Damped Beam Research Articles

Wave dispersion in a damped beam supported by cubic nonlinear springs: A multiscale freewave approach

Wave dispersion in a damped beam supported by cubic nonlinear springs: A multiscale freewave approach

Controllability with one scalar control of a system of interaction between the Navier–Stokes system and a damped beam equation

Controllability with one scalar control of a system of interaction between the Navier–Stokes system and a damped beam equation

Active Constrained Layer Damping of Beams with Natural Fiber Reinforced Viscoelastic Composites

Abstract Viscoelastic composite reinforced with natural fibers is proposed here for the damping treatment of structures for vibration attenuation. The constrained layer damping (CLD) method is found a prominent structural vibration damping method and as a result of the inclusion of natural fiber, substantial damping could be achieved due to the development of shear as well as extensional strains in the viscoelastic material. Finite element analysis of a substrate beam combined with CLD treatment is used to investigate the corresponding improvement in damping. The influence of various geometry of presently considered natural viscoelastic composite layers on the CLD treatment's damping performance has been evaluated and discussed. Damping performances of natural as well as synthetic fibers have been assessed and compared for using natural fibers in viscoelastic composite for various structural applications. The addition of natural fiber has resulted from enhanced damping capacity due to substantial friction at the natural fiber-matrix interface compared to the case of interface with synthetic fiber. Natural fiber attributes less stiffness and a higher loss factor as compared to synthetic fiber and the insertion of stiff fibers affects the polymer chain's movements, decreasing the matrix phase's damping behavior. The dynamic mechanical analysis test is performed to evaluate the damping characteristics in the form of loss factor, storage, and loss modulus over the temperature range from frozen state to melting state.

General Theory for Damped Beams with Elastic Supports Subjected to a Moving Damped Sprung Mass

Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

Measuring the internal damping of thin metal beams in flexure

Since the internal damping of most metals is a relatively small quantity, precise measurement requires that both acoustic radiation damping and boundary condition damping be either eliminated or highly minimized. In this talk, an experiment setup will be described which can measure the internal damping of metal beams in flexure. To perform a measurement, a beam specimen is placed on wire supports within a vacuum chamber and excited by an autonomous force hammer located within the chamber. The wire supports are located exactly at the nodal lines of the beam mode of interest, so that the frictional losses created by the wire supports are minimized. Once excited, the beam deflection is measured with a single point laser vibrometer. The signal decay measured by the vibrometer is used to estimate damping of the mode of interest using either the Hilbert transform or the half- power method. The wire supports are then moved, and the procedure is repeated for the next mode of interest. Results for several metal specimens will be presented and compared to Zener’s thermoelastic model, which describes the losses created by thermal currents for a beam in flexure, to demonstrate the accuracy of the measurement method.

Internal damping measurements of additive manufactured metal beams

Modern additive manufacturing (AM) techniques have created an endless number of new design possibilities. Naturally, it is important to understand how the material properties, including internal damping, differ for AM structures. An experimental procedure has been developed to measure an upper bound for the internal damping of metal beams by minimizing the effects of energy dissipation at the supports and acoustic radiation into the surrounding air. In this talk, measurements for the loss factor of several common metals will be compared to equivalent samples constructed by powder bed fusion at varying angles. The results will also be compared to Zener’s thermoelasticity model, developed for isotropic Euler beams in flexure. Reasons for deviation from theory which arise from the manufacturing technique will be explored.

Development and Whirl Flutter Test of the Maryland Tiltrotor Rig

The first whirl flutter test of the Maryland Tiltrotor Rig (MTR) was recently completed in the Naval Surface Warfare Center Carderock Division 2.44 m by 3.05 m (8- by 10-ft) large subsonic wind tunnel. The MTR is a 1.45 m (4.75-ft) diameter, three-bladed, semispan, floor-mounted, optionally powered, flutter rig. This paper describes the major features of the MTR and the results obtained from the first successful flutter tests. Parametric variations of rig features include wing profile on and off, gimbal free and gimbal locked hub, powered and freewheeling rotor, and straight and swept-tip blades. For the freewheeling rotor condition, the rotor speed is trimmed to 1050 RPM by setting blade collective. The gimbal is trimmed to zero first harmonic apping by setting blade cyclics. Model con gurations were tested up to 100 kt windspeed. The model was excited by oscillating the swashplate at the wing-pylon natural frequencies. Eight speed sweeps were carried out to acquire frequency and damping data on different model con gurations. Frequency and damping of the wing beam and chord modes were extracted using the moving-block method.

Whirl Flutter Test of Swept-Tip Tiltrotor Blades

The first whirl flutter test of the Maryland Tiltrotor Rig (MTR) was recently completed in the Naval Surface Warfare Center Carderock Division (NSWCCD) 8- by 10-ft large subsonic wind tunnel. The MTR is a Froude-scaled, 4.75 ftdiameter, threebladed, semispan, floor-mounted, optionally powered, flutter rig. This paper focuses on the swept-tip blades developed for this test. The swept-tip begins at 80%R and sweeps aft 20° to alleviate whirl flutter. Whirl flutter testing was performed in four con gurations for both blade geometries. The frequency and damping of the wing beam and chord bending were collected at wind speeds up to 100 kt. This paper directly compares whirl utter results between the straight and swept-tip blades and demonstrates higher wing chord damping using swept-tip blades in freewheel conditions, even at lower flight speeds.

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Long time behavior of weak solutions for the nonlinear damped beam equation

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data.

Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation

Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation

Feedback boundary stabilization for a viscous incompressible fluid with Navier slip boundary conditions in interaction with a damped beam

We show the stabilization by a finite number of controllers of a fluid–structure interaction system where the fluid is modeled by the Navier–Stokes system into a periodical canal and where the structure is an elastic wall localized on top of the fluid domain. The elastic deformation of the structure follows a damped beam equation. We also assume that the fluid can slip on its boundaries and we model this by using the Navier slip boundary conditions. Our result states the local exponential stabilization around a stationary state of strong solutions by using dynamical controllers in order to handle the compatibility conditions at initial time. The proof is based on a change of variables to write the fluid–structure interaction system in a fixed domain and on the stabilization of the linearization of the corresponding system around the stationary state. One of the main difficulties consists in handling the nonlinear terms coming from the change of variables in the boundary conditions.

Hysteretic behaviours of a novel lead-viscoelastic coupling beam damper: Numerical analysis and Bouc-Wen based model

The replaceable coupling beam damper (RCBD) is widely recognized as an effective device to improve the seismic performances of shear wall structures. However, the extensively studied metallic-type RCBD has poor fatigue performance, while the viscoelastic-type RCBD is sensitive to the loading frequency. To overcome the drawbacks of the RCBDs mentioned above, the novel lead-viscoelastic coupling beam damper (LVCBD), which is mainly composed of lead rods, composite viscoelastic layers and steel plates, was proposed by the authors recently. Cyclic loading tests revealed that the lead rods of the LVCBD can overcome the drawback of frequency dependency of the conventional viscoelastic damper, making this innovative damper frequency-independent, fatigue resistant and stable in energy dissipation. In order to clarify the influence of the lead rod on the LVCBD, this paper performs numerical studies to investigate the effects of the lead rod number and lead rod diameter on the hysteretic behaviours of the LVCBD. Theoretical formulae are derived to calculate the shear-bearing force and stiffness of the damper and validated by the test and numerical results. Finally, by using the genetic algorithm (GA), a Bouc-Wen based hysteretic model is developed to depict the hysteretic behaviours of the LVCBD. Results show that the properly designed LVCBD with four lead rods exhibits a favorable shear-bearing capacity with uniform stress distribution. Increasing the lead rod diameter can effectively enhance the shear-bearing capacity of the damper. The developed formulae and hysteretic model yield good predictions of the hysteretic behaviours of the LVCBD, which can be used in the design and analysis of the LVCBD.

Material-geometry interplay in damping of biomimetic scale beams

Biomimetic scale-covered substrates are architected meta-structures exhibiting fascinating emergent nonlinearities via the geometry of collective scales contacts. Despite much progress in understanding their elastic nonlinearity, their dissipative behavior arising from scales sliding is relatively uninvestigated in the dynamic regime. Recently discovered is the phenomena of viscous emergence, where dry Coulomb friction between scales can lead to apparent viscous damping behavior of the overall multi-material substrate. In contrast to this structural dissipation, material dissipation common in many polymers has never been considered, especially synergistically with geometrical factors. This aspect is addressed here, where material viscoelasticity is introduced via a simple Kelvin–Voigt model for brevity and clarity. The results contrast the two damping sources in these architectured systems: material viscoelasticity and geometrical frictional scales contact. It is discovered that although topically similar in effective damping, viscoelastic damping follows a different damping envelope than dry friction, including starkly different effects on damping symmetry and specific damping capacity.

Effect of porosity and viscoelastic boundary conditions on FG sandwich beams in thermal environment: Buckling and vibration studies

The present study is carried out to investigate the combined effect of porosity and temperature on the buckling and vibration attributes of FG sandwich beams in the thermal environment using FE formulation. The modeled sandwich beam consisting of the viscoelastic core material is restrained by viscoelastic boundary conditions (VBCs). The FG face layers and core are subjected to temperature-dependent material properties. Complex stiffness model is adopted for VBCs. Porosity patterns such as H, V, X, and O are incorporated into FG face layers. The Lagrange equation is used to derive the sandwich beam’s equilibrium equations of motion in static and dynamic conditions. The derived equilibrium equations are solved for buckling and vibration of the beam using the FE solution. Lagrange and Hermite shape functions are assumed for axial and transverse displacements. Critical buckling temperature (CBT), natural frequency (NF), and loss factors (LF) are obtained for various temperatures and boundary stiffness values (BSVs). Transverse buckling and vibration mode shapes are extracted for changing BSVs. The behavior of NF and LF at buckling temperature is also discussed. The existence of porosities ameliorates the buckling characteristics of the sandwich beam. VBCs expedite the vibration damping of sandwich beams alongside the viscoelastic core. The natural frequency and loss factor reach zero and infinity, respectively, when the temperature reaches CBT.

Nonlinear Damping and Vibration Assessment of Sandwich Beam with Composite Faces and Viscoelastic Core Layer

This work deals with the damping and vibration characteristics evaluation of sandwich beams based on higher-order shear deformation theories (HSDT’s). The studied sandwich beams are composed of composite materials at faces and homogeneous, isotropic viscoelastic material in the core layer. This latter (viscoelastic material) is described by complex Young’s modulus in the frequency domain. Indeed, the real part represents the stiffness, and the imaginary one describes the damping property. Nonlinear damping parameters and frequency curves are obtained using the nonlinear equation of frequency amplitude governed by multiple complex coefficients. In conjunction with a one-mode Galerkin’s procedure, the harmonic balance method is employed to obtain the nonlinear loss factor and the response frequency curves. The nonlinear frequency value and, in turn, the system loss factor ratio are obtained for various amplitudes to examine the impacts of the ply angle of fiber-reinforced composite faces, the viscoelastic shear modulus, the shear deformation model, for considering different geometric and material parameters. In the framework of this study, one attempt is to describe a better way to achieve satisfactory passive vibration control of the laminated beams.

Free vibration of viscoelastic nonlocally damped tapered axially functionally graded beams using the state-space formulation

For the first time, this paper presents an efficient solution for the vibration analysis of damped axially functionally graded beams with non-uniform cross-sections subjected to the axial force based on the Galerkin method and state-space formulation. Euler-Bernoulli and Rayleigh’s theories are considered to derive the governing equations of motion. Currently, simply supported beams are under investigation due to their easy fabrication and typical applications in various bridge components, buildings, and machine tools. Likewise, the proposed method is applicable to other continuous systems without any limitation on boundary conditions, material properties, or arbitrary representation of cross-sections. Moreover, complex and purely real eigenvalues are determined by outlining the state-space framework. Compared to the conventional method, the present formulation reduces computational complexity and simplifies computer implementation, especially in the case of inhomogeneous materials. Then, an extensive parametric study is performed to evaluate the effect of various characteristics on the dynamic behavior of the viscoelastic nonlocally damped beams. Results reveal that the inhomogeneity index in material constituents has a considerable influence on the eigensolutions. Additionally, it is shown that when the relaxation constant and length parameter tend to infinity there would be a decline in the increasing rate of eigenvalues.

Development and Validation of a LabVIEW Automated Software System for Displacement and Dynamic Modal Parameters Analysis Purposes

The structural health monitoring (SHM) technique is a highly competent operative process dedicated to improving the resilience of an infrastructure by evaluating its system state. SHM is performed to identify any modification in the dynamic properties of an infrastructure by evaluating the acceleration, natural frequencies, and damping ratios. Apart from the vibrational measurements, SHM is employed to assess the displacement. Consequently, sensors are mounted on the investigated framework aiming to collect frequent readings at regularly spaced time intervals during and after being induced. In this study, a LabVIEW program was developed for vibrational monitoring and system evaluation. In a case study reported herein, it calculates the natural frequencies as well as the damping and displacement parameters of a cantilever steel beam after being subjected to excitation at its free end. For that purpose, a Bridge Diagnostic Inc. (BDI) accelerometer and a displacement transducer were parallelly mounted on the free end of the beam. The developed program was capable of detecting the eigenfrequencies, the damping properties, and the displacements from the acceleration data. The evaluated parameters were estimated with the ARTeMIS modal analysis software for comparison purposes. The reported response confirmed that the proposed system strongly conducted the desired performance as it successfully identified the system state and modal parameters.

Internal and External Cancellation Conditions for Free Vibration of Damped Simple Beams Traversed by Successive Moving Loads

In this paper, the internal and external cancellation phenomena for damped beams subjected to multi-moving loads are investigated in detail. To start, the theory for the vibration of a simply supported beam is revisited by including the effect of damping. For the first time, a simple expression is derived for the free vibration of the damped beam under multi-moving loads. Based on the concept of local minimum, two cancellation conditions are identified. One is the internal cancellation, which relates to the inherent property of the beam and is conventionally known. The other is the newly formulated external cancellation that relates to the number and spacing of moving loads. For comparison, both the resonant condition and the optimal criterion for span length of the bridge are also briefed. By comparing with the classical solution, the present simple expression for the free vibration of the beam is firstly validated. Then the factors affecting the cancellation are investigated against various load cases and damping levels. The results show that external cancellation occurs more frequently due to the increase in the number and spacing of the moving loads. The damping of the beam has a leaking effect on cancellation, in that nonzero vibration may occur, but it is also quickly damped out by damping itself.