Theoretical and Experimental Studies of Chatter in Turning for Uniform and Stepped Workpieces

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
S. D. Yu ◽  
V. Shah
Keyword(s):  
Coupling Effect ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Beam Theory ◽  
Summation Method ◽  
Cutting Process ◽  
Mode Shapes ◽  
Coupled Equations ◽  
Modal Summation ◽  
System Equations

This paper presents a method for predicting regenerative chatter onset conditions for uniform and stepped workpieces. The lateral deflections of flexible workpieces are modeled using the Timoshenko beam theory and three-node beam finite elements. The modal summation method is employed in conjunction with an adaptive remeshing scheme to determine the varying natural frequencies and varying mode shapes of workpieces during a cutting process, and to reduce the system equations of motion in terms of nodal variables to coupled equations of motion in terms of the modal coordinates. Various simulations were conducted and presented in this paper for understanding the gyroscopic and cross-coupling effect, and effects of other system and cutting process parameters on chatter onset conditions. Six experiments were carried out on an engine lathe for three uniform and three stepped workpieces to verify the theoretical chatter onset conditions. Good agreement in chatter onset conditions was observed between the simulations and the experiments.

Free Vibration Analysis of Planar Flexible Mechanisms

2003 ◽  
Vol 125 (4) ◽  
pp. 764-772 ◽  
Author(s):  
S. D. Yu ◽  
F. Xi
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Summation Method ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Modal Summation ◽  
Gyroscopic Effects ◽  
Flexible Mechanisms

This paper presents a methodology for accurate free vibration analysis of planar flexible mechanisms. Each flexible body is considered as a beam and modelled using higher-order beam elements for longitudinal and flexural deformations. The global equations of motion for a mechanism consisting of multiple flexible bodies are formulated using the augmented Lagrange equations. Free vibration analyses are conducted at desired fast Fourier configurations to determine instantaneous structural natural frequencies and structural mode shapes. Dynamical frequencies and dynamical mode shapes incorporating the gyroscopic effects and dynamic axial loads are obtained using the modal summation method. Numerical results and comparisons are given for a rotating beam and two four-bar crank-rocker mechanisms.

Free Vibration of Cross Ply Laminated Beams with Multiple Distributed Piezoelectric Actuators

2012 ◽  
Vol 28 (1) ◽  
pp. 217-227 ◽  
Author(s):  
A. A. Khdeir ◽  
E. Darraj ◽  
O. J. Aldraihem
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Piezoelectric Actuators ◽  
Mode Shapes ◽  
Beam Model ◽  
Laminated Beams ◽  
State Space Approach ◽  
Laminated Beam ◽  
Beam Surface

ABSTRACTAnalytical solution is obtained for the free vibration of cross-ply laminated beams with multiple distributed extension piezoelectric actuators. The piezoelectric actuators are bonded at local position on the beam surface. The beam structure can contain one pair or two pairs or n pairs of piezoelectric actuators and it can be symmetric or unsymmetric about its mid-plane. The equations of motion and associated boundary conditions are derived for the beam model using Hamilton's principle. The state-space approach is used to find accurate natural frequencies and mode shapes for arbitrary combinations of boary conditions. The exact analytical solutions obtained are illustrated numerically in a number of figures revealing the influences of varying some parameters for the symmetric and unsymmetric cross-ply laminated beam for different type of piezoelectric actuators cases. The first order shear deformation beam theory (FOBT) is used to present the effect of actuators position and length on the nondimensional frequencies when one pair and two pairs of piezoelectric actuators are bonded at a local position on the beam surface.

Frictional Impact-Contacts in Multiple Flexible Links

2021 ◽  
pp. 2150075
Author(s):  
M. Ahmadizadeh ◽  
A. M. Shafei ◽  
R. Jafari
Keyword(s):  
Differential Equations ◽  
Recursive Algorithm ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Open Loop ◽  
Mode Shapes ◽  
Special Treatment ◽  
Exact Moment ◽  
Impact Phase ◽  
The Impact

Multiple impacts of 2D (planar) open-loop robotic systems composed of [Formula: see text] elastic links and revolute joints are studied in this paper. The dynamic equations of motion for such systems are derived by the Gibbs-Appell recursive algorithm, while the regularized method is employed to model the impact-contact mechanism. The Timoshenko beam theory is used to model the transverse vibrations of the links. Also, both the structural damping and air damping are considered to enhance the modeling accuracy. The system joints are assumed to be frictionless and slack-free, but friction force is included for the links colliding with the ground. The [Formula: see text]-flexible-link system considered goes through a flight phase and an impact phase during its motion. In the impact phase, new equations of motion are derived by including the terms caused by the viscoelastic forces in the system’s differential equations. Owing to the extremely short acting time of the impact force, the related differential equations can be solved only via special treatment, i.e. by detecting the exact moment of impact. To this end, entering or leaving the impact phase is analyzed and controlled with high precision by a special computational algorithm presented in this work. To demonstrate the efficacy and precision of the algorithm developed, computer simulations are conducted to study the dynamic behavior of a 3-link robotic mechanism. To investigate the effect of mode shape on the elastic deformation of links, four different mode shapes are used in the simulations and their results are compared.

Dynamic Response of a Geared Rotor-Bearing System Under Residual Shaft Bow Effect

2006 ◽  
Author(s):  
T. N. Shiau ◽  
E. K. Lee ◽  
Y. C. Chen ◽  
T. H. Young
Keyword(s):  
Steady State ◽  
Natural Frequencies ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Transmission Error ◽  
Mode Shapes ◽  
Steady State Response ◽  
State Response ◽  
Rotor Bearing ◽  
Bearing System

The paper presents the dynamic behaviors of a geared rotor-bearing system under the effects of the residual shaft bow, the gear eccentricity and excitation of gear’s transmission error. The coupling effect of lateral and torsional motions is considered in the dynamic analysis of the geared rotor-bearing system. The finite element method is used to model the system and Lagrangian approach is applied to derive the system equations of motion. The dynamic characteristics including system natural frequencies, mode shapes and steady-state response are investigated. The results show that the magnitude of the residual shaft bow, the phase angle between gear eccentricity and residual shaft bow will significantly affect system natural frequencies and steady-state response. When the spin speed closes to the second critical speed, the system steady state response will be dramatically increased by the residual shaft bow for the in-phase case. Moreover the zero response can be obtained when the system is set on special conditions.

Dynamic Characteristics of Resonant Beams With Passive Thermal Stress Regulators

2018 ◽  
Author(s):  
Tyler Kellar ◽  
Pezhman Hassanpour
Keyword(s):  
Boundary Condition ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Mode Shapes ◽  
Elastic Boundary ◽  
Special Cases ◽  
Angular Displacements

This paper addresses the dynamic characteristics of a beam with a particular elastic boundary condition. In this elastic boundary condition, the lateral and angular displacements of the beam are coupled through the elastic constraints. The dynamic characteristic, namely natural frequencies and mode shapes of vibrations are frequently encountered in the design and modeling of resonant micro-structures. The governing equations of motion of the beam is derived using Euler-Bernoulli beam theory considering the elastic coupling between the transverse and rotational displacements of the beam’s end. The characteristic equation for the natural frequencies and mode shapes of vibration is derived by applying the method of separation of variables to the governing partial differential equation of motion. The natural frequencies and mode shapes of the system are derived for various combinations of compliance values of the elastic support and are compared with those of several special cases, namely clamped-free, clamped-guided, clamped-pinned and clamped-clamped beams.

Water Hammer Response of Flexible Piping by Component Synthesis

1991 ◽  
Vol 113 (1) ◽  
pp. 115-119 ◽  
Author(s):  
F. J. Hatfield ◽  
D. C. Wiggert
Keyword(s):  
Finite Element ◽  
Dynamic Pressure ◽  
Equations Of Motion ◽  
Computer Code ◽  
Water Hammer ◽  
Mode Shapes ◽  
Finite Element Program ◽  
Coupled Equations ◽  
Alternative Approach ◽  
Element Program

Water hammer pressure in piping is modified by the consequent motion of the piping. In general, accurate estimates of dynamic pressure and pipe displacement must account for this interaction. One approach is to formulate and solve the coupled equations of motion for the liquid and pipe structure. Implementation for practical pipe systems would require a computer code comparable in scope to a structural finite element program combined with a hydrodynamics program. This paper presents an alternative approach that utilizes any available finite element program to compute natural frequencies and mode shapes of the piping, and then uses those modes to modify a hydrodynamic analysis and to predict motion of the piping. An example analysis demonstrates application of the method to assess the consequence of removing a brace intended to restrain pipe motion caused by water hammer. Results are compared to those given by analyses that neglect the effect of pipe motion on pressure.

Study the Effect of Tapering on the Nonlinear Behavior of an Exponentially Varying Width Piezoelectric Energy Harvester

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
H. Salmani ◽  
G. H. Rahimi
Keyword(s):  
Energy Harvester ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
High Amplitude ◽  
Mode Shapes ◽  
Geometric Nonlinearities ◽  
Coupled Equations ◽  
Piezoelectric Energy

It has been shown that exponentially tapering the width of a vibration-based piezoelectric energy harvester will result in increasing electric power per mass in a specified frequency. In this paper, a nonlinear solution of an exponentially decreasing width piezoelectric energy harvester is presented. Piezoelectric, inertial, and geometric nonlinearities are included in the presented model, while the exponentially tapered piezoelectric beam's mass normalized mode shapes are utilized in Galerkin discretization. The developed nonlinear coupled equations of motion are solved using method of multiple scales (MMS), and the steady states results are verified by experiment in high amplitude excitation. Finally, the exponentially tapering parameter effect is studied, and it is concluded that the voltage per mass of the energy harvester is improved by tapering at high exciting acceleration amplitudes.

The effect of the neutral point position on the natural frequencies of lateral vibrations of a hanging beam

2010 ◽  
Vol 224 (4) ◽  
pp. 817-825
Author(s):  
R Kazemi ◽  
A Jafari ◽  
M Faraji Mahyari
Keyword(s):  
Axial Force ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Summation Method ◽  
Neutral Point ◽  
Mode Shapes ◽  
Critical Position ◽  
Point Position ◽  
Lateral Vibrations ◽  
Energy Equations

In this article, the effect of axial force variation by position, on the natural frequencies of lateral vibration of a hanging beam, is analysed for different boundary conditions. To this aim, the equations of motion are derived by writing the potential and kinetic energies of the beam, considering the non-linear coupling between the axial and lateral vibrations. These energy equations are discreted by the mode summation method. The effect of the neutral point position on the natural frequencies and the mode shapes of the linearized system are investigated. By increasing the end axial force, the beam's natural frequencies are reduced and, for a critical position of neutral point, the first natural frequency of the beam vanishes and the beam buckles.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

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