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.

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.

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.

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.

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.

scholarly journals On the Effect of Track Irregularities on the Dynamic Response of Railway Vehicles

1974 ◽  
Vol 96 (4) ◽  
pp. 1147-1158 ◽  
Author(s):  
M. A. Dokainish ◽  
J. N. Siddall ◽  
W. Elmaraghy
Keyword(s):  
Steady State ◽  
Mathematical Models ◽  
Equations Of Motion ◽  
Railway Vehicle ◽  
Steady State Response ◽  
Full Model ◽  
Input Frequency ◽  
Moving Vehicle ◽  
State Response ◽  
Track Irregularities

The steady state response for models of a six-axle locomotive running on a sinusoidally irregular track has been investigated. Two mathematical models have been set up, a full model for the “stationary” vehicle in which creep between wheels and rails was neglected, and a full model for the “moving” vehicle in which creep forces, gravity stiffness effects and wheel tread profiles were considered. The use of the generalized method of complex algebra to obtain the steady state response of the railway vehicle components to varying input frequencies was used. The results given in this paper are for the case of sinusoidal lateral track irregularities only, but the method is general enough to allow also for vertical track irregularities. For the “stationary” vehicle the input frequency is increased from zero to 3 cycles per second. For the “moving” vehicle the input frequency is a function of the track wave length and the vehicle forward speed and is given in terms of the vehicle speed. The frequency response curves are computer plotted in each case. For the “moving” vehicle, responses for the cases of both new and worn wheels are obtained. The natural frequencies for the full model are also calculated. The results obtained show the effect of the creep forces and the condition of the wheels on the steady state response. It is recommended that slip and corresponding creep forces, wheel tread and rail profiles, and the gravity stiffness effect be included in the steady state response analysis of railway vehicles to track irregularities. The analysis may be used to check the response of any proposed design for a railway vehicle to economically attractive track irregularities. It may also be used to adjust geometry, spring rates and damping characteristics in order to maximize operating speeds while providing optimum damping for the trucks and body motions. This paper illustrates and describes the mathematical models used; gives generalized form for the differential equations of motion and the methods of solution. The equations of motion for the wheelsets are derived in detail including the creep forces and the wheel tread profiles.

Steady-State Analysis of Mechanical Seals With Two Flexibly Mounted Rotors

1997 ◽  
Vol 119 (1) ◽  
pp. 200-204 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Steady State ◽  
Dynamic Behavior ◽  
Reference Frames ◽  
Rapid Determination ◽  
Equations Of Motion ◽  
Steady State Response ◽  
Mechanical Seals ◽  
State Response ◽  
Forcing Functions ◽  
Flexible Supports

The dynamic behavior of a mechanical face seal with two flexibly mounted rotors is investigated. The equations of motion are derived using linearized rotor dynamic coefficients to model the dynamic behavior of the fluid film. The equations are shown to be linear in the inertial reference with harmonic forcing functions which result from the initial misalignment of the flexible supports. A method for obtaining the steady-state response in the system is derived by transforming the equations of motion into reference frames which rotate with the shafts. The resulting equations contain constant forcing functions and can be readily solved for the magnitude of the steady-state response. The method presented allows a rapid determination of the steady-state misalignment of a seal without resorting to numerical modeling.

scholarly journals An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 149-154 ◽  
Author(s):  
T. M. Cameron ◽  
J. H. Griffin
Keyword(s):  
Steady State ◽  
Frequency Domain ◽  
Dynamic Systems ◽  
Time Domain ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Systems ◽  
System Response ◽  
Steady State Response ◽  
State Response ◽  
The Time Domain

A method is proposed for analyzing the steady-state response of nonlinear dynamic systems. The method iterates to obtain the discrete Fourier transform of the system response, returning to the time domain at each iteration to take advantage of the ease in evaluating nonlinearities there—rather than analytically describing the nonlinear terms in the frequency domain. The updated estimates of the nonlinear terms are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the steady-state response. The method is demonstrated by solving a problem with friction damping in which the excitation has multiple discrete frequencies.

Trigonometric Collocation for Computation of Steady State Response of a Cracked Beam

2013 ◽  
Author(s):  
Bakeer Bakeer ◽  
Oleg Shiryayev
Keyword(s):  
Steady State ◽  
Fatigue Cracks ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Steady State Response ◽  
Cracked Beam ◽  
Vibration Period ◽  
State Response ◽  
Trigonometric Collocation ◽  
Alternative Techniques

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.

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