This paper investigates the parametric resonance of electrostatically actuated MicroElectroMechanicalSystems (MEMS) cantilever resonators. The electrostatic force is modeled to include fringe effect. The MEMS consists of a cantilever over a parallel ground plate and an AC voltage between them. The actuation frequency is near first natural frequency of the cantilever beam. This leads to parametric resonance. It is of interest to investigate the amplitude frequency response of MEMS cantilever resonators. This paper uses the Homotopy Analysis Method (HAM), which is able to capture nonlinear behaviors for higher amplitudes, large parameters, and strong nonlinearities. The base method used for comparison in this work is the method of multiple scales (MMS). MMS is a perturbation method. It requires a relatively short computational time for simulations. Although the CPU time is advantageous, MMS is only accurate for weak nonlinearities and low amplitudes. It is in the interest to compare how well HAM captures the softening behavior of this system as opposed to MMS. In this paper the influences of Casimir forces and Van der Waals effects are included. Electrostatic, Van der Waals and Casimir forces are nonlinear. HAM is a deformation technique that continuously deforms the initial guess, provided to the procedure, to the exact solution. In this work the first and second order deformation equations are constructed for the equation of motion governing the behavior of the MEMS cantilever beam. In the first order deformation, HAM deviates from the solution obtained by MMS. This deviation demonstrates the power of the method to capture the softening behavior more accurately than MMS even at the 1 st order deformation HAM. In the second order deformation construction, the HAM’s solution softens more than the previous, demonstrating that higher order deformation approximations result in higher accuracy. In the second order deformation, HAM contains the convergence control parameter. This parameter is chosen via the c 0 curve approach. Up to 2 nd order HAM deformations are evaluated for this paper. These higher order homotopy deformation solutions were developed and automated symbolically in the software Mathematica and tested numerically using Matlab software.

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.

A 2-dimensional anatomical knee model was developed for aligning knee joint related bone structures with experimental kinematic data. The experimental data was collected using motion capture cameras, which recorded the position of reflective markers placed on the human subject. Velocities were calculated by numerically differentiating the marker position with respect to time. Joints, such as the hip, knee, and ankle, were represented by axes of rotation. These axes were determined by calculating the relative instantaneous center of rotation of a body segment with respect to the adjacent body segment. Body-fixed coordinate systems were set for both thigh and shin. Anatomical bone structures were obtained from an x-ray and represented mathematically as polynomials. The femoral bone surface was aligned with the experimental data by superimposing the center of rotation of the shin with respect to thigh with the geometric center of the femoral condyle. The tibial surface was aligned with the experimental data by aligning the bones at minimum flexion and then superimposing the tibia with a shared point between femur and tibia. Ligaments were modeled as non-linear elastic springs. Cruciate ligaments were divided into a posterior and anterior ligament fiber bundle. Cruciate ligament forces were calculated for the squatting exercise for five different femoral geometric centers. Geometric centers were determined using a nonlinear least squares optimization technique. Cruciate ligament forces are discussed in this paper.

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.

The purpose of this paper is to investigate the nonlinear dynamics governing the behavior of electrostatically actuated micro electro mechanical systems (MEMS) cantilever undergoing parametric resonance. The MEMS consists of a cantilever parallel to a ground plate. The beam is actuated via an A/C voltage with excitation frequency near first natural frequency of the cantilever. The model includes damping, electrostatic, and Casimir (or van der Waals) forces. The electrostatic force is modeled to include the fringe effect. The amplitude-voltage response of the parametric resonance and the effects of varying the magnitudes of the fringe, Casimir (or Van der Waals), and damping forces along with varying the detuning parameter are reported. The response is obtained using two different methods, namely the method of multiple scales (MMS), and the homotopy analysis method (HAM). In this study approximations up to a 2 nd order HAM are used. HAM is a deformation technique that begins with an initial guess and continuously deforms it to the exact answer. For the 1 st Order HAM, a softening effect is reported. The 1 st Order HAM matches the MMS results in low amplitude and begins to soften and deviate away from the MMS solution in higher amplitudes. For the 2 nd Order HAM deformation the softening effect is slightly more pronounced with a slightly lower prediction of the maximum deflection of the cantilever tip. For the 2 nd order deformation solution the stable branch of the amplitude-voltage response obtained by the HAM shifts leftward from the MMS solution with the unstable branches between the two methods continue to agree in low amplitudes and deviate in high amplitudes. As a remark, the higher order HAM solutions are obtained symbolically with the software Mathematica and numerically ran with the software Matlab.

This work investigates the human gait transition, from walking to running and running to walking. The data obtained was later filtered to obtain the displacement recorded from the Greater Trochanter, Lateral Knee, the Tibia Tuberosity protrusion, Lateral Ankle, Lateral heel and the foot Toe. The results show a trend to predict the transition phase as a change when the relative phase from the markers is evaluated.

In this paper, the Reduced Order Method (ROM) and the Method of Multiple Scales (MMS) are used to investigate the influences of dimensionless damping and voltage parameters on the amplitude-frequency response of an electrostatically actuated double-walled carbon nanotube (DWCNT). The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping forces are assumed. Herein, the coaxial case is investigated. In this mode of vibration, the outer and inner carbon nanotubes move synchronously (in-phase) with the same maximum tip deflection. The DWCNT structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the DWCNT is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order MMS solution. The analytical approximate solutions provided by MMS are validated numerically by two-term (2T) Time Reponses and AUTO-07P. The two methods in this paper are found to be in excellent agreement at lower amplitudes. Additionally, the two methods are assessed for their advantages and limitations. The importance of the results in this paper are the effect of damping and voltage on the stability of the DWCNT vibration.

Casimir effect on superharmonic resonance of electrostatically actuated bio-nano-electro-mechanical system (Bio-NEMS) circular plate resonator sensor is investigated. The plate sensor resonator is clamped at the outer end and suspended over a parallel ground plate. The sensor can be used for detecting human viruses. Superharmonic resonance of the second order, frequency near one-fourth the natural frequency of the resonator, is induced using Alternating Current (AC) voltage. The magnitude of the AC voltage is also large enough to be consider hard excitation acting on the resonator. Beside Casimir effect, other external forces (i.e. electrostatic force and viscous air damping) acting on the MEMS resonator create a nonlinear behaviors such as bifurcation and pull-in instability. Hence, numerical models, such as Method of Multiple Scales (MMS) and Reduced Order Model (ROM), are used to predict the frequency-amplitude response for MEMS resonator. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. While, ROM, based on the Galerkin procedure which uses the mode shapes of vibration of the resonator as a basis of functions, transforms the nonlinear partial differential equation of motion into a system of ordinary differential equation with respect to dimensionless time. The frequency-amplitude response allows one to observe the behavior of the system for a range of frequencies near the superharmonic resonance. The effects of parameters such as Casimir effect, voltage, and damping on the frequency-amplitude response are reported.

This paper deals with aligning knee geometrical anatomical data with kinematic data from experimental work in order to develop a two-dimensional inverse dynamics anatomical model of human knee. Motion capture cameras were used to collect the experimental data for a knee extension exercise. Reflective markers were placed on the subjects’ skin during the experiment. In this model, joints such as hip, knee, and ankle are represented by axes of rotation. These axes are determined by calculating the relative instantaneous center of rotation of one body segment with respect to an adjacent body segment. Body-fixed coordinate systems were defined using three reflective markers attached to the subject. The origin of each body fixed-coordinate system was located between the three markers on that body segment, the body-fixed x-axis was pointing towards the marker on the lateral side of the body segment, and the body-fixed y-axis fell on the same plane as the three reflective markers on the body segment. Moreover, the axis of rotation that represents the hip was determined by calculating the instantaneous center of rotation of reflective markers located on the pelvis with respect to a body fixed coordinate system on the thigh. The axis of rotation on the knee was determined by calculating the instantaneous center of rotation of reflective markers on the shin (tibia) with respect to the body-fixed coordinate system on the thigh (femur). The axis of rotation on the ankle was determined by calculating the instantaneous center of rotation of reflective markers on the shin with respect to a body-fixed coordinate system on the foot. Bone anatomical geometries of femur and tibia were represented mathematically as polynomials and superimposed over the experimental data. This was done by matching the center of rotation from experimental data with the geometric center of the femoral condyle. This is necessary for estimating the insertions/origins of knee ligaments. These ligaments are modeled as nonlinear elastic springs. Furthermore, ligaments were divided into separate fiber bundles. Both the posterior and anterior cruciate ligaments were divided into an anterior and posterior fiber bundle. The cruciate ligament forces for both exercises are discussed in this paper.

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

This work investigates the human leg muscle and ligaments forces during a drop-landing exercise. An inverse dynamics 2-D model of human leg is used on this ballistic task in order to predict these forces. The model consists of three bony structures, namely femur, tibia, and patella. The joints of the model are the knee joint and the hip joint. The ligamentous structure of the knee includes the two cruciate ligaments, Anterior Cruciate Ligament (ACL) and the Posterior Cruciate Ligament (PCL), and the two collateral ligaments, Lateral Collateral Ligament (LCL) and Medial Collateral Ligament (MCL). The system of muscles of the system includes muscle such as quadriceps, hamstrings, gastrocnemius are included in the model. Experimental data used show a maximum of 100 degrees of flexion angle and ground reaction forces up to 4 times the body weight. The inverse dynamics 2-D model consists of an objective function to minimize the muscle forces, and a set of constraints consisting of equality constraints which are the dynamics equations of the bony structures, and inequality constraints in which all muscle forces must be positive. All muscle forces show a pattern in which they reach large magnitudes at the beginning of landing, decreasing as the subject end the exercise with a standing position.

Knee joint biomechanics for various ambulatory exercises was investigated using inverse dynamics in OpenSim, a computational platform for accelerated biomechanics research. A motion capture system was used to collect kinematic and kinetic information during walking, squatting, and the one-legged squat. Pre-processing of the data was accomplished using a MATLAB toolbox. Subsequent modeling, simulation, and analysis was performed in OpenSim. Loading in the cruciate ligaments and select muscles were the primary focus for the analysis. The accuracy of the results is compared with other available work in the literature. The impact, limitations, technology, and novelty were also reviewed in the scope of this work.

In this paper, the Method of Multiple Scales is used to investigate the influences of dimensionless damping and voltage parameters on the amplitude-frequency response of an electrostatically actuated double-walled carbon nanotube. The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping forces are assumed. Herein, the noncoaxial case is investigated at near-zero amplitude conditions in the free vibration, which eliminates the influence of the cubic van der Waals in the first-order solution. The DWCNT structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the DWCNT is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order MMS solution. The importance of the results in this paper are the effect of damping and detuning frequency on the stability of the DWCNT vibration.

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.

This paper investigates the dynamics governing the behavior of electrostatically actuated MEMS cantilever resonators. The cantilever is held parallel to a ground plate (electrode) with an AC voltage between the plate and the electrode causing the electrostatic actuation (excitation). For the purposes of this paper this is soft excitation. The frequency of the excitation is near the natural frequency of the cantilever leading to what is known as parametric resonance. The electrostatic force in the problem investigated throughout the paper is nonlinear in nature and includes the fringe effect. Two methods are used in investigating this problem: the method of multiple scales (MMS) and the homotopy perturbation method (HPM). The two methods work well for small non-linearities and small amplitudes. The influence of voltage, fringe, damping, Casimir, and Van der Waals parameters will be investigated in this paper using MMS and HPM as a means of verifying the results obtained.

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

In this paper, the Method of Multiple Scales is used to investigate the influences of damping and detuning frequency parameters on the amplitude-voltage response of an electrostatically actuated double-walled carbon nanotube. The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Herein, the coaxial case is investigated, which eliminates the influence of the cubic van der Waals in the first-order solution. The double-walled carbon nanotube structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the double-walled carbon nanotube is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order Method of Multiple Scales solution. The importance of the results in this paper are the effect of damping and detuning frequency on the Hopf bifurcations, as these define the intervals of voltage for nonzero amplitudes.

This paper investigates the frequency response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the MEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance insinuates that the AC voltage is operating in a frequency near one-fourth the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir force. For the electrostatic force, the AC voltage is in the category of hard excitation in order to induce a bifurcation phenomenon. For Casimir forces to affect the system, the gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm. 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) is 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. The influences of parameters (i.e. Casimir, damping, second voltage and fringe) were also investigated.