This paper describes an experimental and theoretical approach to evaluate cycloid drive reducer efficiency. The tests are carried out on 7.5 kW two-disc cycloid drive with a gear ratio of 19. The torque and speed are measured on the input and output shaft. The efficiency is calculated based on the obtained results. The main goal of the second part of the study is to deduce equations of cycloid reducer in order to predict and analyze experimental results. In this way, the following points are set for the simulation: a working condition in which the input speed and the output load are imposed; then, the output speed is determined by the gear ratio, and finally, the input torque is obtained by solving the dynamic problem. A new model for cycloidal reducers is proposed. This model is based on kinematics and dynamics of rigid bodies and a non-linear stiffness based on contact dynamics. The overall elasticity effects are all condensed between the input shaft and the cycloidal disk. The proposed model allows to predict the efficiency for several operational conditions and offer a drastic reduction of computational costs suitable for the optimization process.

Comparing to linear vibration absorbers, nonlinear energy sinks (NESs) have attracted worldwide attention for their intrinsic characteristic of targeted energy transfer (TET) or energy pumping (EP) in a relatively wide frequency range. Unfortunately, they are highly dependent on the vibration amplitude to be attenuated, and will play its role only if the external load exceeds a specific threshold value. Different from the passive bistable NES, a novel piezoelectric nonlinear energy sink (PNES) is designed by introducing in-phase actuation to compensate or enhance the external vibration loads, thus triggering the NES operating in high attenuation efficiency. The nonlinear mathematic model of PNES is established for investigating the dynamic response and determining the threshold compensation strategy. And the results show that the maximum attenuation efficiency can be improved by 58.16% compared to traditional passive NES. Also, the amplitude dependent coefficient (ADC) can be significantly reduced to 0.33 from 1.0, which means that the PNES can effectively mitigate vibrations even when the excitation amplitude is 67% smaller than the original threshold value. Finally, the feasibility of the in-phase compensation method is experimentally validated, which can further expand the application range of NES.

The inertia representation of a constrained system includes the formulation of the kinetic energy and its corresponding mass matrix, given the coordinates of the system. The way to find a proper inertia representation achieving better numerical performance is largely unexplored. This paper extends the modified inertia representation (MIR) to the constrained rigid multibody systems. Using the orthogonal projection, we show the possibility to derive the MIR for many types of non-minimal coordinates. We present examples of the derivation of the MIR for both planar and spatial rigid body systems. Error estimation shows that the MIR is different from the traditional inertia representation (TIR) in that its parameter γ can be used to reduce the kinetic energy error. With preconditioned γ , numerical results show that the MIR consistently presents significantly higher numerical accuracy and faster convergence speed than the TIR for the given variational integrator. The idea of using different inertia representations to improve the numerical performance may go beyond constrained rigid multibody systems to other systems governed by differential-algebraic equations.

In this paper, we study low-velocity multiple impacts of chains of balls and cylinders. We use three types of materials including steel, wood, and rubber. We perform collision experiments of balls and cylinders of three-, four-, and five-body chains and obtain their corresponding pre- and post-impact velocities. Although the common solution methods such as rigid body and bimodal models can not deal with this type of problems, we use a discretization method presented in Alluhydan et al. (2019, “On Planar Impacts of Cylinders and Balls,” ASME J. Appl. Mech., 86 (7), p. 0710091) to accurately calculate the post-impact velocities of the colliding bodies in the chain. We use the bimodal Hertz–Crook contact force model and employ the ball-ball coefficient of restitution at the contacting segments to analyze the impact dynamics of the colliding objects. A segmentation method is used for ball-cylinder and cylinder-cylinder Collisions. The number of segments for each collision is determined by using a relationship that relates the stiffness ratio of two objects to the number of the segments. In ball-ball collisions, however, we do not use the segmentation method. The outcomes demonstrate that the overall average percentage error of post-impact velocities among experimental results and numerical predictions was less than 6.2% for all results.

This paper investigates the nonlinear static response as well as nonlinear forced dynamics of a clamped–clamped beam actuated by piezoelectric patches partially covering the beam from both sides. This study is the first to develop a high-dimensional nonlinear model for such a piezoelectric-beam configuration. The nonlinear dynamical resonance characteristics of the electromechanical system are examined under simultaneous DC and AC piezoelectric actuations, while highlighting the effects of modal energy transfer and internal resonances. A multiphysics coupled model of the beam-piezoelectric system is proposed based on the nonlinear beam theory of Bernoulli–Euler and the piezoelectric constitutive equations. The discretized model of the system is obtained with the help of the Galerkin weighted residual technique while retaining 32 degrees-of-freedom. Three-dimensional finite element analysis is conducted as well in the static regime to validate the developed model and numerical simulation. It is shown that the response of the system in the nonlinear resonant region is strongly affected by a three-to-one internal resonance.

The size and distribution of particles suspended within a fluid influence the rheology of the suspension, as well as strength and other mechanical properties if the fluid eventually solidifies. An important motivating example of current interest is foamed cements used for carbon storage and oil and gas wellbore completion. In these applications, it is desired that the suspended particles maintain dispersion during flow and do not coalesce or cluster. This paper compares the role of mono- against polydispersity in the particle clustering process. The propensity of hard spherical particles in a suspension to transition from a random configuration to an ordered configuration, or to form localized structures of particles, due to flow is investigated by comparing simulations of monodisperse and polydisperse suspensions using Stokesian dynamics. The calculations examine the role of the polydispersity on particles rearrangements and structuring of particles due to flow and the effects of the particle size distribution on the suspension viscosity. A key finding of this work is that a small level of polydispersity in the particle sizes helps to reduce localized structuring of the particles in the suspension. A suspension of monodisperse hard spheres forms structures at a particle volume fraction of approximately 47 % under shear, but a 47 % volume fraction of polydisperse particles in suspension does not form these structures.

The kinetic energy of a mass moving horizontally can be completely converted into potential energy using a spring as an intermediary. The spring can be used to temporarily store some of the energy of the mass and change the direction of motion of the mass from horizontal to vertical. A nondimensional framework is used to study this problem for a point mass, first with a linear spring and then with a nonlinear spring that is an elastica. Solutions to the problems with the linear spring and elastica show many similarities and some dissimilarities. The dynamics of the point mass and elastica resemble the mechanics of a pole-vault; and therefore, a nonconservative external torque is introduced to parallel the muscle work done by vaulters. For the nonconservative system, the problem is solved for complete transformation of the kinetic energy of the mass and the work done by the external torque into potential energy of the mass. The initial velocities for the two cases, with and without the nonconservative force, are quite similar; and therefore, the maximum potential energy of the mass is higher in the presence of the nonconservative force. A realistic dimensional example is considered; the solution to the problem, despite several simplifying assumptions, is found to be similar to data of elite pole vaulters presented in the literature.

The deployment dynamics of a simplified solar sail quadrant consisting of two Euler–Bernoulli beams and a flexible membrane are studied. Upon prescribing the in-plane motion and modeling the tension field based on linearly increasing stresses assumed on the attached boundaries, the coupled equations of motion that describe the system's transverse deflections are obtained. Based on these equations and their boundary conditions (BCs), deployment stability is studied by deriving simplified analytic expressions for the rate of change of system energy. It is shown that uniform extension and retraction result in decreasing and increasing energy, respectively. The motion equations are discretized using expansions in terms of “time-varying quasi-modes” (snapshots of the modes of a cantilevered beam and a clamped membrane), and the integrals needed for the resulting system matrices are rendered time-invariant via a coordinate transformation. Numerical simulation results are provided to illustrate a sample deployment and validate the analytic energy rate expressions.

The paper addresses the problem of isochronous beams, namely those that oscillate with a frequency that is independent of the amplitude also in the nonlinear regime. The mechanism adopted to obtain this goal is that of having, as a boundary condition, a roller that can slide on a given path. A geometrically exact Euler–Bernoulli formulation is considered, and the nonlinear analysis is done by the multiple time scale method, that is applied directly to the partial differential equations governing the motion without an a priori spatial reduction. The analytical expression of the backbone curve is obtained, up to the third-order, and its dependence on the roller path is addressed. Conditions for having a straight backbone curve, i.e., the isochronous beam, are determined explicitly. As a by-product of the main result, the free and forced nonlinear oscillations of a beam with an inclined support sliding on an arbitrary path have been investigated.

In many applications, coupling between thermal and mechanical domains can significantly influence structural dynamics. Analytical approaches to study such problems have previously used assumptions such as a proscribed temperature distribution or one-way coupling to enable assessments. In contrast, time-stepping numerical simulations have captured more detailed aspects of multiphysics interactions at the expense of high computational demands and lack of insight of the underlying physics. To provide a new tool that closes the knowledge gap and broadens potential for analytical techniques, this research formulates and analytically solves a thermomechanical beam model considering a combination of thermal and mechanical excitations that result in extreme nonlinear behaviors. Validated by experimental evidence, the analytical framework facilitates the prediction of the nonlinear dynamics of multi-degree-of-freedom structures exhibiting two-way thermomechanical coupling. The analysis enables the investigation of mechanical and thermomechanical impedance metrics as a means to forecast future nonlinear dynamic behaviors such as extreme bifurcations. For the first time, characteristics of mechanical impedance previously reported to predict the onset of dynamic bifurcations in discrete systems are translated to illuminate the nearness of distributed parameter structures to bifurcations. In addition, fundamental connections are discovered in the thermomechanical evaluations between nonlinear low amplitude dynamics of the postbuckled beam and the energetic snap-through vibration that are otherwise hidden by studying displacement amplitudes alone.

The application of explicit dynamics to simulate quasi-static events often becomes impractical in terms of computational cost. Different solutions have been investigated in the literature to decrease the simulation time and a family of interesting, increasingly adopted approaches are the ones based on the proper orthogonal decomposition (POD) as a model reduction technique. In this study, the algorithmic framework for the integration of the equation of motions through POD is proposed for discrete linear and nonlinear systems: a low dimensional approximation of the full order system is generated by the so-called proper orthogonal modes (POMs), computed with snapshots from the full order simulation. Aiming to a predictive tool, the POMs are updated in itinere alternating the integration in the complete system, for the snapshots collection, with the integration in the reduced system. The paper discusses details of the transition between the two systems and issues related to the application of essential and natural boundary conditions (BCs). Results show that, for one-dimensional (1D) cases, just few modes are capable of excellent approximation of the solution, even in the case of stress–strain softening behavior, allowing to conveniently increase the critical time-step of the simulation without significant loss in accuracy. For more general three-dimensional (3D) situations, the paper discusses the application of the developed algorithm to a discrete model called lattice discrete particle model (LDPM) formulated to simulate quasi-brittle materials characterized by a softening response. Efficiency and accuracy of the reduced order LDPM response are discussed with reference to both tensile and compressive loading conditions.

This paper studies a new comprehensive model for toppling dynamics of regularly spaced dominoes in an array. The model has unlocked the hypotheses introduced by Stronge and Shu (Stronge, W. J., and Shu, D., 1988, “The Domino Effect: Successive Destabilization by Cooperative Neighbours,” Proc. R. Soc. A, 418 (1854), pp. 155–163), which can provide us some essential insights into the mechanism of domino wave. Extensive comparisons are made between the proposed model and the experimental results studied in the existing literature. Our numerical studies show that the existing theoretical models are special cases of the proposed model, and the fluctuation in the waveform of propagation speed observed from experiments was caused by the irregular multiple impacts between colliding dominoes. The influence of physical parameters of domino on the natural speed of toppling dominoes is also considered, and it is found that the coefficients of friction and restitution between colliding dominoes have more effects due to the energy dissipation during impact.

A comprehensive study is reported herein for the evaluation of Lagrangian functions for linear systems possessing symmetric or nonsymmetric coefficient matrices. Contrary to popular beliefs, it is shown that many coupled linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally cannot be determined by system decoupling unless further restriction such as classical damping is assumed. However, a scalar function that plays the role of a Lagrangian function can be determined for any linear system by decoupling. This generalized Lagrangian function produces the equations of motion and it contains information on system properties, yet it satisfies a modified version of the Euler–Lagrange equations. Subject to this interpretation, a solution to the inverse problem of linear Lagrangian dynamics is provided.

A majority of dielectric elastomers (DE) developed so far have more or less viscoelastic properties. Understanding the dynamic behaviors of DE is crucial for devices where inertial effects cannot be neglected. Through construction of a dissipation function, we applied the Lagrange's method and theory of nonequilibrium thermodynamics of DE and formulated a physics-based approach for dynamics of viscoelastic DE. We revisited the nonlinear oscillation of DE balloons and proposed a combined shooting and arc-length continuation method to solve the highly nonlinear equations. Both stable and unstable periodic solutions, along with bifurcation and jump phenomenon, were captured successfully when the excitation frequency was tuned over a wide range of variation. The calculated frequency–amplitude curve indicates existence of both harmonic and superharmonic resonances, soft-spring behavior, and hysteresis. The underlying physics and nonlinear dynamics of viscoelastic DE would aid the design and control of DE enabled soft machines.

The nonlinear response of a flexible structure, subjected to generally supported conditions with nonlinearities, is investigated for the first time. An analytical procedure is proposed first. Moreover, a simulation technique usually employed in static analysis is developed for confirmation. Generally, ordinary perturbation methods could analyze dynamics of flexible structures with linear boundary conditions. As nonlinear boundaries are taken into account, they are out of operation for the modal shape that is hardly to be obtained, which is the key to the analysis. In order to overcome this, nonlinear boundary conditions are rescaled and the technique of modal revision is employed. Consequently, each governing equation with different time-scales could be analyzed exactly according to corresponding rescaled boundary conditions. The total response of any point at the flexible structure will be composed by harmonic responses yielded by the analytical method. Furthermore, the differential quadrature element method (DQEM), a numerical simulation technique could satisfy boundary conditions strictly, is introduced to certify analytical results. The comparison shows a reasonable agreement between these two methods. In fact, the accuracy of the analytical method for nonlinear boundaries could be explained in theory. Based on the certification, boundary nonlinearities are discussed in detail analytically and found to play an important role in responses. Because of the important role played by the nonlinear factors in the vibration and control of the flexible structure, this paper will open the vibration analysis and numerical study of the flexible structure with nonlinear constraints.

The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.

We study cross-flow vortex-induced vibration (VIV) of a linearly sprung circular cylinder equipped with a dissipative oscillator with cubic stiffness nonlinearity, restrained to move in the direction of travel of the cylinder. The dissipative, essentially nonlinear coupling between the cylinder and the oscillator allows for targeted energy transfer (TET) from the former to the latter, whereby the oscillator acts as a nonlinear energy sink (NES) capable of passively suppressing cylinder oscillations. For fixed values of the Reynolds number (Re = 48, slightly above the fixed-cylinder Hopf bifurcation), cylinder-to-fluid density ratio, and dimensionless cylinder spring constant, spectral-element simulations of the Navier–Stokes equations coupled to the rigid-body motion show that different combinations of NES parameters lead to different long-time attractors of the dynamics. We identify four such attractors which do not coexist at any given point in the parameter space, three of which lead to at least partial VIV suppression. We construct a reduced-order model (ROM) of the fluid–structure interaction (FSI) based on a wake oscillator to analytically study those four mechanisms seen in the high-fidelity simulations and determine their respective regions of existence in the parameter space. Asymptotic analysis of the ROM relies on complexification-averaging (CX-A) and slow–fast partition of the transient dynamics and predicts the existence of complete and partial VIV-suppression mechanisms, relaxation cycles, and Hopf and Shilnikov bifurcations. These outcomes are confirmed by numerical integration of the ROM and comparisons with spectral-element simulations of the full system.

Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and transport of energy in essentially nonlinear oscillatory systems. In this paper, we are going to demonstrate that the structure of the resonance manifold can be conveniently described in terms of canonical action–angle (AA) variables. Such formalism has important theoretical advantages: all resonance manifolds may be described at the same level of complexity, appearance of additional conservation laws on these manifolds is easily proven both in autonomous and nonautonomous settings. The harmonic balance-based complexification approach, used in many previous studies on the subject, is shown to be a particular case of the canonical formalism. Moreover, application of the canonic averaging allows treatment of much broader variety of dynamical models. As an example, energy exchanges in systems of coupled trigonometrical and vibro-impact oscillators are considered.

Lagrangians for classically damped linear multi-degrees-of-freedom dynamical systems are obtained using simple and elementary methods. Such dynamical systems are very widely used to model and analyze small amplitude vibrations in numerous naturally occurring and engineered systems. An invariant of the motion is also obtained.