Nonlinear Span Assessment by Amplitude-Dependent Linearization

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
Ralf Peek
Keyword(s):  
Eigenvalue Problem ◽  
Current Practice ◽  
Nonlinear Eigenvalue Problem ◽  
Damping Ratio ◽  
Mode Shapes ◽  
Inelastic Behavior ◽  
Modal Damping Ratio ◽  
Linear Behavior ◽  
Modal Damping ◽  
Vortex Induced Vibrations

Abstract Although it has long been recognized that vortex-induced vibrations of subsea pipeline spans involve nonlinear and inelastic behavior, the current practice to assess such spans for fatigue and ultimate loading conditions is based on the modal analysis assuming linear behavior. Nevertheless, nonlinearity can be captured approximately by making the linearization amplitude dependent. The eigenvalue problem to be solved for the natural frequencies and mode shapes then involves a stiffness matrix that depends on the mode shape and amplitude of vibration. An important part of the nonlinearity comes from the soil, which is generally represented by springs. This paper presents a simple and particularly effective algorithm to solve this nonlinear eigenvalue problem by using the same algorithm that serves to track the bifurcated solution branches in quasi-static structural stability (buckling) analyses. This method is applied to an example in which the nonlinearity comes from the soil springs. The results demonstrate the importance of the nonlinearity, even at relatively low vortex-induced vibrations (VIV) amplitudes typical of the pure inline response. The inelasticity of the soil springs is also used to calculate the associated contribution to the modal damping ratio.

scholarly journals Dynamics of a Pipeline Span on an Elastic Seabed

2021 ◽  
Author(s):  
Ralf Peek ◽  
Matt Witz ◽  
Knut Vedeld
Keyword(s):  
Axial Force ◽  
Exact Analytical Solution ◽  
Damping Ratio ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Element Analysis ◽  
Modal Damping Ratio ◽  
Modal Damping ◽  
Vortex Induced Vibrations ◽  
Arch Action

Natural frequencies, mode shapes and modal damping ratios must be estimated to assess subsea pipeline spans for vortex-induced vibrations (VIV) and response to direct wave loading. Several approximate solutions exist for a linearly elastic pipe under constant axial force supported by linearly elastic springs beyond the span’s shoulders. An exact analytical solution has only recently been published. That solution is used here in a Rayleigh-Ritz approximation to account for arch action arising from combined effects of sag under gravity loads and axial restraint at the shoulders. The method allows survey data to be used directly to quantify arch action. Its accuracy is confirmed by finite element analysis. Further, the modal damping ratio is estimated based on the fractions of the potential energy in bending, the axial force, and the soil springs, all of which are determined analytically. Thus, it is found that the effective modal damping ratio increases without a bound as the axial load approaches the buckling load in compression.

ANALYTICAL RANDOM VIBRATION ANALYSIS OF BOUNDARY-EXCITED THIN RECTANGULAR PLATES

2013 ◽  
Vol 13 (03) ◽  
pp. 1250062 ◽  
Author(s):  
ASHKAN HAJI HOSSEINLOO ◽  
FOOK FAH YAP ◽  
NADER VAHDATI
Keyword(s):  
Random Vibration ◽  
Printed Circuit Boards ◽  
Damping Ratio ◽  
Excitation Frequency ◽  
Jet Impingement ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Frequency Range ◽  
Modal Damping Ratio ◽  
Modal Damping

Fatigue life, stability and performance of majority of the structures and systems depend significantly on dynamic loadings applied on them. In many engineering cases, the dynamic loading is random vibration and the structure is a plate-like system. Examples could be printed circuit boards or jet impingement cooling systems subjected to random vibrations in harsh military environments. In this study, the response of thin rectangular plates to random boundary excitation is analytically formulated and analyzed. In the presented method, closed-form mode shapes are used and some of the assumptions in previous studies are eliminated; hence it is simpler and reduces the computational load. In addition, the effects of different boundary conditions, modal damping and excitation frequency range on dynamic random response of the system are studied. The results show that increasing both the modal damping ratio and the excitation frequency range will decrease the root mean square acceleration and the maximum deflection of the plate.

scholarly journals Dynamic behaviour of a gymnasium floor

1986 ◽  
Vol 13 (3) ◽  
pp. 270-277 ◽  
Author(s):  
J. H. Rainer ◽  
J. C. Swallow
Keyword(s):  
Concrete Slab ◽  
Damping Ratio ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Modal Damping Ratio ◽  
Modal Damping ◽  
Load Factors ◽  
Resonance Frequencies ◽  
Floor Vibrations ◽  
Vibration Tests

Ten mode shapes, natural frequencies, and modal damping values have been measured for a steel-joist concrete-slab floor spanning 32.1 m. From ambient vibrations and steady-state shaker tests the frequency of the fundamental mode was determined to be 3.5 Hz, and the modal damping ratio to be approximately 1% of critical. A comparison of vibration criteria in Appendix G of CAN3-S16.1-M84 confirms satisfactory performance for walking, but for other rhythmic exercises disturbing vibrations developed. These occurred primarily at the forcing frequency of the exercises and not at floor resonance frequencies. Values of dynamic load factors, α, for rhythmic loadings of this floor were evaluated in accordance with the guidelines on floor vibrations in the Commentary to the National Building Code of Canada 1985. Key words: floors, gymnasiums, vibration tests, resonant frequencies, mode shapes, dynamic loads, dynamic response.

Effect of Damping on VIV Response in Umbilicals and Flexible Risers

2015 ◽  
Author(s):  
Kay Hansen-Zahl ◽  
Veronica Henøen ◽  
Rolf Baarholm ◽  
Farzan Parsinejad ◽  
Yiannis Constantinides
Keyword(s):  
Fatigue Life ◽  
Internal Pressure ◽  
Contact Force ◽  
Damping Ratio ◽  
Life Assessment ◽  
Stick Slip ◽  
Modal Damping Ratio ◽  
Modal Damping ◽  
Vortex Induced Vibrations ◽  
Flexible Risers

The fatigue life assessment of umbilicals and flexible risers due to vortex-induced vibrations (VIV) require special attention to the structural properties of the flexible bundle due to stick/slip behavior of helical elements in bending. The VIV response on umbilicals and flexible risers is complex, and it is controlled by several parameters. Structural damping is one of these parameters, but this is typically calculated based on the material damping of the structure. Similar to flexible risers and umbilicals, hysteresis damping due to the stick/slip behavior in the helical layers may be significant for cable structures. The purpose of this study is to evaluate the effect of the hysteresis damping on VIV response, and to demonstrate a procedure for consistent VIV fatigue analysis of flexible risers and umbilicals. The stick/slip hysteresis varies with varying contact force, and the contact force is dependent on internal pressure and effective tension. The effect of varying these parameters is included in the study. Furthermore, the study includes different riser system configurations and current profiles. In this paper, the stick/slip hysteresis has been established for two umbilicals and two flexible risers, with varying internal pressure and effective tension. Combined with five different configurations (both shallow and deep water) and 16 different current profiles, the VIV modes have been calculated and the corresponding stick/slip modal damping ratio has been established. Finally, using fatigue current profiles, fatigue life estimates have been made based on different damping ratios, including the estimated stick/slip damping ratio. The study presented in this paper has been carried out as part of the Norwegian Deepwater Programme (NDP).

Experimental Study the Vibro-Acoustic Performance of a Double Glazed Window

2012 ◽  
Vol 503-504 ◽  
pp. 1129-1132
Author(s):  
Qi Bo Mao
Keyword(s):  
Damping Ratio ◽  
Normal Wave ◽  
Sound Transmission ◽  
Least Square ◽  
Structural Vibration ◽  
Mode Shapes ◽  
Reverberation Chamber ◽  
Modal Damping Ratio ◽  
Modal Damping ◽  
Complex Exponential

This paper presents an experimental investigation of the sound transmission and structural vibration characteristics of the double glazed window. The laboratory experiments were performed placing the window between reverberation chamber and anechoic chamber. The window was subject to diffuse field, approximate normal wave and oblique wave acoustic excitations. The sound transmission performances at far-field were measured. Furthermore, experimental modal analysis has been performed. The Least square complex exponential algorithm is used to extract the modal parameters, i.e. mode shapes, natural frequencies and modal damping ratio of the structure. The results also show that the highest sound transmission of this experimental double glazed window appears around the mass-air-mass resonance frequency.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

scholarly journals Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Axial Force ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Compressive Force ◽  
Mode Shapes ◽  
Dynamic Finite Element ◽  
Starting Point

Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.

Modal damping ratio analysis of dynamical system with non-stationary responses

2016 ◽  
Vol 59 ◽  
pp. 138-146 ◽  
Author(s):  
Da Tang ◽  
Ran Ju ◽  
Qianjin Yue ◽  
Shisheng Wang
Keyword(s):  
Dynamical System ◽  
Damping Ratio ◽  
Ratio Analysis ◽  
Modal Damping Ratio ◽  
Modal Damping

scholarly journals Coupling of Flexural and Longitudinal Damped Vibration in a Two-Layered Beam

1998 ◽  
Vol 5 (5-6) ◽  
pp. 337-341
Author(s):  
F. Pourroy ◽  
S. Shakhesi ◽  
P. Trompette
Keyword(s):  
Damping Ratio ◽  
Modal Damping Ratio ◽  
Modal Damping ◽  
Resonance Frequencies ◽  
Layered Beam ◽  
Flexural Vibrations

In dynamics, the effect of varying the constitutive materials’ thickness of a two-layered beam is investigated. Resonance frequencies and damping variations are determined. It is shown that for specific thicknesses the coupling of longitudinal and flexural vibrations influences the global modal damping ratio significantly.

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