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.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

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.

Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude-Henri Lamarque
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Forced Response ◽  
Stochastic Methods ◽  
Steady State Response ◽  
State Response ◽  
Internal Excitation ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
Support Motion

The aim of this work is to apply stochastic methods to investigate uncertain parameters of rotating machines with constant speed of rotation subjected to a support motion. As the geometry of the skew disk is not well defined, randomness is introduced and affects the amplitude of the internal excitation in the time-variant equations of motion. This causes uncertainty in dynamical behavior, leading us to investigate its robustness. Stability under uncertainty is first studied by introducing a transformation of coordinates (feasible in this case) to make the problem simpler. Then, at a point far from the unstable area, the random forced steady state response is computed from the original equations of motion. An analytical method provides the probability of instability, whereas Taguchi’s method is used to provide statistical moments of the forced response.

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.

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.

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

1993 ◽  
Vol 115 (4) ◽  
pp. 822-828 ◽  
Author(s):  
D. A. Tortorelli
Keyword(s):  
Steady State ◽  
Vibration Amplitude ◽  
Equations Of Motion ◽  
Steady State Response ◽  
Analysis Techniques ◽  
Direct Differentiation ◽  
State Response ◽  
Nonproportional Damping ◽  
The Time Domain ◽  
General Response

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 period 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 separately 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 nonproportional 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.

Optimum inelastic tuned mass dampers without using viscous elements for minimizing steady-state vibration of base-excited systems

2020 ◽  
pp. 107754632095132
Author(s):  
Saman Bagheri ◽  
Vahid Rahmani-Dabbagh
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Analytical Framework ◽  
Tuned Mass Damper ◽  
Steady State Response ◽  
State Response ◽  
Controlled Systems ◽  
Mass Damper ◽  
Fourier Series Approximation ◽  
Direct Numerical Integration

A special type of a tuned mass damper, which consists of a mass and an elasto-plastic spring without using any viscous damper, is used to reduce the steady-state response of structures to base excitation. Previous work of the authors showed that the elasto-plastic tuned mass damper (P-TMD) could help reduce the seismic responses, and a method based on energy equalization was proposed to design it. In this study, the effectiveness of the P-TMD is investigated under harmonic support motion, and a direct approach is developed to find its optimum parameters. To estimate the nonlinear steady-state response of P-TMD-controlled systems, an analytical framework is established using the Fourier series approximation, which is validated by direct numerical integration of the equations of motion. The obtained results for the optimum P-TMD are discussed and compared with those of the optimum elastic tuned mass damper.

Static and Dynamic Analysis of Carbon Nanotube Cantilever Based on Molecular Dynamics Simulation

2008 ◽  
Vol 375-376 ◽  
pp. 631-635 ◽  
Author(s):  
Na Gong ◽  
Ying Chun Liang ◽  
Ying Xue Yao ◽  
Bing Guo Liu
Keyword(s):  
Steady State ◽  
Carbon Nanotube ◽  
Molecular Dynamic Simulation ◽  
Molecular Dynamic ◽  
Dynamic Simulation ◽  
Dynamics Simulation ◽  
System Response ◽  
Driving Voltage ◽  
Steady State Response ◽  
State Response

Static and dynamic characteristic of carbon nanotube is analyzed by molecular dynamic simulation. Static analysis show that maximum deflection of carbon nanotube appears at 0.033nm distance from free end of cantilever. When the value of driving voltage is less, the flexibility is more important, then the stiffness increase with the voltage increasing, and the collapse structure or hole appear on carbon nanobute. Under low pressure condition, deflection difference on equi-increase of pressure between no damping and damping simulation is larger than that on high pressure. The results of forced vibration indicate that system response consist of transient and steady-state response. With steps of molecular dynamic simulation increasing, transient response disappears gradually, so system response includes only steady-state response, which is simple harmonic vibration with the same frequency as excitation force. These results are in accord with the classical vibration theory. Moreover they will provide theoretic foundation for designing of nanostructure device.

Dynamic Response of a Geared Rotor-Bearing System Under Residual Shaft Bow Effect

2006 ◽  
Author(s):  
T. N. Shiau ◽  
E. K. Lee ◽  
Y. C. Chen ◽  
T. H. Young
Keyword(s):  
Steady State ◽  
Natural Frequencies ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Transmission Error ◽  
Mode Shapes ◽  
Steady State Response ◽  
State Response ◽  
Rotor Bearing ◽  
Bearing System

The paper presents the dynamic behaviors of a geared rotor-bearing system under the effects of the residual shaft bow, the gear eccentricity and excitation of gear’s transmission error. The coupling effect of lateral and torsional motions is considered in the dynamic analysis of the geared rotor-bearing system. The finite element method is used to model the system and Lagrangian approach is applied to derive the system equations of motion. The dynamic characteristics including system natural frequencies, mode shapes and steady-state response are investigated. The results show that the magnitude of the residual shaft bow, the phase angle between gear eccentricity and residual shaft bow will significantly affect system natural frequencies and steady-state response. When the spin speed closes to the second critical speed, the system steady state response will be dramatically increased by the residual shaft bow for the in-phase case. Moreover the zero response can be obtained when the system is set on special conditions.

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