, and has different sensitivity to them.

Self-cannibalism and cannibalism: a perverse defence against depression

This observational article describes three cases of self-cannibalism and cannibalism by patients in a secure setting. These are studies of severe personality disordered patients, each of whom have perpetrated terrible offences, usually on others. In our view, the inner world of the cannibalistic patient is committed to a narcissistic biting grip on his objects and is bent upon appropriating their good aspects. In our practice, we have observed a countertransference enactment that may be characterised as a roadblock to explore the past: to feel, to understand, and to repair. The aim of the perverse pact is to remain in a stuck state to protect the self from any awareness of feelings of depression. This is an enactment of the inherently deathly, inner relationship of the individual perpetrator. We believe it is essential to attempt to open up a space to explore the unthinkable act of cutting and eating flesh in the context of the person’s developmental history. Failure to do so within systems of care sets in motion a repeating cycle of a traumatic victim–victimiser dynamic associated with ever increasing risk.

ILS-MPM: An implicit level-set-based material point method for frictional particulate contact mechanics of deformable particles

Finite element simulations of frictional multi-body contact problems via conformal meshes can be challenging and computationally demanding. To render geometrical features, unstructured meshes are often generated via a trial-and-error procedure. This treatment also unavoidably increases the degrees of freedom and makes the construction of slave/master pairs cumbersome. In this work, we introduce an implicit material point method designed to bypass meshing of bodies by employing level set functions to represent boundaries at structured grids. This implicit function representation provides an elegant mean to link an unbiased intermediate reference surface with the true boundaries by closest point projection as shown in Leichner et al. (2019). We then enforce the contact constraints by a penalty method where the Coulomb friction law is implemented such that a return mapping algorithm can be used to provide constitutive updates for both the stick and slip states. To evolve the geometry of the contacts properly, the Hamilton–Jacobi equation is solved incrementally such that the level set and material points are both updated according to the deformation field. To improve the accuracy and regularity of the numerical integration of the material point method, a moving least square method is used to project numerical values of the material points back to the standard locations for Gaussian-Legendre quadrature. Several benchmarks are used to verify the proposed model. Comparisons with discrete element simulations are made to analyze the importance of stress fields on predicting the macroscopic responses of granular assemblies.

Stick-slip vibration of a moving oscillator on an axially flexible beam

This study investigates the stick-slip vibration between an axially flexible beam fixed at both ends and an oscillator moving on the beam. After deriving the equations of motion for the stick and slip states, the stick-slip vibrations between the oscillator and the beam are analyzed. In addition, to obtain the irregularly changed contact position due to the axial deformation of the beam and oscillator movement, a mathematical expression for the contact position is derived. It is found that the long-period stick-slip vibration is influenced mainly by the oscillator and the short-period vibration is influenced mainly by axial deformation of the beam. Furthermore, the dynamic responses show that even if a high damping ratio is applied to the oscillator, stick-slip vibration due to axial deformation of the beam can occur. Finally, the analysis shows that a kind of the internal resonance occurs between the oscillator and the beam when the harmonics of the natural frequency of the oscillator match the natural frequencies of the beam.

Dynamic Analysis of the Stick-Slip Phenomenon in an Automotive Water Pump

In this paper, we present a study on the stick-slip torsional vibration of the spinning shaft in an automotive water pump. We propose a dynamic rotor model with a pulley, an impeller, a mechanical seal and a shaft to analyze the stick-slip vibration of a water pump. The dynamic states of the mechanical seal are classified into stick and slip states depending on whether the seal and mating rings of the mechanical seal have relative motion or not. We derived the partial differential equations of motion and the corresponding boundary conditions for each state of the seal. In addition, we established a judgement criterion for the stick and slip states of the seal. After the equations of motion were discretized by a proposed finite element procedure, the dynamic responses of the rotor system of a water pump were computed by using the generalized-a time integration method. Furthermore, we analyzed the effects of various system parameters on stick-slip vibration, including the mass moment of inertia for the pulley, the drag torque at the impeller, the external torque at the pulley, and the normal force between the seal and mating rings.

Energetically consistent model of slipping and sticking frictional impacts in multibody systems

A unifying slipping and sticking frictional impact model for multibody systems in contact with a frictional surface is presented. It is shown that the model can lead to energetic consistency in both slip and stick states upon imposing specific constraints on the coefficient of friction (CoF) and the coefficient of restitution (CoR). A discriminator in the form of a quadratic function of the pre-impact velocity is introduced based on an isotropic Coulomb constraint such that its sign determines whether the impact occurs in the sticking mode or in the slipping mode just prior to the contact. Solving for the zero-crossings of such a function in terms of the CoF and the CoR variables leads to another discriminator called Critical CoF, which is the lowest static CoF required to prevent the subsequent impulse vector violating the isotropic friction cone constraint. Investigating conditions for the energetically consistent impact model reveals that the maximum values of either CoR or CoF should be limited depending on the stick or slip state. Furthermore, it is shown that these upper-bound limits in conjunction with the introduced Critical CoF variable can be used to specify the admissible set of CoR and CoF parameters, which can be represented by two distinct regions in the plane of CoF versus CoR. Finally, a case study using the Kane’s example for impact in an MBS involving frictional impacts occurring in both slip and stick states are presented to support the analytical results.

Painlevé paradox and dynamic self-locking during passive walking of bipedal robot

When analyzing the passive walking of bipedal robot using the traditional methods, it usually assumes the contact foot and the slope maintain stick state. However, in the actual stable walking, the sliding contact caused by structural vibration is inevitable. Some special configuration of bipedal robot can result in dynamic self-locking or “jam” at the sliding contact if the coefficient of friction (COF) is sufficiently large; this has been termed, Painlevé paradox in rigid body dynamics. The objective of this paper is to propose a rigid body dynamic combined with Linear Complementary Problem to calculate the occurrence conditions of Painlevé paradox, and develop a completely flexible body model (CFBM) to analyze its bouncing motion induced by dynamic self-locking during passive walking of bipedal robots. The structural deformation field and inertial field of CFBM are discretized by finite element method. The contact forces and transient waves propagating in the CFBM are calculated. In the range of configurations and COF where Painlevé paradox occurs, analysis based on rigid body dynamics gives results indicating that either there are multiple solutions or the solution doesn't exist; analysis using CFBM finds that the bouncing motion will cause walking unstable and the normal contact force increases abruptly. In addition, increasing structural compliance will decrease the critical COF of dynamic self-locking roughly by 46%.

A general model to predict torsion and curvature increments of tensile armors in unbonded flexible pipes

An analytical model is proposed to investigate the local torsion and bending behavior of tensile armors in unbonded flexible pipes. The general expressions of torsion and curvature increments, including the effects of axial strain and twisting rotation of tendon cross-section, are firstly derived based on the generalized Frenet–Serret equations. Then the expressions corresponding to the strained helix assumption are obtained, which are further simplified by ignoring the twisting rotation. Afterwards, explicit expressions are given for axisymmetric case and tendon stick and slip states when bending is included. The developed model is finally verified with a finite element simulation. The results show that the model prediction of normal curvature increment is in good agreement with the numerical solution while the changes in torsion and transverse curvature cannot be obtained accurately through simple analytical formulas. Several earlier models are also discussed briefly and are demonstrated to be special cases of the present general model.

The effect of high-frequency torsional impacts on the dynamic response of a drill string in a stick state

Stick–slip vibration is common in the oil well drilling process and is detrimental to down-hole equipment and drilling efficiency. In recent years, a new type of drilling technology, torsional impact drilling, has been developed to mitigate the stick–slip phenomena, particularly in the drilling of deep or abrasive formations. With this drilling technique, high-frequency torsional impacts are generated and applied to the drill bit, providing the drill bit with auxiliary energy. By mitigating or suppressing the stick–slip vibration, part of the energy wasted as a result of vibration can be regained. However, the effect of these impact loads on the dynamic response of a drill string in a stick state is unknown. In order to address this issue, a continuous system model of a drill string that includes torsional impact load was constructed. In the model, a Fourier series approach was used for the impact load, and the mechanical model was resolved with the mode superposition method. Case studies were done to understand the drill string dynamics, with and without the impact. The case study results demonstrate that high-frequency torsional impacts have little influence on the dynamic response of a drill string in a stick state.

Injection‐Induced Shear Slip and Permeability Enhancement in Granite Fractures

AbstractShear stimulation, so‐called hydroshearing, is believed to reactivate critical or near‐critical preexisting fractures by pressurized fluid injection, causing them to slip and dilate for economic production from geothermal and unconventional petroleum reservoirs. However, little or no experiments have been conducted to directly probe the coupled hydromechanical responses and permeability evolution during fracture shearing. In this paper, we present the results of novel injection‐induced shear tests on cylindrical granite samples each containing a single tensile or saw cut fracture. The shear test on rough fractures demonstrates that retainable permeability enhancement can be achieved through dilatant shear slip. The characteristic stick‐slip behavior of the fracture during shearing is also observed. Fracture aperture controlled by effective normal stress dominates the fluid flow in stick state, while in slip state, the major contributor to production increase is the irreversible normal dilation caused by fracture shear slip. The experimental data show that depending on the fracture roughness, the fracture slip process includes two quasi‐static slip intervals with a slip velocity of ~10−9 to ~10−6 m/s and a dynamic slip interval with a slip velocity of ~10−7 to ~10−5 m/s. The fracture slip correlates well with the associated stress relaxation: a faster fracture slip induces a quicker stress relaxation, and vice versa. The influence of surface roughness on the hydromechanical responses during fracture shearing and the shear‐induced asperity degradation are also studied. The various experimental observations on rough and smooth fractures clearly indicate that the key component to shear stimulation is fracture self‐propping by asperities.

Stick-slip vibration of a cantilever beam subjected to harmonic base excitation

Stick-slip vibration of a cantilever beam subjected to harmonic base excitation is analyzed in this paper. To model the stick-slip vibration, considering nonlinearity and discontinuity of the friction force, two kinds of equations of motion are used: one for motion in the stick state and the other for motion in the slip state. Using the finite element method, dynamic responses of stick-slip vibration are calculated by iteratively solving the equations and connecting the stick- and slip-state motions; such calculations are conducted for various excitation conditions. The dynamic responses show that the occurrence of stick-slip vibration depends on excitation amplitude and frequency, and an occurrence of stick-slip vibration is associated with the modal characteristic of a beam. Furthermore, to formulate an analytical expression for the criterion of stick-slip occurrence, the static friction force caused by the base excitation is determined analytically. Finally, excitation conditions for stick-slip occurrence are obtained and validated in a parametric plane of the excitation amplitude and frequency, and design guidelines for avoiding stick-slip occurrence are presented.

Analysis and optimization of clutch judder based on a hybrid uncertain model with random and interval variables

ABSTRACTClutch judder has serious impacts on the noise, vibration and harshness performance. In this article, a simplified dynamic model with nonlinear friction torque is developed to simulate clutch judder, and the stability and dynamic response of the clutch are analysed. The real part of the judder modal eigenvalue, the moment when the clutch enters the stick state and the fluctuation level of the driving part of the clutch are treated as the evaluation indices. An uncertain hybrid model with random and interval variables is used to describe the uncertainty of parameters and a hybrid perturbation vertex method is formulated to compute the uncertainty. Furthermore, parameters with high sensitivities are used as design variables and uncertainty-based optimization is conducted to reduce clutch judder. The optimization results strongly validate that the proposed method is very effective in improving the robustness of the clutch judder performance.

Numerical analysis for the stick-slip vibration of a transversely moving beam in contact with a frictional wall

This paper presents a numerical analysis for the stick-slip vibration of a transversely moving beam, considering both stick-slip transition and friction force discontinuity. The dynamic state of the beam was separated into the stick state and the slip state, and boundary conditions were defined for both. By applying the finite element method, two matrix-vector equations were derived: one for stick state and the other for slip state. However, the equations have different degrees of freedom depending on whether the end of a beam sticks or slips, so we encountered difficulties in time integration. To overcome the difficulties, we proposed a new numerical technique to alternatively use the matrix-vector equations with different matrix sizes. In addition, to eliminate spurious high-frequency responses, we applied the generalized-α time integration method with appropriate value of high-frequency numerical dissipation. Finally, the dynamic responses of stick-slip vibration were analyzed in time and frequency domains: the dynamic behavior of the beam was explained to facilitate understanding of the stick-slip motion, and frequency characteristics of the stick-slip vibration were investigated in relation to the natural frequencies of the beam. The effects of the axial load and the moving speed upon the dynamic response were also examined.

Some factors affecting the loosening failure of bolted joints under vibration using finite element analysis

Finite element analysis has been regarded as an effective research method for analyzing the loosening failure of bolted joints under vibration. However, there exist some factors, which influence the accuracy and reliability of loosening results, thus determining the explanations of the loosening mechanism. In this study, a 3D finite element model of a typical bolted joint was built to investigate the effects of several different factors on the loosening under transverse vibration loading. These influencing factors include preload generation, vibration parameter, and material model. Based on the simulation results, it was found that applying the method of pretension element to generate preload instead of the actual method of torque was reliable and efficient. For the vibration parameter, it showed that the decrease rate in preload was higher for a larger vibration amplitude. But once the bearing surface reached complete slip, the loosening rate would keep constant. This was because the thread surface at that time reached a sticking state. Vibration frequency was proved to have no effect on the loosening behavior. This result demonstrated that the quasi-static assumption for vibration frequency was reasonable. Additionally, it also indicated that plastic material models only affected the preload loss in the initial several vibration cycles and had no influence on the loosening rate of preload after several vibration cycles. Finally, experiments were conducted to confirm qualitatively the results obtained based on finite element analysis.

Analysis of the behavior of a wiper blade around the reversal in consideration of dynamic and static friction

Reducing noise generated by automobile windshield wipers during reversals is a desirable feature. For this purpose, details of the behavior of the wiper blade need to be ascertained. In this study, we present theoretical and experimental clarification of this behavior during reversals. Using simulation algorithms to consider exactly the effects of dynamic and static friction, we determined theoretical predictions for the vibrational response caused by friction and the response frequency and compared these results with experimental ones obtained from a mock-up incorporating an actual wiper blade. We introduce an analytical link model with two degrees of freedom and consider two types of states at the blade tip. In the stick and the slip states, static friction and dynamic friction, respectively, act on the blade tip. In the theoretical approach, the static friction is expressed by a set-valued function. The transition between the two states is repeated and an evaluation of an exact transition time leads to an accurate prediction of the behavior of the wiper system. In the analysis, the slack variable method is used to find the exact transition time. Assuming low blade speeds during reversal, a parameter study indicates that the blade tip transitions between slip and stick states and the frequency of the vibration caused by this transitions is close to the natural frequency of the neck of the wiper blade. The theoretical predictions are in good agreement with experimental observations.

Application of a special class of smooth models of the resultant friction force and moment occurring on a circular contact area

Some special cases of a larger class of smooth models of the resultant friction force occurring on a planar contact area are developed and presented. These models are constructed under assumptions of classical Coulomb’s law of friction on each element of contact and instantaneous transition between two modes: fully developed sliding and rigid stick state without local slips and deformations of the contact area. They are able to model cases of different values of static and kinetic friction factors. The considered models are very effective tools for fast and reliable numerical simulations of mechanical systems with frictional contacts. They are applied in modelling and numerical analysis of a special kind of mechanical system, i.e. a kind of a pendulum with special frictional driving. The rotational joint of the pendulum is elastically suspended in the motion plane. The pendulum is driven by circular frictional contact with a rotating disk. Examples of self-excited bifurcation and chaotic dynamics as well as stick–slip behaviour of the pendulum are presented.