Dynamic Analysis of a Rotating Planar Timoshenko Beam Using an Accurate Global Spatial Discretization Method

2019 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Zhu
Keyword(s):  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Slope Angle ◽  
Dynamic Responses ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Dependent Variables ◽  
Spatial Derivatives ◽  
Shear Strains

Abstract A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Dynamic responses of an engine front-end accessory belt drive system with pulley eccentricities via two spatial discretization methods

2017 ◽  
Vol 232 (4) ◽  
pp. 482-498 ◽  
Author(s):  
Hao Zhu ◽  
Yumei Hu ◽  
Weidong Zhu ◽  
Haiqiang Long
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bending Stiffness ◽  
Drive System ◽  
Dynamic Responses ◽  
Belt Drive ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Galerkin Discretization ◽  
Front End

In this study, a generic mathematical model for calculating the natural frequencies and the dynamic responses of a typical front-end accessory drive system with any number of pulleys and arbitrary configurations of the tensioner and pulleys is established. The belt bending stiffness and the pulley eccentricities are considered in this model, and their influences on the natural frequency and the dynamic responses of the front-end accessory drive system are examined. A generic spatial discretization method and a Galerkin discretization method, which uses Lagrange multipliers to enhance the boundary conditions, are presented to discretize the continuous belt spans and to transform the governing partial differential equations into ordinary differential equations. The accuracies of the generic spatial discretization method and the Galerkin discretization method are validated by modal tests, and the advantages of the generic spatial discretization method with respect to the efficiency and the convenience of implementation are assessed by comparing the generic spatial discretization method with the Galerkin discretization method and the two-layer iteration approach. The dynamic responses of the typical front-end accessory drive system at different operational velocities are calculated from the governing ordinary differential equations derived from these two methods. It is shown that large vibration amplitudes occur in certain belt spans owing to the resonance conditions or the beat phenomena in certain operational conditions and that the belt bending stiffness has a negligible influence on the vibrations of the belt drive system because its value is small.

An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part I: Methodology

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
W. D. Zhu ◽  
H. Ren
Keyword(s):  
Differential Algebraic Equations ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Substructure Method ◽  
Axial Forces ◽  
Geometric Matching ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to accurately calculate dynamic responses of one-dimensional structural systems, which consist of length-variant distributed-parameter components, such as strings, rods, and beams, and lumped-parameter components, such as point masses and rigid bodies. The dependent variable of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge if the dependent variables are smooth enough. Spatial derivatives of the dependent variables, which are related to internal forces/moments of the distributed-parameter components, such as axial forces, bending moments, and shear forces, can be accurately calculated. Combining component equations that are derived from Lagrange's equations and geometric matching conditions that arise from continuity relations leads to a system of differential algebraic equations (DAEs). When the geometric matching conditions are linear, the DAEs can be transformed to a system of ordinary differential equations (ODEs), which can be solved by an ODE solver. The methodology is applied to several moving elevator cable-car systems in Part II of this work.

A New Global Spatial Discretization Method for Two-Dimensional Continuous Systems

2017 ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Bending Moment ◽  
Continuous System ◽  
Dynamic Responses ◽  
Continuous Systems ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
One Dimensional ◽  
Dimensional Boundary

A new global spatial discretization method is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a two-dimensional continuous system is separated into a two-dimensional internal term and a two-dimensional boundary-induced term; the latter is interpolated from one-dimensional boundary functions that are further divided into one-dimensional internal terms and one-dimensional boundary-induced terms. The two- and one-dimensional internal terms are chosen to satisfy predetermined boundary conditions, and the two- and one-dimensional boundary-induced terms use additional degrees of freedom at boundaries to ensure satisfaction of all boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a two-dimensional continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply-supported boundaries and one free boundary with an attached Euler-Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Natural frequencies and dynamic responses that include the displacement, the velocity, rotational angles, a bending moment, and a transverse shearing force are calculated using both the developed method and the assumed modes method, and compared with results from the finite element method and the finite difference method, respectively. Advantages of the new method over local spatial discretization methods are much fewer degrees of freedom and much less computational effort, and those over the assumed modes method are better numerical property, a faster calculation speed, and much higher accuracy in calculation of the bending moment and the transverse shearing force that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Dynamic analysis of a helical gear reduction by experimental and numerical methods

2020 ◽  
Vol 68 (1) ◽  
pp. 48-58
Author(s):  
Chao Liu ◽  
Zongde Fang ◽  
Fang Guo ◽  
Long Xiang ◽  
Yabin Guan ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Numerical Models ◽  
Helical Gear ◽  
Operating Conditions ◽  
Dynamic Responses ◽  
Lumped Mass ◽  
Gear Mesh

Presented in this study is investigation of dynamic behavior of a helical gear reduction by experimental and numerical methods. A closed-loop test rig is designed to measure vibrations of the example system, and the basic principle as well as relevant signal processing method is introduced. A hybrid user-defined element model is established to predict relative vibration acceleration at the gear mesh in a direction normal to contact surfaces. The other two numerical models are also constructed by lumped mass method and contact FEM to compare with the previous model in terms of dynamic responses of the system. First, the experiment data demonstrate that the loaded transmission error calculated by LTCA method is generally acceptable and that the assumption ignoring the tooth backlash is valid under the conditions of large loads. Second, under the common operating conditions, the system vibrations obtained by the experimental and numerical methods primarily occur at the first fourth-order meshing frequencies and that the maximum vibration amplitude, for each method, appears on the fourth-order meshing frequency. Moreover, root-mean-square (RMS) value of the acceleration increases with the increasing loads. Finally, according to the comparison of the simulation results, the variation tendencies of the RMS value along with input rotational speed agree well and that the frequencies where the resonances occur keep coincident generally. With summaries of merit and demerit, application of each numerical method is suggested for dynamic analysis of cylindrical gear system, which aids designers for desirable dynamic behavior of the system and better solutions to engineering problems.

Exact Dynamic Analysis of Space Structures Using Timoshenko Beam Theory

AIAA Journal ◽  
2004 ◽  
Vol 42 (4) ◽  
pp. 833-839 ◽  
Author(s):  
Jen-Fang Yu ◽  
Hsin-Chung Lien ◽  
B. P. Wang
Keyword(s):  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Space Structures
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