Properties of Higher Order Instant Center: A Case Study of Classical Motions

2013 ◽  
Vol 5 (2) ◽  
Author(s):  
Amit Kulkarni ◽  
Delbert Tesar
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Physical Nature ◽  
Higher Order ◽  
Second Order ◽  
Body Motion ◽  
General Point ◽  
Mobile Platforms ◽  
Wheel Rolling ◽  
Fifth Order

A generalized algebraic formulation using instant centers (IC) was developed for the motion description of a general point in the rigid body under a planar 3-DOF (degrees-of-freedom) motion for up to the fifth order kinematics. This motion theory is being applied to planar wheeled mobile platforms. Though the first order and second order instant centers have been previously studied, the properties of higher order instant centers are yet to be understood. Also the expressions for third and higher order motion are highly coupled and more complex than the first and second order motion descriptions. To this effect, this paper studies some special case scenarios of planar rigid body motion that involve well documented 1-DOF motions such as a circle (cylinder /disk/wheel) rolling on a straight line (plane/flat, smooth surface), a circle rolling inside another circle, a circle rolling on another circle, etc. This study will help us understand the physical nature of the kinematic formulation using instant centers. Otherwise, numerical specifications for the higher order properties will have little known physical reference as to the meanings of their magnitudes.

Instant Center Based Kinematic Formulation for Planar Wheeled Platforms

2010 ◽  
Vol 2 (3) ◽  
Author(s):  
Amit Kulkarni ◽  
Delbert Tesar
Keyword(s):  
Rigid Body ◽  
Input Parameter ◽  
Radial Distance ◽  
Planar Motion ◽  
Mobile Platform ◽  
General Point ◽  
Kinematic Modeling ◽  
Mobile Platforms ◽  
General Order ◽  
Operational Criteria

For a general J wheeled mobile platform capable of up to three-degrees-of-freedom planar motion, there are up to two J independent input parameters yet the output of the platform is completely represented by three independent variables. This leads to an input parameter resolution problem based on operational criteria, which are in development just as they have been developed for n input manipulator systems. To resolve these inputs into a meaningful decision structure means that all motions at the wheel attachment points must have clear physical meaning. To this effect, we propose a methodology for kinematic modeling of multiwheeled mobile platforms using instant centers to efficiently describe the motion of all system points up to the nth order using a generalized algebraic formulation. This is achieved by using a series of instant centers (velocity, acceleration, jerk, and jerk derivative), where each point in the system has a motion property with its magnitude proportional to the radial distance of the point from the associated instant center and at a constant angle relative to that radius. The method of instant center provides a straightforward and physically intuitive way to synthesize a general order planar motion of mobile platforms. It is shown that a general order motion property of any point on a rigid body follows two properties, namely, directionality and proportionality, with respect to the corresponding instant center. The formulation presents a concise expression for a general order motion property of a general point on the rigid body with the magnitude and direction separated and identified. The results are summarized for up to the fifth order motion in the summary table. Based on the initial formulation, we propose the development of operational criteria using higher order properties to efficiently synthesize the motion of a J wheeled mobile platform.

On Higher Order Point-Line and the Associated Rigid Body Motions

2006 ◽  
Vol 129 (2) ◽  
pp. 166-172 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Higher Order ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Dual Quaternion ◽  
Displacement Operator ◽  
Screw Motion ◽  
Rigid Body Motions ◽  
Point Line

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

Modeling the Dynamics of Five-Axis Machine Tool Using the Multibody Approach

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Hoai Nam Huynh ◽  
Yusuf Altintas
Keyword(s):  
Rigid Body ◽  
Machine Tool ◽  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Dynamic Performance ◽  
Body Motion ◽  
Proposed Model ◽  
Multibody Dynamic ◽  
Machining Operations ◽  
Five Axis

Abstract A systematic modeling of multibody dynamics of five-axis machine tools is presented in this article. The machine is divided into major subassemblies such as spindle, column, bed, tool changer, and longitudinal and rotary drives. The inertias and mass center of each subassembly are calculated from the design model. The subassemblies are connected with elastic springs and damping elements at contact joints to form the complete multibody dynamic model of the machine that considers the rigid body kinematics and structural vibrations of the machine at any point. The unknown elastic joint parameters are estimated from the experimental modal analysis of the machine tool. The resulting position-dependent multibody dynamic model has the minimal number of degrees-of-freedom that is equivalent to the number of measured modes, as opposed to thousands used in finite element models. The frequency response functions of the machine can be predicted at any posture of the five-axis machine, which are compared against the directly measured values to assess the validity of model. The proposed model can predict the combined rigid body motion and vibrations of the machine with computational efficiency, and hence, it can be used as a digital twin to simulate its dynamic performance in machining operations and tracking control tests of the servo drives.

scholarly journals Modeling Legged Microrobot Locomotion Based on Contact Dynamics and Vibration in Multiple Modes and Axes

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Jinhong Qu ◽  
Clark B. Teeple ◽  
Kenn R. Oldham
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Beam Theory ◽  
Contact Dynamics ◽  
Body Motion ◽  
Small Scale ◽  
Robot Locomotion ◽  
Actuation Scheme ◽  
Dynamics And Vibration

A dynamic model is developed for small-scale robots with multiple high-frequency actuated compliant elastic legs and a rigid body. The motion of the small-scale robots results from dual-direction motion of piezoelectric actuators attached to the legs, with impact dynamics increasing robot locomotion complexity. A dynamic model is developed to describe the small-scale robot motion in the presence of variable properties of the underlying terrain. The dynamic model is derived from beam theory with appropriate boundary and loading conditions and considers each robot leg as a continuous structure moving in two directions. Robot body motion is modeled in up to five degrees-of-freedom (DOF) using a rigid body approximation for the central robot chassis. Individual modes of the resulting multimode robot are treated as second-order linear systems. The dynamic model is tested with two different centimeter-scale robot prototypes having an analogous actuation scheme to millimeter-scale microrobots. In accounting for the interaction between the robot and ground, a dynamic model using the first two modes of each leg shows good agreement with experimental results for the centimeter-scale prototypes, in terms of both magnitude and the trends in robot locomotion with respect to actuation conditions.

Dynamic Analysis of Flexible Supercavitating Vehicles Using Modal Based Elements

2003 ◽  
Author(s):  
Jou-Young Choi ◽  
Massimo Ruzzene ◽  
Olivier A. Bauchau
Keyword(s):  
Numerical Model ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Structural Characteristics ◽  
Rigid Body Motion ◽  
Operating Conditions ◽  
Flight Mechanics ◽  
Body Motion ◽  
Supercavitating Vehicles ◽  
Elastic Modes

This presents a numerical model for the simulation of the flight mechanics behavior of flexible supercavitating vehicles. Supercavitating vehicles exploit supercavitation as a means to reduce drag and increase the underwater speed. In the proposed formulation, the vehicle’s rigid body motion is described by 6 degrees of freedom, which define pitch, yaw and roll motion and the displacement of the center of gravity with respect to a fixed inertial reference system. The forces applied to the vehicle include the control actions at the nose and at the fins, propulsion, gravity and cavity/vehicle periodic interactions associated to typical operating conditions. The elastic displacements are superimposed to the rigid body motion through a modal superposition technique. The mode synthesis is performed using Herting’s Transformation, which provides maximum flexibility in the selection of the elastic modes to be used for the used for the superposition, and the possibility of easily handling free-free modes. The developed numerical model predicts the dynamic response of the considered class of supercavitating vehicles resulting from assigned maneuvers. The analysis is motivated by the need of accurately modeling the structural characteristics of supercavitating vehicles in order to estimate vibrations in the structure and to envision and design systems that improve their guidance and control efficiency.

Calculations of non-linear waves generated by complex body motion

2000 ◽  
Vol 214 (6) ◽  
pp. 771-780 ◽  
Author(s):  
T Chen ◽  
A T Chwang
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Grid Method ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Promising Alternative ◽  
Potential Flows ◽  
Linear Waves ◽  
Non Linear ◽  
Marine Engineering

A structured and unstructured hybrid overlapping grid method is developed for simulating free-surface waves generated by submerged arbitrary bodies undergoing rigid body motion in multi-degrees of freedom. Exact boundary conditions are applied to the transient free and body surfaces. The accuracy, efficiency and generality of the present two-dimensional code for potential flows are validated by comparisons with available theories and experiments. Numerical experiments are reported in this paper to investigate the non-linear behaviour of waves due to the complex rigid body motion, in terms of wave patterns and the pressure distribution. Combining the best features of both grid systems for finite elements and finite differences, the present method provides a promising alternative in computational fluid dynamics for the design and analysis in marine engineering.

Dynamic Model of the Piston-Ring Assembly Using Curved Beam Finite Elements

2011 ◽  
Author(s):  
Mohannad Hakeem ◽  
Nabil G. Chalhoub ◽  
Peter Schihl
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Piston Ring ◽  
Translational Motion ◽  
Total Duration ◽  
Variable Structure ◽  
Body Motion ◽  
Curved Beam ◽  
Connecting Rod

A dynamic model for the crankshaft/connecting-rod/piston-assembly for a single cylinder engine is developed. The model considers the rigid body motion of the crank-slider mechanism including the piston secondary motions such as the piston-slap and piston-tilting. The formulation considers the ring to have three rigid body degrees of freedom in addition to its longitudinal and in-plane transverse deformations. The structural flexibility terms are approximated by using curved beam finite element method. The dynamic model has a variable structure whereby the number of degrees of freedom depends on the piston-liner and piston-ring interactions. Its formulation does not include frictional losses. The simulation results illustrate the piston secondary motions along with the ring tilting angles relative to the piston orientation for the total duration of the engine cycle. In addition, they exhibit the translational motion of the ring within the piston groove.

Qualified Geometric Stiffness for Linear Buckling and Second-Order Nonlinear Analysis of Framed Structures

2019 ◽  
Vol 19 (08) ◽  
pp. 1950094
Author(s):  
Y. Wen ◽  
Q. Tan ◽  
Z. L. Chen
Keyword(s):  
Rigid Body ◽  
Nonlinear Analysis ◽  
Kinematic Model ◽  
Second Order ◽  
Body Motion ◽  
Benchmark Problems ◽  
Framed Structures ◽  
Geometric Stiffness ◽  
Linear Buckling ◽  
Post Buckling

There exist various potential energy formulations dealing with the linear buckling and second-order nonlinear analysis of framed structures with different degrees of refinement in the kinematic model. However, the geometric stiffnesses derived often give rise to different structural behaviors, which indeed represents a confusion regarding their qualified usage in bifurcation and post-buckling analysis. This study aims to carry out a comprehensive evaluation of the validity of the geometric stiffness for use at the predictor and corrector phrases of an incremental analysis based on the rigid-body motion test. To remove the unbalanced element forces caused by the nonqualified geometric stiffness, a supplementary correction matrix is developed according to simple kinematic and static analysis in the context of rigid rotations. An updated Lagrangian approach-based force recovery procedure (FRP) is presented for updating the element forces with improved reliability and efficiency, when using relatively large step size. Some benchmark problems which exhibit compound three-dimensional nonlinear behavior of framed structures are solved to clarify the capabilities of rigid-body qualified and nonqualified geometric stiffnesses along with the existing FRPs for different mesh sizes, step sizes and load patterns. It is shown that the proposed procedure can be adopted to predict correct buckling loads and post-buckling equilibrium paths without adding extra computational costs.

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