Harmonic balance method (HBM) is a popular computational tool for the nonlinear dynamic analysis of structural elements in the frequency domain. Its application in conjunction with the finite element ...(FE) procedure involves complexity in the formulation of the geometrically nonlinear equation of motion. Further complexity arises in the case of a viscoelastic structure as its constitutive model involves temporal derivative/integral of stress/strain. In this concern, the consideration of a few harmonic terms in HBM poses somewhat simplified formulation, but it may not provide a good theoretical estimation of nonlinear dynamics. Therefore, a large number of harmonic terms in HBM are to be considered despite the corresponding complexity, as well as a high computational cost. In this view, presently, two new formulation strategies are introduced toward a generalized FE formulation, especially for the consideration of an arbitrary number of harmonic terms in HBM. The first strategy lies in the formulation of the geometrically nonlinear stiffness matrix through a special factorization of the nonlinear strain–displacement matrix, while the second one lies in the analytical integration of system matrices/vectors over a time period by exploiting the orthogonality of Fourier basis functions. These formulation strategies provide not only the equation of motion with a reduced number of terms in the HBM-based expanded forms of system matrices/vectors but also a significantly reduced computational time. Additionally, various time–domain viscoelastic constitutive models are reduced into a generalized form for the periodic stress/strain to achieve a common HBM-based FE formulation for any of these viscoelastic material models.
For improved stability of fluid-conveying pipes operating under the thermal environment, functionally graded materials (FGMs) are recommended in a few recent studies. Besides this advantage, the ...nonlinear dynamics of fluid-conveying FG pipes is an important concern for their engineering applications. The present study is carried out in this direction, where the nonlinear dynamics of a vertical FG pipe conveying hot fluid is studied thoroughly. The FG pipe is considered with pinned ends while the internal hot fluid flows with the steady or pulsatile flow velocity. Based on the Euler–Bernoulli beam theory and the plug-flow model, the nonlinear governing equation of motion of the fluid-conveying FG pipe is derived in the form of the nonlinear integro-partial-differential equation that is subsequently reduced as the nonlinear temporal differential equation using Galerkin method. The solutions in the time or frequency domain are obtained by implementing the adaptive Runge–Kutta method or harmonic balance method. First, the divergence characteristics of the FG pipe are investigated and it is found that buckling of the FG pipe arises mainly because of temperature of the internal fluid. Next, the dynamic characteristics of the FG pipe corresponding to its pre- and post-buckled equilibrium states are studied. In the pre-buckled equilibrium state, higher-order parametric resonances are observed in addition to the principal primary and secondary parametric resonances, and thus the usual shape of the parametric instability region deviates. However, in the post-buckled equilibrium state of the FG pipe, its chaotic oscillations may arise through the intermittent transition route, cyclic-fold bifurcation, period-doubling bifurcation and subcritical bifurcation. The overall study reveals complex dynamics of the FG pipe with respect to some system parameters like temperature of fluid, material properties of FGM and fluid flow velocity.
A new 1-3 viscoelastic composite material (VECM) layer is designed for improved active constrained layer damping (ACLD) treatment of vibration of a functionally graded (FG) circular cylindrical ...shell. Besides this improved active damping treatment, another objective of this study is to control all the modes of vibration of the shell effectively using the treatment (active constrained layer damping) in layer-form throughout the outer shell-surface. In this design of active constrained layer damping treatment in layer-form, its (active constrained layer damping) necessary conformability with the curved host shell-surface is ensured by the use of a vertically reinforced 1-3 piezoelectric composite (PZC) constraining layer, whereas the effective control of several modes of vibration of the shell is achieved by the use of electrode-patches over the surfaces of the constraining layer. A fruitful strategy in the arrangement of electrode-patches is proposed for effective control of several modes of vibration of the shell using one configuration of the electrode-patches. An electric potential function is assumed for this use of electrode-patches and a geometrically nonlinear coupled electro-visco-elastic incremental finite element model of the overall shell is developed for its analysis in the frequency-domain. The analysis reveals significant improvement of active damping characteristics of the active constrained layer damping layer for the use of the present 1-3 viscoelastic composite material layer instead of the traditional monolithic viscoelastic material (VEM) layer. The analysis also reveals the suitability of the present strategy of arrangement of electrode-patches for achieving aforesaid control-activity of the ACLD layer. The effects of temperature in the host functionally graded shell and different geometric parameters in the design of the 1-3 viscoelastic composite material layer on the damping characteristics of overall shell are also presented.
This work presents the active control of parametrically excited porous functionally graded (FG) geometrically nonlinear beam integrated with extensional-mode piezoelectric actuators. The porous FG ...smart beam is parametrically excited by applying a compressive harmonic load along its axis. The extensional-mode piezoelectric actuators are actuated by applying an extrinsic electric field though negative velocity-feedback control strategy to resist bending deformation in beam. For the corresponding study of active vibration control, a two-dimensional incremental electro-elastic finite element model is derived. The von Karman geometric nonlinearity is accounted for large bending deformation of beam. The corresponding nonlinear finite element governing equations of motion of smart beam are solved in the frequency domain and time domain using harmonic balance method and Bathe time integration method, respectively. The results revealed that the porosity mainly reduces the critical buckling load due to the reduction in flexural rigidity. Further, it induces higher vibration amplitudes of beam or leads to the requirement of higher control electric field. In contrast, the control capability of piezoelectric actuator in controlling parametric instability increases with the porosity. Thus, extensional-mode piezoelectric actuator exhibits the adequate control capability for flexible beams compared to stiffer beams.
In this work, the damping characteristics of an actively constrained viscoelastic material layer are examined because of the inclusion of dispersed graphite particles within the viscoelastic material ...layer. The study is carried out by analysing the active–passive damping in a layered plate made of a substrate layer, a constrained viscoelastic particulate composite layer and a thin constraining piezoelectric actuator layer. The effective properties of the viscoelastic particulate composite are estimated using a differential scheme and the elastic–viscoelastic correspondence principle. The piezoelectric layer is activated according to the velocity feedback control law, and a closed-loop finite element model of the overall plate is derived for the analysis. The results reveal that the inclusion of graphite particles not only causes an improved transfer of active action from the piezoelectric layer to the substrate plate but also enhances the energy dissipation capability of the constrained viscoelastic layer. It is found that the maximum transfer of active action and the maximum passive damping capability of the viscoelastic particulate composite layer arise almost at the same volume fraction of inclusion. So, an optimal volume fraction of inclusion is obtained for significantly improved active–passive damping in the overall plate. The overall study presents a potential means of improved active–passive damping treatment of structural vibration.
In this work, a new 0-3 viscoelastic composite (VEC) layer is presented for augmented constrained layer damping of plate vibration. The 0-3 VEC layer comprises a rectangular array of the thin ...rectangular graphite-wafers embedded within the viscoelastic matrix. The inclusions of graphite-wafers in the constrained 0-3 VEC layer confine the motion of the viscoelastic phase for its reasonable in-plane strains along with the enhanced transverse shear strains. This occurrence of coincidental shear and extensional strains within the viscoelastic phase is supposed to cause augmented damping capacity of the constrained layer, and it is investigated by integrating the constrained 0-3 VEC layer over the top surface of a substrate plate. A finite element (FE) model of the overall plate is developed based on the layer-wise shear deformation theory. Using this FE model, first, a bending analysis of the overall plate is performed to investigate the mechanisms of damping in the use of 0-3 VEC layer. Next, the damping in the overall plate is quantified for different sets of values of the geometrical parameters of the 0-3 VEC layer. These results reveal significant improvement of damping in the plate due to the inclusions of graphite-wafers within the constrained viscoelastic layer. But, the augmentation of damping indicatively depends on the geometrical parameters in the arrangement of the graphite-wafers. So, the 0-3 VEC layer is configured appropriately through an optimization algorithm, and finally, the forced frequency responses of the overall plate are evaluated to demonstrate the augmented attenuation of vibration-amplitude via the inclusions of graphite-wafers within the constrained viscoelastic layer in an optimal manner.
Present investigation provides an insight into fluoride uptake by immobilized beads of two fluoride-resistant bacterial species
Providencia vermicola
(KX926492) (PV) and
Staphylococcus lentus
...(KX941068) (SL) from simulated waste water in packed bed column reactor. Experiments have been carried out to assess the effects of parameters like influent fluoride concentration, feed flow rate and bed height on fluoride removal. The column data were fitted to Adams-Bohart and Yoon-Nelson models to determine the influence on biosorption performance. Different influential parameters were studied such as fluoride concentration (15–25 mg L
−1
), rate of feed flow (2, 4 and 6 mL min
−1
) and height of bed (4, 6 and 8 cm). Optimum operating conditions were observed to be: concentration (conc) = 20 mg L
−1
, influent flow rate = 2 mL min
−1
and bed height = 8 cm. A maximal fluoride uptake ability of 32.48 mg g
−1
has been achieved under these conditions. The experimental results of the breakthrough curves fit well with the Yoon-Nelson model. Therefore, immobilized beads of PV and SL could serve as potent remediating agents towards water defluoridation in large-scale operations.
Graphical abstract
This work deals with the damping capability of an actively constrained viscoelastic particulate composite (VEPC) layer in attenuation of vibration of a beam element. The overall beam is composed in ...the layered form where VEPC layer is constrained between the substrate beam and extension actuated piezoelectric layer. The VEPC layer is a graphite particle-filled viscoelastic composite layer and its effective properties are estimated using differential scheme and elastic–viscoelastic correspondence principle. Based on the theoretically estimated effective properties of VEPC and the velocity feedback control law, a closed-loop finite element (FE) model of the layered beam is derived, and the active–passive damping characteristics of the overall beam are analysed by varying the volume fraction of inclusion (VFI) of graphite particles within constrained viscoelastic layer. It is found that the inclusion of graphite particles not only causes an improved transfer of active action from the piezoelectric layer to the substrate beam but also enhances the passive action through constrained viscoelastic layer. However, maximum active and passive actions appear at an optimal VFI so that a significantly augmented hybrid active–passive damping is achieved. The overall study presents the augmented damping capability of the hybrid damping treatment through the proposition of a VEPC layer.
For improved flexibility and conformability of piezoelectric fiber–reinforced composite actuator, it is reconstructed in a recent study by the use of short piezoelectric fibers (short piezoelectric ...fiber–reinforced composite) instead of continuous fibers (continuous piezoelectric fiber–reinforced composite). This modification facilitates its application in short piezoelectric fiber–reinforced composite layer form instead of continuous piezoelectric fiber–reinforced composite patch form particularly in case of host structures with highly curved boundary surfaces. But the corresponding change in actuation capability is a major issue for potential application of short piezoelectric fiber–reinforced composite that is studied in this work through the control of vibration of a functionally graded circular cylindrical shell under thermal environment. First, an arrangement of continuous piezoelectric fiber–reinforced composite actuator patches over the host shell surface is presented with an objective of controlling all modes of vibration. Next, the use of short piezoelectric fiber–reinforced composite actuator layer for similar control activity is demonstrated through an arrangement of electrode patches over its surfaces. Subsequently, an electric potential function is assumed for the consideration of electrode patches and a geometrically nonlinear coupled thermo-electro-mechanical incremental finite element model of the harmonically excited overall functionally graded shell is developed. The numerical results reveal actuation capability of short piezoelectric fiber–reinforced composite actuator layer with reference to that of the existing continuous piezoelectric fiber–reinforced composite/monolithic piezoelectric actuator patches. The effects of temperature, size of electrode patches, properties of piezoelectric fiber–reinforced composite, and functionally graded properties on the control activity of short piezoelectric fiber–reinforced composite/continuous piezoelectric fiber–reinforced composite actuator are also presented.
A new 1-3 viscoelastic composite material (VECM) layer is designed in order to achieve improved passive damping characteristics of the unconstrained (UCLD)/constrained (PCLD) layer damping treatment ...of structural vibration. The present 1-3 VECM layer is used instead of traditional monolithic viscoelastic material (VEM) layer within the UCLD/PCLD treatment. The corresponding change in damping is studied through the static as well as dynamic finite element (FE) analyses of a substrate beam integrated with the UCLD/PCLD treatment. The static analysis reveals augmentations of damping in the use of 1-3 VECM layer. The dynamic analysis quantifies improvement in the passive unconstrained/constrained layer damping through the present design.
