Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin. If spun in, say, the clockwise direction, the rattleback will exhibit stable rotary motion. If spun in the counter-clockwise direction, the rattleback’s rotary motion will transition to a rattling motion, and then reverse its spin resulting in clockwise rotation. This counter-intuitive dynamic behavior has long been a favored subject of study in graduate-level dynamics classes. Previous literature on rattleback dynamics offer insight into a myriad of advanced topics, including three-dimensional motion, sliding and rolling friction models, stability regions, nondimensionalization, etc. However, it is the current authors’ view that focusing on these advanced topics clouds the students’ understanding of the fundamental kinetics of the body. The goal of this paper is to demonstrate that accurately simulating rattleback behavior need not be complicated; undergraduate engineering students can accurately model the behavior using concepts from introductory dynamics and numerical methods. The current paper develops an accurate dynamic model of a rattleback from first principles. All necessary steps are discussed in detail, including computing the mass moment of inertia, choice of reference frame, conservation of momenta equations, and application of kinematic constraints. Basic numerical techniques like Gaussian quadrature, Newton-Raphson root-finding, and Runge-Kutta time-stepping are employed to solve the necessary integrals, nonlinear algebraic equations, and ordinary differential equations. Since not all undergraduate engineering students are familiar with 3D dynamics, a simpler 2D rocking semi-ellipse example is first introduced to develop the transformation matrix between an inertial reference frame and a body-fixed reference frame. This provides the framework to transition seamlessly into 3D dynamics using roll, pitch, and yaw angles, concepts that are widely understood by engineering students. In fact, when written in vector notation, the governing equations for the rocking ellipse and the spin-biased rattleback are shown to be the same, enforcing the concept that 3D dynamics need not be intimidating. The purpose of this paper is to guide a typical undergraduate engineering student through a complex dynamic simulation, and to demonstrate that he or she already has the tools necessary to simulate complex dynamic behavior. Conservation of momenta will account for the dynamics, intimidating integrals and differentials can be tackled numerically, and classic time-stepping approaches make light work of nonlinear differential equations.

With the advance of microfluidic platforms and due to the need to solve different implications that still exist on the transport of electrically conducting fluids, the analysis on strategies in micropumps that involve a simplicity in its structure, absence of mechanical moving parts, flow reversibility and low power requirement is current. Therefore, the present investigation contributes with the analysis of the combined magnetohydrodynamic/pressure driven flow of multilayer immiscible fluids in a microchannel formed by two parallel flat plates. The mathematical model is based in a steady fully developed flow and the pumped fluids follow the power law model to describe the pseudoplastic fluids rheology, while magnetic effects on the flow are given from the Lorentz forces. The velocity profiles and flow rate are obtained in the limit of small Hartmann numbers by solving analytically a closed system of ordinary differential equations, together to the corresponding boundary conditions at the solid-liquid interfaces in the channel walls and at the liquid-liquid interfaces between the fluid layers. The results show that the flow field is controlled by the dimensionless parameters that arise from the mathematical modeling being a parameter that indicates the competition between pressure to the magnetic forces, magnetic parameters related to Hartmann numbers, viscosities ratios between the fluids, flow behavior indexes and the dimensionless position of the liquid-liquid interfaces.

In this paper, the effect of absorption of antigens to the functionalized surface of a biosensor is modeled using a single degree-of-freedom mass-spring-damper system. The change in the mass of the system due to absorption is modeled with an exponential function. The governing equations of motion is derived considering the change in the mass of the system as well as the impact force due to absorption. It has been demonstrated that this equation is a linear second-order ordinary differential equation with time-varying coefficients. The solution of this differential equation is approximated by expanding the exponential function with a Taylor series and applying the method of multiple scales. The advantage of using the method of multiple scales to derive an approximate solution is in the insight it provides on the effect of each parameter on the response of the system. The free vibration response of the biosensor is derived using the approximate solution under different conditions, namely, with and without viscous damping, with and without considering the impact force, and for different binding rates.

Balance control naturally deteriorates with age, so it comes as no surprise that nearly 30% of the elderly population in the United States report stability problems that lead to difficulty performing daily activities or even falling. Postural stability is an integral task to daily living which is reliant upon the control of the ankle and hip. To this end, the estimation of ankle and hip parameters in quiet standing can be a useful tool when analyzing compensatory actions aimed at maintaining postural stability. Using an analytical approach, this work builds upon the results obtained by the authors and expands it to a two degrees of freedom system where the first two modes of vibration of a standing human are considered. The physiological parameters a second-order Kelvin-Voigt model were estimated for the actuation of the ankle and hip. Estimates were obtained during quiet standing when healthy volunteers were subjected to a step-like perturbation. This paper presents the analysis of a second-order nonlinear system of differential equations representing the control of lumped muscle-tendon units at the ankle and hip. This paper utilizes motion capture measurements to obtain the estimates of the control parameters of the system. The dynamic measurements are utilized to construct a simple time-dependent regression that allows calculating the time-varying estimates of the control and body segment parameters with a single perturbation. This work represents a step forward in estimating the control parameters of human quiet standing where, usually, the analysis is either restricted to the first vibrational mode of an inverted pendulum model or the control parameters are assumed to be time-invariant. The proposed method allows for the analysis of hip related movement in the control of stability and highlights the importance of core muscle training.

Nearly everyone, throughout their life, is at risk of being involved in a serious traumatic event, such as motor vehicle accidents, sports and occupational injuries, or natural disaster related injuries. Twenty-eight percent of trauma patients precipitously develop abnormalities in their blood coagulation system. These coagulopathies increase their mortality rate by 5fold. The current coagulopathy diagnosis protocol collects basic patient information, vital signs, and performs traditional lab and point-of-care (POC) blood testing. A high-stakes decision must then be made by the trauma surgeon, using their intuition, training, and the results from the blood drawn at least 15 minutes prior, to determine the requirement for a resuscitation treatment through coagulation inhibitors or activators. Computational modeling and system analysis of the human blood coagulation are integral to developing superior decision support tools for trauma surgeons. In short, the coagulation system consists of the following functional subsystems: 1) blood flow, 2) platelet function, 3) diffusion, 4) advection, and 5) biochemical kinetics. We utilize a combined approach of both 0-D and 3-D model development with the overarching goal of developing a validated, near real-time decision support system. The biochemical kinetics of the coagulation system is implemented in the 0-D model with a set of 113 nonlinear, coupled ordinary differential equations (ODEs), describing the time rate of change of the numerous chemical concentrations and their interaction with one another. 0-D models provide a fast, efficient means of simulating the coagulation biochemical kinetics, but these ODEs lack the ability to describe the global effects of fluid flow, advection, and diffusion. Hence, the set of 113 ODEs are modeled as source terms and combined with the Navier-Stokes and chemical advection/diffusion equations in a three-dimensional finite volume computational domain, providing a global coagulation model. Model validation studies employ parallel experimental POC blood testing and 3-D computational modeling. Results from the 0-D model are consistent with testimonials from expert trauma surgeons, whom verify the model provides appropriate reasoning for their difficulties in predicting patient outcome. Thus, validated computational models have potential as a hypothesis generator used for developing new approaches for providing trauma surgeons with sufficient information to make better informed clinical decisions, “the decision support tool,” leading to decreased mortality.

In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

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.

In this paper, period motions in a periodically forced, damped, double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the period motions are predicted through eigenvalue analysis. Numerical simulation of the period motions in the double pendulum is completed from analytical predictions.

To study the impact of compliant terrains on the biomechanics of rapid legged movements, a well-known spring loaded inverted pendulum (SLIP) model is deployed. The model is a three-degrees-of-freedom system (3 DOF), inspired by galloping greyhounds competing in a racing condition. A single support phase of hind-leg stance in a galloping gait is taken into consideration due to its primary function in powering the greyhounds locomotion and higher rate of musculoskeletal injuries. To obtain and solve the nonlinear second-order differential equation of motions, the Lagrangian method and MATLABb R2017b (ode45 solver), which is based on the Runge-Kutta method, has been used, respectively. To get the viscoelastic behavior of compliant terrains, a Clegg hammer test was developed and performed five times on each sample. The effective spring and damping coefficients of each sample were then determined from the hysteresis curves. The results showed that galloping on the synthetic rubber requires more muscle force compared with wet sand. However, according to the Clegg hammer test, wet sand had a higher impact force than synthetic rubber which can be a risk factor for bone fracture, particularly hock fracture, in greyhounds. The results reported in this paper are not only useful for identifying optimum terrain properties and injury thresholds of an athletic track, but also can be used to design control methods and shock impedances for legged robots performing on compliant terrains.

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.

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.

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.

The demand of using artificial muscle similar to the human muscle is significantly increased during past decades. Recently, silver-plated Twisted and Coiled Polymer (TCP) muscle was employed in many research projects. A first order differential equations (1st ODE) was used to predict the force of this muscle, assuming that the TCP muscle acts similar to a mechanical spring that has variable stiffness depending on the electrical power supplied. Thus, extensive study should be performed on different types of TCP muscles to reach a conclusion. In this paper, a black box system identification method is used to examine the behavior of TCP muscles under different input conditions. Different order differential equations are compared with experimental results. Prediction error method (PEM) is used for estimation of the force of silver-plated TCP muscle with several linear time invariant (LTI) discrete time state space system. In addition, we suggest a fast method (rule of thumb) to model a TCP muscle. Moreover, two key parameters have been introduced to compare the quality of the TCP muscle from force perspective.

One of the main objectives of the structural health monitoring by piezoelectric wafer active sensor (PWAS) using electromechanical impedance method is continuously damage detection applications. In present work impedance method of beam structure is considered and the effect of early crack using breathing crack modeling is studied. In order to model the effect of a crack in beam, the beam is connected with a rotational spring in crack location. The Rayleigh–Ritz method is used to generate ordinary differential equation of cracked beam. Firstly, only open crack is considered that this is leads to linear system equation. In linear system, time domain system equations are converted to frequency domain, and then impedance of PWAS in frequency domain is calculated. Secondly, the breathing crack is modeled to be fully open or fully closed. This phenomenon leads to the nonlinear system equations. These nonlinear equations are solved using pseudo-arc length continuation scheme and collocation method for any harmonic voltage applied to actuator. Then impedance of PWAS is calculated. Two methods are used to detect early crack using breathing crack modeling on PWAS impedance. At the first, frequency response of breathing crack in the frequency range with its sub-harmonics is calculated. Second, only frequency response of one harmonic is computed with its super-harmonics. Finally, the detection method of linear is compared with nonlinear model.

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

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.

Twisted and Coiled Polymers (TCP) muscles are actuators that generate force and linear displacement in response to thermal stimuli. Their length changes significantly by heating due to a high negative coefficient of thermal expansion (CTE). A mathematical model for predicting the behavior of TCP muscles is essential for exploiting maximum advantage from these actuators and also controlling them. In this work, a simple, practical, and accurate model for predicting the displacement of TCP muscles, as a function of input electrical actuation and load, is derived. The problem is broken down into two, i.e. modeling of the thermal and thermo-elastic part. For the first part, a differential equation with changing electrical resistance term is derived. In the next step, by using a temperature-dependent modulus of elasticity and CTE as well as taking the geometry of muscles into account, an expression for displacement as a function of temperature and load is proposed. Experimental actuation data of a TCP muscle is used for verifying the model and investigating its accuracy. The thermal part shows a good agreement between the simulation and experimental result. The displacement part also has a good accuracy for medium and high actuation currents but there is a mismatch in very high current magnitudes. The cause of the discrepancy is explained and recommendations are made for the best performance of TCP muscles.

Structures such as beams and plates can produce unwanted noise and vibration. An emerging technique can reduce noise and vibration without any additional weight or cost. This method focuses on creating two dimples in the same and opposite direction on a beam’s surface where the effect of dimples on its natural frequencies is the problem of interest. The change in the natural frequency between both cases have a different trend. The strategic approach to calculate natural frequencies is as follows: first, a boundary value model (BVM) is developed for a beam with two dimples and subject to various boundary conditions using Hamilton’s Variational Principle. Differential equations describing the motion of each segment are presented. Beam natural frequencies and mode shapes are obtained using a numerical solution of the differential equations. A finite element method (FEM) is used to model the dimpled beam and verify the natural frequencies of the BVM. Both methods are also validated experimentally. The experimental results show a good agreement with the BVM and FEM results. A fixed-fixed beam with two dimples in the same and opposite direction is considered as an example in order to compute its natural frequencies and mode shapes. The effect of dimple locations and angles on the natural frequencies are investigated. The natural frequencies of each case represent a greater sensitivity to change in dimple angle for dimples placed at high modal strain energy regions of a uniform beam.

Analysis of local entropy generation is an effective means to investigate sources of efficiency loss in turbulent combustion from the standpoint of the second law of thermodynamics. A methodology, termed the entropy filtered density function (En-FDF), is developed for large eddy simulation (LES) of turbulent reacting flows to include the transport of entropy, which embodies the complete statistical information about entropy variations within the subgrid scale. The modeled En-FDF contains a stochastic differential equation (SDE) for entropy which is solved by a Lagrangian Monte Carlo method. In this study, a numerical study has been done on effectiveness of SDE to model entropy variation using a partially stirred reactor (PaSR). This provides a computationally affordable case to compare different effects of entropy generation source terms and fine tune mixing coefficients. In this equation, turbulent mixing is modeled with Interaction by Exchange with the Mean (IEM). Combustion source terms are provided by direct integration of a GRI3.0 mechanism for methane/air system. Evolution of entropy was calculated from stochastic model and then compared with the one obtained directly by integrating the chemical mechanism. It was shown that results of both calculations have very good agreement versus different mixture fractions.