Investigation of multi-layer sandwich beams through single degree-of-freedom transformation

The dynamic behaviors of multi-layer sandwich beams are investigated through single degree-of-freedom (SDOF) transformation. The frequency response of the multi-layer sandwich beam is obtained using finite element code COMSOL and is transformed to a SDOF system with the same frequency response. Hence, the mass, spring constant and damping coefficient of the sandwich beams with different lengths and number of visco-elastic layers can be investigated. Further, viscous damping and structural damping models are individually employed to simulate the damping effect of the sandwich beam. The frequency responses from both models are compared with that from COMSOL and experiment. The resonant peak and resonant frequency of the SDOF system using structural damping model is more consistent with that from COMSOL. The experimental result demonstrates that the response of the sandwich beam can be predicted through COMSOL and SDOF transformation.     

Vibration isolation using extreme geometric nonlinearity

A highly deformed, slender beam (or strip), attached to a vertically oscillating base, is used in a vibration isolation application to reduce the motion of a supported mass. The isolator is a thin strip that is bent so that the two ends are clamped together, forming a loop. The clamped ends are attached to an excitation source and the supported system is attached at the loop midpoint directly above the base. The strip is modeled as an elastica, and the resulting nonlinear boundary value problem is solved numerically using a shooting method. First the equilibrium shapes of the loop with varying static loads and lengths are studied. The analysis reveals a large degree of stiffness tunability; the stiffness is dependent on the geometric configuration, which itself is determined by the supported mass, loop length, and loop self-weight. Free vibration frequencies and mode shapes are also found. Finally, the case of forced vibration is studied, and the displacement transmissibility over a large range of forcing frequencies is determined for varying parameter values. Experiments using polycarbonate strips are conducted to verify equilibrium and dynamic behavior.     

THE DYNAMIC STIFFNESS MATRIX METHOD IN FORCED VIBRATION ANALYSIS OF MULTIPLE-CRACKED BEAM

The dynamic behaviour of a beam with numerous transverse cracks is studied. Based on the equivalent rotational spring model of crack and the transfer matrix for beam, the dynamic stiffness matrix method has been developed for spectral analysis of forced vibration of a multiple cracked beam. As a particular case, when the excitation frequency is close to zero, the solution for static response of beam with an arbitrary number of cracks has been obtained exactly in an analytical form. In general case, the effect of crack number and depth on the dynamic response of beam was analyzed numerically.     

Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: A theoretical and experimental study

This paper concerns the vibration isolation characteristics of a nonlinear isolator using Euler buckled beams as negative stiffness corrector. Both analytical and experimental studies are carried out. The Harmonic Balance Method (HBM) is used to determine the primary resonance response for the single degree of freedom (SDOF) nonlinear system composed by a loaded mass and the nonlinear isolator. The distuning of the loaded mass is taken into consideration, resulting in a Helmoholtz–Duffing equation. The performance of the nonlinear isolator is evaluated by the defined two kinds of transmissibility and compared with that of the linear isolator without the stiffness corrector. The study shows that the asymmetric SDOF nonlinear system can behave like a purely softening, a softening–hardening or a purely hardening system, depending on the magnitude of the excitation level. An experimental apparatus is set up to validate the analytical results. The transmissibility results of the SDOF nonlinear system under base excitation with both discrete sinusoidal frequencies and slowly forward and backward sweeps are given and discussed. The complex jump phenomena under different excitation levels are identified. By introducing the stiffness corrector, the starting frequency of isolation of the nonlinear isolator is found to be lower than that of the linear one with the same support capacity. The proposed nonlinear isolator performs well in applications where the excitation amplitude is not too large.     

Forced vibration of a damped sandwich beam subjected to moving forces

The response of a sandwich beam subjected to moving forces (constant as well as pulsating) is analyzed by the use of Fourier and Laplace transforms and compared with the response of an equivalent elastic beam. The results indicate that the critical speed of force on a sandwich beam is always greater than that on an elastic beam of identical mass per unit length and flexural rigidity, and depends on its geometric and shear parameters. For subcritical speeds, the maximum deflection of a sandwich beam is shown to occur earlier than that of an equivalent elastic beam. An increase in the core shear stiffness is shown to be beneficial in reducing the dynamic magnification of the central deflection of the sandwich beam.     

A sensitivity-based structural damage identification method with unknown input excitation using transmissibility concept

Unknown input excitation and local damages universally coexist in a practical situation. Therefore, in this paper a structural damage identification method based on the transmissibility concept in state space domain is proposed without the need for input measurements. On the basis of the transformation matrix which is computed using the system Markov parameters in state space, the relationship between two different sets of acceleration response measurements can be formulated under the same input excitation. A sensitivity-based model updating approach is applied to identify the local damages by minimizing the difference between the measured response and the reconstructed response. The sensitivity of the dynamic acceleration response with respect to the elemental stiffness factors is derived analytically in the state space domain, which accelerates the process of damage identification. A numerical cantilever beam is employed to validate that the variation of structural parameters induced by the local damages can be accurately and effectively identified without the input excitation information by the proposed method even with measurement noise considered. A laboratory test is further carried out to verify the proposed structural damage identification method based on the response reconstruction technique.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Vibration and dynamic stability of a traveling sandwich beam

The vibration and dynamic stability of a traveling sandwich beam are studied using the finite element method. The damping layer is assumed to be linear viscoelastic and almost incompressible. The extensional and shear moduli of the viscoelastic material are characterized by complex quantities. Complex-eigenvalue problems are solved by the state-space method, and the natural frequencies and modal loss factors of the composite beam are extracted. The effects of stiffness and thickness ratio of the viscoelastic and constrained layers on natural frequencies and modal loss factors are reported. Tension fluctuations are the dominant source of excitation in a traveling sandwich material, and the regions of dynamic instability are determined by modified Bolotin's method. Numerical results show that the constrained damping layer stabilizes the traveling sandwich beam.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.     

Transient response of an elastic beam with discrete impedances from decelerating boundaries

Analyzed is transient response of an elastic beam from decelerating boundaries. It is shown that passive attenuation of peak transient deceleration is possible by coupling the beam to a tuned one-degree-of-freedom (1-dof) oscillator. For a fixed oscillator mass, optimum spring stiffness yields the lowest deceleration response. Furthermore, over a finite range of spring stiffness, the second coupled system mode disappears yielding an additional sharp reduction in transient deceleration response. The disappearance of the second mode is independent of boundary conditions from the weakest simple supports to the strongest clamped ends.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.     

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.     

A tunable high-static-low-dynamic stiffness vibration isolator

In this study, a novel vibration isolator is developed. The developed isolator possesses the characteristics of high-static-low-dynamic stiffness (HSLDS) and can act passively or semi-actively. The HSLDS property of the isolator is obtained by connecting a mechanical spring, in parallel with a magnetic spring that is constructed by a pair of electromagnets and a permanent magnet. The mechanical spring is a structural beam whose stiffness exhibits a hardening behavior. The stiffness of the magnetic spring can be positive or negative, depending on the polarity of the current to the electromagnets. A passive HSLDS isolator is obtained when the electromagnet current is zero. In the stiffness characterization study, the analytical model for each of the springs is established and the tuning parameters are identified. Using the stiffness models, the design optimization issues are investigated. In the experimental study, the effectiveness of the isolator for vibration isolation is tested. The analytical natural frequencies of the isolator are validated experimentally. The relationships between the displacement transmissibility and the exciting frequency are measured both under the passive mode and under the semi-active mode. The on-line tuning capability of the isolator is investigated.     

Dynamic properties of the human head

Driving point mechanical impedance measurements were used to determine the dynamic response of the human head to sinusoidal vibration in the frequency range between 30 Hz and 5000 Hz at excitation levels of 0·98 m/s² and 3·4 m/s2. Because of the low excitation levels, the weight of the head was sufficient to couple the head to the vibration source.At 20 Hz the impedance magnitude was about 790 N-s/m but increased at approximately 6dB/octave to a peak near 3500 N-s/m at 70–90 Hz. Between 100 Hz and 2000 Hz impedance decreased by about two orders of magnitude while the apparent mass decreased by three orders of magnitude indicating good vibration isolation at higher frequencies. The impedance response contains the information for modelling the head as a dynamic system.The response of the head to vibration can be simulated by a two degree-of-freedom, mass-excited system consisting of a series connection of a small driving mass, a damper, a spring and damper in parallel and a large final mass. Parameter values, derived by computer techniques, suggest that the large mass represents the total head, the small mass the tissue in contact with the vibration input and the spring the skull stiffness.     

A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of-freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not.     

On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets

The frequency range over which a linear passive vibration isolator is effective is often limited by the mount stiffness required to support a static load. This can be improved upon by incorporating a negative stiffness element in the mount such that the dynamic stiffness is much less than the static stiffness. In this case, it can be referred to as a high-static-low-dynamic stiffness (HSLDS) mount. This paper is concerned with a theoretical and experimental study of one such mount. It comprises two vertical mechanical springs between which an isolated mass is mounted. At the outer edge of each spring, there is a permanent magnet. In the experimental work reported here, the isolated mass is also a magnet arranged so that it is attracted by the other magnets. Thus, the combination of magnets acts as a negative stiffness counteracting the positive stiffness provided by the mechanical springs. Although the HSLDS suspension system will inevitably be nonlinear, it is shown that for small oscillations the mount considered here is linear. The measured transmissibility is compared with a comparable linear mass-spring-damper system to show the advantages offered by the HSLDS mount.     

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Free vibrations of a beam-mass system with elastically restrained ends

The vibration problem of a beam with an arbitrarily placed concentrated mass and elastically restrained against rotation at either end is solved by using Laplace transforms. The effects on eigenfrequencies of the system produced by varying the ratios of the concentrated mass to the mass of the beam, stiffness of the end spring to the stiffness of the beam and position of the mass to the total length of the beam are presented. The effect of neglecting the mass of the beam is considered.