scholarly journals Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach

2021 ◽  
Vol 11 (4) ◽  
pp. 1389
Author(s):  
Piotr Wielgos ◽  
Robert Geryło
Keyword(s):  
Degrees Of Freedom ◽  
Research Work ◽  
System Optimization ◽  
Degree Of Freedom ◽  
Primary System ◽  
Single Degree Of Freedom ◽  
Modal Approach ◽  
Single Degree ◽  
Novel Approach ◽  
The Impact

The research paper presents a novel approach toward constructing motion equations for structures with attached MTMDs (multiple tuned mass dampers). A primary system with MDOF (multiple dynamic degrees of freedom) was reduced to an equivalent system with a SDOF (single degree of freedom) through the modal approach, and equations from additional MTMDs were added to a thus-created system. Optimization based on ℌ2 and ℌ∞ for the transfer function associated with the generalized displacement of an SDOF system was applied. The research work utilized GA (genetic algorithms) and SA (simulated annealing method) optimization algorithms to determine the stiffness and damping parameters for individual TMDs. The effect of damping and stiffness (MTMD tuning) distribution depending on the number of TMDs was also analyzed. The paper also reviews the impact of primary system mass change on the efficiency of optimized MTMDs, as well as confirms the results of other authors involving greater MTMD effectiveness relative to a single TMD.

scholarly journals Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) Through the Modal Approach

2020 ◽  
Author(s):  
Piotr Wielgos ◽  
Robert Geryło
Keyword(s):  
Degrees Of Freedom ◽  
Research Work ◽  
System Optimization ◽  
Degree Of Freedom ◽  
Primary System ◽  
Single Degree Of Freedom ◽  
Modal Approach ◽  
Single Degree ◽  
Stiffness And Damping ◽  
The Impact

The research paper presents a new approach towards constructing motion equations for structures with attached MTMDs (multiple tuned mass dampers). A primary system, with MDOF (multiple dynamic degrees of freedom) was reduced to an equivalent system with a SDOF (single degree of freedom) through the modal approach, and equations from additional MTMDs were added to a thus-created system. Optimization based on H2 and H∞ for the transfer function associated with the generalized displacement of an SDOF system. The research work utilized GA (genetic algorithms) and SA (simulated annealing method) optimization algorithms to determine the stiffness and damping parameters for individual TMDs. The effect of damping and stiffness (MTMD tuning) distribution depending on the number of TMDs was also analyzed. The paper also reviews the impact of primary system mass change on the efficiency of optimized MTMDs, as well as confirms the results of other authors involving greater MTMD effectiveness relative to a single TMD.

Predicting the Impact Response of a Nonlinear Single-Degree-of-Freedom Shock-Absorbing System From the Measured Step Response

1997 ◽  
Vol 119 (3) ◽  
pp. 221-227 ◽  
Author(s):  
S. N. Robinovitch ◽  
W. C. Hayes ◽  
T. A. McMahon
Keyword(s):  
High Energy ◽  
Degree Of Freedom ◽  
Step Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Impact Forces ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
The Impact ◽  
Measured Response

We measured the step response of a surrogate human pelvis/impact pendulum system at force levels between 50 and 350 N. We then fit measured response curves with four different single-degree-of-freedom models, each possessing a single mass, and supports of the following types: standard linear solid, Voigt, Maxwell, and spring. We then compared model predictions of impact force during high-energy collisions (pendulum impact velocity ranging from 1.16 to 2.58 m/s) to force traces from actual impacts to the surrogate pelvis. We found that measured peak impact forces, which ranged from 1700 to 5600 N, were best predicted by the mass-spring, Maxwell, and standard linear solid models, each of which had average errors less than 3 percent. Reduced accuracy was observed for the commonly used Voigt model, which exhibited an average error of 10 percent. Considering that the surrogate pelvis system used in this study exhibited nonlinear stiffness and damping similar to that observed in simulated fall impact experiments with human volunteers, our results suggest that these simple models allow impact forces in potentially traumatic falls to be predicted to within reasonable accuracy from the measured response of the body in safe, simulated collisions.

scholarly journals Symmetric waterbomb origami

2016 ◽  
Vol 472 (2190) ◽  
pp. 20150846 ◽  
Author(s):  
Yan Chen ◽  
Huijuan Feng ◽  
Jiayao Ma ◽  
Rui Peng ◽  
Zhong You
Keyword(s):  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Solar Panels ◽  
Deployable Structures ◽  
Single Degree Of Freedom ◽  
Folding Process ◽  
Multiple Degrees Of Freedom ◽  
Single Degree ◽  
Flat Roofs ◽  
Additional Constraints

The traditional waterbomb origami, produced from a pattern consisting of a series of vertices where six creases meet, is one of the most widely used origami patterns. From a rigid origami viewpoint, it generally has multiple degrees of freedom, but when the pattern is folded symmetrically, the mobility reduces to one. This paper presents a thorough kinematic investigation on symmetric folding of the waterbomb pattern. It has been found that the pattern can have two folding paths under certain circumstance. Moreover, the pattern can be used to fold thick panels. Not only do the additional constraints imposed to fold the thick panels lead to single degree of freedom folding, but the folding process is also kinematically equivalent to the origami of zero-thickness sheets. The findings pave the way for the pattern being readily used to fold deployable structures ranging from flat roofs to large solar panels.

A guide to classical flutter

1988 ◽  
Vol 92 (919) ◽  
pp. 339-355 ◽  
Author(s):  
L. T. Niblett
Keyword(s):  
Degrees Of Freedom ◽  
Normal Modes ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Structural Coupling ◽  
Two Degrees Of Freedom ◽  
Single Degree ◽  
Lifting Surface ◽  
Classical Flutter ◽  
Comprehensive Study

Summary First essentials of classical flutter are demonstrated by a comprehensive study of the behaviour of a lifting surface with two degrees of freedom under the action of airforces limited to those in phase with displacement. Structural coupling between the coordinates is eliminated by taking the normal modes to be the deflection coordinates, and this results in conditions for stability with particularly concise forms. It is shown that the flutter stability can be seen to be very much a matter of the relative amplitudes of heave and pitch in the normal modes. In-quadrature airforces are then introduced and it is shown that they have little effect when the flutter is severe. They are of more importance in the milder forms of flutter, the extreme of which are shown to be little different from instabilities in a single degree of freedom.

Tunable-with-energy intense modal interactions induced by geometric nonlinearity

2020 ◽  
pp. 095440622090862
Author(s):  
Alireza Mojahed ◽  
Lawrence A Bergman ◽  
Alexander F Vakakis
Keyword(s):  
Geometric Nonlinearity ◽  
Degree Of Freedom ◽  
Primary System ◽  
Energy Relation ◽  
Initial Angle ◽  
Modal Interactions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Energy Transfers ◽  
Wave Tailoring

Modal interactions are distinct features of nonlinear systems that can be exploited in applications such as vibration and shock mitigation, targeted (irreversible) energy transfers (TET), and acoustic/stress wave tailoring. For such applications, different types of nonlinearities, e.g. hardening, softening, smooth, non-smooth, material or geometric, have been considered. In this work, we examine the geometric nonlinearity resulting from an initially inclined element consisting of a linear spring and a viscous damper connected in parallel, having an initial angle of inclination, [Formula: see text]. Because of its inclined configuration, this element possesses strong (and doubly tunable with respect to [Formula: see text] and energy) geometrically nonlinear stiffness and damping effects, despite the linear constitutive laws governing its constituent components. First, we consider a single-degree-of-freedom linearly grounded oscillator attached to the nonlinear inclined element. Omitting dissipative effects, we investigate the frequency–energy relation of this system by employing the canonical action-angle transformation and show that, depending on the initial angle of inclination and the energy-level, the resulting nonlinearity can be tuned to be softening, hardening or a combination of both. Next, we explore the efficacy of the geometric nonlinearity to induce strong modal interactions by considering a three-degree-of-freedom lightly damped primary system that is weakly coupled to a single-degree-of-freedom lightly damped attachment with the inclined nonlinear element, subjected to impulsive excitation. Varying [Formula: see text] and the input energy, we demonstrate strong modal energy-exchanges between the modes of the primary system and the nonlinear attachment over broad energy-dependent spans of [Formula: see text]. We show that the passive self-adaptiveness of the nonlinear damping and the hardening–softening geometric nonlinearity can induce narrowband or broadband frequency TET, including high-to-low frequency energy transfers. Interestingly, over a definitive range of [Formula: see text], these modal interactions may be limited only between the nonlinear mode of the attachment and the highest-frequency linear mode of the primary system, inducing strong high-frequency targeted energy transfer to the primary system.

Expedient Estimation of the Maximum Amplification Factor in Damped Mistuned Bladed Disks

2007 ◽  
Author(s):  
Yun Han ◽  
Bing Xiao ◽  
Marc P. Mignolet
Keyword(s):  
Upper Bound ◽  
Amplification Factor ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Computational Technique ◽  
Bladed Disks ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Novel Approach ◽  
Maximum Amplification

This paper focuses on the formulation and validation of a novel approach for the expedient estimation of the maximum amplification factor induced by mistuning in damped bladed disks. This computational approach is based on earlier analytical results yielding an upper bound of the maximum amplification factor in the limit of zero damping. Extensions of these results are derived first to broaden the applicability of the methodology. Next, the computational technique is described: it involves the components of one of the mode shapes of the mistuned disk and the associated frequency as the variables over which the optimization is carried out. Further, the initial guess for these variables is obtained from the analytical estimates of the upper bound. This approach removes the limitations of earlier analytical efforts, i.e. damping is considered, and the actual value of the maximum amplification factor and the corresponding mistuning of the blade properties are obtained. Limitations on the magnitude of the mistuning could also be considered in the algorithm if desired. This novel approach was applied for the parametric study of the maximum amplification factor as a function of damping in two single-degree-of-freedom per blade disk models as well as in a reduced order model of a blisk. The results obtained in connection with the two single-degree-of-freedom systems very closely match the global maxima predicted by an existing, more tedious algorithm introduced earlier to avoid convergence to one of the many local maxima known to often exist.

Nonlinearities in Experimental System Identification of a Piezocomposite Beam Model

2017 ◽  
Author(s):  
Patrick S. Heaney ◽  
Onur Bilgen
Keyword(s):  
System Identification ◽  
Electromechanical Coupling ◽  
Degree Of Freedom ◽  
Parameter Estimates ◽  
Beam Model ◽  
Single Degree Of Freedom ◽  
Testing Procedures ◽  
Single Degree ◽  
Constitutive Relationships ◽  
The Impact

Piezocomposite beams are often modeled using linear constitutive equations describing the electromechanical coupling of the material. In nearly all experimental identification processes, nonlinearities in these equations are ignored, which can lead to significant errors in the identified models. Following a common practice in the literature, a piezocomposite cantilever beam is modeled as a single degree of freedom system, with strain induced harmonic excitation governed by linear piezoelectric constitutive relationships. The validity of the linear property assumptions is investigated. It is experimentally demonstrated that the relationship between input and response of the beam is significantly nonlinear. The impact of this nonlinear behavior on the parameter identification of the system is shown for three different testing methods, (1) Open Loop Excitation, (2) Constant Input, and (3) Constant Response. For each method, the command amplitude is varied which yields different parameter estimates for the single degree of freedom beam model. These results demonstrate that the assumed linear constitutive relationships lead to parameter estimates which are only accurate for the specific testing method and the specific commanded input or response amplitude, even under highly controlled testing procedures. The paper concludes with comments on the system identification of a single degree of freedom model given this nonlinear system behavior.

The Effect of the Impact Model on Vibration Response of Discrete and Continuous Systems

2012 ◽  
Author(s):  
M. R. Brake
Keyword(s):  
Constitutive Model ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Impact Dynamics ◽  
Impact Model ◽  
Single Degree Of Freedom ◽  
Elastic Plastic ◽  
Single Degree ◽  
Contact Models ◽  
The Impact

Impact is a phenomenon that is ubiquitous in mechanical design; however, the modeling of impacts in complex systems is often a simplified, imprecise process. In many high fidelity finite element simulations, the number of elements required to accurately model the constitutive properties of an impact event is impractical. As a result, rigid body dynamics with approximate representations of the impact dynamics are commonly used. These approximations can include a constant coefficient of restitution, an artificially large penalty stiffness, or a single degree of freedom constitutive model for the impact dynamics that is specific to the type of materials involved (elastic, plastic, viscoelastic, etc.). In order to understand the effect of the impact model on the system’s dynamics, simulations are conducted to investigate a single degree of freedom, two degrees of freedom, and continuous system each with rigid stops limiting the amplitude of vibration. Five contact models are considered: a coefficient of restitution method, a penalty stiffness method, two similar elastic-plastic constitutive models, and a dissimilar elastic-plastic constitutive model. Frequency sweeps show that simplified contact models can lead to incorrect assessments of the system’s dynamics and stability. In the worst case, periodic behavior can be predicted in a chaotic regime. Additionally, the choice of contact model can significantly affect the prediction of wear and damage in the system.

Time to seismic failure induced by repeating SSEs in a single-degree-of-freedom spring-slider model

2020 ◽  
Vol 224 (2) ◽  
pp. 1242-1255
Author(s):  
Makiko Ohtani ◽  
Masao Nakatani ◽  
Nobuki Kame
Keyword(s):  
Return Period ◽  
Degree Of Freedom ◽  
Seismogenic Zone ◽  
Cumulative Probability ◽  
Seismic Cycle ◽  
Time Intervals ◽  
Single Degree Of Freedom ◽  
Megathrust Earthquakes ◽  
Single Degree ◽  
The Impact

SUMMARY Some large earthquakes may have been triggered by large slow slip events (SSEs). This paper studies the time it takes from the most recent SSE to seismic failure in a situation where earthquakes are influenced by periodic SSEs, unlike in earlier studies that investigated the impact of a single SSE imposed at an arbitrary timing during a seismic cycle. A single-degree-of-freedom (SDOF) spring-slider model obeying a rate- and state-dependent friction (RSF) law is used to study the time to failure. The results are compared with those in Ohtani et al., which simulates megathrust earthquakes triggered by large, spontaneous SSEs along the deeper extension of a seismogenic zone in a 2-D elastic medium. Suppose a model fault under steady-rate tectonic loading is also impacted by stress steps induced by periodic SSEs. In the absence of SSEs, there is only a unique value of the characteristic slip distance L of RSF for a given seismic return period T. In the presence of SSEs, by contrast, synchronization occurs, and there exists a finite range of L values that corresponds to the same T. The timing (phase difference) of earthquakes relative to the SSEs varies continuously with L within that range. This study focuses on the case where T is triple the SSE return period, TSSE, (T = 3TSSE) to allow comparison with Ohtani et al. For each value of L in that range, disturbance tests that assign random values to T0, the timing of the third SSE within the seismic cycle, are conducted to obtain a probability distribution for tf, the time from that SSE to seismic failure. The distribution is converted into P(t; L), the cumulative probability that seismic failure occurs within time t of the SSE. The P(t; L) is averaged over the different L values to produce $\bar{P}$(t), which takes account of variability in the strength excess on the seismic fault at the time of the SSE, a parameter that is generally unknown. These numerical disturbance tests are conducted for three different values of an SSE size parameter. It is found that the larger the SSEs, the more intensely seismic failure is concentrated within short time intervals following an SSE. For the largest SSE, whereby a single SSE accounts for 3/10 of the total stress accumulated during a seismic cycle, there is a 50 per cent probability that seismic failure occurs within 132 d. These results contrast with those of Ohtani et al., in which tf was found to be concentrated more intensely within even shorter time intervals following an SSE (80 per cent probability that seismic failure occurs within 2.78 d of the SSE). It is suggested the timing of SSE-triggered seismic failure is concentrated more strongly on a fault embedded in a continuum because of a factor that cannot be taken into account by an SDOF model—a spatial structure of stress concentration that is already there on the fault even before the triggering event, the SSE.

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