This paper addresses the dynamic characteristics of a beam with a particular elastic boundary condition. In this elastic boundary condition, the lateral and angular displacements of the beam are coupled through the elastic constraints. The dynamic characteristic, namely natural frequencies and mode shapes of vibrations are frequently encountered in the design and modeling of resonant micro-structures. The governing equations of motion of the beam is derived using Euler-Bernoulli beam theory considering the elastic coupling between the transverse and rotational displacements of the beam’s end. The characteristic equation for the natural frequencies and mode shapes of vibration is derived by applying the method of separation of variables to the governing partial differential equation of motion. The natural frequencies and mode shapes of the system are derived for various combinations of compliance values of the elastic support and are compared with those of several special cases, namely clamped-free, clamped-guided, clamped-pinned and clamped-clamped beams.

A nonlinear dynamics investigation is conducted on the frequency-amplitude response of electrostatically actuated micro-electro-mechanical system (MEMS) clamped plate resonators. The Alternating Current (AC) voltage is operating in the realm of superharmonic resonance of second order. This is given by an AC frequency near one-fourth of the natural frequency of the resonator. The magnitude of the AC voltage is large enough to be considered as hard excitation. The external forces acting on the MEMS resonator are viscous air damping and electrostatic force. Two proven mathematical models are utilized to obtain a predicted frequency-amplitude response for the MEMS resonator. Method of Multiple Scales (MMS) allows the transformation of a partial differential equation of motion into zero-order and first-order problems. Hence, MMS can be directly applied to obtain the frequency-amplitude response. Reduced Order Model (ROM), based on the Galerkin procedure, uses mode shapes of vibration for undamped circular plate resonator as a basis of functions. ROM is numerically integrated using MATLAB software package to obtain time responses. Also, ROM is used to conduct a continuation and bifurcation analysis utilizing AUTO 07P software package in order to obtain the frequency-amplitude response. The time responses show the movement of the center of the MEMS circular plate as a function of time. The frequency-amplitude response allows one to observe bifurcation and pull-in instabilities within the nonlinear system over a range of frequencies. The influences of parameters (i.e. damping and voltage) are also included in this investigation.

We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous media flow. The covariance matrix of pressure is constructed via Monte Carlo sampling. The truncated Karhunen–Loève (KL) expansion technique is used to decompose the log-permeability field, as well as the random pressure field resulting from random permeability. We find that although very high-dimensional representation is needed to recover the permeability field when the correlation length is small, the pressure field is not sensitive to high-oder KL terms of input parameter, and itself can be modeled using a low-dimensional model. Thus a low-rank representation of the pressure field in a low-dimensional parameter space is constructed using the truncated KL expansion technique.

In advanced heat transfer courses, a technique exists for reducing a partial differential equation, where the dependent variable is a function of two independent variables, to an ordinary differential equation where that same dependent variable becomes a function of only one. The key to this technique is finding out what the functional form of the similarity variable is to make such a transformation. The difficulty is that the form of the similarity variable is not intuitive, and many heat transfer textbooks do not reveal how this variable is found in classical problems such as viscous and thermal boundary layer theory. It turns out that one way to find this variable is by utilizing the integral technique. By employing the integral technique to boundary layer theory, it will be shown that when the approximate functional relationship for the dependent variable (temperature, velocity, etc) can be represented by an nth order polynomial, the similarity variable can be found very simply. This is seen to be a good tool especially in heat transfer education, but may have applications in research as well. The approach described here is a variation of a well-known technique used for isothermal momentum boundary layer consideration.

For the laminated piezoelectric rectangular plate with large deflection and large rotation, the nonlinear equilibrium differential equations are derived and solved. Firstly, the global Cartesian coordinate system to describe the undeformed geometry and the local orthogonal curvilinear coordinate system to describe the deformed geometry are established respectively on the mid-plane of the plate before and after the deformation, and the relationship between the two coordinates is expressed by transformation matrix. For the convenience of calculation, the expressions of the nonlinear curvatures and inplane strains are obtained by Taylor series expansion. Considering the piezoelectric effect, three equilibrium partial differential equations describing nonlinear bending problems are obtained by the principle of virtual work. Furthermore, in order to simplify the solution process, the stress function is introduced to automatically satisfy the first two equations for the large deformation of the cantilever plate, and the relationship between stress function, the mid-plane internal force and shear force is also given for the first time. Therefore, the stress function and the transversal displacement are the main unknowns of the governing equation and compatibility equation. Additionally, the approximate deflection function and stress function are given which can satisfy all the displacement boundary conditions and only part of the force boundary conditions. Thereby, the generalized Galerkin method is used to obtain the approximate solution of the nonlinear bending problem. Finally, the results in the study are verified by comparison with the results obtained from the finite element method. It also provides a theoretical basis for the engineering application of the large deformation of the piezoelectric cantilever plate.

The superharmonic resonance of second order of microelectro-mechanical system (MEMS) circular plate resonator under electrostatic actuation is investigated. The MEMS resonator consists of a clamped circular plate suspended over a parallel ground plate under an applied Alternating Current (AC) voltage. The AC voltage is characterized as hard excitation, i.e. the magnitude is large enough, and the operating frequency is near one-fourth of the natural frequency of the resonator. Reduced Order Model (ROM), based on the Galerkin procedure, transforms the partial differential equation of motion into a system of ordinary differential equations in time using mode shapes of vibration of the circular plate resonator. Three numerical methods are used to predict the voltage-amplitude response of the MEMS plate resonator. First, the Method of Multiple Scales (MMS) is directly applied to the partial differential equation of motion which is this way transformed into zero-order and first-order problems. Second, ROM using two modes of vibration is numerical integrated using MATLAB to predict time responses, and third, the AUTO 07P software for continuation and bifurcation to predict the voltage-amplitude response. The nonlinear behavior (i.e. bifurcation and pull-in instability) of the system is attributed to the inclusion of viscous air damping and electrostatic force in the model. The influences of various parameters (i.e. detuning frequency and damping) are also investigated in this work.

Nonlinear vibration of a simply-supported Euler-Bernoulli microbeam with fractional Kelvin-Voigt viscoelastic model subjected to harmonic excitation is investigated in this paper. For small scale effects the modified strain gradient theory is used. For take into account geometric nonlinearities the Von karman theory is applied. Beam equations are derived from Hamilton principle and the Galerkin method is used to convert fractional partial differential equations into fractional ordinary differential equations. Problem is solved by using the method of multiple scales and amplitude-frequency equations are obtained for primary, super-harmonic and sub-harmonic resonance. Effects of force amplitude, fractional parameters and nonlinearity on the frequency responses for primary, super-harmonic and sub-harmonic resonance are investigated. Finally results are compared with ordinary Kelvin-Voigt viscoelastic model.

Numerical simulation is a very useful tool for the prediction of physical quantities in two-phase flows. One important application is the study of oil-gas flows in pipelines, which is necessary for the proper selection of the equipment connected to the line during the pipeline design stage and also during the pipeline operation stage. The understanding of the phenomena present in this type of flow is more crucial under the occurrence of undesired effects in the duct, such as hydrate formation, fluid leakage, PIG passage, and valve shutdown. An efficient manner to model two-phase flows in long pipelines regarding a compromise between numerical accuracy and cost is the use of a one-dimensional two-fluid model, discretized with an appropriate numerical method. A two-fluid model consists of a system of non-linear partial differential equations that represent the mass, momentum and energy conservation principles, written for each phase. Depending on the two-fluid model employed, the system of equations may lose hyperbolicity and render the initial-boundary-value problem illposed. This paper uses an unconditionally hyperbolic two-fluid model for solving two-phase flows in pipelines in order to guarantee that the solution presents physical consistency. The mathematical model here referred to as the 5E2P (five equations and two pressures) comprises two equations of continuity and two momentum conservation equations, one for each phase, and one equation for the transport of the volume fraction. A priori this model considers two distinct pressures, one for each phase, and correlates them through a pressure relaxation procedure. This paper presents simulation cases for stratified two-phase flows in horizontal pipelines solved with the 5E2P coupled with the flux corrected transport method. The objective is to evaluate the numerical model capacity to adequately describe the velocities, pressures and volume fraction distributions along the duct.

In recent years, piezoelectric actuators have been extensively utilized in novel technologies such as insect-sized micro air vehicles. Utilization of common piezoceramics and high performance nanocomposite materials coupled with special geometry like trapezoidal plates which are driven at high electric field yields suitable actuators for use in such applications. First, in this paper the nonlinear vibrations of the carbon nanotube reinforced composites cantilever trapezoidal plate with two surface-bonded piezoelectric layers is modeled in accordance to classical laminate plate theory and large deflection Von Karman type equations for the geometric nonlinearity by using the Hamilton’s principle. The geometry of trapezoidal plate is mapped into rectangular computational domain. Second, the Galerkin discretization method is used for changing the partial differential equations into ordinary differential equations. Finally, the governing equations of the motion of piezoelectric laminated carbon nano-tube reinforced composite trapezoidal actuator with cubic nonlinearities under the external excitation and strong electric field with considering electroelastic and electrostrictive effects, is modeled and the linear natural frequency of transversal deflection is obtained.

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

In this study, we present a 1D method to predict the droplet ejection of a drop-on-demand (DoD) inkjet which includes the drop breakup, coalescence, and the meniscus movement at nozzle orifice. A simplified 1D slender-jet analysis based on the lubrication approximation is used to study the drop breakup. In this model, the free-surface (liquid-air interface) is represented by a shape function so that the full Navier-Stokes (NS) equations can be linearized into a set of simple partial differential equations (PDEs) which are solved by method of lines (MOL). The shape-preserving piecewise cubic interpolation and third-order polynomial curve are employed to merge approaching droplets smoothly. The printhead is simplified into a circular tube, and a 2D axisymmetric unsteady Poiseuille flow model is adopted to acquire the relationship between the time-dependent driving pressure and velocity profile of the meniscus. Drop breakup and meniscus movement are coupled together by a threshold of meniscus extension to complete a full simulation of droplet ejection. These algorithms and simulations are carried out using MATLAB code. The result is compared with a high fidelity 2D simulation which was previously developed [10], and good agreement is found. This demonstrates that the proposed method enables rapid parametric analysis of DoD inkjet droplet ejection as a function of nozzle dimensions, driving pressure and fluid properties.

Recent research has identified Adini’s rectangular element as an efficient higher order element for solving second order elliptic partial differential equations such as Poisson’s equation, which governs the steady state heat conduction problem. This type of element includes both the primary field variable and its spatial derivatives as nodal degrees of freedom. Compared to the conventional cubic elements of the Serendipity and Lagrange families, Adini’s element includes the minimum number of nodes per element and has the advantage that the nodal values of the spatial derivatives of the temperature field are directly retrieved from the FEM solution. As a result, the differentiation and averaging procedures that are typically used to obtain the nodal values of the temperature gradients are avoided. In this paper a generalized version of Adini’s element for solving two-dimensional steady state heat transfer problems in non-rectangular geometries is presented. Also, the traditional finite element formulation is modified to allow the application of essential boundary conditions without having to constrain the nodal values of the tangential derivative of the temperature. The resulting higher order element and modified FEM formulation are used to solve an example problem and the accuracy of the solution is compared with solutions obtained using the traditional linear, quadratic, and cubic Serendipity elements to show the efficiency, in terms of accuracy per number of degrees of freedom, of the proposed approach for finding the nodal values of the temperature gradients, which are required to compute the nodal values of the heat flux vector.

The main goal of the current study is developing an advanced and robust numerical tool for accurate capturing heat front propagation. In some applications such as impermeable medium, Heat transfer in the surrounding domain of fracture acts just as a conduction process but the heat transfer through the fractures appears as a convection process. From a mathematical point of view, a parabolic partial differential equation (PDE) should be solved in the surrounding domain whereas a hyperbolic PDE should be solved in the domain of fractures. In fact, they have completely different treatments and this is one of the complicated problems in this area. In this paper, the presence of fractures and discontinuities are considered with the aim of eXtended Finite Element Method (X-FEM). In the proposed numerical approach, the domain is decomposed into local and global scales. Global and local domains are solved by the X-FEM and Least Square Method (LSM) techniques, respectively. As a final result, it is determined that the treatment of coupling term between two scales is one of the most important factors for system performance. Increasing its effect can significantly improve the efficiency of the whole system.

All undergraduate heat transfer textbooks available today give an overriding role to the method of separation of variables to the point of excluding other frameworks and views for solving the underlying governing PDEs. However, the transition from a linear to a nonlinear heat transfer model not only makes the method of separation of variables inapplicable, but also introduces additional mathematical and computational difficulties that must be studied further and overcome. Yet, none of these textbooks discuss integral methods for solving the governing PDEs in heat transfer which are at least as good as the common separation of variables and finite difference techniques taught in the classroom. In this paper, we extend our new methodology from the linear to the nonlinear heat conduction problems by bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs [1]. Integral methods of Von Kármán together with Ritz and Kantorovich methods are used to show our students in the undergraduate heat transfer course how to find approximate analytic solutions to nonlinear multidimensional steady and unsteady conduction problems involving surface radiation and temperature-dependent thermo-physical properties under distinct temperature profiles. The approach has a certain elegance in the sense that it expresses the complete physical effect of the system in terms of a single integral representing the first law of thermodynamics; moreover, the implications of using integral methods in this undergraduate course show the value of mathematical simplification in reducing the order of the governing PDEs and/or the number of associated independent variables. No knowledge of separation of variables or transform methods is needed to obtain an approximate analytic solution to such nonlinear multidimensional steady or unsteady problems with accuracy acceptable by most engineering standards.

In the development of space craft design index, the requirements of hypersonic space craft control accuracy has been increasingly rigorous. Thin-walled structure is often employed in hypersonic craft to reduce the weight of the load and to save the room. During the flight of the craft, temperature field is produced along the surface and the dynamic properties of the craft structure are obviously changed. The decreasing elastic modulus of the structure material and the appearance of thermal stress lead to the decrease of integral rigidity and stability of the structure, then the thermal flutter appears and control difficulties increase. Shape Memory Alloy (SMA) has the advantages of the considerable driving force in the compact volume and the simple driving method. By the combination of actuator structure design and stiffness control, the smart structure is able to make active control to the thermal stiffness variation. In this paper, the apex high-temperature area is equivalent to a ring structure. Finite difference method is employed firstly to transform the governing partial differential equation into discrete finite difference equations. Then the elastic modulus change, thermal stress and tension along the circumference are considered comprehensively to propose the calculation formulas of equivalent young’s modulus. The discrete dynamic matrix model is obtained containing the control terms of SMA. To solve the big-matrix calculation and multiple iterated large data problem, hybrid program is developed with C++ and MATLAB. Finite element software is employed to make optimization analysis to design an expanding loop actuator containing SMA as driving source, variable thickness loops of spring steel as expanding units, and universal-ball pre-loading units. On the basis of that, the thermal stiffness variation active control system with smart structure is developed based on expanding loop SMA actuator. After the analysis of examples, the variation law of the needed SMA driving force is obtained. The distribution position and quantity of the driving source is optimized. This research provides reference for the Theoretical Analysis and Simulation of structure stiffness active control and adaptive control of the aircraft employing smart material. The research results have guiding significance for the smart structure design of hypersonic aircraft in the future.

In this research, boundary control is developed at the upper end of the riser based on the Lyapanuv’s direct method to reduce top angle and transverse vibration of the riser subjected to time-varying disturbance. First, ocean surface current is assumed to be linearly declined to zero from the ocean surface to the ocean floor and then it is assumed to be exponentially declined to zero. The riser is modeled as a distributed parameter system with one partial differential equation (PDE) coupled with boundary conditions (ODE). Since all of the control signals can be measured by sensors or can be calculated by a backwards difference algorithm, so the boundary control is practical and implementable with existing instrumentation. The Lyapunov’s direct method is applied to stability analysis of the closed-loop system. Finally, efficiency of the controller is verified and results of linear and exponential profiles are compared to each other.

A concurrent methodology to design a PID controlled four-bar mechanism with a flexible coupler is proposed. From a mathematical point of view, the design problem is posed as a multi objective dynamic optimization problem constrained by Partial Differential Equations modeling the flexible mechanism. As performance objectives path generation error, tangential velocity error and vibrations at the end effector are considered. We consider the link lengths and PID gains as independent variables. It has been found that mechanical vibrations can be minimized with the proper dimensional synthesis of the mechanism.

Today’s strict fuel economy requirement produces the need for the cars to have really optimized shapes among other characteristics as optimized cooling packages, reduced weight, to name a few. With the advances in automotive technology, tight global oil resources, lightweight automotive design process becomes a problem deserving important consideration. It is not however always clear how to modify the shape of the exterior of a car in order to minimize its aerodynamic resistance. Air motion is complex and operates differently at different weather conditions. This gap can be covered by the use of adjoint solvers which predict the sensibility of the aerodynamic forces to changes of the geometry. Alternatively, Computational Fluid Dynamics (CFD) solvers can be partnered with optimization software which guide model design changes and evaluate the corresponding results. Design changes can be executed by modifying a parameterized geometry or using mesh morphing techniques. With the advances in computational fluid dynamics, design optimization methods in the aerodynamic design are more important than ever. In the present paper, ANSYS Fluent will be used in conjunction with the optimization software ANSYS DesignXplorer to study ways of reducing drag and lift for a simplified car body. ANSYS simulation software allows one to predict, with confidence, the impact of fluid flows on the product throughout design and manufacturing as well as during end use. CFD is a complex technology involving strongly coupled non-linear partial differential equations which attempt to computationally simulate theoretical and experimental models in a discrete domain of complex geometric shape. A detailed assessment of errors and uncertainties has to concern itself with the three roots of CFD: theory, experiment, and computation. Further, the application of CFD is rapidly expanding with the growth in computational resources. The body in question in this study is the Ahmed body [1] which has been used numerous times for CFD code validation. This geometry represents a road legal car which is used to study the effect of different forces like, aerodynamic drag force, lift force, and some other major forces which affect a car’s motion significantly. Despite being a simple body, accurate prediction of its aerodynamic performance often requires very accurate and computationally expensive calculations. We would like to investigate if meaningful optimizations can be achieved by using reduced resources, by analyzing how air at different velocity affect the body and what changes might be necessary for a further optimized performance. The purpose here is not to predict the absolute values of drag for this body, but to demonstrate that optimization can be performed with limited resources relying on information about drag deltas rather than absolute values. Keeping limiting resources in mind, a grid independence study wasn’t done.

This paper deals with electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) circular plates. The system consists of a flexible circular plate with the edge fixed above a parallel ground plate. The forces acting on the MEMS plate are the electrostatic, damping, and elastic restoring force. In this work Casimir and/or van der Waals forces are neglected, since they are significant for gaps less than one micron and/or 50 nanometers, respectively. The electrostatic force is given by a soft Alternate Current (AC) harmonic voltage between the two plates. The electrostatic force leads the flexible plate into vibration. The assumption of axisymmetrical vibrations is valid in this work. The AC frequency is near natural frequency of the flexible circular plate. Since the electrostatic force is proportional to the square of the voltage, this results in an electrostatic force of frequency twice the natural frequency. Therefore the system experiences a parametric resonance. The partial differential equation of motion is non-dimensionalized. Next the resulting lumped parameters are found from a typical MEMS circular plate resonator. Reduced Order Model (ROM) is the method of investigation used in this work, specifically two terms ROM. Numerical simulations are conducted to predict the voltage response of the system. The effects of frequency and damping on the response are predicted as well. MEMS circular plate systems are excellent candidates for resonator sensors, i.e. sensors functioning at resonance. They are capable of detecting cells, viruses, as well as any microparticles provided the plates are coated for such recognition.

High speed machining is widely used in manufacturing. For its high cutting speed, high feeding rate, and high machining accuracy, its requirements for cutting trajectory in high speed machining are so strict that only continuous and smooth trajectories with even cutting loads can lead to high machining efficiency and accuracy. The traditional row or ring machining trajectory fails to meet these requirements. In order to acquire the continuous and smooth machining trajectories, and to avoid load mutation in the machining process, some researchers developed a curvilinear cutting trajectory generating method based on partial differential equation. This trajectory, however, is still made up from free form curve segment, and unable to completely eliminate the effects on smoothness by line segment interpolation, which has a very adverse effect on the efficiency, tool life and machining accuracy. A new strategy to generate a continuous and smooth cavity cutting trajectories in high speed machining is introduced in this article. The new trajectory determines the necessity-nodes from outside (the boundary) to inside in the way of spiral cutting. It starts from the cavity center with spiral expanding, and each cutting loop adopts end connection between straight line and the tangent arc, meeting continuous first-order constraint satisfaction. The smooth trajectory reduces the amplitude and directing mutation of cutting force, thus effectively avoids the impact on the machining efficiency and machining accuracy by speeding down in the corner. The strategy also, by controlling the row space, ensures that there is no cutting residual. A cavity machining programming system based on this strategy is developed on Siemens UG-CAM module. The manufacturing of triangle cavities is studied as a case. It turns out that the new trajectory improves efficiency by 26.23% compared with the traditional one. It ensures a stable operation of the cutting tool in machining, therefore effectively extends the tool life. The main advantages are that the new strategy adopts the geometrical drawing strategy and the trajectories are all made up from the straight lines and the tangent arcs. The trajectory can greatly reduce NC code. It thoroughly removes the effort to mind the smooth, continuity and even cutting load of the tool-path.