Analytic Solution for Planar Indeterminate Multiple Point Impact Problems With Coulomb Friction

2014 ◽  
Author(s):  
Adrian Rodriguez ◽  
Alan Bowling
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Equations Of Motion ◽  
Algebraic Approach ◽  
Multiple Point ◽  
Stick Slip ◽  
Rocking Block ◽  
Consistent Solution ◽  
Post Impact ◽  
The Impact

This work analyzes the effects of the stick-slip transition of planar rigid body systems undergoing simultaneous, multiple point impact with Coulomb friction. A discrete, algebraic approach is used in conjunction with an event-driven scheme which detects impact events. The system equations of motion for the examples considered are indeterminate with respect to the impact forces. Constraints consistent with rigid body assumptions are implemented to overcome the indeterminacy. The post-impact velocities of a system are determined by exploiting the work-energy relationship of a collision and using an energetic coefficient of restitution to model energy dissipation. These developments lead to a unique and energetically consistent solution to the post-impact velocities. A frictionless rocking block example is analyzed as a benchmark case and compared to experimental results to demonstrate the accuracy of the proposed method. Simulation results are also presented for a planar ball example with friction.

Study of the Stick-Slip Transition of Newton’s Cradle With Friction

2013 ◽  
Author(s):  
Adrian Rodriguez ◽  
Alan Bowling
Keyword(s):  
Impact Behavior ◽  
Multiple Point ◽  
Stick Slip ◽  
Kinematic Relationship ◽  
Short Time Span ◽  
Post Impact ◽  
Newton’S Cradle ◽  
Six Degree Of Freedom ◽  
Slip Transition ◽  
The Impact

This work uses a new discrete approach to analyze the stick-slip transition of Newton’s cradle with frictional contact. The consideration of friction here leads to a simultaneous, multiple point, indeterminate collision. This work strictly adheres to the assumptions of rigid body modeling in conjunction with the notion that the configuration of the system are constant in the short time span of the collision, which enforces a kinematic relationship between the impact points. The post-impact velocities are determined by using the work-energy relationship of a collision and an energetic coefficient of restitution (ECOR) to model energy dissipation. A three and six degree-of-freedom (DOF) model of the system is considered in this work to examine the stick-slip transition and simulate the post-impact behavior. Simulations are conducted for each model using different coefficients of friction (COFs). The results obtained are compared to theoretical and experimental results reported in other works.

Solution to Three-Dimensional Indeterminate Contact and Impact With Friction Using Rigid Body Constraints

2015 ◽  
Author(s):  
Adrian Rodriguez ◽  
Abhishek Chatterjee ◽  
Alan Bowling
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Impact Behavior ◽  
Impact Event ◽  
Slip Direction ◽  
Rocking Block ◽  
Post Impact ◽  
Contact And Impact ◽  
The Impact

This work analyzes three-dimensional multibody systems undergoing indeterminate contact and impact in the presence of Coulomb friction. A discrete approach is used to analyze the impact behavior upon detection of the impact events during simulation. During an impact event, the velocities of the impact points describe the systems state and can be studied to characterize the nature of impact and determine the post-impact behavior of the system. The velocities of the impact points during an impact event can be described in terms of the impulses at those points. This work uses Amontons-Coulombs law of friction and rigid body constraints to develop a technique for reducing the number of impulses required to compute the velocities of the impact point during the impact event. Indeterminacies associated with slip direction arise, when Coulombs friction is considered. Therefore, a numerical approach is used to evolve the slip direction along with the slip velocity, with respect to a normal impulse. The work-energy theorem is used to detect the end of the impact event, and determine the post-impact velocities of the system. Examples of a three-dimensional rocking block and a sphere impacting a corner are analyzed to demonstrate the proposed methodology.

Resolving the Unique Invariant Slip-Direction in Rigid Three-Dimensional Multi-Point Impacts at Stick-Slip Transitions

2018 ◽  
Author(s):  
Abhishek Chatterjee ◽  
Alan Bowling
Keyword(s):  
Single Point ◽  
Three Dimensional ◽  
Impact Behavior ◽  
Slip Direction ◽  
Stick Slip ◽  
Impact Forces ◽  
Post Impact ◽  
Impact Problems ◽  
Slip Transition ◽  
The Impact

This work presents a new approach for resolving the unique invariant slip direction at Stick-Slip Transition during impact. The solution method presented in this work is applicable to both single-point and multi-point impact problems. The proposed method utilizes rigid body constraints to resolve the impact forces at all collision points in terms of a single independent impact forces parameter. This work also uses an energetic coefficient of restitution to terminate impact events, thereby yielding energetically consistent post-impact behavior.

Seismic Analysis of Structures With Coulomb Friction

1983 ◽  
Vol 105 (2) ◽  
pp. 171-178 ◽  
Author(s):  
V. N. Shah ◽  
C. B. Gilmore
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Seismic Event ◽  
Seismic Analysis ◽  
Equations Of Motion ◽  
Relative Displacement ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition Method

A modal superposition method for the dynamic analysis of a structure with Coulomb friction is presented. The finite element method is used to derive the equations of motion, and the nonlinearities due to friction are represented by pseudo-force vector. A structure standing freely on the ground may slide during a seismic event. The relative displacement response may be divided into two parts: elastic deformation and rigid body motion. The presence of rigid body motion necessitates the inclusion of the higher modes in the transient analysis. Three single degree-of-freedom problems are solved to verify this method. In a fourth problem, the dynamic response of a platform standing freely on the ground is analyzed during a seismic event.

On Some Problems With Modeling of Coulomb Friction in Self-Locking Speed Reducers

2014 ◽  
Author(s):  
Marek Wojtyra
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Equations Of Motion ◽  
Velocity Ratio ◽  
Friction Forces ◽  
Friction Law ◽  
Rigid Body Model ◽  
Feasible Sets ◽  
Unstable Solutions ◽  
Coulomb Friction Law

A simple mathematical model of friction in speed reducers is presented and discussed. A rigid body approach, typical for multibody simulations, is adopted. The model is based on the Coulomb friction law and exploits the analogy between reducers and wedge mechanisms. The first version of the model is purely rigid, i.e. no deflections of the mechanism bodies are allowed. Constraints are introduced to maintain the ratio between input and output velocity. It is shown that when friction is above the self-locking limit, paradoxical situations may be observed when kinetic friction is investigated. For some sets of parameters of the mechanism (gearing ratio, coefficient of friction and inertial parameters) two distinct solutions of normal and friction forces can be found. Moreover, for some combinations of external loads, a solution that satisfies equations of motion, constraints and Coulomb friction law does not exist. Furthermore, for appropriately chosen loads and parameters of the mechanism, infinitely many feasible sets of normal and friction forces can be found. Examples of all indicated paradoxical situations are provided and discussed. The second version of the model allows deflection of the frictional contact surface, and forces proportional to this deflection are applied to contacting bodies (no constraints to maintain the input-output velocity ratio are introduced). In non-paradoxical situations the obtained results are closely similar to those predicted by the rigid body model. In previously paradoxical situations no multiple solutions of friction force are found, however, the amended model does not solve all problems. It is shown that in regions for which the paradoxes were observed only unstable solutions are available. Numerical examples showing behavior of the model are provided and analyzed.

scholarly journals Transitions and singularities during slip motion of rigid bodies

2018 ◽  
Vol 29 (5) ◽  
pp. 778-804 ◽  
Author(s):  
P. L. VÁRKONYI
Keyword(s):  
Coulomb Friction ◽  
Rigid Bodies ◽  
Two Dimensions ◽  
Multiple Point ◽  
Point Contacts ◽  
Stick Slip ◽  
Unilateral Contacts ◽  
Rigid Rod ◽  
Body Theory ◽  
Slip Motion

The dynamics of moving solids with unilateral contacts are often modelled by assuming rigidity, point contacts, and Coulomb friction. The canonical example of a rigid rod with one endpoint slipping in two dimensions along a fixed surface (sometimes referred to as Painlevé rod) has been investigated thoroughly by many authors. The generic transitions of that system include three classical transitions (slip-stick, slip reversal, and liftoff) as well as a singularity called dynamic jamming, i.e., convergence to a codimension 2 manifold in state space, where rigid body theory breaks down. The goal of this paper is to identify similar singularities arising in systems with multiple point contacts, and in a broader setting to make initial steps towards a comprehensive list of generic transitions from slip motion to other types of dynamics. We show that – in addition to the classical transitions – dynamic jamming remains a generic phenomenon. We also find new forms of singularity and solution indeterminacy, as well as generic routes from sliding to self-excited microscopic or macroscopic oscillations.

LYAPUNOV'S EXPONENTS FOR NONSMOOTH DYNAMICS WITH IMPACTS: STABILITY ANALYSIS OF THE ROCKING BLOCK

2005 ◽  
Vol 15 (06) ◽  
pp. 2015-2039 ◽  
Author(s):  
ALESSIO AGENO ◽  
ANNA SINOPOLI
Keyword(s):  
Stability Analysis ◽  
Coulomb Friction ◽  
Equations Of Motion ◽  
Nonsmooth Dynamics ◽  
Rocking Block ◽  
Fundamental Solution Matrix ◽  
Unilateral Contacts ◽  
Discontinuous Nature ◽  
The Subject ◽  
Solution Matrix

The plane dynamics of a rigid block simply supported on a harmonically moving rigid ground is a problem which still needs investigating, although the matter has been the subject of much research since the last century. Unilateral contacts, Coulomb friction and impacts make the system hybrid as it reveals a mixed continuous and discontinuous nature. Thus, stability analysis requires the extension and adaptation of concepts with regard to variational-perturbative procedures. In particular, discontinuous systems exhibit discontinuities or "saltations" in the fundamental solution matrix which must be analyzed carefully. In this paper, the adaptation of numerical methods that permit us to obtain characteristic multipliers and Lyapunov's exponents for the rocking mode of the block will be tackled. Analytical methods are used for the linearized equations of motion; the results are compared with those in the scientific literature.

Rigid Body Motion Conversion due to Collision

2013 ◽  
Vol 705 ◽  
pp. 540-545
Author(s):  
Svetlana Polukoshko
Keyword(s):  
Rigid Body ◽  
Cylindrical Body ◽  
Plane Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Impact Theory ◽  
Post Impact ◽  
Task Oriented ◽  
The Impact ◽  
Impact Motion

The impact phenomenon may be used for task-oriented changing of rigid body motion. When moving body encounters with some obstacle all parameters of motion are changing as a result of impact and trajectory and type of motion are also changing. In this work the conversion of translatory motion of prismatic rigid body into plane or rotation and conversion of plane motion of cylindrical body due to impact are considered. The conditions of conversion of one type of motion into another and parameters post-impact motion are studied. Problems are solved in the framework of rigid body motion, using rigid body impact theory. Studying of such phenomena is important for location of parts on industrial conveyors, feeders, etc.

scholarly journals Kinematics and dynamics analysis of new rotary-percussive PDM drilling tool

2021 ◽  
pp. 146134842198915
Author(s):  
Jialin Tian ◽  
Xuehua Hu ◽  
Liming Dai ◽  
Lin Yang ◽  
Yi Yang ◽  
...  
Keyword(s):  
Drill String ◽  
Structure Design ◽  
Drilling Process ◽  
Analysis Model ◽  
Drilling Tool ◽  
Stick Slip ◽  
Rate Of Penetration ◽  
Bottom Hole ◽  
Bottom Hole Assembly ◽  
The Impact

This paper presents a new drilling tool with multidirectional and controllable vibrations for enhancing the drilling rate of penetration and reducing the wellbore friction in complex well structure. Based on the structure design, the working mechanism is analyzed in downhole conditions. Then, combined with the impact theory and the drilling process, the theoretical models including the various impact forces are established. Also, to study the downhole performance, the bottom hole assembly dynamics characteristics in new condition are discussed. Moreover, to study the influence of key parameters on the impact force, the parabolic effect of the tool and the rebound of the drill string were considered, and the kinematics and mechanical properties of the new tool under working conditions were calculated. For the importance of the roller as a vibration generator, the displacement trajectory of the roller under different rotating speed and weight on bit was compared and analyzed. The reliable and accuracy of the theoretical model were verified by comparing the calculation results and experimental test results. The results show that the new design can produce a continuous and stable periodic impact. By adjusting the design parameter matching to the working condition, the bottom hole assembly with the new tool can improve the rate of penetration and reduce the wellbore friction or drilling stick-slip with benign vibration. The analysis model can also be used for a similar method or design just by changing the relative parameters. The research and results can provide references for enhancing drilling efficiency and safe production.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

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