Modal Truncation Technique Research Articles

On the use of component mode synthesis methods for the model reduction of flexible multibody systems within the floating frame of reference formulation

This paper is focused on the application of the fundamental model reduction techniques used in structural dynamics to flexible multibody systems. In particular, an effective computational approach is developed and tested in this work for adapting the conventional structural model condensation strategies to the dynamics of flexible multibody systems. Thereby, the synthesis methods considered in this work can be readily implemented in the principal computational framework used for modeling flexible multibody systems subjected to large reference displacements and small deformations, namely the Finite Element (FE) Floating Frame of Reference Formulation (FFRF). Three component mode synthesis methods are of interest for this study: the Guyan-Iron condensation method, the well-known modal truncation technique, and the Craig-Bampton synthesis approach. Employing the computational approach proposed in this paper, it is demonstrated by means of numerical experiments that the three fundamental reduction methods considered in this investigation lead to correct numerical results when used for obtaining condensed mechanical models of flexible multibody systems based on the FE-FFRF. In the case of flexible multibody systems, the computational approach developed in this paper consists of two fundamentally distinct steps. First, the reference conditions are formulated in the FE-FFRF in a way consistent with the geometric properties of the mechanical joints that connects the bodies of the flexible multibody system under consideration. This is a key step that allows the analyst to model multibody systems in complex geometric configurations. Subsequently, a reduction technique is selected and used for reducing the dimensionality of the problem at hand. It is remarked that the reference conditions assumed in the first step of the proposed computational procedure modify all the component modes resulting from the application of the reduction strategy chosen in the second step of the computational algorithm and may or may not delete some of these. In the second step of the computational approach mentioned before, the conventional model reduction techniques can be effectively used leading to consistent numerical results. In this work, three numerical examples are employed for demonstrating the effectiveness of the computational procedure developed in the paper. More importantly, considerations on the choice of either isostatic or hyperstatic reference conditions associated with a particular choice of component modes are discussed in detail throughout the manuscript. The numerical simulations demonstrate that these choices can have a significant influence also on the efficiency of the integration schemes.

A Structural Pounding Formulation Using Systematic Modal Truncation

The evaluation of the response function of the structural pounding problem is generally time‐consuming if high‐order systems are applied. The well‐known modal truncation strategy is outstandingly efficient for a single linear ground‐accelerated structure. However, for the analysis of the structural pounding problem, the classical modal truncation technique turns out to be ineffective as additional higher frequency motion due to possible contact impact occurs. This makes the determination of how many modes should be taken into account in order to obtain a required level of accuracy more difficult. Therefore, in this paper, a systematically controlled modal truncation strategy adapted to the seismic pounding formulation under consideration of high nonlinearity and nonsmoothness of contact problems is introduced. A comparative study of the classical and the controlled modal truncation technique is presented and a comparison with the commercial software package ABAQUS© is provided. It is shown that the computational accuracy is significantly improved when applying the new systematically controlled modal truncation strategy.

Real-time simultaneous identification of structural systems and unknown inputs without collocated acceleration measurements based on MEKF-UI

Identifications of excitations and structural systems are two crucial factors in structural health monitoring. There have been some research works focusing on simultaneous identification of structural systems and unknown inputs. Among them, the approach based on the extended Kalman filter with unknown inputs (EKF-UI) has the superiority of implementing identification with limited measurements and in real-time fashion. However, in previous EKF-UI approaches, it is necessary to have the measurements of acceleration responses at the locations where unknown inputs applied. In this paper, it is proposed to circumvent the above limitations for real time simultaneous identification of structural systems and unknown inputs without using collocated acceleration measurements. The proposed method is based on a novel modal extended Kalman filter with unknown inputs (MEFK-UI), which is the combination of extended Kalman filter in modal domain and the recent EKF-UI developed by the authors. Simultaneous identification of structural modal co-ordinates and unknown modal inputs are first conducted in the reduced dimension by modal truncation technique. Moreover, data fusion of displacement and acceleration measurements is used to prevent in real time the low-frequency drifts in the identification results. Subsequently, structural systems including structural parameters and unknown external inputs are identified in real time. Such an approach is not available in the literature. Several numerical examples are used to demonstrate the effectiveness of the proposed method.

An integrated prediction model including the cutting process for virtual product development of machine tools

Recently activities for shortening the development time of machine tools and demands for high-value products are consistently being increased. As a result, many studies for predicting the performance in design stage have been conducted to solve these issues. So far, most manufacturers of machine tools have verified designs excluding the cutting process, even though the main performance is the machining quality of the workpiece. Therefore, it has been required to consider together the cutting process model for analyzing the machining quality before the manufacturing of the prototype. This paper presents an integrated dynamic model considering the all interaction of the machine structure, the control system, and the cutting process. The modal truncation technique using the steady-state dynamic analysis in a selected location and frequency range of interest is applied to increase the efficiency of mechatronics simulations including the cutting process. In particular, this proposed prediction model allows accurate design verifications for high-speed machines because it reflects the influences of the high inertia forces and the contour errors caused by high feed rate. Several peripheral milling applications were demonstrated in order to show the feasibility of the proposed prediction model, and we confirmed the superiority of prediction by comparing the simulated and measured results in high feed milling.

Virtual collocation method for a flexible rotor supported by active magnetic bearings

Local vibration control systems of mechanical structures can be collocated or non-collocated. If a sensor is placed at the same location as an actuator the system is said to be collocated. Otherwise, the system is non-collocated. It is not always possible to collocate the vibration control actuator at the location of interest, e.g. in active magnetic bearings (AMBs) eddy current sensors are used to measure displacements of the supported shaft. Due to inevitable electromagnetic interferences the sensors cannot be placed exactly at locations of electromagnetic coils of the AMB. Non-collocation complicates the control problem because the dynamics of the structure between the control actuator and sensor disturbs the performance of the vibration control system. Non-collocated systems have non-minimum-phase zeros at the right half-plane, which may even destabilize the control system. Non-collocation effects were suppressed by several methods including phase shifting, time delay or passive vibration absorber methods. However, none of these methods ensured the required stability, nor a good performance. The present paper recommends the virtual (recalculated) collocation method for local control loops of the AMB. The idea is to calculate displacements at actuators' locations given the displacements at sensors' locations. The approach is illustrated with numerical results of the flexible rotor supported by AMBs and controlled with four proportional integral derivative (PID) controllers. The finite element model of the rotor is developed and transformed to the state-space. The model is reduced by the modal truncation technique. Then, an observer for the reduced system is developed. This way the shaft displacements at any discretization node along the shaft axis are simply estimated and introduced to the inputs of PID controllers. Calculation results demonstrate a good performance of the collocated PID control systems, confirming the potential of the method and giving a rationale for its further development.

Fuzzy Sliding Mode Control of Plate Vibrations

In this paper, fuzzy logic is meshed with sliding mode control, in order to control vibrations of a cantilevered plate. Test plate is instrumented with a piezoelectric sensor patch and a piezoelectric actuator patch. Finite element method is used to obtain mathematical model of the test plate. A design approach of a sliding mode controller for linear systems with mismatched time-varying uncertainties is used in this paper. It is found that chattering around the sliding surface in the sliding mode control can be checked by the proposed fuzzy sliding mode control approach. With presented fuzzy sliding mode approach the actuator voltage time response has a smooth decay. This is important because an abrupt decay can excite higher modes in the structure. Fuzzy rule base consisting of nine rules, is generated from the sliding mode inequality. Experimental implementation of the control approach verify the theoretical findings. For experimental implementation, size of the problem is reduced using modal truncation technique. Modal displacements as well as velocities of first two modes are observed using real-time kalman observer. Real time implementation of fuzzy logic based control has always been a challenge because a given set of rules has to be executed in every sampling interval. Results in this paper establish feasibility of experimental implementation of presented fuzzy logic based controller for active vibration control.

Fuzzy Sliding Mode Control of Plate Vibrations

In this paper, fuzzy logic is meshed with sliding mode control, in order to control vibrations of a cantilevered plate. Test plate is instrumented with a piezoelectric sensor patch and a piezoelectric actuator patch. Finite element method is used to obtain mathematical model of the test plate. A design approach of a sliding mode controller for linear systems with mismatched time-varying uncertainties is used in this paper. It is found that chattering around the sliding surface in the sliding mode control can be checked by the proposed fuzzy sliding mode control approach. With presented fuzzy sliding mode approach the actuator voltage time response has a smooth decay. This is important because an abrupt decay can excite higher modes in the structure. Fuzzy rule base consisting of nine rules, is generated from the sliding mode inequality. Experimental implementation of the control approach verify the theoretical findings. For experimental implementation, size of the problem is reduced using modal truncation technique. Modal displacements as well as velocities of first two modes are observed using real-time kalman observer. Real time implementation of fuzzy logic based control has always been a challenge because a given set of rules has to be executed in every sampling interval. Results in this paper establish feasibility of experimental implementation of presented fuzzy logic based controller for active vibration control.

ハイブリッド空力振動法に適用する無条件安定と等価な陽的積分法―多自由度系弾塑性挙動に対する検討―

This paper examines accuracy and applicability of the modal-explicit technique in numerical experiments and the New Hybrid Aerodynamic Vibration Technique. The modal-explicit technique is a numerical integration technique which is combination of Newmark Explicit method with modal truncation technique in order to secure stability of solution and simple algorithm. In the numerical experiments, the accuracy of solution is verified by using three types of lumped-mass model. It is difficult for a numerical integration technique to obtain stable solution because all the models include high frequency vibration modes. The hysteretic curve in the each story has non-linearity. The random vibration behavior is followed by the modal-explicit technique, the accuracy and stability is satisfied. In the New Hybrid Aerodynamic Vibration system, that is set in the modal-explicit technique as a numerical integration technique, unstable wind response of a base-isolated high rise building is simulated. The accuracy is satisfied, too.

Numerical Integration Scheme for Hybrid Vibration Technique

This paper describes an application of an explicit integration scheme for a control algorithm to the Hybrid Vibration Technique. The applied scheme is a combination of explicit Newmark β method with the modal truncation technique. Recently, it is considered in the structural design that some structures behave not only in elastic range also in elasto-plastic range even under wind loading. Therefore, the scheme must be able to follow non-linear dynamic behavior. On the other hand, the modal analysis can be performed only for a linear system. In order to perform the modal analysis in a non-linear system, it is assumed that restoring force of the structures consists of the component for assumed linear stiffness and unbalanced force. And, on the equation of motion, it is considered that assumed external force, which is equal to actual force added to unbalanced force, acts on the linear vibration system with the assumed linear stiffness. From the assumption, step-by-step integration can be carried out in the each mode on the generalized coordinate system for the assumed stiffness. The higher mode components are rejected by using the orthogonality of the generalized coordinate system because the components have the severe condition for the stability of the solution and little influence on the accuracy of the total response. As a result, the magnitude of integral time interval can be determined without considering the severe stable condition even though the response is calculated by means of an explicit scheme with conditional stable.Numerical simulations are conducted for examining the performance of the applied scheme. The vibration model is multi-degree-of-freedom vibration system including the highest mode with much higher natural frequency than the other modes. Bi-linear model and elastic linear model are adapted to the restoring model. Assumed linear stiffness is utilized for the modal analysis in the every model. From the simulation, the applied scheme has the enough performance for conducting step-by-step integration in the Hybrid Vibration System.

Seismic Analysis of a Combined System Composed of Equipment and Structure Using Modal Truncation Technique

Seismic Analysis of a Combined System Composed of Equipment and Structure Using Modal Truncation Technique

Finite Element Modal Formulation for Hypersonic Panel Flutter Analysis with Thermal Effects

A finite element time-domain modal formulation for analyzing nonlinear flutter of panels subjected to hypersonic airflow has been developed. Von Karman large deflection plate theory is used for description of the structural nonlinearity, and third-order piston theory is employed to consider the aerodynamic nonlinearity. The thermal loadings of uniformly distributed surface temperatures and temperature gradients through the panel thickness are considered. By the application of the modal truncation technique, the number of governing equations of motion is reduced dramatically so that the computational costs are reduced significantly. All possible types of panel behavior, including flat and stable, buckled but dynamically stable, limit cycle oscillation (LCO), periodic motion, and chaotic motion were observed. Examples of the applications of the proposed methodology were flutter responses of isotropic and composite panels. Special emphasis was placed on the boundary between LCO and chaos and on the route to chaos. Time history, phase plane plot, Poincare map, bifurcation diagram, and Lyapunov exponent are employed in the chaos study. It is found that at low or moderately high dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos, whereas, at high dynamic pressures, the route generally involves bursts of chaos and rejuvenations of periodic motions.

Finite Element Simulation of Smart Structures for Active Vibration andAcoustic Control

AbstractThe design process of engineering smart structures using piezoelectric patches as distributed sensors and actuators requires a virtual overall model, which includes the main functional parts. In the paper a suitable software tool based on the FEM is presented to simulate coupled electro‐mechanical‐acoustical problems regarding interior noise problems. The modal truncation technique is used to design a LQ‐controller as well as to study the time‐depending behaviour of a clamed smart plate.

Stochastic Response Cumulants of MDOF Linear Systems

This study addresses the complexities that arise in developing algorithms to formulate and subsequently solve the very large system of equations describing the stochastic response cumulants of multi-degree-of-freedom (MDOF) linear systems subjected to nonnormal filtered delta-correlated processes. A general method is presented that takes advantage of the symmetry of the cumulant tensors to simplify the analysis. New computationally efficient algorithms based on modal analysis of the large linear system of cumulants are proposed for calculating the nonstationary response cumulants. A modal truncation technique is subsequently used in which a selected number of modes for the system of cumulants is retained in the analysis. With the exception of a small number of modes corresponding to low eigenvalues, most of the higher contributing modes are treated statically in the modal expansion. The methodology provides insight into the conditions for neglecting a mode, as well as treating a contributing mode statically or dynamically. An example structure is used to illustrate various aspects concerning the computational efficiency of the methodology and the accuracy of the modal truncation technique.