A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multi-mode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid In the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, non-linear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

A pseudo-similarity solution is obtained for the flow of an incompressible fluid of second grade past a wedge with suction or blowing at the surface. The non-linear differential equation is solved using quasi-linearization and orthonormalization. The numerical method developed for this purpose enables computation of the flow characteristics for any value of the parameters K , a , and b , where K is the dimensionless normal stress modulus of the fluid, a is related to the wedge angle, and b is the suction or blowing parameter. A significant effect of suction or blowing on the wall shear stress is observed. The present results match exactly with those from an earlier perturbation analysis for Kx 2 a ≤ 0.01 but differ significantly as Kx 2 a increases.

The dynamic equations of motion obtained in the modeling of flexible structural systems with unknown rigid body gross motion are often highly nonlinear and possess time-varying coefficient matrices. The inherent characters of nonlinear large overall rigid body motion and linear small vibration are also involved in the system equations. Neither the implicit nor the explicit algorithm seems optimally suited and efficient by itself in dealing with these kinds of equations. This paper, therefore, presents a sequential implicit-explicit method in which it is attempted to achieve the attributes of both classes of algorithms. The equation system expressed in matrix form is first mapped to a subsystem in which the specified generalized coordinates are eliminated. The subsystem is then partitioned into two sets of coupled equations. One set of equations, describing the elastic motion, is linear with respect to the elastic generalized coordinates and is integrated implicitly. The other set of equations, governing the rigid body motion, contains the highly nonlinear coupling terms and is integrated explicitly with the back substitutions of the elastic kinematic properties already calculated in solving the first set of equations. A Newmark algorithm is employed to integrate the second order system of differential equations directly. A predictor-corrector scheme also coming from the Newmark algorithm is applied to the explicit integration. The procedures developed in the current paper are applied to simulating dynamic response of a complicated flexible system with mutually dependent rigid body unconstrained spherical motion and small elastic deformation.

This paper addresses modeling issues that arise in the formulation of the equations of motion for the flexible multibody mechanical systems intended for space applications and designed according to ground test results. A planar multibody system consisting of two flexible links interconnected by two revolute joints and a payload at its free end is proposed for the investigations. In addition to the gravity and transverse deflections (most common two conditions adopted for the research in this field), the foreshortening effects, the axial deflections and the work done by the system’s own weight on the elastic deflections are also taken into consideration. Since the slender link assumption is made, the Euler-Bernoulli Beam theory is considered sufficient and satisfactory for describing the behavior of the deformed link components. The Lagrangian formulation in conjunction with assumed displacement field method is then implemented to develop the equations of motion for the system. After achieving the analytical model for the system, a linearization about various system configurations transforms the fully coupled nonlinear differential equations into standard eigenvalue problems. In doing so, the roles played by gravity, foreshortening and system’s own weight (‘weight-load’) on the dynamic behavior of the system undergoing ground testing are examined. For analysis, the fundamental frequency of the system is chosen as a measurement index. Finally, parametric studies focusing on the mass properties of payload, lower and upper links, and actuators are undertaken to address the stability problems. Results indicate that the ‘weight-load’ exhibits interesting effects on the ‘foreshortening and stability’, hence, merits further investigation.

Periodic steady state solutions are generated from the nonlinear differential equation of an Euler-Bernoulli beam rod, constant speed crank, slider crank mechanism. The system is approximated by a single ordinary differential equation. To facilitate Harmonic Balance, the response and difficult nonlinear and linear portions of the ODE are expanded into two separate Fourier Series. Harmonic Balance/Fast Fourier Transform method is utilized to determine the harmonic content of each Fourier Series and generate periodic solutions which compared favorably with Runge-Kutta numerical integration. Stability of periodic solutions was examined using the monodromy matrix method. Nondimensional amplitude versus nondimensional crank speed was plotted to examine the effect on dynamic response of external force, piston mass, and crank length. Jump and period doubling bifurcations and regions of amplified response were found.

A variable-structure system, consisting of a thin horizontal elastic plate, loaded by a heavy layer of granular material is considered. The system is excited by prescribed vertical vibrations of the plate boundaries. An essentially nonlinear vibrational rheological model with unilateral constraints is proposed for simulation of the layer behavior. During the coupled motion stages the layer dynamic response on the interface between plate and granular layer is described by the Volterra integro-operator relationship with nonlinear isochronic force-displacement characteristic. The governing set of equations includes the plate dynamics differential equation, an integro-differential equation of the layer dynamics, and also appropriate boundary and initial conditions. Due to utilization of the Galerkin method and Kroosh approximation, the initial set is reduced to the set of nonlinear differential equations. The latter is integrated numerically. Model identification is carried out by the computer processing of the experimental bending strain histories. Influence of the excitation parameters, height of the layer, and damping on the bending strains is analyzed. Adequate simulation of the layer behavior for one-period modes is achieved with regard to the motion periodicity, the values of the strain peaks, the critical time points and the duration of the main time stages. Analysis of numerical phase trajectories and Poincaré sections predicts transition to stochastic vibration modes and the existence of stable two-period limit cycles, when the maximum forcing acceleration is increased above 3 g .

This paper presents the dynamic response and analysis of a dynamic resistor in a rotating system which is sensitive to the vibration of the rotating system. The sensitivity of the dynamic resistor is adjusted in order to stabilize the rotating system when acted upon by various excitation input functions. The equations of motion of the dynamic resistor and the rotating system, derived by the use of the energy method, result in coupled nonlinear differential equations, and the response of the dynamic resistor is highly correlated to the vibration of the rotating system. In this paper, the derivation of equations of motion of the dynamic resistor and the rotating system and their dynamic responses are analyzed in order to determine the optimal design parameters that provide stability for the rotating system when it is excited by external input forcing functions. The dynamic response is obtained both by experimental data and computer simulation. The experimental data indicate that the sensitivity of the dynamic resistor is related to the vibrating response of the rotating system and the rotational system parameters including the natural frequency. The dynamic response obtained by computer simulation is compared with the experimental data. The computer simulation results are to be used to determine the optimal values of the parameters of the dynamic resistor with respect to the rotating system. The uniqueness of this system is that the dynamic resistor is completely isolated from other inputs except the rotational vibration of the rotating system. The conclusions include recommendations and areas for future research.

An efficient method has been developed to determine the steady state response of an internal combustion engine, which is governed by a nonlinear differential equation with discontinuous forcing function due to the nature of the pressure-volume (P-V) diagram. Unlike previously reported methods for dynamically driven mechanical systems, this approach can accurately and efficiently predict the sensitivities of an engine’s output to dimensional tolerances. Hence, this technique is well suited for the development of an optimal tolerance allocation scheme based on nonlinear programming. The validity of this method which is based on the alternating frequency/time (AFT) domain technique has been established by comparison with results from numerical integration of differential equation of motion. This method directly yields the mean crank speed and Fourier coefficients of temporal variations in engine’s outputs such as crank’s speed and the driving torque.

Hydraulic lash adjusters can eliminate clearance between components in the automotive valve trains. A two-mode dynamic model for the adjuster is derived by including check valve motions at two operating modes. The oil refill mechanism controlled by the check valve is experimentally identified. The oil compressibility in the oil chamber and the oil leakage through the annular gap between the plunger and the body are also considered. It is shown that motion of the adjuster plunger must satisfy a third-order, non-linear differential equation of motion. The differential equation is numerically integrated, and the experimental data in leak-down test provided by the manufacturer is shown to be in good agreement with the model. The annular gap is found to be the key factor in controlling leak-down speed, and the effective orifice diameter is shown to be the key factor in describing the pump-up phenomenon. It is also found that one percent air entrainment in the oil chamber of the adjuster seriously compromises overall adjuster rigidity and reduces the natural frequency of the adjuster almost 50 percent.

Parametric stability in beams with combined quadratic and cubic material nonlinearities is examined. A general mathematical model is developed for parametrically excited beams accounting for their nonlinear material characteristic. Second- and forth-order nonlinear differential equations are found to govern the axial and transverse motions, respectively. Expansions for displacements are assumed in terms of the linear undamped free-oscillation modes. Boundary conditions are applied to the expansions for displacements to determine the mode shapes. Multiplying the equations of motion by the corresponding shape functions, accounting for their orthogonal properties, and integrating over the beam length, a set of coupled nonlinear differential equations in the time-dependent modal coefficients is obtained. Utilizing the method of multiple scales, frequency response as well as response versus excitation amplitude are obtained for two beams of different cross sectional areas. Results are presented for three boundary conditions. It is found that, qualitatively, the response is similar for all the boundary conditions. Quantitative comparison of the cases considered indicate that the highest response amplitude occurs for the cantilever beam with the end mass. The bifurcation points for simply supported beam occur at lower excitation parameter value. It is apparent that more slender columns have larger response amplitude.

Parametric oscillations formulated by linear or nonlinear differential equations of order two are often encountered in the textbooks on the theory of nonlinear oscillations. Mathieu equation or its modified nonlinear version are seen in many papers as the typical examples. Parametric oscillations of the higher order than two, however, are less investigated, and the fundamental form is little known, especially the knowledge on the engineering viewpoint is little published in relation with mechanical or physical models. This paper deals with the parametirc oscillations governed by periodic differential equations with the higher order than two, presenting the fundamental mathematical form with many physical, mechanical, and electrical examples. General nonlinear systems with parametric excitation are not easy to treat with, because they have many characters depending on the specific forms of the systems. In this paper we confine systems which have three conditions, first excitation frequency region is limited, second symmetry condition or reciprocity is kept, third no energy condition of nonlinear functions. These conditions are partly mathematical and partly physical. In order to treat with parametric oscillation we think that the causes of oscillations are only parametric instability. Without parametric excitation no significant oscillations are assumed to exist.

A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

A nonlinear model describing the dynamic, fluid dynamics and thermodynamics of a dynamically balanced Gifford-McMahon (G-M) cryorefrigerator was developed. The cryorefrigerator consists of two horizontally opposed first- and second-stage displacers, a first- and a second-stage thermal regenerator, and a rotary valve that cycles the compressor supply gas pressure. The model accurately describes the motion of the displacers and the refrigeration produced at the two stages of the cold head. The analytical model consists of a set of twelve first order nonlinear differential equations. The numerical solution of the system is presented in the form of pressure histories in each chamber, position and velocity histories of each displacer, and P-V diagrams of the two stages of the cold head. The performance of the cryorefrigerator is calculated under various conditions. It is clearly shown that the device can achieve non-impacting displacer motion with sufficient cooling capacities of the two stages. This non-impacting operation creates a very low vibration device, which is valuable in applications where low vibration is required, such as magnetic resonance imaging (MRI) systems. It was also shown that the cooling capacity of the device can be optimized over the operating frequency.

Dynamics of multi-cylinder engines have been performed by the conventional methods using lumped pure torsional systems. This article offers a finite element method of performing elastodynamic analysis of multi-cylinder engines considering them as spatial linkage systems with true spatial geometries of the crankshaft and the linkage loops. An engine can have any number of cylinders with linear offsets and angular orientations relative to each other. A three-dimensional finite-line element with isoparametric joint irregularity freedoms is developed and used. Consistent or lumped mass systems can be used. Elastodynamics of engines is considered in two forms: (a) kinetoelastostatics (KES) where all forces and torques acting on the system are considered except the vibratory motion effect; (b) kinetoelastodynamics (KED) where the forced and damped equations of motion of the system are solved. Matrix exponential method of solving equations of KED motions are presented and used. It is proven to be a very efficient and stable technique for the solutions of large systems of linear and nonlinear differential equations of any order. After solving for the generalized coordinates, time histories of the neutral coordinate displacements, forces, moments, stresses, bearing forces, and generated torque are determined for as many work cycles as desired. A generalized computer program performing KED and KES studies of any multi-cylinder engine is made available for industrial use. KED and KES analyses of a four-cylinder automobile engine are performed.

Experimental and numerical investigations are carried out on an autoparametric system consisting of a composite pendulum attached to a harmonically base excited mass-spring subsystem. The dynamic behavior of such a mechanical system is governed by a set of coupled nonlinear equations with periodic parameters. Particular attention is paid to the dynamic behavior of the pendulum. The instability threshold of the pendulum is determined from the semi-trivial solution of the linearized equations. The set of nonlinear differential equations is then integrated with respect to time using step-by-step methods. The developed experimental setup makes it possible to analyze the angular displacement of the pendulum using an opto-electronic sensor. The motion of the pendulum is numerically analyzed via phase-plane portraits and Poincaré maps. A subcritical bifurcation is plotted and chaos is observed. The predicted results are experimentally validated.

Control analysis and design for robot manipulators require the knowledge of their dynamic model. In the first part of this paper, the Lagrange-Euler (L-E) formulaion has been developed to separate the generalized forces/torques due to rotational motion from that due to the translational motion. Therefore a new form for the dynamic equations of the robot manipulators has been obtained. In the second part, the generalized d’Alembert (GD) method has been used to develop the equations of motion for robot manipulators with revolute and/or prismatic joints. Also, the rotational effect and the translational effect in the new form of the GD method are separated. The develpoed equations, by both methods, when applied to robot manipulators result in an efficient and explicit set of closed form second order nonlinear differential equations. They give fairly well structured equations of motion suitable for control analysis and manipulator design. Using the proposed models, the rotational effects and translational effects have been studied separately and a simplified model has been obtained, a computational algorithm has been established and a computer software has been developed to perform the necessary calculations to obtain the generalized force required for each joint to follow any pre-specified trajectory. As an application of the proposed methods, a simplified model for a Stanford manipulator has been obtained and it is found that the results of the normal and the simplified models are very close and the simplified model can be accepted even with end-effector load variation and with different trajectories. The simplified model has the advantage of less computational time which is more appropriate for control purposes. Also, the results obtained by the developed L-E and GD methods are the same.

A new method for analyzing periodic response of rotor dynamic system with nonlinear supports is presented in this paper. Based on a finite element formulation in the time domain, this method transforms nonlinear differential equations governing the dynamic behavior of rotor-bearing system into a set of nonlinear algebraic equations that can be reduced and calculated by the characteristic set of Wu elimination method. The analytic solution of the nodal displacement has been obtained finally. According to this result the behavior of periodic response is analyzed. The feasibility and advantage of the proposed method are illustrated with an example of flexible Jeffcott rotor-bearing system with nonlinear supports.

In the present study a wavelet-based procedure is applied for the parametric identification of a nonlinear discrete cable model on the basis of experimental data obtained from free-vibration dynamic tests. The 3D cable dynamics is described by two partial integro-differential equations in the transversal planar and nonplanar displacement components. By expanding such variables in series of linear eigenfunctions, a discrete model is obtained by applying a classical Galerkin procedure. The resulting set of nonlinear ODEs contains quadratic and cubic coupling terms. The core of the identification technique is the discretization of such nonlinear differential equations by means of orthogonal Daubechies scaling functions. The Wavelet-Galerkin solution of the nonlinear equation involving the scaling function expansion of nonlinear terms leads to an algebraic problem that has to be inverted in order to solve for the vector of unknown parameters. A sdof model able to describe the in-plane unimodal oscillations is considered. Numerical and experimental identification tests are presented. Accurate estimates of the linear and quadratic parameters are achieved from the moderate oscillation amplitude response of an experimental discrete cable-mass model. The accuracy of the cubic parameter estimate is shown to depend on the level of both the oscillation amplitude and noise.

The collective nonlinear dynamics of electrostatically coupled nanobeams under parametric excitation is modeled and investigated, while including the main sources of nonlinearities up to the fifth order. The normalized nonlinear differential equations are solved using secular perturbation theory. Numerical simulations have been performed using the Asymptotic Numerical Method for two and three coupled nanobeams. It is shown that, the fifth order nonlinearity leads to an increase in the number of multimodal solutions and their complexity in term of bifurcation topology. This model can be exploited to design arrays of NEMS with high performances for sensing applications.

Nonlinear free vibration analysis of stiffened plates is presented in this paper. The von Karman theory is employed to model the rectangular stiffened steel plate. The first two symmetric and asymmetric modes are taken into consideration and the coupled nonlinear differential equations of system are derived using the Galerkin approach. The Variational Iteration Method (VIM) is considered as the solution technique and an integral iterative formulation is presented to obtain the nonlinear natural frequencies. Subsequently a parametric sensitivity study is carried out and the effect of different initial amplitudes on the frequency responses is investigated.