Improving the convergence rate of a transient substructuring iterative method using the rigid body modes of its static equivalent

1995 ◽  
Author(s):  
C Farhat ◽  
F Hemez ◽  
J Mandel
Keyword(s):  
Convergence Rate ◽  
Iterative Method ◽  
Rigid Body ◽  
Rigid Body Modes

Generalized Modes in Time-Domain Seakeeping Calculations

2012 ◽  
Vol 56 (04) ◽  
pp. 215-233
Author(s):  
Johan T. Tuitman ◽  
Šime Malenica ◽  
Riaan van't Veer
Keyword(s):  
Rigid Body ◽  
Transient Response ◽  
Time Domain ◽  
Large Amplitude ◽  
Degrees Of Freedom ◽  
Mode Shapes ◽  
Nonlinear Load ◽  
Large Amplitude Motions ◽  
Rigid Body Modes ◽  
The Time Domain

The concept of "generalized modes" is to describe all degrees of freedom by mode shapes and not using any predefined shape, like rigid body modes. Generalized modes in seakeeping computations allow one to calculate the response of a single ship, springing, whipping, multibody interaction, etc., using a uniform approach. The generalized modes have already been used for frequency-domain seakeeping calculations by various authors. This article extents the generalized modes methodology to be used for time-domain seakeeping computations, which accounts for large-amplitude motions of the rigid-body modes. The time domain can be desirable for seakeeping computations because it is easy to include nonlinear load components and to compute transient response, like slamming and whipping. Results of multibody interaction, two barges connected by a hinge, whipping response of a ferry resulting from slamming loads, and the response of a flexible barge are presented to illustrate the theory.

Receptance Series for Systems Possessing “Rigid Body” Modes

1962 ◽  
Vol 66 (618) ◽  
pp. 394-397 ◽  
Author(s):  
G. M. L. Gladwell ◽  
R. E. D. Bishop ◽  
D. C. Johnson
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Series Representation ◽  
Forced Vibrations ◽  
Rigid Bodies ◽  
Points Of Interest ◽  
Elastic Systems ◽  
Rigid Body Modes ◽  
Simple Systems

SummaryCertain elastic systems may not only vibrate freely at proper (non-zero) natural frequencies, but may also move as rigid bodies. Such systems have “rigid body” modes which behave like principal modes corresponding to zero natural frequencies. These modes may be disregarded in the series representation of static distortions of such systems but must be taken into account in the representation of forced vibrations.This note is concerned with the series representation of receptances of certain simple systems of this type, namely strings, bars, shafts and beams. These systems were discussed in reference 1, but there the rigid body modes were omitted. As the matter appears to raise some points of interest, a discussion of it seems to be called for. A similar analysis to that presented here may be applied to other unsupported, or partially supported systems, such as an unsupported plate.

Validation of Measured Dynamic Data Using Rigid Body Response

2012 ◽  
Vol 55 (1) ◽  
pp. 25-39
Author(s):  
David Smallwood
Keyword(s):  
Rigid Body ◽  
Low Frequency ◽  
Linear Combinations ◽  
Resonant Frequencies ◽  
Low Frequencies ◽  
Spectral Density Matrix ◽  
Rigid Body Modes ◽  
Linear Fit ◽  
Value Decomposition ◽  
Body Response

As multiple axis vibration testing has become more widespread, it has become increasingly important to ensure the instrumentation is accurately portrayed in the instrumentation table. However, errors do occur. The method used in this paper to help uncover these errors is based on the condition that at low frequencies (below any resonant frequencies of the object being studied) the response is essentially rigid body. The spectral density matrix (SDM) at a low frequency, of many more than six response measurements, is decomposed using singular value decomposition (SVD). Under the assumption of rigid body response, it is assumed that the first six singular vectors are linear combinations of the six rigid body modes. The best linear fit is then calculated for this fit. The measurements are then removed one at a time, and the reduction in the fit error is calculated. It is assumed that if the removal of a measurement reduces the error significantly, that measurement is likely in error.

In-Plane Inertial Coupling in Tuned and Severely Mistuned Bladed Disks

1982 ◽  
Author(s):  
E. F. Crawley
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Analytic Model ◽  
Angular Rotation ◽  
Structural Coupling ◽  
Nodal Diameter ◽  
Rigid Disk ◽  
Compressor Rotor ◽  
Rigid Body Modes ◽  
Bending Modes

A model has been developed and verified for blade-disk-shaft coupling in rotors due to the in-plane rigid body modes of the disk. An analytic model has been developed which couples the in-plane rigid body modes of the disk on an elastic shaft with the blade bending modes. Bench resonance tests were carried out on the M.I.T. Compressor Rotor, typical of research rotors with flexible blades and a thick rigid disk. When the rotor was carefully tuned, the structural coupling of the blades by the disks was confined to zero and one nodal diameter modes, whose modal frequencies were greater than the blade cantilever frequency. In the case of the tuned rotor, and in two cases where severe mistuning was intentionally introduced, agreement between the predicted and observed natural frequencies is excellent. The analytic model was then extended to include the effects of constant angular rotation of the disk.

scholarly journals Iterative method for solving the second boundary value problem (BVP) for Biharmonic type equation

2016 ◽  
Vol 14 (4) ◽  
pp. 66-72
Author(s):  
Đặng Quang Á
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Type Equation ◽  
Second Order ◽  
Second Order Equations

Solving BVPs for the fourth order differential equations by the reduction of them to BVPs for the  second order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by  ourselves in recent works, we construct iterative method for the second BVP for  biharmonic type equation. The convergence rate of  the method is established.

Modeling and analytical methods for a mounting system with double stage isolation

2021 ◽  
pp. 095440702110632
Author(s):  
Yawei Zheng ◽  
Wen-Bin Shangguan ◽  
Yingzi Kang
Keyword(s):  
Free Vibration ◽  
Rigid Body ◽  
Motion Control ◽  
Three Dimensional ◽  
Vibration Equation ◽  
Proposed Model ◽  
Solution Methods ◽  
Rigid Body Modes ◽  
Simulation Results ◽  
Solutions Of Equations

A calculation method for obtaining the displacements and rigid body modes of a Powertrain Mounting System (PMS) with double stage isolation is proposed in this paper. Firstly, the PMS with double stage isolation is modeled as a 12 Degree of Freedoms (DOFs) model, which includes six DOFs for the powertrain and the subframe respectively. The mounts are simplified as a three-dimensional spring along each axis of its Local Mount Coordinate System (LMCS), which takes the non-linear relation of the force versus the displacement of each spring into account. Secondly, the quasi-static equilibrium equation and the free vibration equation as well as the forced vibration equation of the proposed model are derived and the solutions of equations are presented. Then, the calculation and solution methods are validated by the simulation results. The differences of rigid body modes and displacements of the powertrain between single and double stage isolation are estimated, which demonstrates that the proposed model is more accurate, especially when powertrain mounts are stiff. Also, the effect of locations for powertrain mounts on car body is investigated, which shows that is beneficial for motion control of powertrain.

Use of experimental dynamic substructuring to predict the low frequency structural dynamics under different boundary conditions

2017 ◽  
Vol 23 (11) ◽  
pp. 1444-1455
Author(s):  
Walter D’Ambrogio ◽  
Annalisa Fregolent
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Frequency Response ◽  
Response Function ◽  
Frequency Response Function ◽  
Low Frequency ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Different Boundary Conditions ◽  
Rigid Body Modes

Flexible structural components can be attached to the rest of the structure using different types of joints. For instance, this is the case of solar panels or array antennas for space applications that are joined to the body of the satellite. To predict the dynamic behaviour of such structures under different boundary conditions, such as additional constraints or appended structures, it is possible to start from the frequency response functions in free-free conditions. In this situation, any structure exhibits rigid body modes at zero frequency. To experimentally simulate free-free boundary conditions, flexible supports such as soft springs are typically used: with such arrangement, rigid body modes occur at low non-zero frequencies. Since a flexible structure exhibits the first flexible modes at very low frequencies, rigid body modes and flexible modes become coupled: therefore, experimental frequency response function measurements provide incorrect information about the low frequency dynamics of the free-free structure. To overcome this problem, substructure decoupling can be used, that allows us to identify the dynamics of a substructure (i.e. the free-free structure) after measuring the frequency response functions on the complete structure (i.e. the structure plus the supports) and from a dynamic model of the residual substructure (i.e. the supporting structure). Subsequently, the effect of additional boundary conditions can be predicted using a frequency response function condensation technique. The procedure is tested on a reduced scale model of a space solar panel.

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