Although stiffness analyses of specific parallelogram-type parallel manipulators (PTPMs) have been presented by several researchers, an algebraic expression is still needed to obtain the stiffness of ...a general PTPM. To address this issue, this paper uses a strain energy method considering the compliances of the mobile platform, the limb and the actuator of a PTPM. In this method, the deformation of the mobile platform, which has typically been ignored by many researchers, is integrated in the total deformation of the PTPM. After comparison with a FEA method, it is found that the proposed algebraic method is a comparable alternative to the FEA method to be used in the pre-design stage. Additionally, a new stiffness index is proposed to evaluate the stiffness property. Compared with other stiffness indices, the new index uses virtual work to unify the units of translation and orientation and relates the index value to the direction of the wrench experienced by a parallel manipulator in a task. With this index, the resistance of a PTPM to deformation under a given wrench can be measured easily.
•Stiffness analysis of a general PTPM using an algebraic method.•Result comparison between the proposed method and a finite element analysis method.•A new stiffness index relating the stiffness property to the wrench experienced in a task.
In this paper, we investigate the decay properties of suspension bridge with memories in one dimension. To prove our results, we use the energy method to build some very delicate Lyapunov functionals ...that give the desired results.
•This paper proposed a modified TVMS model to quantitatively evaluate the influence of the tooth wear on the mesh stiffness.•The analytical expression of the modified mesh stiffness considering the ...tooth wear is theoretically derived.•The modified TVMS model is applied in a planetary gear set.•Most tooth wear models can be integrated into the modified TVMS model.
Time-varying mesh stiffness (TVMS), one of the most important parameters in gear dynamics, is usually used to analyze the influence of gear fault in gear transmission systems. In this study, a modified TVMS model is proposed to quantitatively evaluate the influence of the tooth wear on the mesh stiffness. In this model, the potential energy method is employed to calculate the mesh stiffness and the Archard's wear equation is adopted to calculate the tooth wear depth. Furthermore, the analytical expression of the modified mesh stiffness considering the tooth wear is theoretically derived, which can be used for a variety of tooth wear models. The modified TVMS model is applied in a planetary gear set, including the sun-planet mesh and the planet-ring mesh. The results of numerical simulation demonstrate that the tooth wear can result in the reduction of the mesh stiffness; for a planetary gear set, in the initial wear process, the mesh stiffness decreases proportionally when the tooth wear depth increases.
We present a deep energy method for finite deformation hyperelasticitiy using deep neural networks (DNNs). The method avoids entirely a discretization such as FEM. Instead, the potential energy as a ...loss function of the system is directly minimized. To train the DNNs, a backpropagation dealing with the gradient loss is computed and then the minimization is performed by a standard optimizer. The learning process will yield the neural network's parameters (weights and biases). Once the network is trained, a numerical solution can be obtained much faster compared to a classical approach based on finite elements for instance. The presented approach is very simple to implement and requires only a few lines of code within the open-source machine learning framework such as Tensorflow or Pytorch. Finally, we demonstrate the performance of our DNNs based solution for several benchmark problems, which shows comparable computational efficiency such as FEM solutions.
•First machine learning based solution of large deformation hyperelasticity problems.•Highly efficient method once the network is trained.•The minimization of the loss function is similar to the classical principle of minimum potential energy of finding the stationary points. .•Traction free boundary conditions are fulfilled automatically.•Lower requirement on the differentiability compared to deep collocation methods.•The constrained optimization is replaced by an unconstrained optimization problem.
Time-varying mesh stiffness, caused by the change of tooth contact number and contact position, is one of the main sources of vibration of a gear transmission system. In order to comprehensively ...understand the vibration properties of a planetary gear set, it is necessary to evaluate the mesh stiffness effectively. When a crack happens in one gear, the mesh stiffness will decrease and consequently the vibration properties of the gear system will change. This change of vibration can be characterized through dynamic simulation of a gearbox and processed further to detect the crack severity and location. In this paper, the potential energy method is used to analytically evaluate the mesh stiffness of a planetary gear set. A modified cantilever beam model is used to represent the external gear tooth and derive the analytical equations of the bending, shear and axial compressive stiffness. A crack propagation model is developed and the mesh stiffness reduction is quantified when a crack occurs in the sun gear or the planet gear.
•A beam model is developed for the perfect external gear tooth.•Advantages of the proposed beam model are emphasized.•A crack propagation model is proposed for the external gear with crack.•Mesh stiffness equations are derived analytically for the external gear.•Mesh stiffness reduction is quantified when a crack occurs.
The mining of underground coal seam often induces the dynamic disasters of coal seam, such as rock burst and gas outburst. The determination of evolution characteristics of mechanical parameters and ...seepage parameters of coal mass after excavation is the basis for analyzing the formation mechanism of dynamic disasters. In this study, a coupled hydromechanical numerical approach is developed and their complex interactions are analyzed using the energy method to study the influence of outburst factors during underground excavation. A finite element model of porous media, which represents the gas flow field, the coal deformation field, the coal damage field and the cross-couplings between them, is established and the reliability of the model is ensured by numerical and experimental validation. Subsequently, during underground excavation, the coal deformation and gas flow properties around the borehole associated with different factors, including gas sorption, the matrix-fracture characteristics of coal and the seepage characteristics, are analyzed quantitatively using numerical simulations. Furthermore, an energy index is developed based on the energy method to analyze the comprehensive influence of in situ stress and gas pressure on coal mass. The results revealed that in the damage zone, coal permeability increases greatly by one to two orders of magnitude and the gas pressure remains low. Gas adsorption not only affects the stress and deformation of coal, but also affects the extent of the damage zone. High in situ stress, high gas pressure and low coal strength cause the failure range to increase.
Based on the weighted and shifted Grünwald operator, a new high-order compact finite difference scheme is derived for the fractional sub-diffusion equation. It is proved that the difference scheme is ...unconditionally stable and convergent in
L
∞
-norm by the energy method. The convergence order is
O
(
τ
3
+
h
4
)
, where
τ
is the temporal step size and
h
is the spatial step size. Although the unconditional stability and convergence of the difference scheme are obtained for all
α
∈
(
0
,
α
∗
,
where
α
∗
=
0.9569347
,
some numerical experiments show that they are valid for all
α
∈
(
0
,
1
)
.
Finally, some numerical examples are given to confirm the theoretical results.
The specifications of runoff entered the pavement has an essential role in the severity of moisture damage of asphalt concretes. Changing the pH of runoff has a negative impact on moisture damage and ...intensifies this damage in asphalt concretes. Accordingly, in this study, the impact of nanographene oxide (NGO) as a contaminant adsorbent and bitumen modified to decrease the destructive impacts of acidity changes in runoff has been investigated. For this purpose, limestone and siliceous aggregates, PG 64–16 bitumen, and NGO in two different contents (0.3 % and 0.5 % of the weight of bitumen) were utilized. To asses the strength of asphalt concretes with different runoff pH conditions, hydrochloric acid and sodium oxide were used to simulate acidic and basic runoffs, respectively. The indirect tensile strength (ITS) of asphalt concretes in different environmental states was tested to compare the severity of moisture sensitivity. Moreover, the surface free energy (SFE) elements of bitumen and aggregates were calculated utilizing the Wilhelmy Plate (WP) and the Universal Sorption Device (USD), respectively. The parameters of SFE theory were used to assess the mechanisms of water damage potential. The SFE results showed that NGO decreased the debonding energy (DE) of the mixtures. Changes in DE in neutral environments were minor, but with the polarization of environmental conditions, especially in basic states, these changes intensified. The application of NGO improved the total bitumen SFE, which reduced the possibility of moisture sensitivity as a cohesive failure. The results of mechanical methods indicated that the use of NGO in specimens with limestone and siliceous aggregate increased the indirect tensile strength ratio (TSR) in moisture conditions with different degrees of acidity. The mechanical experiment results also confirmed the SFE method's results that the effect of using NGO on increasing the TSR index at pH equal to 7 was minor, and it was intensified in acidic and basic waters.
•NGO enhances the properties of asphalt, its non-polar and adhesive characteristics.•NGO reduces the debonding energy and the stripping potential of asphalt mixes.•Acidic and basic conditions in water have a more stripping detrimental.•NGO increases the resistance of HMA, especially under harsh moisture conditions.
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