The Discretization Methods of a Rotating Flexible Cantilever Beam

2012 ◽  
Vol 226-228 ◽  
pp. 697-707
Author(s):  
Ji Hua Fan ◽  
Ding Guo Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cantilever Beam ◽  
Interpolation Method ◽  
Flexible Beam ◽  
Transverse Deformation ◽  
Longitudinal Deformation ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The dynamics of a rotating flexible cantilever beam is investigated by using assumed mode method, finite element method and Bezier curve interpolation method in this paper. The longitudinal deformation and the transverse deformation of the flexible beam are considered and the coupling term of the deformation which is caused by transverse deformation is included in the expression of longitudinal deformation. The assumed mode, finite element and Bezier interpolation are used to discretize the deformation of the flexible beam, and then the dynamics equations are built by Lagrange equation, and a software package for the dynamic simulation of the rotating cantilever beam is developed. Two cases are considered in the simulations. One is the dynamics study which the external rotating torque is known, another one is that a rotating fixed axis flexible beam falls with gravity. According to the simulation results, assumed mode, finite element and Bezier interpolation method can be well used for discretizing the deformation of the flexible beam; the computational efficiency of finite element method is the lowest, and Bezier interpolation method is the highest; the calculation accuracy of the assumed mode method is lower than the Bezier interpolation method.

Squeal Analysis of a Disc Brake Considering Damping between a Disc and a Pad Lining

2012 ◽  
Vol 157-158 ◽  
pp. 1000-1003
Author(s):  
Ke Wei Zhou ◽  
Cheol Kim ◽  
Min Ok Yun ◽  
Ju Young Kim
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Element Model ◽  
Disc Brake ◽  
Vibration Modes ◽  
Assumed Mode Method ◽  
Frictional Damping ◽  
System Components ◽  
Mode Method ◽  
Assumed Mode

The improved equations of motion for a friction-engaged brake system have been newly derived on the basis of the assumed mode method and frictional damping. The equations of motion with a finite element model were constructed by a set of vibration modes found from FE modal analysis on all system components. Consequently, the modal information of system components are combined with equations of motion derived from the analytical model. Numerical analysis showed the mode which was unstable in an undamped case became stable in a damped case.

Complex Modal Analysis of a Non-Modally Damped Continuous Beam

2013 ◽  
Author(s):  
Xing Xing ◽  
Brian F. Feeny
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Separation Of Variables ◽  
Mode Shapes ◽  
Modal Assurance Criterion ◽  
Assumed Mode Method ◽  
State Variable ◽  
Mode Method ◽  
Cantilevered Beam ◽  
Assumed Mode

This work represents an investigation of the complex modes of continuous vibration systems with nonmodal damping. As an example, a cantilevered beam with damping at the free end is studied. Traditional separation of variables for this problem leads to a differential eigenvalue problem which requires a numerical solution. In this paper, assumed modes are applied to discretize the eigenvalue problem in state-variable form, to then obtain estimates of the frequencies and modes. The finite-element method (FEM) is also utilized to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenproblem. A comparison between the assumed-mode and finite-element eigenvalues and modal vectors shows that the methods produce consistent results. The comparison of the modes was done visually and also by using the modal assurance criterion (MAC) on the modal vectors. The assumed-mode method is then used to study the effects of the damping coefficient on mode shapes and modal damping.

Simplified dynamic analysis of stepped thickness rectangular plate structures by the assumed mode method

2016 ◽  
Vol 231 (1) ◽  
pp. 177-187
Author(s):  
Dae-Seung Cho ◽  
Byung Hee Kim ◽  
Jin-Hyeong Kim ◽  
Nikola Vladimir ◽  
Tae-Muk Choi
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Rectangular Plate ◽  
Forced Response ◽  
Plate Structures ◽  
Assumed Mode Method ◽  
Frequency Responses ◽  
Mode Method ◽  
Assumed Mode

In this article, the assumed mode method is applied to simplified dynamic analysis of stepped thickness rectangular Mindlin plates and stiffened panels with arbitrary boundary conditions. The natural and frequency responses of stepped thickness plate structures subjected to harmonic point excitation force and enforced acceleration at boundaries, respectively, are considered. Potential and kinetic energies of the system are formulated and used to derive eigenvalue problem utilizing Lagrange’s equation of motion, and mode superposition method is further used for forced response assessment. Characteristic orthogonal polynomials having the property of Timoshenko beam functions are used for the assumed modes. Numerical examples analysing vibration of stepped thickness plate structures with different topologies and various sets of boundary conditions are provided. Numerical results are compared with the results from the relevant literature and finite element solutions obtained by a general finite element tool, and a very good agreement is achieved. Hence, it is expected that stepped rectangular plate structures satisfying the prescribed criteria regarding natural and frequency responses can be efficiently designed based on the proposed method.

scholarly journals DYNAMIC ANALYSES OF A FLEXIBLE VEHICLE MOVING ALONG A FLEXIBLE GUIDEWAY CONSIDERING THE TIP-OFF EFFECT

2013 ◽  
Vol 13 (01) ◽  
pp. 1350009 ◽  
Author(s):  
S. N. CHOU ◽  
F. P. CHENG ◽  
C. S. HUANG
Keyword(s):  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Flexible Beam ◽  
Differentiation Formula ◽  
Assumed Mode Method ◽  
Dynamic Analyses ◽  
Mode Method ◽  
Assumed Mode ◽  
Bdf Method ◽  
Free Beam

A semi-analytical solution for the tip-off response of a vehicle moving along a guideway is obtained, considering the dynamic interaction between the two subsystems. The guideway is modeled as an inclined simply-supported uniform flexible beam, and the vehicle as a flexible free-free beam under a pre-specified thrust force. The equations of motion for the vehicle and guideway are developed using the Lagrangian approach and the assumed mode method based on the Euler–Bernoulli hypothesis. In the form of nonlinear differential equations, they are solved by the Petzold-Gear backward differentiation formula (BDF) method. The solutions obtained are validated by comparing them with the published results for the models with a rigid vehicle running over a rigid guideway or a flexible guideway. Comparisons of the present solutions with the existing ones for the vehicle and guideway reveal the advantages of the approach proposed herein. Other effects on the tip-off responses of the vehicle that are investigated include the length of the guideway, distance between the shoes of the vehicle, and mass and rigidity ratios of the vehicle to the guideway. The results presented herein provide valuable information for the design of the vehicle launch system.

A Method to Improve the Modal Convergence for Structures With External Forcing

1987 ◽  
Vol 54 (4) ◽  
pp. 904-909 ◽  
Author(s):  
Keqin Gu ◽  
Benson H. Tongue
Keyword(s):  
Spatial Distribution ◽  
Free Vibration ◽  
Convergence Rate ◽  
Traditional Approach ◽  
Vibration Modes ◽  
External Forcing ◽  
Slow Convergence ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The traditional approach of using free vibration modes in the assumed mode method often leads to an extremely slow convergence rate, especially when discete interactive forces are involved. By introducing a number of forced modes, significant improvements can be achieved. These forced modes are intrinsic to the structure and the spatial distribution of forces. The motion of the structure can be described exactly by these forced modes and a few free vibration modes provided that certain conditions are satisfied. The forced modes can be viewed as an extension of static modes. The development of a forced mode formulation is outlined and a numerical example is presented.

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