A Numerical Evaluation of the Quadratic Transfer Function for a Floating Structure

2019 ◽  
Author(s):  
Zhitian Xie ◽  
Yujie Liu ◽  
Jeffrey Falzarano
Keyword(s):  
Transfer Function ◽  
Frequency Domain ◽  
Transfer Functions ◽  
Second Order ◽  
Wave Frequency ◽  
Wave Forces ◽  
Floating Structure ◽  
Transfer Function Matrix ◽  
Wave Induced ◽  
Function Matrix

Abstract The second order force of a floating structure can be expressed in terms of a time independent quadratic transfer functions along with the incident wave elevation, through which it is possible to evaluate the second order wave exciting forces in the frequency domain. Newman’s approximation has been widely applied in approximating the elements of the quadratic transfer function matrix while numerically evaluating the second order wave induced force. Through Newman’s approximation, the off-diagonal elements can be numerically approximated with the diagonal elements and thus the numerical calculation efficiency can be enhanced. Newman’s approximation assumes that the off-diagonal elements do not change significantly with the wave frequency and that hydrodynamic phenomenon regarding the low difference frequency are usually of interest. However, it is obviously less satisfying when an element that is close to the diagonal line in the quadratic transfer function matrix shows an extremum if the corresponding wave frequency is close to the natural frequency of the certain motion. In this paper, the full derivation and expression of the second order wave forces and moments applied to a floating structure have been presented, through which the numerical results of the quadratic transfer function matrix including the diagonal and the off-diagonal elements will be illustrated. This work will present the basis of numerically evaluating the second order forces in the frequency domain. The comparisons among various approximations regarding the second order forces in deep water will also be presented as a meaningful reference.

Error analyses of a Multi-Input-Multi-Output cantilever beam test

2021 ◽  
pp. 107754632110337
Author(s):  
Arup Maji ◽  
Fernando Moreu ◽  
James Woodall ◽  
Maimuna Hossain
Keyword(s):  
Transfer Function ◽  
Frequency Domain ◽  
Cantilever Beam ◽  
Transfer Functions ◽  
Linear Superposition ◽  
Transfer Function Matrix ◽  
Error Analyses ◽  
Pseudo Inverse ◽  
Function Matrix

Multi-Input-Multi-Output vibration testing typically requires the determination of inputs to achieve desired response at multiple locations. First, the responses due to each input are quantified in terms of complex transfer functions in the frequency domain. In this study, two Inputs and five Responses were used leading to a 5 × 2 transfer function matrix. Inputs corresponding to the desired Responses are then computed by inversion of the rectangular matrix using Pseudo-Inverse techniques that involve least-squared solutions. It is important to understand and quantify the various sources of errors in this process toward improved implementation of Multi-Input-Multi-Output testing. In this article, tests on a cantilever beam with two actuators (input controlled smart shakers) were used as Inputs while acceleration Responses were measured at five locations including the two input locations. Variation among tests was quantified including its impact on transfer functions across the relevant frequency domain. Accuracy of linear superposition of the influence of two actuators was quantified to investigate the influence of relative phase information. Finally, the accuracy of the Multi-Input-Multi-Output inversion process was investigated while varying the number of Responses from 2 (square transfer function matrix) to 5 (full-rectangular transfer function matrix). Results were examined in the context of the resonances and anti-resonances of the system as well as the ability of the actuators to provide actuation energy across the domain. Improved understanding of the sources of uncertainty from this study can be used for more complex Multi-Input-Multi-Output experiments.

scholarly journals An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger

Energies ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 4174
Author(s):  
Krzysztof Bartecki
Keyword(s):  
Heat Exchanger ◽  
Transfer Function ◽  
Differential Equations ◽  
Transfer Functions ◽  
Counter Flow ◽  
Transfer Function Matrix ◽  
Rational Transfer ◽  
Flow Heat ◽  
Double Pipe ◽  
Function Matrix

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

Frequency-Domain Calculations of Moored Vessel Motion Including Low Frequency Effect

2008 ◽  
Author(s):  
C. Le Cunff ◽  
Sam Ryu ◽  
Jean-Michel Heurtier ◽  
Arun S. Duggal
Keyword(s):  
Frequency Domain ◽  
Drag Force ◽  
Low Frequency ◽  
Domain Analysis ◽  
Frequency Domain Analysis ◽  
Second Order ◽  
Wave Frequency ◽  
Diffraction Radiation ◽  
Floating Structure ◽  
Low Frequency Motion

Frequency-domain analysis can be used to evaluate the motions of the FPSO with its mooring and riser. The main assumption of the frequency-domain analysis is that the coupling is essentially linear. Calculations are performed taking into account first order wave loads on the floating structure. Added mass and radiation damping terms are frequency dependent, and can be easily considered in this formulation. The major non-linearity comes from the drag force both on lines and the floating structure. Linearization of the non-linear drag force acting on the lines is applied. The calculations can be extended to derive the low frequency motion of the floating structure. Second order low frequency quadratic transfer function is computed with a diffraction/radiation method. Given a wave spectrum, the second order force spectrum can then be derived. At the same time frequency-domain analysis is used to derive the low frequency motion and wave frequency motion of the floating system. As an example case, an FPSO is employed. Comparison is performed with time domain simulation to show the robustness of the frequency-domain analysis. Some calculations are also performed with either low frequency terms only or wave frequency terms only in order to check the effect of modeling low and wave frequency terms, separately. In the case study it is found that the low frequency motion is reduced by the wave frequency motion while the wave frequency motion is not affected by the low frequency motion.

Internal Fluid Effect Inside a Floating Structure: From Frequency Domain Solution to Time Domain Solution

2017 ◽  
Author(s):  
Jingzhe Jin ◽  
Allan Ross Magee ◽  
Mengmeng Han ◽  
Jan Roger Hoff ◽  
Elin Marita Hermundstad ◽  
...  
Keyword(s):  
Transfer Function ◽  
Frequency Domain ◽  
Time Domain ◽  
Transfer Functions ◽  
Wave Excitation ◽  
Floating Structure ◽  
Floating Structures ◽  
Force Transfer ◽  
Hydrodynamic Response ◽  
Excitation Force

Liquid inside a floating structure influences both hydrostatic and hydrodynamic response of the floating structure. Some examples of floating structure with internal liquid are: vessels with roll damping tanks, floating hydrocarbon storage facility, LNG tankers and etc. A floating oil storage tank is considered in this study, which has a cuboid-shaped external wall, cylinder-shaped internal wall and simple structure configurations for its roof and bottom. Influence of internal liquid on the hydrostatic response of floating structures is well established and must be taken into consideration. The internal liquid reduces the stability of floating structures. The focus of this study is the influence of internal liquid on the hydrodynamic response of floating tank. Frequency domain analysis is performed with WAMIT, for partially filled tank with both solid mass and liquid mass. By comparing the different cases, the force induced by the internal liquid on the floating tank is illustrated. Based on the WAMIT calculated radiation damping force for the external flow, Impulse Response Function (IRF) connecting frequency domain and time domain solution is constructed and the force and moment induced by internal liquid is considered as an excitation force. By assuming linear tank motion, the internal liquid induced force is related to the incoming wave by a set of force transfer functions and it is moved to the right hand side of the tank motion equation. In practice, this set of force transfer function due to internal liquid has to be combined with the wave excitation force transfer function and it is the total force transfer function (internal liquid plus wave excitation) to be imported to SIMO. SIMO time domain simulation is performed in regular waves. Motion transfer functions from WAMIT frequency domain and SIMO time domain calculations are compared and reasonable agreement is achieved.

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