Nonlinear Forced Response Research Articles

Multimodal Resonance Response of Incompressible Hyperelastic Moderately Thick Cylindrical Shells

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Nonlinear axial-lateral coupled vibration of functionally graded-fiber reinforced composite laminated (FG-FRCL) beams subjected to aero-thermal loads

This paper studies the nonlinear axial-lateral coupled vibration of functionally graded-fiber reinforced composite laminated (FG-FRCL) cantilever beams subjected to aero-thermal loads. The nonlinear partial differential equations governing the problem are determined based on the Euler-Bernoulli beam theory using the von Kármán-type geometrical nonlinearities. The Galerkin method is then applied to discretize the differential equations and to transform them into a nonlinear ordinary differential equation. The coupled ordinary differential equations are solved analytically using the method of multiple time scales (MTS) to study the nonlinear forced vibration time response of the selected FG-FRCL structures, where a large parametric study checks for the effect of power-index, uniform temperature rise, velocity of the free stream air and Mach number on their axial and lateral response, with interesting insights from a scientific and design purposes.

Synchronization phenomena observed in glacial–interglacial cycles simulated in an Earth system model of intermediate complexity

Abstract. The glacial–interglacial cycles of the Quaternary exhibit 41 kyr periodicity before the Mid-Pleistocene Transition (MPT) around 1.2–0.8 Myr ago and ∼ 100 kyr periodicity after that. From the viewpoint of dynamical systems, proposed mechanisms generating these periodicities are broadly divided into two types: (i) nonlinear forced responses of a mono- or multi-stable climate system to the astronomical forcing or (ii) synchronization of internal self-sustained oscillations to the astronomical forcing. In this study, we investigate the dynamics of glacial cycles simulated by the Earth system model of intermediate complexity CLIMBER-2 with a fully interactive carbon cycle, which reproduces the MPT under gradual changes in volcanic-CO2 degassing and regolith cover. We report that, in this model, the dominant frequency of glacial cycles is set in line with the principle of synchronization. It is found that the model exhibits self-sustained oscillations in the absence of astronomical forcing. Before the MPT, glacial cycles synchronize to the 41 kyr obliquity cycles because the self-sustained oscillations have periodicity relatively close to 41 kyr. After the MPT the timescale of internal oscillations becomes too long to follow every 41 kyr obliquity cycle, and the oscillations synchronize to the 100 kyr eccentricity cycles that modulate the amplitude of climatic precession. The latter synchronization occurs with the help of the 41 kyr obliquity forcing, which enables some terminations and glaciations to occur robustly at their right timing. We term this phenomenon vibration-enhanced synchronization because of its similarity to the noise-enhanced synchronization known in nonlinear science. While we interpret the dominant periodicities of glacial cycles as the result of synchronization of internal self-sustained oscillations to the astronomical forcing, the Quaternary glacial cycles show facets of both synchronization and forced response.

Nonlinear Forced Vibration of Functionally Graded Graphene-Reinforced Composite (FG-GRC) Laminated Cylindrical Shells under Different Boundary Conditions with Thermal Repercussions

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid

In this paper, nonlinear static and dynamic behaviors of slightly curved cantilevered pipes conveying fluid under transverse base excitations are investigated. Incorporating geometrical and inertial nonlinearities, a new three-dimensional dynamic model is developed for the inextensible cantilevered pipe based on the Euler–Bernoulli beam theory. The resultant partial differential equations are discretized using the Galerkin method and solved utilizing the pseudo-arclength continuation technique and a time-integration method. The static equilibrium configuration, linear stability, and nonlinear response of the pipe are calculated for various cases, which allow us to investigate the effects of flow velocity, gravity coefficient, initial curvature amplitude and shape, and base-excitation amplitude and frequency. An emphasis is put on the importance of modal interactions on frequency-response and force-response relationships, evidencing the possible in-plane and out-of-plane transverse vibrations and identifying the preferred forms of pre- and post-instability behaviors. This study offers an efficient theoretical model for exploring the nonlinear forced response of slightly curved cantilevered pipes conveying fluid and provides a better insight for further investigations on the modal interaction in cantilevered pipe devices.

Reliability of Response-Controlled Stepped Sine Testing for Experimental Detection of Nonlinear Structure

Nonlinear structural dynamic analysis is required for mechanical structures experiencing nonlinearity through large force-vibration response ranges. Nonlinearities can be caused by large vibration displacements, material properties, or joints. Experimental modal analysis for nonlinear detection is achieved using conventional force-controlled stepped sine testing. However, this approach often encounters premature jumps in frequency response curves before reaching actual resonance peaks. In recent years, response-controlled stepped sine testing (RCT) has been introduced to precisely quantify resonant peaks. This approach, however, has only been limitedly utilised to detect and analyse nonlinearity in jointed structures and structures experiencing large displacement. In this paper, the reliability of the RCT approach is assessed for detecting nonlinearity from different sources. The experimental setup involves placing two magnets on opposite sides of a plate's free end to induce localised nonlinearity through magnet attraction. A low force magnitude of random excitation is employed to identify the frequency range of the first vibration mode using an electromagnetic shaker. Subsequently, RCT is performed within this range to measure the nonlinear forced response. Frequency response functions are measured at ten different controlled displacement amplitudes at the driving point. The analysis observed a symmetry curve of response in the measured FRFs. The results indicate that nonlinear hardening is detected at structures with localised magnet attraction. In conclusion, the reliability of applying the RCT approach for detecting nonlinearity from magnet attraction is achieved due to the absence of a jump issue in FRFs.

Nonlinear forced response of doubly-curved laminated panels composed of cnt patterned layers within first order shear deformation theory

In this study, the solution of the nonlinear forced vibration (NFV) problem of double curvature panels consisting of inhomogeneous (INH) nanocomposite (NC) layers is discussed within first order shear deformation theory (FSDT), taking into account the viscous damping effect. First, the mechanical properties of laminated double curvature panels composed of carbon nanotube patterned layers are modeled mathematically. Then, based on FSDT with a von Kármán-type nonlinearity derived the nonlinear basic relationships and partial differential equations (PDEs). The expression for the NFV frequency of multilayer panels made of inhomogeneous nanocomposite layers (INHNCLs) is obtained within FSDT, using the multi-scale method to the nonlinear PDEs for the first time. Finally, the influences of the external excitation, nonlinearity, transverse shear strains, number and arrangement of layers and CNT-models on the forced vibration frequencies are studied in detail.

Nonlinear Dynamic Characteristics of Rotating Thick Ring on the Elastic Foundation Based on Plane Stress Theory

Radial tires are widely used in transportation machinery, such as large aircraft, new energy vehicles, and engineering vehicles. The ring on the elastic foundation (REF) model is often used for dynamic analysis of radial tires, but the influence of elastic boundary on the dynamic characteristics of radial tires is ignored in this model. In this paper, based on the absolute nodal coordinate formulation (ANCF) and plane stress theory, a rotating thick ring on the elastic foundation (RTREF) model is established to explore the influence of elastic boundary position and other factors on dynamic characteristics. Because existing rectangular and triangular elements in ANCF can’t accurately describe radial/circumferential mesh of a thick ring, a novel ANCF-annular-sector element (AASE) is proposed for the RTREF model. By analyzing the influences of elastic boundary position, thick wall, and rotational speed on the vibration characteristics and nonlinear forced transient response of the RTREF model, it is demonstrated that even if the ring wall is thin, the influence of elastic boundary positions on the dynamic characteristics of the model can’t be ignored.

Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using the nonlocal Euler-Bernoulli beam theory and multiple time scales method are investigated. Lorentz force equation is used to obtain the vertical force due to the applied voltage to the system. The surface effects as well as a combinational non-linear viscoelastic foundation are considered, and finally, the dynamic equilibrium equations are used, and non-linear equations of motion are extracted. In the following, the Galerkin and multiple time scales methods are used, and finally, analytical solutions are extracted as the non-linear free and forced vibrational responses of the system. The relevant coefficients of the extracted analytical solutions are discovered for two both simple support (S-S) and clamped (C-C) boundary conditions. In the following, , and the effects of the different parameters such as non-local parameter as well as electric-magnetic fields, effect of hardness-linear damping parameters of nonlinear considered viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied. As one the results of this study, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. Also, it is predictable that as the load affect position moves away from the center of the single-walled piezoelectric carbon nanotube toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak, and the degree of nonlinearity of the vibrational response of the system.

Correction to: A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

Correction to: A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

Influence of Fiber Angle on the Steady State Periodic Response of Annular Sectorial Plates

Structural elements in the form of annular sectorial plates are widely used in aeronautical, biomedical, and marine engineering. When these components are exposed to dynamic load, large-amplitude vibrations occur. The vibration response analysis based on the linear strain-displacement relation typically yields a conservative estimate and can be used as a first approximation to the actual solution for thin structural elements. As a result, geometric nonlinearity must be incorporated for the efficient and fail-proof design of such elements. This paper shows non-linear and linear steady-state forced vibration responses of the annular sectorial plates. The governing equations of motion have been solved in the time domain by using a modified shooting method and an arc-length/pseudo-arc length continuation technique at the bifurcation point to obtain the complete response curve comprised of stable and unstable branches. This work investigates the effect of fibre angle on the non-linear steady-state forced vibration response of annular sectorial plates. The strain/stress fluctuation throughout the thickness of the laminated annular sectorial plate is determined to explain why hardening nonlinearity has increased. The cyclic fluctuation of the non-linear steady-state normal stress during a time period at the centre of the top and bottom surface is also provided in relation to the forcing frequency ratio of peak amplitude in the non-linear response. Due to the change in restoring forces, the frequency spectra reveal much increased harmonic involvement along with the fundamental harmonic for all fibre angles.

Continuation methods for lab experiments of nonlinear vibrations

AbstractIn this work, we will give an overview of our recent progress in experimental continuation. First, three different approaches are explained and compared which can be found in scientific papers on the topic. We then show S‐Curve measurements of a Duffing oscillator experiment for which we derived optimal controller gains analytically. The derived formula for stabilizing PD‐controller gains makes trial and error search for suitable values unnecessary. Since feedback control introduces higher harmonics in the driving signal, we consider a harmonization of the forcing signal. This harmonization is important to reduce shaker‐structure interaction in the treatment of nonlinear frequency responses. Finally, the controlled nonlinear testing and harmonization is enhanced by a continuation algorithm adapted from numerical analysis and applied to a geometrically nonlinear beam test rig for which we measure the nonlinear forced response directly in the displacement‐frequency plane.

Nonlinear vibration of annular radially graded plate subjected to temperature at one edge

An annular plate graded in radial direction made of ceramic and metal having intrinsic geometric imperfection subjected to mechanical and thermal loading is studied for nonlinear vibration response. The gradation of material in radial direction is represented by a simple power law. The Hamilton principle along with Von Karman nonlinear strain displacement relation & FSDT employed for developing governing nonlinear equation of motion. The finite element formulation of radially graded imperfect heated plate is solved by displacement control direct iteration method. The analysis reveals a significant reduction in the hardening stiffness of the plate because of the increase of plate-edge temperature. The results also shows that the plate undergoes initial bending deformation due to the plate-edge temperature that substantially influence the dynamic behaviour of the plate. The geometric imperfection shows substantial impact on the dynamic response of the plate when its inner-edge is heated as compared to that for heated outer-edge of the plate. The nonlinear free and forced vibration response for different volume fraction of involved materials is also explored.

Geometrically Non-Linear Dynamic Behavior of Simply Supported Rectangular Plates Carrying a Concentrated Mass

Simply supported plates carrying an added point mass are encountered in many engineering fields, like circuit boards or slabs carrying machines at different locations. Determination of the plate modified dynamic characteristics is a quite laborious task, especially in the non-linear regime, which is rarely treated in the literature. The added mass effect on the plate linear parameters was first examined using Hamilton’s principle and spectral analysis. The modified plate's non-linear fundamental mode was then calculated and its non-linear response to high levels of harmonic excitation was determined. The non-linear formulation, involving a fourth order tensor due to the membrane forces induced in the plate mid-plane by large vibration amplitudes, led to a non-linear algebraic amplitude equation. The iterative solution gave the free vibration case a better qualitative understanding and a quantitative evaluation of the effect of the added mass. The non-linear forced response of the modified plate, examined for a wide frequency range, shows that the added eccentric mass induces changes in the area between the mass location and the simple supports and decreases the non-linear hardening effect. The numerical results, covering new situations, are expected to be useful in engineering applications necessitating for some reason the addition to the plate of a point mass or an adaptation of the plate frequencies in order to avoid the occurrence of undesirable resonances.

A test rig for the full characterization of the dynamics of shrouded turbine blades

Highly stressed forced vibrations experienced by Low Pressure Turbine (LPT) blades during their operation can result in high cycle fatigue (HCF) which can ultimately lead to failure. To avoid this occurrence, the vibration amplitudes must be anticipated and reduced. This is generally done by employing friction damping devices like under-platform dampers, shrouds and snubbers. In case of blades with shrouds at their tips, the adjacent shrouded blades are coupled to each other and three-dimensional periodic shroud contact forces acting on shrouds significantly influence the vibration amplitudes as energy is dissipated due to friction. Hence, for the experimental validation of the numerical contact models that are used for the prediction of nonlinear forced response of shrouded blades, it is equally necessary to measure the contact forces acting at the shrouds. Moreover, present-day requirement of more accurate and detailed models requires thorough and comprehensive experimental validation calling for the measurement of three-dimensional shroud contact forces from purposely designed test rigs.This study presents the design, development and operation of an experimental test rig that allows full characterization of the dynamics of shrouded turbine blade, i.e., simultaneous measurement of three-dimensional shroud contact forces and shrouded blade forced response. The starting point was the design of the test rig components based on the design requirements. This is followed by the description of the major components of the proposed test rig i.e., the three-dimensional contact force measurement system and the torque screw mechanism. In the subsequent section, the details of the experimental setup to measure the forced response and three-dimensional shroud contact forces simultaneously are highlighted as the test campaign is performed for different normal preloads and excitation force levels. Consequently, the effects of the variation in the normal preload and excitation force on the measured response and shroud contact forces are discussed. Finally, it is shown how the results establish the efficacy of the proposed test rig in providing a comprehensive depiction of the dynamic response of the shrouded blade and 3D shroud contact forces that will result in more accurate and reliable experimental validation of numerical tools.

Size-dependent nonlinear vibration and instability of a damped microplate subjected to in-plane parametric excitation

The semi-analytical framework for nonlinear vibration and dynamic instability of a damped microplate under periodic parametric excitation is presented. The microplate is modeled using the higher-order shear deformation theory (HSDT) in conjugation with the modified strain gradient theory (MSGT). The governing partial differential equations of motion are obtained using Hamilton’s principle and further solved using Galerkin’s method. The ordinary differential equations without the nonlinear terms are solved using Bolotin’s method to obtain the dynamic instability region. A combination of the incremental harmonic balance (IHB) and the arc-length continuation methods is used to obtain the nonlinear forced vibration response. The effect of initial displacements on the steady-state response of the microplate is discussed. The Newmark-β method is used to obtain the time-history response plots. A comparison of results with those obtained from modified couple stress theory (MCST) and classical continuum theory (CCT) are examined. The effect of various parameters, such as the size of a plate, damping coefficient, static and dynamic loading factors, different boundary conditions, and different loading profiles, on the width of linear and nonlinear instability regions, are also studied.

Highly nonlinear hyperelastic shells: Statics and dynamics

Investigated in this paper are comprehensive static, dynamic and internal-resonance analyses on a wide range of hyperelastic shell structures, including cylindrical, spherical, doubly-curved, and hyperbolic hyperelastic shallow shells. Donnell's nonlinear shell theory and the Mooney-Rivlin strain energy density model are used to formulate the hyperelastic shell structure. Coupled equations of motion are obtained using Hamilton's principle with highly nonlinear terms due to the curvature in the structure, together with the material nonlinearity and large deformations. The coupled equations of motion are converted to a large set of equations using a two-dimensional Galerkin technique and are solved by employing the Newton-Raphson approach and dynamic equilibrium technique. The strength of the current methodology and model is first verified by comparing the static response of the structure with those obtained by using a finite element software. After verifying the model, a detailed analysis of the bending behaviour of the structure under a time-independent pressure is presented. Moreover, the free and forced vibration responses of the shell structure are presented for different cases, showing that the curvature terms play a significant role in changing the mechanical response of the hyperelastic shell. Furthermore, it is shown that for specific sets of curvatures, internal resonances are present which leads to a complicated, rich nonlinear responses. The nonlinear forced vibration response of the hyperelastic shell is also presented for different shell types and resonances.

Relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates

In this paper, the relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates is studied based on Reddy higher order shear deformation theory and von Kármán geometric nonlinearity assumption. Kelvin–Voigt viscoelastic model is adopted to describe the viscoelasticity of the plate. According to the geometric equations and constitutive equations, the elastic potential energy, kinetic energy and virtual work done by the external force and viscous damping force are obtained. Furthermore, the nonlinear governing differential equations of the system are achieved by using Hamilton’s principle. On the basis of the harmonic balance method and Runge–Kutta method, the nonlinear free vibration responses and nonlinear forced vibration responses of the system are presented, respectively. The major innovation of the manuscript is that the nonlinear frequency ratio derived from the nonlinear free vibration of the system, is in good agreement with the resonant frequency ratio obtained from the nonlinear forced vibration of the system. Also, the viscous damping coefficient has a little effect on the nonlinear frequency ratio.

Spin-induced internal resonance in circular cylindrical shells

Internal resonances in circular cylindrical shells, induced by the spinning motion, are explored in this research. The mathematical model of spinning cylindrical shells is established by employing Donnell’s nonlinear shell theory and the Lagrange equations of motion. Then, the nonlinear forced vibration responses of the shells are solved by using the pseudo-arclength continuation technique. Some comparisons are conducted to verify the accuracy of the present model and method The results show that internal resonances can be caused by the spinning motion of the shells, which cover a wide range of spinning speeds. Different types of internal resonances can occur, such as 1:1, 1:1:1, and 1:2 internal resonances. Energy exchanges between different modes with internal resonance relationships. Both softening and hardening nonlinearities exist in the frequency response of the spinning cylindrical shells. The excitation amplitude has a threshold to excite the internal resonances. It is found that the internal resonances can be avoided by tuning the spinning speed of the shells.