The dynamics of a DC-AC self-oscillating LC resonant inverter with a zero current switching strategy is considered in this paper. A model that includes both the series and the parallel topologies and accounts for parasitic resistances in the energy storage components is used. It is found that only two reduced parameters are needed to unfold the bifurcation set of this extended system: one is related to the quality factor of the LC resonant tank, and the other accounts for the balance between serial and parallel losses. Through a rigorous mathematical study, a complete description of the bifurcation set is obtained and the parameter regions where the inverter can work properly is emphasized.
Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for two-member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with (Formula presented.) and (Formula presented.) species revealed, respectively, coexistence via strongly and weakly damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of four-member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability cannot be fully characterized by linearizing near the fixed point. In this letter, we determine the stability of a non-elementary four-member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point cannot undergo a Hopf bifurcation, as supported by numerical simulations previously reported.
Electrostatic parallel-plate actuators are a common way of actuating microelectromechanical systems, both statically and dynamically. Nevertheless, actuation voltages and oscillations are limited by the nonlinearity of the actuator that leads to the pull-in phenomena. This work presents a new approach to obtain the electrostatic parallel-plate actuation voltage, which allows to freely select the desired frequency and amplitude of oscillation. Harmonic Balance analysis is used to determine the needed actuation voltage and to choose the most energy-efficient actuation frequency. Moreover, a new two-sided actuation approach is presented that allows to actuate the device in all the stable range using the Harmonic Balance Voltage.
This is the peer reviewed version of the following article: “Fargas Marques, A., Costa Castelló, R. (2017) Energy-efficient full-range oscillation analysis of parallel-plate electrostatically actuated MEMS resonators, 1-13.” which has been published in final form at [doi: 10.1007/s11071-017-3633-8]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."
The dynamic analysis and simulation of human gait using multibody dynamics techniques has been a major area of research in the last decades. Nevertheless, not much attention has been paid to the analysis and simulation of robotic-assisted gait. Simulation is a very powerful tool both for assisting the design stage of active rehabilitation robots and predicting the subject–orthoses cooperation and the resulting aesthetic gait. This paper presents a parameter optimization approach that allows simulating gait motion patterns in the particular case of a subject with incomplete spinal cord injury (SCI) wearing active knee–ankle–foot orthoses at both legs. The subject is modelled as a planar multibody system actuated through the main lower limb muscle groups. A muscle force-sharing problem is solved to obtain optimal muscle activation patterns. Furthermore, denervation of muscle groups caused by the SCI is parameterized to account for different injury severities. The active orthoses are modelled as external devices attached to the legs, and their dynamic and performance parameters are taken from a real prototype. Numerical results using energetic and aesthetic objective functions, and considering different SCI severities are obtained. Detailed discussions are given related to the different motion and actuation patterns both from muscles and orthoses. The proposed methodology opens new perspectives towards the prediction of human-assisted gait, which can be very helpful for the design of new rehabilitation robots.
The aim of this paper was to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d'Alembert-Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.
The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(¿)=U(¿ 1,¿ 2,¿ 3). This motion subject to the constraint <¿,¿>=0 with ¿ is a constant vector is known as the Suslov problem, and when ¿=¿ is the known Veselova problem, here ¿=(¿ 1,¿ 2,¿ 3) is the angular velocity and <¿,¿> is the inner product of R3 .
We provide the following new integrable cases.
(i) The Suslov’s problem is integrable under the assumption that ¿ is an eigenvector of the inertial tensor I and the potential is such that
where I 1,I 2, and I 3 are the principal moments of inertia of the body, µ 1 and µ 2 are solutions of the first-order partial differential equation
(ii) The Veselova problem is integrable for the potential
where ¿ 1 and ¿ 2 are the solutions of the first-order partial differential equation where p=I1I2I3(¿21I1+¿22I2+¿23I3)--------------v .
Also it is integrable when the potential U is a solution of the second-order partial differential equation where t2=I1¿21+I2¿22+I3¿23 and t3=¿21I1+¿22I2+¿23I3 .
Moreover, we show that these integrable cases contain as a particular case the previous known results.
The hysteretic behavior is an essential feature of many physical systems (e.g. mechanical structures, buildings dampers). Such a feature is conveniently accounted for in hysteretic systems’ modeling through the well-known Bouc–Wen equations. But these involve several unknown parameters and internal signals that are not all accessible to measurements. Furthermore, not all parameters come in linearly. All these difficulties make the identification of hysteretic systems a challenging problem. To cope with these issues, previous works have simplified the problem by supposing that the system displacements are large, the restoring force (and other internal signals) are accessible to measurements, the displacement is the actual control signal, the unknown parameter entering nonlinearly is known or is an integer, etc. In fact, these restrictive assumptions amount to supposing, among others, that the Bouc–Wen equations describe an isolated physical element in which ‘hysteretis’ is the only dynamic feature. The point is that the control input should be an external driving force and not the displacement. In this paper, the hysteretic equations are let to be what they really are in most practical situations: just a part of the system dynamics. Such a more realistic problem statement has a cost, which is in additional unknown parameters. A multi-stage parametric identification scheme is designed in this paper and shown to recover consistently the unknown system parameters. The proposed solution is suitable for systems not tolerating large displacements (e.g. buildings) as well as for situations where force, velocity and acceleration sensors are not available.
The Bouc–Wen model for smooth hysteresis has received an increasing interest in the last few years due to the ease of its numerical implementation and its ability to represent a wide range of hysteresis loop shapes. This model consists of a first-order nonlinear differential equation that contains some parameters that can be chosen, using identification procedures, to approximate the behavior of given physical hysteretic system. Despite a large body of literature dedicated to the Bouc–Wen model, the relationship between the parameters that appear in the differential equation and the shape of the obtained hysteresis loop is little understood. The objective of this paper is to fill this gap by analytically exploring this relationship using a new form of the model called the normalized one. The mathematical framework introduced in this study formalizes the vague notion of “loop shape" into precise quantities whose variation with the Bouc–Wen model parameters is analyzed. In light of this analysis, the parameters of Bouc–Wen model are re-interpreted.
This paper deals with the problem of characterizing analytically the limit cycle of the Bouc–Wen model. This question arises often in parameter identification issues where the input is chosen to be periodic and the experimentally obtained limit cycle is then used to determine the model parameters. However, it has never been proved analytically that a T-periodic input leads to a T-periodic output for the Bouc–Wen model. Furthermore, an analytical expression of the limit cycle is lacking. The objective of this paper is to fill this gap by proving that the response of the Bouc–Wen model to a class of T-periodic inputs of practical interest in identification procedures is T-periodic. We also provide an exact explicit description of the limit cycle which will be used in the companion paper to derive an identification method for the Bouc–Wen model parameters.
This paper deals with the problem of identifying the parameters of the hysteretic Bouc–Wen model. In the existing literature, the methods devoted to this problem rely mainly on numerical simulations and do not have, to a very large extent, a rigorous mathematical justification. Our method consists in exciting the hysteretic system with a periodic input and obtain the desired parameters from the resulting limit cycle. The identification method that we propose has a rigorous mathematical basis as it based on the analytic description of the limit cycle, and, unlike existing identification methods for the Bouc–Wen model, gives guaranteed relative errors between the (unknown) exact model parameters and their corresponding estimates. We also prove that this method is robust with respect to constant and T-periodic disturbances commonly present in any laboratory experiment. A numerical simulation example illustrates the use of our identification method.