Multiple Degree Of Freedom Systems Research Articles

Brain Injury and Impact Characteristics

Almost all studies of traumatic brain injuries have only focused on the maximum acceleration associated with the impact. However, it has been noted that other impact characteristics should also be considered. This study has examined the effect on brain deformation [maximum principal strain (MPS)] associated with changing three characteristics of an isosceles trapezoid acceleration profile: initial slope (jerk), maximum acceleration, and impact energy (maximum velocity squared). This involved using a preexisting finite element model of the brain and applying the trapezoidal acceleration impact to the center of the forehead. The results showed the largest values of MPS were found along a line between the coup and contrecoup positions, and that a strong direct relationship existed between maximum acceleration and brain deformation as measured using MPS. In addition, a strong direct relationship was found to exist between impact energy and brain deformation as measured using MPS. However, it was found that there was almost no observable change in MPS with respect to different jerk values, and in fact there was a slight decrease in MPS as jerk values increased. This comported with a shock spectrum analysis of a simple one dimensional multiple degree of freedom system exposed to similar trapezoidal impulses.

Critical damping in nonviscously damped linear systems

• Viscoelastic structures are characterized by a frequency dependent damping matrix. • Critically damped modes of nonviscous systems are characterized. • Mathematical conditions to find critical surfaces in nonviscous systems are presented. • A new numerical method to compute critical curves in nonviscous systems is developed. • Critical surfaces of single and multiple degrees of freedom systems are presented. In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are named critical damping surfaces (or critical manifolds, since several parameters can be involved). In this paper the theoretical foundations to determine critical damping surfaces in nonviscously damped systems are established. In addition, a numerical method to obtain critical curves is developed. The approach is based on the transformation of the algebraic equations , which define implicitly the critical curves, into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.

Modeling dynamic behavior of MDOF systems with multiple bilinear springs

A computational scheme is presented in this paper to simulate dynamical behavior of multiple degrees of freedom (MDOF) systems with multiple bilinear springs. In the proposed scheme, a bilinear spring is modeled using by two parallel springs - a primary spring and a secondary spring. The primary spring is an ordinary linear spring having identical stiffness in tension and compression, and is active for tension and compression. The secondary spring, whose stiffness characterizes the bilinearity, is active only during compression. It is employed in connection with the Newmark integration method and the linear complementarity problem (LCP) formulation to obtain time-domain responses of dynamical systems with bilinear springs due to initial disturbances and harmonic excitations. The scheme described in this paper is effective in dealing with the sudden transition from tension to compression and vice versa simultaneously for all bilinear springs. Numerical results for bilinear oscillators with finite bilinearity ratios and impact oscillators with an infinite bilinearity ratio show that the proposed bilinear spring model is accurate, generic and valid for bilinearity ratios ranging from zero to infinity. Orderly and chaotic behavior of viscously damped 3-DOF system under harmonic excitation is studied for a wide range of excitation frequencies and bilinear ratios to demonstrate the effectiveness and applicability of the proposed model for MDOF bilinear systems.

A novel technique to predict harmonic response of Particle-damping structure based on ANSYS® secondary development technology

Since the particle damping is highly developed and widely used to suppress the structure vibration and a more efficient and accurate technique for predicting the harmonic responses of particle-damping structure is in need, this paper presents a novel technique to predict the harmonic response of particle-damping structure based on iterative solution method and ANSYS® secondary development technology. In order to establish the iteration solution method for predicting the harmonic response of particle-damping structure, iteration scheme of particle-damping structure (system) is derived from the mechanical dynamic formulae and particle damping formulae. For single or multiple degree of freedom system, basic solutions of each iteration cycle are solved by mathematical model, while the solutions of continuous systems are solved by ANSYS® in this paper. To conduct harmonic analysis of highly nonlinear particle-damping system in ANSYS®, iterative solution method (i.e. iteration scheme) is coded into program by ANSYS® Parametric Design Language (APDL). Based on the code, module of particle damping is developed and interfaces are created in ANSYS® using User Interface Design Language (UIDL). To verify the accuracy of the novel technique, two research cases are detailed introduced and frequency response functions (FRFs) of acceleration are obtained by simulations and experiments. It's worthwhile to note that simulation results have excellent consistency with the experiment results. In addition, the simulations conducted by ANSYS® using iteration solution method developed in this paper is more efficient and consumes less computer resources than previous methods. Consequently, both accuracy and higher efficiency of the novel technique are guaranteed.

Seismic control performance for Pounding Tuned Massed Damper based on viscoelastic pounding force analytical method

This paper focuses on vibration control performance of Pounding Tuned Massed Damper (PTMD) with viscoelastic pounding layers, which can dissipate energy during collision process. By using of viscoelastic pounding force analytical method, two types of pounding force models for different contact geometries are developed to illustrate interactions among pounding components in PTMD. A shake table test based on a simplified experimental structure is conducted to verify the effectiveness of single PTMD as well as the viscoelastic pounding force model. Furthermore, a parametric study for a 14-storeys steel structure is presented to investigate the performance of PTMD device in multiple degree of freedom (MDOF) system. Results from numerical simulations demonstrated that properties of viscoelastic material and gaps between mass block and the limiters in PTMD are key factors for vibration control performance. By inputting a series of seismic excitations into MDOF system, comparisons between PTMD and traditional TMD reveals that the optimized PTMD has better performance than the traditional TMD in vibration suppression in certain cases.

Dynamic behaviour of MDOF oscillators subjected to multiple visco-elastic contact constraints

Dynamic behaviour of multiple degree-of-freedom (MDOF) oscillators subjected to multiple gap-activated visco-elastic (VE) constraints is investigated in this paper. The complex problem is formulated to be a Linear Complementarity Problem (LCP) by means of the Newmark integration scheme and the incremental displacements. The proposed method is tested for a 2-mass oscillator with one mass subjected to a gap-activated VE constraint. Excellent agreement with the results available in the literature is achieved. The dynamical behaviour of a 2-mass and a 5-mass oscillator with all masses subjected to gap-activated VE constraints are then investigated. The bifurcation diagram for a range of frequency and excitation amplitude ratios is presented. The harmonic and sub-harmonic resonances and chaotic motion are accurately captured. The method is free from any prior assumption and can be used to simulate the dynamic behaviour of MDOF system with arbitrary number of constraints.

Mixed-sensitivity design of a dynamic controller for systems pre-compensated by input shapers

We employ recent control design algorithms for delay systems to address the mixed sensitivity design of a dynamic controller for multiple degree of freedom systems interconnected with an inverse signal shaper. A delay based inverse shaper is included within a feedback loop in order to eliminate oscillatory modes due to an attached flexible substructure, which has been shown to be advantageous to inducing directly the filtering property by an appropriate controller design. Due to the presence of the inverse shaper, the closed loop dynamics become infinite dimensional. Furthermore we design fixed-order controllers, aiming at an easy implementation. As a consequence of these two features standard techniques from H-infinity control can no longer be used. We present a two-stage method, which firstly makes the closed-loop system asymptotically stable and, secondly, computes a low-order controller that minimizes the H-infinity norm of the weighted closed-loop transfer functions. The approach is grounded in recent methods for fixed-order H-infinity control of delay systems, and the delay-differential algebraic equations (DDAE) modelling and control framework.

Dynamic analysis of a cantilever beam with a multiple degrees of freedom system

A beam attached to a multiple degrees of freedom system is a common and complex structure in a broad field of engineering and science. The paper presents a novel and efficient computer model for dynamic analysis of a cantilever beam structure attached to a multiple degrees of freedom system by means of the recurrence matrix method. In this model, the forced responses and natural characteristics of the coupled beam are achieved and numerical simulations of the established model agree well with the early results. Finally, the vibration reductions of a beam by a dynamic vibration absorber are numerically studied.

Experiment-based identification of time delays in linear systems

This paper presents an identification approach to time delays in single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) systems. In an SDOF system, the impedance function of the delayed system is expressed by the system parameters, the feedback gain, and the time delay. The time delay can be treated as the “frequency” of the difference between the impedance function of the delayed system and that of the corresponding uncontrolled system. Thus, it can be identified from the Fourier transform of the difference between the two impedance functions. In an MDOF system, the pseudo-impedance functions are defined. The relationships between the time delay and the pseudo-impedance functions of the delayed system and uncontrolled system are deduced. Similarly, the time delay can be identified from the Fourier transform of the difference between the two pseudo-impedance functions. The results of numerical examples and experimental tests show that the identification approach to keeps a relatively high accuracy.

New approach for identifying boundary characteristics using transmissibility

A novel approach is proposed for identifying boundary properties as a response model using transmissibility. This approach differs from those proposed in previous studies dealing with frequency response functions (FRFs) for joint identification. Transmissibility includes only response data, unlike FRFs that include force measurements. The boundary properties can be estimated by comparing the characteristics of the components under the free condition and connected to boundary conditions. When analyzing the components assembled compactly in the system for setting the shaker or measuring the impact force exerted on the component correctly, the proposed method could reduce the errors caused by an incorrectly measured force. The derived equation is verified using a discrete multiple degrees of freedom system with single boundary and multiple boundary conditions and by application to a beam, which is the simplest continuous structural form to validate the feasibility of the theory. The transmissibility defined by the apparent mass matrix is used for verifying the derived equation for identifying the boundary properties in the discrete system. However, when applying the equation to practical cases, as is the purpose of this research, the transmissibility matrix should be defined using only the response data. For this purpose, the accelerance matrix is modified slightly to the response matrix using the input as a unit force. This transmissibility matrix composed of response data is used for validating the equation in a continuous system. Furthermore, the effects of measurement noise are also investigated to assess the robustness of the method for application under practical conditions. Consequently, the proposed method could show reliable results by properly extracting the boundary properties in both cases. In many practical cases, this research is expected to contribute toward identifying the boundary properties in a complex system more conveniently compared to the method using FRFs.

Application of Cauchy wavelet transformation to identify time-variant modal parameters of structures

This work proposes a procedure for accurately identifying instantaneous modal parameters of a linear time-varying system using a time-varying autoregressive with exogenous input (TVARX) model with the continuous Cauchy wavelet transform (CCWT). An appropriate TVARX model is established using the velocity and displacement responses of the system under consideration. The time-varying coefficients of the TVARX are expanded as piecewise polynomial functions. CCWTs with various scale parameters are then applied to the TVARX model to evaluate the instantaneous modal parameters of different modes. The CCWTs of the velocity and displacement responses are analytically obtained from the CCWT of the measured acceleration responses. The effectiveness and accuracy of the proposed procedure are validated by numerical simulations of single and multiple degrees of freedom systems that have periodically varying and sharply varying stiffness and damping coefficients. The effects of noise, the Cauchy wavelet function and the order of the polynomial on the evaluation of the modal parameters are explored in processing the numerically simulated acceleration responses of systems with a single degree of freedom subjected to base excitation. Finally, the proposed procedure is adopted to determine the modal parameters of a five-story symmetric steel frame from its measured acceleration responses in a shaking table test. The measured strains reveal the yielding of columns in the first story. The variations of the identified instantaneous natural frequencies and modal damping ratios with time are consistent with the physical phenomena that are observed from the measured strains and base excitation acceleration.

Location identification of nonlinearities in MDOF systems through order determination of state-space models

This paper presents a new time-domain approach to identify locations of nonlinearities in multiple-degree-of-freedom (MDOF) systems. The approach is based on the well-known subspace method and utilizes the analysis of all the subsystems in the original MDOF system to determine locations of nonlinearities. The advantage of the proposed method is that it requires little prior knowledge about the system and does not need sampling under varying force amplitudes. Moreover, the method only needs the sampling of displacement data and thus avoids repetitive direct measurements or indirect numerical differentiation or integration to achieve velocity and acceleration data. In addition, benefitting from the subspace method, the approach is robust in noisy environment. However, one inconvenient drawback of the method is that it may misjudge linearity as nonlinearity under two specific circumstances. Several numerical simulations are carried out to test the performance of the approach, and the results demonstrate the consistency with the theoretical analysis.

Eigensolutions of non-proportionally damped systems based on continuous damping sensitivity

The viscous damping model has been widely used to represent dissipative forces in structures under mechanical vibrations. In multiple degree of freedom systems, such behavior is mathematically modeled by a damping matrix, which in general presents non-proportionality, that is, it does not become diagonal in the modal space of the undamped problem. Eigensolutions of non-proportional systems are usually estimated assuming that the modal damping matrix is diagonally dominant (neglecting the off-diagonal terms) or, in the general case, using the state–space approach. In this paper, a new closed-form expression for the complex eigenvalues of non-proportionally damped system is proposed. The approach is derived assuming small damping and involves not only the diagonal terms of the modal damping matrix, but also the off-diagonal terms, which appear under higher order. The validity of the proposed approach is illustrated through a numerical example.

Local variation detection in MDOF system using wavelet based transmissibility and its application in cracked beam

Since the local stiffness or damping variation happens when damage occurs in engineer structures, it is useful to detect the local variation as a way for structural damage inspection. As a vibration based approach, transmissibility has attracted considerable interest because of its convenience and effectiveness in damage detection. However, using the traditional Fourier transform, it should be very careful to select the frequency bands in transmissibility calculation. Inappropriate choice of frequency band could cause a complete inaccurate result. For unknown damage detection, it is difficult to select the frequency band which eigen-frequency should be included. This paper proposes a novel method using wavelet based transmissibility for local variation detection. Benefiting from the ability in subtle information acquisition of wavelet transform, it is useful in reducing the influence of frequency bands to the indicators. Analytical derivation using wavelet balance method and numerical studies of a multiple degree of freedom (MDOF) system are carried out to verify the effectiveness of the proposed method. In the last section, the method is applied for detecting crack position in cantilever beam with analysis of its sensitivity to frequency bands and measurement inaccuracies.

Identification of Clearance and Contact Stiffness in a Simplified Barrel-Cradle Structure of Artillery System

In gun barrel-cradle structure, the presence of clearance usually changes the dynamic response of muzzle and results in shooting dispersion (under continuous firing condition). The parameter estimation of such clearance nonlinear system is the prerequisite for establishing quantitative relation between the clearance and muzzle disturbance. In this paper, the restoring force surface (RFS) method and the nonlinear identification through feedback of outputs (NIFO) method are first combined for parameter identification in a simplified barrel-cradle structure. With the RFS method, clearance value can be obtained by analyzing the restoring force plot. Then the contact stiffness can be identified by using NIFO method. This identification process is verified in a single-degree-of-freedom (SDOF) system with clearance. To adapt to the rigid-flexible coupled beam system with clearances which is simplified from the barrel-cradle structure, a modification for the combined method mentioned above is proposed. The core idea of the modification is reducing the continuous system to multiple-degree-of-freedom (MDOF) system to reserve the nonlinear characteristics through modal transformation matrix. The advantage of this transformation is that the linear parts of the MDOF systems are decoupled, which greatly reduces the difficulty of identification. The simulation results have shown the effectiveness of current method.

A novel displacement-based seismic design method for framed structures considering P-Delta induced dynamic instability

An innovative displacement-based seismic design procedure for regular planar framed structures considering sidesway collapse prevention is presented. The method proposed, based on the characterization of a multiple degree of freedom system by means of a single degree of freedom system, allows the design of structures with a P-Delta induced negative stiffness to satisfy an interstorey drift threshold and/or prevent dynamic instability. The design procedure relies on the use of elastic analysis of simplified models and collapse or constant ductility spectra, hence, non-linear dynamic analyses are not required. To illustrate the potential of the method proposed, example applications and validation of the results obtained via incremental dynamic analysis are shown.

Estimating floor spectra in multiple degree of freedom systems

As the desire for high performance buildings increases, it is increasingly evident that engineers require reliable methods for the estimation of seismic demands on both structural and non-structural components. To this extent, improved tools for the prediction of floor spectra would assist in the assessment of acceleration sensitive non-structural and secondary components. Recently, a new procedure was successfully developed and tested for the simplified construction of floor spectra, at various levels of elastic damping, atop single-degree-of-freedom structures. This paper extends the methodology to multi-degree-of-freedom (MDOF) supporting systems responding in the elastic range, proposing a simplified modal combination approach for floor spectra over upper storeys and accounting for the limited filtering of the ground motion input that occurs over lower storeys. The procedure is tested numerically by comparing predictions with floor spectra obtained from time-history analyses of RC wall structures of 2- to 20-storeys in height. Results demonstrate that the method performs well for MDOF systems responding in the elastic range. Future research should further develop the approach to permit the prediction of floor spectra in MDOF systems that respond in the inelastic range.

Accuracy of one‐step integration schemes for damped/forced linear structural dynamics

SUMMARYThis paper proposes an energy‐based measure for the evaluation of the local truncation error of two‐level one‐step integration schemes. The measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model; it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses, and it segregates the error associated with the free and forced response components of the problem. To illustrate the approach, two examples associated with the application of the trapezoidal scheme and of a high‐order scheme proposed in the literature are analyzed. The latter reveals the shortcomings of the standard approach that is based on the undamped/unforced linear oscillator and therefore highlights the need for the proposed framework. Indeed, the scheme order of accuracy is below expectation when structural damping or external forcing is considered, in the numerically dissipative setting. Developments on the basis of the time discontinuous Galerkin (TDG) method are then proposed to recover the scheme high‐order accuracy. Additionally, they show the similarity that exists between schemes related to the TDG method and the ones obtained by integration by parts of the equation of motion. Copyright © 2014 John Wiley & Sons, Ltd.

Weakly nonlinear wave interactions in multi-degree of freedom periodic structures

This work presents a multiple time scales perturbation analysis for analyzing weakly nonlinear wave interactions in multi-degree of freedom periodic structures. The perturbation analysis is broadly applicable to (discretized) periodic systems in any dimensional space and with a wide range of constitutive nonlinearities. Specific emphasis is placed on cubic nonlinearity, as dispersion shifts typically arise from the cubic components in nonlinear restoring forces. The procedure is first presented in general. Then, application to the diatomic chain and monoatomic two-dimensional lattice demonstrates, individually, the treatment of multiple degree of freedom systems and higher dimensional spaces. The dispersion relations are modified by weakly nonlinear wave interactions and lead to additional opportunities to control wave propagation direction, band gap size, and group velocity. Numerical simulations validate the expected dispersion shifts. An amplitude-tunable focus device demonstrates the viability of utilizing dynamically-introduced dispersion to produce beam steering that may, ultimately, lead to a phononic superprism effect as well as multiplexing/demultiplexing behavior.

State Estimate-Based Robust Adaptive Control of a Linked Mass-Spring System With Nonlinear Friction

This paper aims to develop a state estimate-based friction fuzzy modeling and robust adaptive control techniques for controlling a class of multiple degrees of freedom (MDOF) mechanical systems. A fuzzy state estimator is proposed to estimate the state variables for friction modeling. Under some conditions, it is shown that such a state estimator guarantees the uniformly ultimate boundedness (UUB) of the estimate error. Based on system input–output data and our proposed state estimator, a robust adaptive fuzzy output-feedback control scheme is presented to control multiple degrees of freedom system with friction. The adaptive fuzzy output-feedback controller can guarantee the uniformly ultimate boundedness of the tracking error of the closed-loop system. A typical mass-spring system is employed in our simulation studies. The results demonstrate that our proposed techniques in this paper have good potential in controlling nonlinear systems with uncertain friction.