Periodic Motions of a Two-Body System Subjected to Repetitive Impact

1969 ◽  
Vol 91 (4) ◽  
pp. 931-938 ◽  
Author(s):  
David L. Sikarskie ◽  
Burton Paul
Keyword(s):  
Steady State ◽  
High Speed ◽  
Equations Of Motion ◽  
Point Of View ◽  
State Behavior ◽  
State Response ◽  
Stability Of Solution ◽  
Straightforward Method ◽  
Parametric Studies ◽  
Repetitive Impact

The dynamics of a widely used class of hammer impact machines are investigated on the basis of a two-degree-of-freedom idealization. The difficulty in the problem is due to the repetitive impact which introduces a nonlinearity in the system. It is the purpose of the analysis to develop a solution for the steady-state behavior of the system. There are several ways this can be done. One of the most efficient ways, from the point of view of ease of parametric studies of the system, is to convert the problem to a “boundary” value problem. With this technique, the system is governed by the equations of motion between impacts, and further satisfies additional conditions at the beginning and end of each impact cycle. Since the solution is obtained in only one cycle, it thus represents a straightforward method of studying the effect of various system parameters. A fundamental assumption in the analysis is that the steady-state response of the system has a period equal to the forcing period. This is verified for one set of parameters through the use of high-speed movies of an actual machine. There are several other interesting features in the analysis, including multivaluedness of the solution, allowable solution domain, and stability of solution, which have not been completely resolved to date.

Suspended Decoupler Design of Hydraulic Engine Mount

2011 ◽  
Author(s):  
Reza N. Jazar ◽  
M. Mahinfalah ◽  
J. Christopherson
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Equations Of Motion ◽  
State Behavior ◽  
State Response ◽  
Nonlinear Finite Elements ◽  
Lumped Parameter ◽  
Start Up ◽  
Engine Mount ◽  
Static Conditions

A known problem in classical hydraulic engine mount is that because of the density mismatch between the decoupler and surrounding fluid, the decoupler might float, or stick to the cage bounds, assuming static conditions. The problem appears in the transient response of a bottomed up floating decoupler hydraulic engine mount. To overcome the bottomed up problem, a suspended decoupler design for improved decoupler control is introduced. The new design does not noticeably effect the mechanisms steady state behavior, but improves start up and transient response. Additionally, the decoupler mechanism is incorporated into a smaller, lighter, yet more tunable and hence more effective hydraulic mount design. Ususally the elastomechanical components in a hydraulic engine mount are assumed lumped and linear. To have a more realistic modeling, utilizing nonlinear finite elements in conjunction with a lumped parameter modeling approach, we evaluate the resorting characteristics of the components and implement them in the equations of motion. The steady state response of a dimensionless model of the mount is examined utilizing the averaging perturbation method applied to a set of second order nonlinear ordinary differential equations. It is shown that the frequency responses of the floating and suspended decoupled designs are similar and functional.

Sensitivity Analysis for the Steady-State Response of Damped Linear Elastodynamic Systems Subject to Periodic Loads

1990 ◽  
Author(s):  
Daniel A. Tortorelli
Keyword(s):  
Steady State ◽  
Vibration Amplitude ◽  
Equations Of Motion ◽  
Steady State Response ◽  
Analysis Techniques ◽  
Direct Differentiation ◽  
State Response ◽  
Need To Evaluate ◽  
The Time Domain ◽  
General Response

Abstract Adjoint and direct differentiation methods are used to formulate design sensitivities for the steady-state response of damped linear elastodynamic systems that are subject to periodic loads. Variations of a general response functional are expressed in explicit form with respect to design field perturbations. Modal analysis techniques which uncouple the equations of motion are used to perform the analyses. In this way, it is possible to obtain closed form relations for the sensitivity expressions. This eliminates the need to evaluate the adjoint response and psuedo response (these responses are associated with the adjoint and direct differentiation sensitivity problems) over the time domain. The sensitivities need not be numerically integrated over time, thus they are quickly computed. The methodology is valid for problems with proportional as well as non-proportional damping. In an example problem, sensitivities of steady-state vibration amplitude of a crankshaft subject to engine firing loads are evaluated with respect to the stiffness, inertial, and damping parameters which define the shaft. Both the adjoint and direct differentiation methods are used to compute the sensitivities. Finite difference sensitivity approximations are also calculated to validate the explicit sensitivity results.

Periodic Steady State Response and Fatigue Analysis of a Nonlinear City Bus Model

2009 ◽  
Author(s):  
George Valsamos ◽  
Christos Theodosiou ◽  
Sotirios Natsiavas
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
High Stress ◽  
Direct Determination ◽  
Accurate Information ◽  
Steady State Response ◽  
State Response ◽  
Periodic Steady State

Dynamic response related to fatigue prediction of an urban bus is investigated. First, a quite complete model subjected to road excitation is employed in order to extract sufficiently reliable and accurate information in a fast way. The bus model is set up by applying the finite element method, resulting to an excessive number of degrees of freedom. In addition, the bus suspension units involve nonlinear characterstics. A step towards alleviating this difficulty is the application of an appropriate coordinate transformation, causing a drastic reduction in the dimension of the final set of the equations of motion. This allows the application of a systematic numerical methodology leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. Next, typical results were obtained for excitation resulting from selected urban road profiles. These profiles have either a known form or known statistical properties, expressed by an appropriate spatial power spectral density function. In all cases examined, the emphasis was put on investigating ride response. The main attention was focused on identifying areas of the bus suspension and frame subsystems where high stress levels are developed. This information is based on the idea of a nonlinear transfer function and provides the basis for applying suitable criteria in order to perform analyses leading to prediction of fatigue failure.

Trigonometric Collocation for Computation of Steady State Responses of Cracked Structures

2012 ◽  
Author(s):  
Bakeer Bakeer ◽  
Oleg Shiryayev ◽  
Ammaar Tahir
Keyword(s):  
Steady State ◽  
Fatigue Cracks ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Vibration Period ◽  
State Response ◽  
Trigonometric Collocation ◽  
Steady State Responses ◽  
Cracked Structures ◽  
State Responses

Development of vibration-based structural health monitoring techniques requires the use of various computational methods to predict dynamic responses of damaged structures. The method described in this work can be used for prediction of steady state harmonic responses for structures with fatigue cracks and may have several advantages over alternative techniques. The method appears to be relatively easy to implement and computationally inexpensive. The steady state response of the system at a given number of time points distributed over one vibration period is represented in terms of Fourier series containing higher frequency harmonics. Equations of motion are formulated in the form that allows for easy computation of Fourier coefficients for all terms in the series. Iterative procedure is used for determining the time of stiffness change in order to capture bilinear dynamic behavior. We present results of initial investigation by applying the method to a model of a cantilever beam with a crack.

An Efficient Method for Predicting the Vibratory Response of Linear Structures With Friction Interfaces

1986 ◽  
Vol 108 (4) ◽  
pp. 633-640 ◽  
Author(s):  
E. Bazan ◽  
J. Bielak ◽  
J. H. Griffin
Keyword(s):  
Equations Of Motion ◽  
Solution Procedure ◽  
System Response ◽  
Steady State Response ◽  
Friction Forces ◽  
Practical Applications ◽  
Linear Elastic ◽  
State Response ◽  
Vibratory Response ◽  
Parametric Studies

A simple methodology to study the steady-state response of systems consisting of linear elastic substructures connected by friction interfaces is presented. Assuming that only the first Fourier components of the friction forces contribute significantly to the system response, the differential equations of motion are transformed into a system of algebraic complex equations. Then, an efficient linearized procedure to solve these equations for different normal loads in the friction interfaces is developed. As part of the solution procedure, a criterion to determine the slip-to-stuck transitions in the joint is proposed. Within the assumption that the response is harmonic, any desired accuracy can be obtained with this methodology. Selected numerical examples are presented to illustrate practical applications and the relevant features of the methodology. Due to its simplicity, this methodology is particularly appropriate for performing parametric studies that require solutions for many values of normal loads.

On The Steady State and Dynamic Performance Characteristics of Floating Ring Bearings

1981 ◽  
Vol 103 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Chin-Hsiu Li ◽  
S. M. Rohde
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
High Speed ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Journal Bearings ◽  
Transient Problem ◽  
The Stability ◽  
Bearing System

An analysis of the steady state and dynamic characteristics of floating ring journal bearings has been performed. The stability characteristics of the bearing, based on linear theory, are given. The transient problem, in which the equations of motion for the bearing system are integrated in real time was studied. The effect of using finite bearing theory rather than the short bearing assumption was examined. Among the significant findings of this study is the existence of limit cycles in the regions of instability predicted by linear theory. Such results explain the superior stability characteristics of the floating ring bearing in high speed applications. An understanding of this nonlinear behavior, serves as the basis for new and rational criteria for the design of floating ring bearings.

scholarly journals Effects of Stator Stiffness, Gap Size, Unbalance, and Shaft’s Asymmetry on the Steady-State Response and Stability Range of an Asymmetric Rotor with Rub-Impact

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zhaoli Zheng ◽  
Yonghui Xie ◽  
Di Zhang ◽  
Xiaolong Ye
Keyword(s):  
Steady State ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Impact Behavior ◽  
Gap Size ◽  
Steady State Response ◽  
Arc Length ◽  
State Response ◽  
Asymmetric Rotor ◽  
Nonlinear Equations Of Motion

The asymmetric rotor and the rub-impact behavior are important sources of instability and may cause severe vibrations. However, the dynamics of the rotor-bearing system simultaneously considering the two factors has not gained sufficient attention in available investigations. In this paper, the steady-state response and stability of an asymmetric rotor with rub-impact were evaluated. The asymmetric rotor was modeled by beam elements with asymmetric cross section, and the nonlinear equations of motion were established in the rotating frame. The multiharmonic balance (MHB) method was employed to obtain the linearized form of the nonlinear equations of motion. Either the asymmetry of rotor or rub-impact can result in instability and make the problem difficult to solve. Thus, the arc-length method was utilized to trace the branch of the solutions. In order to improve the calculation speed and accurately predict the solution, the alternating frequency/time domain (AFT) was adopted to calculate the iteration of the arc-length method. Based on the proposed method, the effects of stator stiffness, gap size, unbalance, and asymmetric in shaft on the steady-state response and stability were obtained.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

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