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research-article  
Rajan Prasad and Abhijit Sarkar
J. Appl. Mech   doi: 10.1115/1.4042011
The alternating stop band characteristics of periodic structures have been widely used for narrow band vibration control applications. The objective of the present work is to extend this idea for broadband excitations. Towards this end, we seek to synthesize a longitudinal and a flexural periodic structure having the largest fraction of the frequencies falling in the attenuation bands of the structure. Such a periodic structure when subjected to broadband excitation has minimal transmission of the response away from the source of excitation. The unit cell of such a periodic structure is constituted of two distinct regions having different inertial and stiffness properties. We derive guidelines for suitable selection of inertial and stiffness properties of the two regions in the unit cell, such that the maximal frequency region corresponds to attenuation bands of the periodic structure. It is found that maximal dissimilarity between the neighbouring regions of the unit cell leads to maximal attenuating frequencies. In the extreme case, it is found that more than 98% of the frequencies are blocked. For seismic excitations, it is shown that large, finite periodic structures corresponding to the optimal unit cell derived using the infinite periodic structure theory has significant vibration isolation benefits in comparison to a homogeneous structure or an arbitrarily chosen periodic structure.
TOPICS: Vibration isolation, Periodic structures, Rods, Excitation, Stiffness, Vibration control
research-article  
Hamed Farokhi and Mergen Ghayesh
J. Appl. Mech   doi: 10.1115/1.4041964
The nonlinear large-amplitude oscillation of a cantilever subject to motion constraints is examined for the first time. In order to be able to model the large-amplitude oscillations accurately, the equation governing the cantilever centreline rotation is derived. This allows for analysing motions of very large amplitude even when tip angle is larger than p/2. The Euler-Bernoulli beam theory is employed along with the centreline inextensibility assumption, which results in nonlinear inertial terms in the equation of motion. The motion constraint is modelled as a spring with a large stiffness coefficient. The presence of a gap between the motion constraint and the cantilever causes major difficulties in modelling and numerical simulations, and results in a non-smooth resonance response. The final form of the equation of motion is discretised via the Galerkin technique, while keeping the trigonometric functions intact to ensure accurate results even at large-amplitude oscillations. Numerical simulations are conducted via a continuation technique, examining the effect of various system parameters. It is shown that the presence of the motion constraints widens the resonance frequency band effectively which is particularly important for energy harvesting applications.
TOPICS: Cantilevers, Oscillations, Resonance, Computer simulation, Equations of motion, Electromagnetic spectrum, Modeling, Energy harvesting, Springs, Stiffness, Euler-Bernoulli beam theory, Rotation
research-article  
Gaurav Singh and Tanmay K. Bhandakkar
J. Appl. Mech   doi: 10.1115/1.4041965
The present work proposes a novel strategy to render mixed boundary conditions on circular linear elastic homogeneous domain to displacement based condition all along the surface. With Michell solution as the starting point, the boundary conditions and extent of the domain is used to associate the appropriate type and number of terms in the Airy stress function. Using the orthogonality of sine and cosine functions, the modified boundary conditions lead to a system of linear equations for the unknown coefficients in the Airy stress function. Solution of the system of linear equations provides the Airy stress function and subsequently stresses and displacement. The effectiveness of the present approach in terms of ease of implementation, accuracy and versatility to model variants of circular domain is demonstrated through excellent comparison of the solution of following problems, (i) annulus with mixed boundary conditions on outer radius and prescribed traction on the inner radius, (ii) cavity surface with mixed boundary conditions in an infinite plane subjected to far-field uni-axial loading and (iii) circular disc constrained on part of the surface and subjected to uniform pressure on rest of the surface.
TOPICS: Elasticity, Boundary-value problems, Stress, Displacement, Traction, Cavities, Disks, Annulus, Pressure
research-article  
Sami F. Masri and John Caffrey
J. Appl. Mech   doi: 10.1115/1.4041910
Exact steady-state solutions are obtained for the motion of an SDOF system that is provided with a highly-nonlinear auxiliary mass damper which resembles a conventional dynamic vibration neutralizer (DVN) whose relative motion with respect to the primary system is constrained to remain within a specified gap, thus operating as a ``pounding DVN." This configuration of a conventional DVN with motion-limiting stops could be quite useful when a primary structure with a linear DVN is subjected to transient loads (e.g., earthquakes) that may cause excessive relative motion between the auxiliary and primary systems. Under the assumption that the motion of the nonlinear system under harmonic excitation is undergoing steady-state motion with two impacts per period of the excitation, an exact, closed-form solution is obtained for the system motion. This solution is subsequently used to develop an approximate analytical solution for the stationary response of the pounding DVN when subjected to random excitation with white spectral density and Gaussian probability distribution. Comparison between the analytically-estimated rms response of the primary system and its corresponding response obtained via numerical simulation shows that the analytical estimates are quite accurate when the coupling (tuning parameters) between the primary system and the damper are weak, but only moderately accurate when the linear components of the tuning parameters are optimized. It is also shown that under nonstationary, the pounding DVN provides slightly degraded performance compared to the linear one, but simultaneously limits the damper free motion to specified design constraints.
TOPICS: Vibration absorbers, Random excitation, Dampers, Steady state, Excitation, Statistical distributions, Design, Nonlinear systems, Earthquakes, Computer simulation, Stress, Spectral energy distribution, Transients (Dynamics)
research-article  
Glenn Sinclair, Jeffrey R. Beisheim and Ajay Kardak
J. Appl. Mech   doi: 10.1115/1.4041766
Finite element analysis has become the method of choice for the stress analysis of many of the complex configurations encountered in practice. Such configurations can contain stress singularities. Then it is critical for the necessarily finite estimates from finite elements to be rejected as valid results for the infinite stresses present. There is an extensive literature devoted to the asymptotic identification of stress singularities that can often, but not always, provide a means for such rejection. The present study seeks to offer a further means of rejection: mesh refinement with divergence checks. These divergence checks are a natural counterpart to the convergence checks of ASME. The two are used together on 269 finite element stresses at 32 different singularities: all of these finite element stresses are thus rejected.
TOPICS: Finite element analysis, Stress singularity, Stress, Stress analysis (Engineering)

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