The effect of mesh stiffness on the dynamic response of face gear transmission system combining with backlash nonlinearity is studied. First, a nonlinear time-varying (NLTV) and a nonlinear time-invariant (NLTI) dynamic models of face gear transmission system with backlash nonlinearity are formulated. The 6DOF motion equations of the face gear pair considering the mesh stiffness, backlash, contact damping and supporting stiffness are proposed. Second, the effect of mesh stiffness on the dynamic response of the face gear drive system is analyzed with the numerical method, where the mesh stiffness is expressed in two patterns as time-varying form and time-invariant form. According to the comparative study, some significant phenomena as bifurcation, chaos, tooth separation and occurrence of multijump are detected. The results show that different forms of mesh stiffness generate an obvious change on the dynamic mesh force.

A synthesized gear mesh and dynamic model assuming line contact that is derived from a set of manufacturing parameters is formulated for analyzing the beveloid gear mesh-coupling mechanism. Using the proposed model, the effect of the dominant geometry design parameter that is the crossed angle between the first principal directions of the tooth surface curvatures (FPD-angle) on gear mesh characteristic and dynamic response is investigated. Also, the analysis of the gear mesh characteristic and dynamic response subject to torque load variation is performed. It is shown that the dynamic transmission error and dynamic mesh force worsen as the geometry FPD-angle increases for a specific torque load level. Furthermore, even though higher torque load can produce larger contact area, which is desirable, it also increases the gear mesh stiffness and transmission error that tend to aggravate dynamic response.

This paper demonstrates a novel quadstable monolithic mechanism (QsMM), which provides four stable equilibrium positions within its planar operation range. The QsMM has been realized from the use of both X - and Y -directional bistable structures, which utilize curved snapping beams. Two pairs of curved beams were attached to an inner frame in both X and Y directions to present an independent bistable behavior in the directions. It was found out that the design of the inner frame is crucial for the quadstability and dynamic responses of the mechanism. A millimeter-scale brass mechanism was actually fabricated by ultraprecision milling to test the quadstability and the force-displacement behavior. The prototype clearly demonstrates four distinct stable positions in its millimeter-scale operation range. The design concept, finite element simulation, fabrication, and experimental measurement of the proposed multistable mechanism have been presented. The mechanical multistability of the proposed QsMM can be utilized for multiple switching and optical networking applications, yielding low power consumption operations.

In this work, the problem of shape optimization of flexible robotic manipulators of circular cross sections is studied. Two different manipulators are considered—a manipulator with revolute joint and a roller supported Cartesian manipulator. The finite element method is used to find the natural frequency and dynamic response of a flexible manipulator by treating it as an Euler-Bernoulli beam. The cross-sectional diameter is varied along the length keeping the constraint on the mass of the manipulator and static tip deflection in order to maximize the fundamental frequency of the beam. This optimization problem is compared with other optimization problems (with different objective functions and constraints). It is observed that the proposed optimization problem is superior to other optimization problems.

Dynamic equations of the intermittent-motion of a globoidal cam driven system were derived in this study. The effects of roller mesh flexibility and cam profile curve on the residual vibration of a globoidal cam system were studied experimentally and numerically. Time varying roller mesh stiffness and damping coefficients were used to account for the periodic variation of the mesh stiffness in the dwell and the active periods respectively. Dynamic responses of a globoidal cam system in the active and the dwell periods were simulated and measured. The effects of cam profile and input shaft speed on the residual vibration were also studied in this work. Results indicated that the proposed model was feasible for the dynamic simulation of a globoidal cam system.

Clearances exist in different kinds of joints in multibody mechanical systems, which could drastically affect the dynamic behavior of the system. If the joint is dry with no lubricant, impact occurs, resulting in wear and tear of the joint. In practical engineering design of machines, joints are usually designed to operate with some lubricant. Lubricated journal bearings are designed so that even when the maximum load is applied, the joint surfaces do not come into contact with each other. In this paper, a general methodology for modeling lubricated long journal bearings in multibody mechanical systems is presented. This modeling utilizes a method of solving for the forces produced by the lubricant in a dynamically loaded long journal bearing. A perfect revolute joint in a multibody mechanical system imposes kinematic constraints, while a lubricated journal bearing joint imposes force constraints. As an application, the dynamic response of a slider-crank mechanism including a lubricated journal bearing joint between the connecting rod and the slider is considered and analyzed. The dynamic response is obtained by numerically solving the constraint equations and the forces produced by the lubricant simultaneously with the differential equations of motion and a set of initial conditions numerically. The results are compared with the previous studies performed on the same mechanism as well a dry clearance joint. It is shown that in a multibody mechanical system, the journal bearing lubricant introduces damping and stiffness to the system. The earlier studies predict that the order of magnitude of the reaction moment is twice that of a perfect revolute joint. The proposed model predicts that the reaction moment is within the same order of magnitude of the perfect joint simulation case.

In two recent papers (Chen, J.-S., and Chen, K.-L., 2001, “The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod,” ASME Journal of Mechanical Design, 123 , pp. 542–548; Chen, J.-S., and Huang, C.-L., 2001, “Dynamic Analysis of Flexible Slider-Crank Mechanisms With Nonlinear Finite Element Method,” Journal of Sound and Vibration, 246 , pp. 389–402) we reported that previous researches of others on the dynamic response of a flexible connecting rod may have overestimated the deflections by ten folds when the crank rotates near the bending natural frequency of the connecting rod because terms of significant order of magnitude were ignored inadequately. While the findings in (Chen, J.-S., and Chen, K.-L., 2001, “The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod,” ASME Journal of Mechanical Design, 123 , pp. 542–548; Chen, J.-S., and Huang, C.-L., 2001, “Dynamic Analysis of Flexible Slider-Crank Mechanisms With Nonlinear Finite Element Method,” Journal of Sound and Vibration, 246 , pp. 389–402.) were obtained via numerical simulations, the present paper emphasizes the analytical approach with an aim to exploring the physical insights behind these numerical results. The equations of motion are first derived by applying Hamilton’s principle with all high order terms in the strain energy function being retained. After careful examination of the order of magnitude of each term, the coupled equations are simplified to a single one in terms of the transverse deflection, which turns out to be a Duffing equation under parametric and external excitations simultaneously. Closed-form approximations of the dynamic response are then derived by using multiple scale method. It is found that the combined effects of parametric and external excitations dominate the response when Ω is close to 0.5 and 1. Away from these two speed ranges, on the other hand, the response is dominated by the external excitation alone.

The influence of gear tooth flank modifications in the form of linear involute tip relief on the torsional vibration behavior of a spur gear pair is investigated by using an experimental test stand. Measured dynamic transmission error (DTE) values are compared and a family of forced response curves is presented. Guidelines for the design of quiet spur gear sets are also given.

A model was used, where the total gear mesh stiffness was approximated by two constant stiffness levels, in order to analyze the influence of the contact ratio on the dynamic response of spur gears. Due to the stiffness variation there is parametric excitation of the transmission error, which generally causes tooth separation at certain critical rotational speeds. The present paper discloses a method to analytically calculate which contact ratio to use in order to avoid tooth separation near a specific critical rotational speed. [S1050-0472(00)02604-0]

The dynamic response of a helicopter planetary gear system is examined over a wide range of operating speeds and torques. The analysis tool is a unique, semianalytical finite element formulation that admits precise representation of the tooth geometry and contact forces that are crucial in gear dynamics. Importantly, no a priori specification of static transmission error excitation or mesh frequency variation is required; the dynamic contact forces are evaluated internally at each time step. The calculated response shows classical resonances when a harmonic of mesh frequency coincides with a natural frequency. However, peculiar behavior occurs where resonances expected to be excited at a given speed are absent. This absence of particular modes is explained by analytical relationships that depend on the planetary configuration and mesh frequency harmonic. The torque sensitivity of the dynamic response is examined and compared to static analyses. Rotational mode response is shown to be more sensitive to input torque than translational mode response. [S1050-0472(00)00403-7]

Previous researches on the dynamic response of a flexible connecting rod can be categorized by the ways the axial load in the rod is being formulated. The axial load may be assumed to be (1) dependent only on time and can be obtained by treating the rod as rigid, (2) related to the transverse displacement by integrating the axial equilibrium equation, and (3) proportional to linear strain. This paper examines the validity of these formulations by first deriving the equations of motion assuming the axial load to be proportional to the Lagrangian strain. In order for the dimensionless displacements to be in the order of O(1), different nondimensionalization schemes have to be adopted for low and high crank speeds. The slenderness ratio of the connecting rod arises naturally as a small parameter with which the order of magnitude of each term in the equations of motion, and the implication of these simplified formulations can be examined. It is found that the formulations in previous researches give satisfactory results only when the crank speed is low. On the other hand when the crank speed is comparable to the first bending natural frequency of the connecting rod, these simplified formulations overestimate considerably the dynamic response because terms of significant order of magnitude are removed inadequately.

The transport of hazardous materials in truck cargo tanks can cause severe environmental damage as a result of the tank’s failure during a collision. Impact due to collision involves the transient dynamic response of the tank, fluid and their interaction. This paper develops a design oriented computational approach to predict the dynamic transient response of the tank shell structure subjected to impact loads during crash accidents. In order to compute the fluid and structural interaction, the finite element formulations for the added mass to the structure are developed and integrated with DYNA3D, a nonlinear dynamic structural finite element code, and they are validated by pendulum impact experiment. This paper presents the lumping process required by the added mass approach for cargo tanks under impact conditions. Thus, due to its efficiency the computer based approach provides a design tool for fluid filled thin walled structures in general and cargo tanks subjected to an impact situation. The structural performance of cargo tank shell construction is investigated. This research will contribute to improvement in design, modeling, and analysis techniques for crashworthiness and integrity of liquid mechanical structure systems which are subjected to impulsive loads like those found in vehicle collisions.

Based on the analytical solution of the equation of motion for a single degree-of-freedom model of a spring, the relation between the dynamic behavior and the kinematic features of input cam motions is discussed in this paper. A simple expression for the dynamic response spectrum of the vibration excited by the input motion is presented. It provides a useful tool to estimate the effect of cam motions on the dynamic behavior of springs. A method for the selection of cam motion curves based on this response spectrum is also presented in the paper. Examples are given to illustrate the method.

Computer models of the human body are robust tools for gaining insight into the gross motion of ground vehicle or aircraft occupants and evaluating the loads and, deformations of their critical parts. The knowledge of occupant responses will help in the determination of the type and probable causes of injuries that may he sustained during a crash. An important aspect in crash analysis is how the large motion of the relatively rigid segments of an occupant, such as the limbs, and the small deformations of flexible segments, such as the spine column, are interrelated. To this end, a general methodology for kineto-static analysis of multibody systems with flexible structures undergoing large motion and structural deformations is developed. Rigid multibody dynamics is used to predict the gross motions and displacements at the boundaries between the relatively bulky (rigid) bodies and relatively flexible ones. A mixed boundary-condition finite-element analysis is formulated and solved at every numerical integration time to determine the corresponding reaction forces and moments at the boundaries and also the structural deformations. Based on this methodology, a multibody model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II anthropomorphic crash test dummy. The analytical results obtained are compared with the experimental results from the impact sled tests. Comparison of the results has shown better correlation between the analyses and the experiments compared with earlier studies.

The joint European X-ray Telescope, JET-X, is one of the core instruments in the scientific payload of the Russian Spectrum Roentgen-Gamma-RG which is a high energy astrophysics mission. The JET-X structure is constructed from three carbon fibre tubular sections, and contains the instruments and acts as an optical bench. The forward section of the structure has a deployable door, which remains closed until the satellite has reached orbit. The function of the door is to protect sensitive optical surfaces from contamination and moisture from ground operations, during launch and early orbits operations. The door is fabricated from a honeycomb core faced with aluminium alloy skins. The door is approximately 1 m in diameter and weighs about 6 Kg only. The door body, the cutter assembly and the hinge assembly were designed to withstand the mechanical stresses arising during the launch environment. The door assembly was integrated to the main structure and vibration tests which simulate the conditions during the launch were conducted. These tests were: Sine, Random and Shock. This paper presents the door assembly design, the finite element modelling and the dynamic responses during the tests.

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. This paper presents a Bernstein-Be´zier form of harmonic trajectory patterns for synthesizing low-harmonic trajectories. Similar to Bernstein-Be´zier polynomial curves, Bernstein-Be´zier harmonic trajectories can be defined either explicitly using Bernstein-Be´zier basis harmonics or recursively using the harmonic deCasteljau algorithm. The second part of the paper demonstrates how a Bernstein-Be´zier trajectory can be combined with the inverse dynamics of a robot manipulator for synthesizing a joint trajectory that demands “minimal” dynamic response from its actuators. An example involving a planar 2R robot is also presented.

A lumped/distributed-parameter dynamic model is developed to investigate the dynamic responses of a finger-follower cam system by considering a hydraulic lash adjuster with an oscillating pivot, and frictional forces between sliding surfaces. The measured force data at low speed are employed to derive an algorithm to determine the dynamic Coulomb friction coefficients at contact points. The contact position between the cam and the follower with moving pivot is determined by a constraint equation method. A hydraulic lash adjuster acting as the pivot of the follower is also modeled with the effects of oil compressibility and oil refill mechanism. Simulated contact forces at three different speeds are shown to have good agreement with experimental data. The separation between the hydraulic lash adjuster and the follower is predicted at a camshaft speed of 2535 rpm, and experiment indicates at 2520 rpm.

An extended three-dimensional model is used for calculating dynamic tooth loads on a planetary gear set. Time dependent mesh stiffnesses are determined and an original Ritz method aimed at solving large parametrically excited differential systems is proposed. Results from the Ritz method compare favorably with those given by direct integrations for highly reduced computation times. The difference between local critical speeds (for one individual mesh) and global critical speeds (for sun or ring gear-planet meshes) on a sequential spur gear train is pointed out. Finally, it is shown that, for linear behaviors, mesh stiffnesses are largely controlling dynamic tooth loads while the influence of a floating sun or ring gear is less important.

A new linearized two-degree-of-freedom model of an industrial press feed mechanism and its experimental verification are presented. By introducing a double cantilever model of the coupler with an assumed quarter sine shape function and replacing the nonlinear clutch spring by a linear torsional spring with a deflection dependent stiffness, it was possible to develop a set of two linear differential equations, which could be fully solved in an analytic manner for the dynamic responses of the coupler strain and the clutch windup angle during the feeding stroke. An associated parametric study shows insights pertaining to the relationship between the system and component frequencies and allows the formulation of certain design rules regarding the magnitudes of the coupler and clutch windup angle deflections.

In a previous paper (Ehrich, 1994), the author has cataloged a variety of unique rotordynamic responses which have been observed in a computer model of a simple Jeffcott rotor in a nonlinear anisotropic mounting system; that is, operating eccentrically within a clearance and in local intermittent contact with the stator. In addition to the critical synchronous resonant response similar to that found in linear systems, unique responses are found at subcritical operating speeds—superharmonic pseudo-resonances, and regions of chaotic and periodic response in transition zones between successive superharmonic orders. In the transcritical operating regime on both the subcritical and supercritical sides of the critical peak, spontaneous sidebanding is found when the system is very lightly damped. At supercritical operating speeds, subharmonic pseudoresonances, and regions of chaotic response and periodic response in transition zones between successive subharmonic orders are identified. These phenomena are characterized by their unique signature in the response curve, derived from a simple numerical model of a Jeffcott rotor with a bilinear stiffness in the direction normal to the plane of contact. Each phenomenon is further characterized by a typical example of the wave form and the wave form’s spectral analysis. All the waveforms display the tendency of the nonlinear system to have a significant asynchronous response component at its natural frequency irrespective of the rotational or stimulus frequency. In a more recent publication (Ehrich, 1995), recorded observations of rotordynamic response, in the format of “waterfall” or “cascade” charts, of operational high-speed turbomachinery in two typical instances have been compared with equivalent data from the same computer model used to illustrate the cataloged phenomena. The instances are representative of several of the cataloged phenomena. Excellent correspondence between the computed waterfall charts and the data from the actual operational machinery are achieved. The results may be of considerable interest to the community of vibrations engineers who deal with prevention of and remedial action for deleterious asynchronous vibration in operational high speed rotating machinery.