Characteristic Equation-Based Dynamic Analysis of a Three-Revolute Prismatic Spherical Parallel Kinematic Machine

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Jun Zhang ◽  
Jian S. Dai ◽  
Tian Huang
Keyword(s):  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Eigenvalue Decomposition ◽  
Mode Shapes ◽  
Design Parameters ◽  
Parallel Kinematic Machine ◽  
Characteristic Equations ◽  
Parallel Kinematic ◽  
Compliance Model

A three-revolute prismatic spherical (3-RPS) parallel kinematic machine (PKM) module is proposed as an alternative solution for high-speed machining (HSM) tool. Considering the PKM as a typical compliant parallel device, whose three limb assemblages have bending, extending, and torsional deflections, this paper applies screw theory to establish an analytical compliance model for the device. The developed compliance model is then combined with the energy method to deduce a comprehensive dynamic model of the 3-RPS module. The solution for the characteristic equations of the dynamic model leads to the modal properties of the PKM module. Based on the eigenvalue decomposition of the characteristic equations, a modal analysis is conducted. The natural frequencies and corresponding mode shapes at typical and nontypical configurations are analyzed and compared with finite element analysis (FEA) results. With an algorithm of workspace partitions combining with eigenvalue decompositions, the distributions of natural frequencies throughout the workspace are predicted to reveal a strong dependency of dynamic characteristics on mechanism's configurations. At the last stage, the effects of some design parameters on system dynamic characteristics are investigated with the purpose of providing useful information for the conceptual design and performance improvement for the PKM.

scholarly journals Rapid Evaluation for Position-Dependent Dynamics of a 3-DOF PKM Module

2014 ◽  
Vol 6 ◽  
pp. 238928 ◽  
Author(s):  
Hai-wei Luo ◽  
Hui Wang ◽  
Jun Zhang ◽  
Qi Li
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Parallel Kinematic Machine ◽  
Computationally Efficient ◽  
Element Analysis ◽  
Reduction Techniques ◽  
Modal Reduction ◽  
Governing Differential Equation ◽  
Parallel Kinematic ◽  
Analytical System

Based on the substructure synthesis and modal reduction technique, a computationally efficient elastodynamic model for a fully flexible 3-RPS parallel kinematic machine (PKM) tool is proposed, in which the frequency response function (FRF) at the end of the tool can be obtained at any given position throughout its workspace. In the proposed elastodynamic model, the whole system is divided into a moving platform subsystem and three identical RPS limb subsystems, in which all joint compliances are included. The spherical joint and the revolute joint are treated as lumped virtual springs with equal stiffness; the platform is treated as a rigid body and the RPS limbs are modelled with modal reduction techniques. With the compatibility conditions at interfaces between the limbs and the platform, an analytical system governing differential equation is derived. Based on the derived model, the position-dependent dynamic characteristics such as natural frequencies, mode shapes, and FRFs of the 3-RPS PKM are simulated. The simulation results indicate that the distributions of natural frequencies throughout the workspace are strongly dependant on mechanism's configurations and demonstrate an axial-symmetric tendency. The following finite element analysis and modal tests both validate the analytical results of natural frequencies, mode shapes, and the FRFs.

Elastodynamic Modeling and Analysis for an Exechon Parallel Kinematic Machine

2015 ◽  
Vol 138 (3) ◽  
Author(s):  
Jun Zhang ◽  
Yan Q. Zhao ◽  
Yan Jin
Keyword(s):  
Differential Equations ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Structural Features ◽  
Parallel Kinematic Machine ◽  
Dynamic Behaviors ◽  
Elastodynamic Modeling ◽  
Parallel Kinematic ◽  
Moving Platform

As a newly invented parallel kinematic machine (PKM), Exechon has attracted intensive attention from both academic and industrial fields due to its conceptual high performance. Nevertheless, the dynamic behaviors of Exechon PKM have not been thoroughly investigated because of its structural and kinematic complexities. To identify the dynamic characteristics of Exechon PKM, an elastodynamic model is proposed with the substructure synthesis technique in this paper. The Exechon PKM is divided into a moving platform subsystem, a fixed base subsystem and three limb subsystems according to its structural features. Differential equations of motion for the limb subsystem are derived through finite element (FE) formulations by modeling the complex limb structure as a spatial beam with corresponding geometric cross sections. Meanwhile, revolute, universal, and spherical joints are simplified into virtual lumped springs associated with equivalent stiffnesses and mass at their geometric centers. Differential equations of motion for the moving platform are derived with Newton's second law after treating the platform as a rigid body due to its comparatively high rigidity. After introducing the deformation compatibility conditions between the platform and the limbs, governing differential equations of motion for Exechon PKM are derived. The solution to characteristic equations leads to natural frequencies and corresponding modal shapes of the PKM at any typical configuration. In order to predict the dynamic behaviors in a quick manner, an algorithm is proposed to numerically compute the distributions of natural frequencies throughout the workspace. Simulation results reveal that the lower natural frequencies are strongly position-dependent and distributed axial-symmetrically due to the structure symmetry of the limbs. At the last stage, a parametric analysis is carried out to identify the effects of structural, dimensional, and stiffness parameters on the system's dynamic characteristics with the purpose of providing useful information for optimal design and performance improvement of the Exechon PKM. The elastodynamic modeling methodology and dynamic analysis procedure can be well extended to other overconstrained PKMs with minor modifications.

Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine

2007 ◽  
Vol 129 (5) ◽  
pp. 623-630 ◽  
Author(s):  
Zili Zhou ◽  
Chris K. Mechefske ◽  
Fengfeng Xi
Keyword(s):  
Natural Frequencies ◽  
Kinematic Model ◽  
Inertial Force ◽  
Mode Shapes ◽  
Absolute Amount ◽  
Hard Material ◽  
Parallel Kinematic Machine ◽  
Damping Matrix ◽  
Nonstationary Vibration ◽  
Parallel Kinematic

This paper studies the problem of the nonstationary vibration of a fully flexible parallel kinematic machine (PKM) that has flexibilities both in links and in joints. In the stationary case, the PKM was treated as a varying structure and the natural frequencies and mode shapes changed with the changes in the PKM configuration, without consideration of the PKM nominal motion. In the nonstationary case as studied in this paper, the nominal motion is included to investigate how it would affect the natural frequencies and mode shapes. To do so, a nonstationary model is developed using the elasto-dynamics method. First, a kinematic model is built based on rigid links and ideal joints, which is used to solve the PKM nominal motion. Second, the kinetic model is developed considering the flexibilities in the links and joints. In this case, the vibration equations would contain the Coriolis and gyroscopic damping matrix and the tangential and normal stiffening matrix, which are the terms resulting from the nominal motion. The instantaneous eigensolutions are obtained from the nonstationary eigenequations. The results show that (i) the slider velocity affects the instantaneous natural frequencies more than the slider acceleration; and (ii) the nominal motion has an effect on the system eigencharacteristics (e.g., the nonstationary frequencies can be higher or lower than the stationary ones) but the effect is small in an absolute amount (within 2.1Hz in natural frequencies presented at set nominal motions of the studied PKM prototype). This is because the extra inertial force from the nominal motion is always much smaller than the stiffness force in the system bodies as long as the bodies are made of hard material. The method presented is more convenient to use for the multibody system with flexible joints than other methods.

Elastodynamic Model-Based Vibration Characteristics Prediction of a Three Prismatic–Revolute–Spherical Parallel Kinematic Machine

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Jun Zhang ◽  
Yan Q. Zhao ◽  
Marco Ceccarelli
Keyword(s):  
Dynamic Characteristics ◽  
Lower Order ◽  
High Speed ◽  
Natural Frequencies ◽  
Eigenvalue Decomposition ◽  
Design Stage ◽  
Response Analysis ◽  
Early Design Stage ◽  
Synthesis Strategy ◽  
Parallel Kinematic

Parallel kinematic machines (PKMs) have been proposed as an alternative solution for high-speed machining (HSM) tool for several years. However, their dynamic characteristics are still considered an issue for practice application. Considering the three prismatic–revolute–spherical (3-PRS) PKM design as a typical compliant parallel device, this paper applies substructure synthesis strategy to establish an analytical elastodynamic model for the device. The proposed model considers the effects of component compliances and kinematic pair contraints so that it can predict the dynamic characteristics of the 3-PRS PKM. Based on eigenvalue decomposition of the characteristic equations, the natural frequencies and corresponding vibration modes at a typical configuration are analyzed and verified by numerical simulations. With an algorithm of workspace partitions combining with eigenvalue decompositions, the distributions of lower-order natural frequencies throughout the workspace are computed to reveal a strong dependency of dynamic characteristics on mechanism's configurations. In addition, the effects of the radii of the platform and the base along with the cross section of the limb on lower-order natural frequencies are analyzed to provide useful information during the early design stage. At last, frequency response analysis for the tool center point (TCP) is worked out based on the elastodynamic model to provide primary guideline for cutting chatter avoidance.

Dynamic model for high-speed rotors based on their experimental characterization

2017 ◽  
Vol 2 (4) ◽  
pp. 25
Author(s):  
L. A. Montoya ◽  
E. E. Rodríguez ◽  
H. J. Zúñiga ◽  
I. Mejía
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Experimental Characterization ◽  
Rotating Systems ◽  
Computing Performance ◽  
The Impact ◽  
Stiffness Distribution ◽  
Good Agreement

Rotating systems components such as rotors, have dynamic characteristics that are of great importance to understand because they may cause failure of turbomachinery. Therefore, it is required to study a dynamic model to predict some vibration characteristics, in this case, the natural frequencies and mode shapes (both of free vibration) of a centrifugal compressor shaft. The peculiarity of the dynamic model proposed is that using frequency and displacements values obtained experimentally, it is possible to calculate the mass and stiffness distribution of the shaft, and then use these values to estimate the theoretical modal parameters. The natural frequencies and mode shapes of the shaft were obtained with experimental modal analysis by using the impact test. The results predicted by the model are in good agreement with the experimental test. The model is also flexible with other geometries and has a great time and computing performance, which can be evaluated with respect to other commercial software in the future.

Split Vibration Modes in Acoustically-Coupled Disk Stacks

1995 ◽  
Author(s):  
Jung-Ge Tseng ◽  
Jonathan Wickert
Keyword(s):  
Natural Frequencies ◽  
Three Dimensional ◽  
System Level ◽  
Thin Plates ◽  
Mode Shapes ◽  
Design Parameters ◽  
Phase System ◽  
Acoustic Coupling ◽  
Annular Plates ◽  
Vibration Model

Abstract Vibration of an array of stacked annular plates, in which adjacent plates couple weakly through an acoustic layer, is investigated through experimental and theoretical methods. Such acoustic coupling manifests itself through split natural frequencies, beating in the time responses of adjacent or separated plates, and system-level modes in which plates in the array vibrate in- or out-of-phase at closely-spaced frequencies. Laboratory measurements, including a technique in which the frequency response function of all in-phase modes but no out-of-phase modes, or visa versa, is measured, demonstrate the contribution of coupling to the natural frequency spectrum, and identify the combinations of design parameters for which it is important. For the lower modes of primary interest here, the natural frequencies of the out-of-phase system modes decrease as the air layer becomes thinner, while those of the in-phase mode remain sensibly constant at the in vacuo values. A vibration model comprising N classical thin plates that couple through the three-dimensional acoustic fields established in the annular cavities between plates is developed, and its results are compared with measurements of the natural frequencies and mode shapes.

Dynamic characteristics of horizontally curved bridges

2017 ◽  
Vol 24 (19) ◽  
pp. 4465-4483 ◽  
Author(s):  
Mohsen Amjadian ◽  
Anil K Agrawal
Keyword(s):  
Free Vibration ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Response Spectrum ◽  
Translational Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Curved Bridges ◽  
Horizontally Curved ◽  
Horizontally Curved Bridges

Horizontally curved bridges have complicated dynamic characteristics because of their irregular geometry and nonuniform mass and stiffness distributions. This paper aims to develop a simplified and practical method for the calculation of the natural frequencies and mode shapes of horizontally curved bridges that would be of interest to bridge engineers for the estimation of the seismic response of these types of bridges. For this purpose, a simple three-degree-of-freedom (3DOF) dynamic model for free vibration equation of this type of bridge has been developed. It is shown that the translational motion of the deck of horizontally curved bridges in the direction that is perpendicular to their axis of symmetry is always coupled with the rotational motion of the deck, regardless of the location of the stiffness center. The model is further exploited to develop closed-form formulas for the estimation of the maximum displacements of the corners of the deck of one-way asymmetric horizontally curved bridges. The accuracy of the model is verified by finite-element model of a horizontally curved bridge prototype in OpenSEES. Finally, the model is utilized to study the influence of the location of the stiffness center with respect to the deck curvature center on the natural frequency and the maximum displacements of the corners of the deck for different curvatures of the deck. The results of free vibration analysis show that the natural frequencies of one-way asymmetric horizontally curved bridges, in general, increase with the increase of the subtended angle of the deck. The results of earthquake response spectrum analysis show that the increase in the subtended angle of one-way asymmetric horizontally curved bridges decreases the radial displacements of the corners of the deck but increases the azimuthal displacement. These two responses both increase with the increase in the distance between the stiffness center and the curvature center.

A Study on the Effect of Bird Hit Damage on the Dynamic Response of Aero-Engine Fan Blade

2013 ◽  
Author(s):  
Hithesh Channegowda ◽  
Raghu V. Prakash ◽  
Anandavel Kaliyaperumal
Keyword(s):  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Foreign Object Damage ◽  
Aero Engine ◽  
Need To Evaluate ◽  
Bird Impact ◽  
Post Impact ◽  
Fan Blade ◽  
Critical Components

Fan blades of an aero-engine assembly are the critical components that are subjected to Foreign Object Damage (FOD) such as bird impact. Bird impact resulting in deformation damage onto set of blades, which in turn alters the blade mass and stiffness distribution compared to undamaged blades. This paper presents the numerical evaluation of dynamic characteristics of bird impact damaged blades. The dynamic characteristics evaluated are the natural frequencies and mode shapes of post impact damaged set of blades and the results are compared with undamaged set of blades. The frequencies and mode shapes are evaluated for the damaged blades, with varying angles of bird impact and three blade rotational speeds. Study reveals that first bending and torsional frequencies of deformed blades are significantly affected compared to undamaged set of blades. Study emphasize the need to evaluate the natural frequencies deformed blades, that has direct bearing on High Cycle Fatigue (HCF) life of the blade, to ensure post damaged blades operate safely for certain time to reduce inflight accidents and safe landing.

Dynamic Modeling of a Ball-Screw Drive and Identification of Its Installation Parameters

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Yu-Jia Hu ◽  
Yaoyu Wang ◽  
Weidong Zhu ◽  
Haolin Li
Keyword(s):  
Dynamic Model ◽  
Natural Frequencies ◽  
Data Fitting ◽  
Ball Screw ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Screw Drive ◽  
Log Normal ◽  
Ball Screw Drive ◽  
Log Normal Distribution

Abstract Parametric expressions of equivalent stiffnesses of a ball-screw shaft are obtained by derivation of its geometric parameters, the finite element method (FEM), and data fitting based on a modified probability density function of log-normal distribution. A dynamic model of a ball-screw drive that considers effects of bearing stiffnesses, the mass of the nut, and the axial pretension is established based on equivalent stiffnesses of its shaft. With the dynamic model and modal experimental results obtained by Bayesian operational modal analysis (BOMA), installation parameters of the ball-screw drive are identified by a genetic algorithm (GA) with a new comprehensive objective function that considers natural frequencies, mode shapes, and flexibility of the ball-screw drive. The effectiveness of the methodology is experimentally validated.

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