An accurate approximate closed-form solution is presented for bending of thin skew plates with clamped edges subjected to uniform loading using the extended Kantorovich method (EKM). Successive application of EKM together with the idea of weighted residual technique (Galerkin method) converts the governing forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of oblique coordinates system. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration step, exact closed-form solutions are obtained for two ODE systems. It is shown that some parameters such as angle of skew plate have an important effect on results. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plates show very good agreement with results of other analytical and numerical analyses. Also, it has been shown that for skew angle less than 30° this method provides more accurate results and when the skew angle becomes greater than 30°, results gradually begin to deviate from those reported using other methods or by finite element softwares.
Polymer foams are commonly used in the protective packaging of fragile products. Cushion curves are commonly used within the packaging industry to characterize a foam's impact performance. These curves are two‐dimensional representations of the deceleration of an impacting mass versus static stress. Cushion curves are currently generated from exhaustive experimental test data. This study represents the first time that the physics of the mass‐cushion impact have been analysed by modelling the foam as nonlinear, continuous rod. Using a single mode of vibration and excluding the effects of damping, the maximum displacement during the impact can be obtained from a polynomial describing the maximum elastic energy in the foam. The displacements can be used to recover the amplitude of the deceleration shock pulse. Numerical and analytical analysis of the model with damping is considered in its ability to predict the shock pulse shape, duration, and amplitude at various static stresses, foam thickness, and drop heights as compared with experimental data. Furthermore, both the analytical and numerical results agree and are primarily within the expected lab‐to‐lab variability of 18% documented in ASTM D1596 ‐ Standard Test Method for Dynamic Shock Cushioning Characteristics of Packaging Material.
Vehicle vibration presents challenges to a packaged product that is inevitable in any distribution environment. Typically products are tested in only a single, vertical axis; researchers have shown that there is energy in all six axes of motion. In this work, we study the recording methods of the six degrees of freedom (6DOF) motion of a transport vehicle. Using a co‐planar sensor array, three tri‐axial linear accelerometers and three angular rate sensors mounted in a L shape are used to calculate the rotational accelerations that occur in the back of a vehicle. Missing from prior work is a scientific study designed to determine the minimum sensor spacing necessary to accurately capture the yaw, pitch, and roll of transport vehicles. A sensitivity study is conducted to determine the effect of the misplacement and misorientation of sensors on the angular acceleration calculation. A laboratory study is used to determine the distance error begins to accumulate in the angular acceleration calculation in response to a sinusoidal input. A field study is conducted to calculate the rotational motions of a vehicle on a rough road. It is found that a mounting fixture is valuable in assuring the necessary sensor placement accuracy needed to accurately determine angular accelerations of a truck. Additionally, laboratory and field analysis show that as the sensor spacing location approaches the origin sensor, angular acceleration calculation error increases due to a loss in distinctiveness. It is desired for a close sensor mounting array, but there is a trade off between measurement accuracy and compactness of the recording array. A limit exists where the sensors can not be mounted in close proximity.
In this article we propose a theoretical investigation of the nonlinear dynamical response of a class of planar resonators dubbed the V-Shaped resonator. The resonators are intended for energy harvesting purpose and are designed to exhibit two-to-one internal resonance. In particular, we navigate the design space for the generalized V-shaped resonator to investigate the influence of shape parameters on the performance of the Vibration Energy Harvester. Notably, we introduce two metrics that help elucidating the role of the shape parameter in dictating the behavior of the system in terms of peak voltage and operational bandwidth width. For simplicity, we consider that the system is subjected to harmonic excitations near its primary resonances.
Elastic metamaterials can be used in a variety of vibration and shock absorbing applications due to having mechanical properties that are not typically seen in nature. This work creates 2-dimensional metamaterial from a tessellation of unit cells composed of array Chi-springs. The Chi-springs exhibit a J shaped force displacement relationship or equivalently an effective stress strain relationship that is characterized by three regions: a linear stiffness/module, a quasi-zero stiffness/modulus region, and region of high tangent force/stress. The springs form the fundamental building block for a 2-dimensional unit cell and are created through a series of isometric transformations while meeting constraints on its effective constitutive behavior. The orientations of each unit cell are analyzed using Cayley diagrams, which are then used to determine various 2 × 2 tessellations of the structure. The resulting unit cells and their tessellations are manufactured through 3D printing. They exhibit a J shaped effective constitutive relationship in 2 material loading directions that resembles the force displacement relationship. Material testing indicates that the resulting structures are anisotropic. Furthermore, they have a dominant loading direction that displays larger values of linear modulus, plateau stress, and toughness. This work represents the first nascent steps toward the systematic design of metamaterial behavior in multiple load directions.
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