Multiple Degree Of Freedom Systems Research Articles

Exploring Practical Acoustic Transduction Attacks on Inertial Sensors in MDOF Systems

In cyber-physical systems, inertial sensors are the basis for identifying motion states and making actuation decisions. However, extensive studies have proved the vulnerability of those sensors under acoustic transduction attacks, which leverage malicious acoustics to trigger sensor measurement errors. Unfortunately, the threat from such attacks is not assessed properly because of the incomplete investigation on the attack's potential, especially towards multiple-degree-of-freedom systems, e.g., drones. To thoroughly explore the threat of acoustic transduction attacks, we revisit the attack model and design a new yet practical acoustic modulation-based attack, named KITE. Such an attack enables stable and controllable injections, even under frequency offset based distortions that limit the effect of prior attacking approaches. KITE exploits the potential threat of transduction attacks without the need of strengthening attackers' abilities. Furthermore, we extend the attack surface to multiple-degree-of-freedom (MDOF) systems, which are more widely deployed but ignored by prior work. Our study also covers the scenario of attacking moving targets. By revealing the practical threat from acoustic transduction attacks, we appeal for both the attention to their harm and necessary countermeasures.

Tracking superharmonic resonances for nonlinear vibration of conservative and hysteretic single degree of freedom systems

Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics — responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

The impacting vibration absorbers

Impacting vibration absorbers (IVA) and impacting inertial amplifier vibration absorbers (IIAVA) are established in this study to control the vibrations of the single and multiple degrees of freedom systems, i.e. SDOF and MDOF. Further, by incorporating the IVA and IIAVA with the base isolator, two hybrid passive vibration absorbers, namely base-isolated impacting vibration absorbers (BI-IVA) and base-isolated impacting inertial amplifier vibration absorbers, are developed and applied to SDOF and MDOF systems. In order to derive the equations of the motion for controlled dynamic systems, Newton's second law has been employed. The impact between the absorber mass and barrier has been modelled using the “signum” function. The standard linearization method has been used to linearise each nonlinear element of the equations of motion. H2 and H∞ optimization techniques are implemented to determine optimal design parameters analytically. The frequency response function is developed to obtain the dynamic system's responses. Accordingly, each proposed absorber's vibration reduction capacity is determined. For SDOF systems, the optimum IIAVA and BI-IVA have 59.44% and 96.63% more vibration reduction capacities than the optimum IVA. In contrast, the optimum IIAVA and BI-IVA have 46.83% and 96.74% more vibration reduction capacities than the optimum IVA when applied to the MDOF systems. In this paper, each one of the results is mathematically accurate.

Numerical simulation of stability and responses of dynamic systems under parametric excitation

The stability and responses of dynamic systems under parametric excitation are often encountered in many fields of science and engineering, such as slender columns and thin plates under axial loadings. This paper proposes a novel numerical simulation method to simultaneously construct stability diagrams and predict the responses of multiple degree-of-freedom (DOF) dynamic systems under arbitrary parametric loadings. The method divides an arbitrary load into discrete segments to approximate the variable excitation function by a step function and then accumulates the system responses of each segment using a matrix method. Dynamic stability and response analysis of undamped and damped 1DOF, 2DOF, 3DOF, and multiple DOF systems are conducted, and numerical examples are compared to conventional methods to show the accuracy and advantage of the proposed numerical simulation method. The instability diagrams are also substantiated by vibration response curves obtained from the same method. The method is applied to calibrate the validity and ranges of applicability of the Hill infinite determinants in the Floquet theory of 1DOF Mathieu-Hill equations. The second order approximation of the first instability region from the Hill determinant is acceptable. However, the fourth order approximation of the second instability region must be used to obtain a relatively accurate result. For 2DOF, 3DOF, and multi-parameter linear periodic systems, the method can simultaneously render four types of instability regions in a single procedure: no resonances, simple parametric resonances, combination additive resonances, and combination differential resonances. These results could provide a better understanding of stability and responses of dynamic systems under parametric excitation.

Damping parameter estimation using topological signal processing

Energy is dissipated in engineering systems through a variety of different, typically complex, damping mechanisms. While there are several common damping models with physical interpretations, it is often necessary to examine noisy, experimental data in order to identify and fit the damping parameters. One class of methods fits the damping parameters based on the decay envelope of the signal’s peaks based on some assumed damping mechanism. While there exist results in the literature, especially for viscous damping, to guide the selection of the spacing between the peaks for the optimum damping ratio identification, these methods often overlook the difficult and classic problem of identifying the signal’s “true” peaks in the presence of noise. Therefore, in this work we utilize tools from Topological Data Analysis (TDA) to address the problem of robust and automatic power-law damping identification from decay or free-response function. The decay function can be obtained from initial excitation, or using random decrement method thus making our approach a viable tool for operational modal analysis. We present our approach using one-dimensional examples, and describe extensions to multiple degree of freedom systems. Our damping identification framework uses the persistent homology of zero dimensional (0D) sub-level sets, one of the main tools of TDA, to separate the true topological features of the signal (i.e. the peaks and valleys) from measurement noise. Using persistent homology we are able to automatically represent the damped response in a noise-robust, two-dimensional summary of the peak–valley pairs called the persistence diagram. The persistence diagram is then analyzed using two methods: (1) a theoretical analysis of the significant persistence pairs and (2) function fitting to the persistence space. We present theoretical results that establish the class of general power-law damping terms where our approach applies, and develop specific tools to identify damping parameters for viscous, coulomb, and quadratic damping. We show that our approach is computationally fast, operates on the raw signal itself without requiring any pre-processing, and reduces the number of decisions needed by the user to perform the needed calculations. The results are validated using a combination of numerical and experimental data.

Detachment Detection in Cam Follower System Due to Nonlinear Dynamics Phenomenon

The detachment between the cam and the follower was investigated for different cam speeds (N) and different internal distance of the follower guide from inside (I.D.). The detachment between the cam and the follower were detected using largest Lyapunov exponent parameter, power density function of Fast Fourier Transform (FFT), and Poincare’ maps due to the nonlinear dynamics phenomenon of the follower. The follower displacement and the contact force between the cam and the follower were used in the detection of the detachment heights. Multi-degrees of freedom (spring-damper-mass) systems at the very end of the follower were used to improve the dynamic performance and to reduce the detachment between the cam and the follower. Nonlinear response of the follower displacement was calculated at different cam speeds, different coefficient of restitution, different contact conditions, and different internal distance of the follower guide from inside. SolidWorks program was used in the numerical solution while high speed camera at the foreground of the OPTOTRAK 30/20 equipment was used to catch the follower position. The friction and impact were considered between the cam and the follower and between the follower and its guide. The peak of nonlinear response of the follower displacement was reduced to (15%, 32%, 45%, and 62%) after using multiple degrees of freedom systems.

1504] PEDAGOGICAL APPROACH FOR STRUCTURAL STABILITY WITH NUMERICAL IMPLEMENTATION

This article aims to present a Stability Theory in a compact pedagogical way, where important concepts, analytical, and numerical procedures are included. The strong point of this article is the simplicity of the presentation of the nonlinear geometric analysis. Numerical computational procedures are presented for geometric nonlinear system analysis in order to facilitate understanding for those who intend to study the subject because it brings the basic concepts and the numerical implementations that can be used for the solution of more complex problems. Computational implementation details and analytical solutions are presented for two examples: single and multiple degree of freedom systems (SDOF and MDOF systems, respectively). The first one, the single degree of freedom scalar problem, is presented because it contains a limit load point, a dynamic jump, and the geometric imperfections influence, besides incorporating additional important concepts. The second one, a MDOF system, represented by a two degrees of freedom problem, is presented to introduce to matrix algebra.

DynNet: Physics-based neural architecture design for nonlinear structural response modeling and prediction

Data-driven models for predicting dynamic responses of linear and nonlinear systems are of great importance due to their wide application from probabilistic analysis to inverse problems such as system identification and damage diagnosis. In this study, a physics-based recurrent neural network model is designed that is able to estimate the dynamics of linear and nonlinear multiple degrees of freedom systems given the ground motions. The model is able to estimate a complete set of responses, including displacement, velocity, acceleration, and internal forces. Compared to the most advanced counterparts, this model requires smaller number of trainable variables while the accuracy of predictions is higher for long trajectories. In addition, the architecture of the recurrent block is inspired by differential equation solver algorithms and it is expected that this approach yields more generalized solutions. In the training phase, we propose multiple novel techniques to substantially accelerate the learning process using smaller datasets, such as hardsampling, utilization of a trajectory loss function, and implementation of a trust-region optimization approach. Numerical case studies are conducted to examine the strength of the network to learn different nonlinear behaviors. It is shown that the network is able to capture different nonlinear behaviors of dynamic systems with high accuracy and with no need for prior information or very large datasets.

An output-only ARX model-based sensor fusion framework on structural dynamic measurements using distributed optical fiber sensors and fiber Bragg grating sensors

The advent of distributed optical fiber sensing technologies has made continuous and dense measurement possible, and has a very broad application prospect in the field of structural health monitoring. However, not all of them are able to guarantee measurement accuracy in a good condition compared to traditional fiber sensors, such as FBGs (Fiber Bragg Grating sensor), strain gauges, etc. In our previous study (Cheng et al., 2019) of the calibration on static measurements for distributed sensors, it turns out that the signal to noise ratio from each single sensing point has proven to be much higher for FBGs than for distributed sensors.This paper presents a novel sensor fusion approach aiming to enhance the measurement quality of distributed fiber optical sensors in dynamic strain measurement. First of all, an output-only common-structured ARX (Auto-Regressive with exogenous input) model for dynamic MDOF (Multiple Degree of Freedom) systems with n degrees of freedom is studied where one of the measurement outputs is utilized as the input rather than actual external force, thus avoiding the inaccessibility of input signals in real structural engineering campaigns. After applying this model into a measurement system using a limited number of sensing points but with high fidelity, the outputs at unobserved points can then be roughly predicted using various curve fitting techniques on the ARX model coefficients against structural positions. However, the quality of the predicted performance cannot be guaranteed. Benefiting from a dense measurement but with low fidelity, the variation trend of the built ARX model coefficients against structural locations can be evaluated. Combined with EKR interpolation technique, the built ARX model from a high-fidelity system is then coupled to the evaluated trend of ARX model coefficients from a low-fidelity system, which leads to a further enhancement of its measurement quality at the unobserved points.To verify the feasibility and effectiveness of the presented sensor fusion approach, numerical studies on a 10-DOF MDOF system are conducted, indicating a satisfactory forecast with very small values of mean square errors for the unobserved DOFs. To further investigate the benefits of the proposed sensor fusion approach, experimental activities on a steel-beam are performed, aiming to enhance/calibrate the measurement performance on distributed fiber sensor (low fidelity) by incorporating four FBG sensors (high fidelity) as a calibration reference. The experimental results show that the intervention of the proposed approach yields significant improvement in the measurement quality for distributed fiber optic sensors.

An improvement over Fourier transform to enhance its performance for frequency content evaluation of seismic signals

The Fourier analysis is conventional technique for approximating the frequency content of earthquake ground motions. These approaches assume that earthquake excitation is a stationary process. Time frequency analysis techniques such as wavelet transform consider an earthquake to be a non-stationary process that shows a variation in frequency content over time, but these techniques are not easy to apply and most engineers are not familiar with their sophisticated mathematics. This paper introduces a new method for investigating of frequency content of earthquake records which is more accurate than Fourier spectrum but yet easy to apply for structural design and seismological studies. The efficiency of this new approach is investigated by analyzing a variety of single and multiple degree of freedom systems against near, mid and far field earthquake components. The results demonstrate the efficiency of the proposed method for a range of systems having different natural fundamental periods.

An Eigen-Based Theory For Structure-Dependent Integration Methods for Nonlinear Dynamic Analysis

An eigen-based theory for structure-dependent integration methods is constructed and it can provide the fundamental basis for the successful development of this type of integration methods. It is proved that a structure-dependent integration method can simultaneously combine unconditional stability and explicit formulation since it is proposed to accurately integrate low-frequency modes while no instability is guaranteed for high-frequency modes. In general, it is promising for solving inertial problems, where the total response is dominated by low-frequency modes. In addition, it can be explicitly and implicitly implemented for time integration although an explicit implementation is of practical significance since it can save many computational efforts due to the combination of unconditional stability and explicit formulation. A typical procedure to develop structure-dependent integration methods is constructed. In general, a coupled equation of motion for a multiple degree of freedom system can be decomposed into a set of uncoupled modal equations of motion by means of an eigen-decomposition technique. Next, an eigen-dependent integration method is developed to solve each modal equation of motion. Consequently, all the eigen-dependent integration methods are combined to form a structure-dependent integration method by employing a reverse procedure of the eigen-decomposition technique.

A dissipative family of eigen-based integration methods for nonlinear dynamic analysis

A novel family of controllable, dissipative structure-dependent integration methods is derived from an eigen-based theory, where the concept of the eigenmode can give a solid theoretical basis for the feasibility of this type of integration methods. In fact, the concepts of eigen-decomposition and modal superposition are involved in solving a multiple degree of freedom system. The total solution of a coupled equation of motion consists of each modal solution of the uncoupled equation of motion. Hence, an eigen-dependent integration method is proposed to solve each modal equation of motion and an approximate solution can be yielded via modal superposition with only the first few modes of interest for inertial problems. All the eigen-dependent integration methods combine to form a structure-dependent integration method. Some key assumptions and new techniques are combined to successfully develop this family of integration methods. In addition, this family of integration methods can be either explicitly or implicitly implemented. Except for stability property, both explicit and implicit implementations have almost the same numerical properties. An explicit implementation is more computationally efficient than for an implicit implementation since it can combine unconditional stability and explicit formulation simultaneously. As a result, an explicit implementation is preferred over an implicit implementation. This family of integration methods can have the same numerical properties as those of the WBZ-α method for linear elastic systems. Besides, its stability and accuracy performance for solving nonlinear systems is also almost the same as those of the WBZ-α method. It is evident from numerical experiments that an explicit implementation of this family of integration methods can save many computational efforts when compared to conventional implicit methods, such as the WBZ-α method.

Response Calculation in Zero Memory Nonlinear System Based on Digital Filter Method

Analysis of dynamic characteristics of structures often relies on linear theory in many practical applications. However, almost all structures show nonlinear characteristics when experienced with large enough excitations. Firstly, calculation of forced response in linear dynamic system using digital filter method is presented. Then, linear system is extended to nonlinear system. The nonlinear restoring force is treated as external loads and used to derive a generalized mathematical model to calculate the forced response in zero memory nonlinear system based on digital filter method. Finally, the ‘Reverse Path’ and multiple coherence function are used to estimate the frequency response function of underlying linear system. Numerical simulation of a multiple degree of freedom system with multiple nonlinear elements is used to verify the presented method. The results indicate that the presented model works well for calculating the forced response in zero memory nonlinear system.

Dynamic amplification factors for a system with multiple-degrees-of-freedom

It is well-known that the responses of a structure are different when subjected to a static load or a sudden step load. The dynamic amplification factor (DAF), which is defined as the ratio of the amplitude of the vibratory response to the static response, is normally used to depict the dynamic effect. For a single-degree-of-freedom system (SDOF) subjected to a sudden dynamic load, the maximum value of DAF is 2. Many design guidelines therefore use 2 as an upper bound to consider the dynamic effect. For a civil engineering structure, which is normally a multiple-degrees-of-freedom (MDOF) system, the DAF may exceed 2 in certain circumstances. The adoption of 2 as the upper bond as suggested by the design guidelines therefore may lead to unsafe structural design. Very limited studies systematically investigate the DAF of a MDOF system. This study theoretically investigates the DAF of a MDOF system when it is subjected to a step load based on the fundamental theory of structural dynamics. The condition on which the DAF may exceed 2 is defined. Two numerical examples and one experimental study of a cable-stayed bridge subjected to sudden cable loss are presented to illustrate the problem.

Effect of Time-Varying Delay on Stability of Real-Time Hybrid Simulation with Multiple Experimental Substructures

ABSTRACT In a real time hybrid simulation (RTHS), the actuator delay might deviate experimental results, or even destabilize the real-time test. Currently, research is very limited on stability of multiple-degree-of-freedom (MDOF) structures with multiple delays. Using the Lyapunov-Krasovskii (L-K) method, this study proposes stability criteria for MDOF systems with multiple time-varying/constant delays, and derives a more accurate result. Application of criteria shows that the stability of time-varying delay system is smaller than the corresponding constant delay system. With number of DOF and delay increasing, the criterion accuracy decreases slowly for constant delay system; meanwhile, the stability difference between constant- and time-varying- delay system increases gradually. Such difference is significant with small stiffness ratio of physical substructure.

Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models

The visco-elastic dampers can be economically designed for response reduction of dynamical systems. Recent developments in fractional calculus have affected the modeling of visco-elastic materials. The fractional models for visco-elastic dampers, which are parsimonious, require fewer parameters in comparison with other models. In this paper, we use the visco-elastic dampers for control of structural responses. The visco-elastic damper utilizes three important parameters including damping coefficient, stiffness, and fractional order. The dynamic model of the building with visco-elastic dampers can be acquired by a system of fractional differential equations, which includes both fractional and ordinary derivatives. We apply spline collocation methods for obtaining the numerical solution of this complex system. The collocation method is implemented in a graded mesh. The discrete data of earthquake are smoothed by spline interpolation of higher degree. The introduced method is used to simulate the response of a structure with active and passive controllers. A comparison of using different dampers is considered. Also, norm of the transform function is used for response analysis. The results of simulations for four-story and 10-story buildings show significant reductions in structural responses. Moreover, our analysis presents that the large variety of values can be used for parameters of the visco-elastic damper to provide flexibility in designing appropriate visco-elastic-dampers.

Parameter Variation on Nonlinear Energy Sink attached to Multiple Degree of Freedom System

AbstractThe importance of lightweight structures has increased in the last years motivated by the goals of improving efficiency of vehicles and conserving resources. In most cases the loss of weight by using fewer or lighter materials is accompanied by a reduction of the structures stiffness having a strong effect on occuring vibrations. Even if the damping ratio of the structure remains unchanged, by reducing weight and stiffness the vibration amplitudes might increase if the vibration excitation stays the same. Damping elements might be introduced to the structure mitigating critical resonance gains of the structure to reduce these vibrations. In this work the dynamics of a linear multiple degrees of freedom structure with a nonlinear damping element are investigated. The damping element is chosen to have a nonlinear characteristic to allow for a situation adaptive design. Numerical parameter studies are carried out to optimize the resulting damping of the entire structure. The nonlinear equation of motion is solved by applying the Harmonic Balance Method. The evaluation of the performance of the nonlinear damping element is done by consideration of the nonlinear frequency response functions of the structure.

Probabilistic Evaluation of Soil–Structure Interaction Effects on Strength Demands of Shear Buildings

This paper revisited the effect of soil–structure interaction (SSI) on the nonlinear response of multistory buildings with a probabilistic approach. In particular, this study showed that the magnitude of SSI effects on the strength demand of a multiple-degree-of-freedom (MDOF) system stems from the effect of SSI on the corresponding equivalent single-degree-of-freedom (eSDOF) system. This seems to negate several past studies on flexible-base MDOF systems. In fact, such studies did not quantify the split between contributions of SSI effects to the corresponding eSDOF system and to those unique aspects of MDOF systems that cannot be represented by the eSDOF system. Doing so in the present paper showed that the effect of SSI on the inelastic response of MDOF systems can be well predicted based on the knowledge of the SSI effect on the response of SDOF systems, i.e., the share of MDOF aspects of the system in this effect is insignificant. Based on this finding, a procedure is proposed to determine the strength demand of a flexible-base shear building given a target ductility demand.

An improved performance-based plastic design method for seismic resilient fused high-rise buildings

The performance-based plastic design (PBPD) method generally relies on the nonlinear response of equivalent elastic-perfectly-plastic single degree of freedom system. It usually cannot achieve the design of three performance objectives simultaneously and may not consider the high mode effect of structure, which is significant for high-rise building. In this paper, a trilinear force-displacement model indicating three prescribed performance objectives at three seismic hazard levels is adopted to improve the PBPD method. The proposed improved PBPD method is derived based on multiple degrees of freedom system, while the high-mode effect and post-yield stiffness of the structure is considered. It can be used for designing seismic resilient fused high-rise buildings. A novel dual system composed of steel energy-dissipative column (EDC) and moment resisting frame (MF) is employed for application of the proposed method. This dual system has its fuse members decoupled from the gravity-resisting system, and the performance-based design of this system is discussed as well as its application for high-rise buildings. To demonstrate the effectiveness of the proposed method, a 20-story EDC-MF structure system is designed using the improved PBPD method. A detailed numerical model of the designed EDC-MF system is then built, and nonlinear dynamic response analyses at different seismic intensities are performed to verify the actual structure performance. Results show that the designed structure can achieve the prescribed yielding mechanism and performance objectives at three seismic hazard levels, and the EDC-MF system can be effectively applied to high-rise building as a seismic resilient fused structure.

Periodic, aperiodic and chaotic motions of harmonically excited SDOF and MDOF nonlinear dynamical systems

Responses of nonlinear dynamical systems with single degree of freedom (SDOF) or multiple degrees of freedom (MDOF) to periodic excitations are investigated in this paper using a numerical scheme. The scheme is developed on the basis of the effective mass, damping and stiffness matrices, the incremental generalized coordinates and the Newmark integration method to solve efficiently and accurately second-order nonlinear ordinary differential equations in the hundreds or more. Using the proposed numerical method, long-term behavior of SDOF and MDOF systems of any type of nonlinearities including the well-known van der Pol nonlinear damping forces, the Duffing type nonlinear spring forces and the time-delayed spring force at any strength level (weak, moderate and strong) can be accurately determined. With the help of an adequate mapping frequency, orders of periodicity of periodic responses can be easily and reliably identified. Numerical results, obtained for three oscillators – an SDOF van der Pol oscillator, a time-delayed SDOF Duffing oscillator, and a five-DOF Duffing oscillator, demonstrate that the proposed scheme is ideally suited for solving large scale nonlinear dynamical problems.