Nonlinear Strain Energy Formulation to Capture the Constraint Characteristics of a Spatial Symmetric Beam

Author(s):  
Shiladitya Sen ◽  
Shorya Awtar

In the past, a beam constraint model (BCM) that captures pertinent geometric nonlinearities associated with large displacements has been proposed for slender spatial beams with uniform and symmetric cross-sections. By providing closed-form parametric relations between the end-loads and end-displacements of the beam, the BCM quantifies the constraint characteristics of the beam in terms of stiffness variations, parasitic error motions, and the cross-axis coupling. This paper presents a nonlinear strain and strain energy formulation for the spatial symmetric beam, based on assumptions that are consistent with the BCM. This strain energy derivation, employing the Principle of Virtual Work, provides a simpler mathematical approach for the analysis of flexure mechanisms with multiple spatial beams. Using this formulation, we obtain the stiffness relations in the transverse bending directions, the constraint relations in the axial and torsional directions, and the overall strain energy expression in terms of the beam end-loads and end-displacements. These expressions, collectively the BCM, are in form that is suitable for the analysis of multi-beam flexure mechanisms.

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Shorya Awtar ◽  
Shiladitya Sen

The beam constraint model (BCM), presented previously, captures pertinent nonlinearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a nonlinear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape, is derived. Using energy-based arguments, new insight into the BCM is elucidated by fundamental relations among its stiffness, constraint, and energy coefficients. The presence of axial load in the geometric constraint and strain energy expressions—a unique attribute of distributed compliance flexures that leads to the elastokinematic effect—is highlighted. Using the principle of virtual work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, of a flexure mechanism comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via nonlinear finite element analysis.


Author(s):  
Shorya Awtar ◽  
Shiladitya Sen

In the past, we have introduced the Beam Constraint Model (BCM), which captures pertinent non-linearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a non-linear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end-displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape is derived. Using the Principle of Virtual Work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, for flexure mechanisms comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via non-linear Finite Element Analysis. Furthermore, the proposed energy formulation leads to new insights into the nature of the BCM.


1970 ◽  
Vol 12 (2) ◽  
pp. 130-134 ◽  
Author(s):  
T. Harrison

Previous studies of the behaviour, in generalized co-ordinates, of thin-walled, prismatic beams of open cross-sectional profile have included, explicitly, only the effects of distributed transverse forces, q x and q y, distributed longitudinal forces, q z, and distributed torsional couples, m z. Using the principle of virtual displacements, the work of previous investigators is extended to include, quite generally, the effects of the hitherto neglected distributed couples, m x and m y. The derivation of the differential equation relating to the twisting of an open-section prismatic beam is presented fully whilst those relating to transverse and axial displacements of cross-sections are merely stated. The kinematic and static boundary conditions for a cantilever are also established from the virtual work equations. These show that the free-end shear boundary condition associated with transverse bending which is usually adopted in engineering calculations is inadequate for such a generalized loading system.


2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a uniform and symmetric cross-section, slender, spatial beam—a basic flexure element commonly used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the nonlinear load–displacement relations at the beam end. Appropriate assumptions are made while formulating the strain and strain energy expressions for the spatial beam to retain relevant geometric nonlinearities. Using the principle of virtual work, nonlinear beam governing equations are derived and subsequently solved for general end loads. The resulting nonlinear load–displacement relations capture the constraint characteristics of the spatial beam in a compact, closed-form, and parametric manner. This constraint model is shown to be accurate using nonlinear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in design and optimization of 3D flexure mechanism geometries, and its elucidation of fundamental performance tradeoffs in flexure mechanism design.


Author(s):  
Akihiro Takezawa ◽  
Shinji Nishiwaki ◽  
Kazuhiro Izui ◽  
Masataka Yoshimura

This paper discuses a new topology optimization method using frame elements for the design of mechanical structures at the conceptual design phase. The optimal configurations are determined by maximizing multiple eigen-frequencies in order to obtain the most stable structures for dynamic problems. The optimization problem is formulated using frame elements having ellipsoidal cross-sections, as the simplest case. Construction of the optimization procedure is based on CONLIN and the complementary strain energy concept. Finally, several examples are presented to confirm that the proposed method is useful for the topology optimization method discussed here.


2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Guimin Chen ◽  
Ruiyu Bai

Modeling large spatial deflections of flexible beams has been one of the most challenging problems in the research community of compliant mechanisms. This work presents a method called chained spatial-beam constraint model (CSBCM) for modeling large spatial deflections of flexible bisymmetric beams in compliant mechanisms. CSBCM is based on the spatial-beam constraint model (SBCM), which was developed for the purpose of accurately predicting the nonlinear constraint characteristics of bisymmetric spatial beams in their intermediate deflection range. CSBCM deals with large spatial deflections by dividing a spatial beam into several elements, modeling each element with SBCM, and then assembling the deflected elements using the transformation defined by Tait–Bryan angles to form the whole deflection. It is demonstrated that CSBCM is capable of solving various large spatial deflection problems either the tip loads are known or the tip deflections are known. The examples show that CSBCM can accurately predict large spatial deflections of flexible beams, as compared to the available nonlinear finite element analysis (FEA) results obtained by ansys. The results also demonstrated the unique capabilities of CSBCM to solve large spatial deflection problems that are outside the range of ansys.


Geosphere ◽  
2021 ◽  
Author(s):  
Jesse E. Robertson ◽  
Karl E. Karlstrom ◽  
Matthew T. Heizler ◽  
Laura J. Crossey

The Surprise Valley landslide complex is the name used here for a group of prominent river-damming landslides in Grand Canyon (Arizona, USA) that has shifted the path of the Colorado River several times in the past 2 m.y. We document a sequence of eight landslides. Three are Toreva-block landslides containing back-rotated but only mildly disrupted bedrock stratigraphy. The largest of these landslides, Surprise Valley landslide, is hypothesized to have dammed the Colorado River, cut off a meander loop through Surprise Valley, and rerouted the river 2.5 km south to near its present course at the Granite Narrows. Another bedrock landslide, Poncho’s runup, involved a mass detachment from the north side of the river that drove a kilometer-scale bedrock slab across the river and up the south canyon wall to a height of 823 m above the river. A lake behind this landslide is inferred from the presence of mainstem gravels atop the slide that represent the approximate spillway elevation. We postulate that this landslide lake facilitated the upriver 133 Mile slide detachment and Toreva block formation. The other five landslides are subsequent slides that consist of debris from the primary slides; these also partially blocked and diverted the Colorado River as well as the Deer Creek and Tapeats Creek tributaries into new bedrock gorges over the past 1 m.y. The sequence of landslides is reconstructed from inset relationships revealed by geologic mapping and restored cross-sections. Relative ages are estimated by measuring landslide base height above the modern river level in locations where landslides filled paleochannels of the Colorado River and its tributaries. We calculate an average bedrock incision rate of 138 m/m.y. as determined by a 0.674 ± 0.022 Ma detrital sanidine maximum depositional age of the paleoriver channel fill of the Piano slide, which has its base 70 m above the river level and ~93 m above bedrock level beneath the modern river channel. This date is within error of, and significantly refines, the prior cosmogenic burial date of 0.88 ± 0.44 Ma on paleochannel cobbles. Assuming steady incision at 138 m/m.y., the age of Surprise Valley landslide is estimated to be ca. 2.1 Ma; Poncho’s runup is estimated to be ca. 610 ka; and diversion of Deer Creek to form modern Deer Creek Falls is estimated to be ca. 400 ka. The age of the most recent slide, Backeddy slide, is estimated to be ca. 170 ka based on its near-river-level position. Our proposed triggering mechanism for Surprise Valley landslides involves groundwater saturation of a failure plane in the weak Bright Angel Formation resulting from large volumes of Grand Canyon north-rim groundwater recharge prior to establishment of the modern Deer, Thunder, and Tapeats springs. Poncho’s and Piano landslides may have been triggered by shale saturation caused by 600–650 ka lava dams that formed 45 river miles (73 river km; river miles are measured along the Colorado River downstream from Lees Ferry, with 1 river mile = 1.62 river kms) downstream near Lava Falls. We cannot rule out effects from seismic triggering along the nearby Sinyala fault. Each of the inferred landslide dams was quickly overtopped (tens of years), filled with sediment (hundreds of years), and removed (thousands of years) by the Colorado River, as is also the potential fate of modern dams.


Author(s):  
Khodayar Javadi ◽  
Aliyar Javadi

A well performance film cooling implies for a high cooling effectives accompanied with a wide cooling coverage. During the past six decades, film cooling effectiveness has been well defined with a specific relation to quantify it. However, despite of numerous film cooling research, there is not an explicit method to quantify the uniformity of a coolant film spread over the hot surfaces. This work introduces a cooling uniformity coefficient (CUC) to evaluate how well a coolant film spreads over a surface being cooled. Four different cases are computationally studied. In the three cases, a single jet is injected into a hot cross flow with different jet exit shapes (i.e. square, spanwise rectangular, and streamwise rectangular). The fourth case is a novel combined triple jet (CTJ) introduced in our previous work. The cross sections of all the systems are equal to maintain the same coolant mass flow rate injection into the hot cross flow. The CUC’s of the different cases are compared with each other at two blowing ratios of 0.5 and 1.5. It is proposed that in addition to the film cooling effectiveness, the CUC is a necessary parameter to evaluate how well a coolant film is spread over a hot surface.


Atoms ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 33
Author(s):  
Jan Hendrik Bredehöft

Electron–molecule interactions have been studied for a long time. Most of these studies have in the past been limited to the gas phase. In the condensed-phase processes that have recently attracted attention from academia as well as industry, a theoretical understanding is mostly based on electron–molecule interaction data from these gas phase experiments. When transferring this knowledge to condensed-phase problems, where number densities are much higher and multi-body interactions are common, care must be taken to critically interpret data, in the light of this chemical environment. The paper presented here highlights three typical challenges, namely the shift of ionization energies, the difference in absolute cross-sections and branching ratios, and the occurrence of multi-body processes that can stabilize otherwise unstable intermediates. Examples from recent research in astrochemistry, where radiation driven chemistry is imminently important are used to illustrate these challenges.


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