Optimal distributed control of a continuous beam with damping

Optimal control of a damped two-span beam is studied with the objective of minimizing its deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transverse distributed force. The multiple objectives of the problem lead to a vector performance criterion which is reduced to a scalar one by using the concept of Pareto optimality. Necessary and sufficient conditions of optimality are expressed in the form of a maximum principle which leads to an explicit expression for the control force. The behaviour of the controlled beam is numerically studied which indicates that optimally controlled distributed forces are effective in damping out the dynamic response. Relations between various objectives are studied by means of optimal trade-off curves showing the best performance of the controlled beam.     

Active vibration control using an inertial actuator with internal damping

Collocated direct velocity feedback with ideal point force actuators mounted on structures is unconditionally stable and generates active damping. When inertial actuators are used to generate the control force, the system can become unstable even for moderate velocity feedback gains due to an additional -180 degree phase lag introduced by the fundamental axial resonant mode of the inertial actuator. In this study a relative velocity sensor is used to implement an inner velocity feedback loop that generates internal damping in a lightweight, electrodynamic, inertial actuator. Simulation results for a model problem with the actuator mounted on a clamped plate show that, when internal relative velocity feedback is used in addition to a conventional external velocity feedback loop, there is an optimum combination of internal and external velocity feedback gains, which, for a given gain margin, maximizes vibration reduction. These predictions are validated in experiments with a specially built lightweight inertial actuator.     

Self-tuning control systems of decentralised velocity feedback

This paper is concerned with decentralised velocity feedback for the control of vibration on a flexible structure. Previous studies have shown that a direct velocity feedback loop with a collocated force actuator produces a damping action. Multiple velocity feedback control loops thus reduce the vibration and sound radiation of structures at low frequency resonances, where the response is controlled by damping. However, if the control gains are too high, so that the response of the structure at the control point is close to zero, the feedback control loops will pin the panel at the control positions and thus no damping action is generated. Therefore, in order to maximise the active damping effect, the feedback gains have optimum values and the loops need to be properly tuned.In this paper, a numerical investigation is performed to investigate the possibility of self-tuning the feedback control gains to maximise the power absorbed by the control loops and hence maximise the active damping. The tuning principle is first examined for a single feedback loop for different excitation signals. The tuning of multiple control loops is then considered and the implementation of a practical tuning algorithm is discussed.     

Plate with decentralised velocity feedback loops: Power absorption and kinetic energy considerations

This paper is focused on the vibration effects produced by an array of decentralised velocity feedback loops that are evenly distributed over a rectangular thin plate to minimise its flexural response. The velocity feedback loops are formed by collocated ideal velocity sensor and point force actuator pairs, which are unconditionally stable and produce ‘sky-hook’ damping on the plate. The study compares how the overall flexural vibration of the plate and the local absorption of vibration power by the feedback loops vary with the control gains. The analysis is carried out both considering a typical frequency-domain formulation based on kinetic energy and structural power physical quantities, which is normally used to study vibration and noise problems, and a time-domain formulation also based on kinetic energy and structural power, which is usually implemented to investigate control problems. The time-domain formulation shows to be much more computationally efficient and robust with reference to truncation errors. Thus it has been used to perform a parametric study to assess if, and under which conditions, the minimum of the kinetic energy and the maximum of the absorbed power cost functions match with reference to: (a) the number of feedback control loops, (b) the structural damping in the plate, (c) the mutual distance of a pair of control loops and (d) the mutual gains implemented in a pair of feedback loops.     

A comparison of decentralized, distributed, and centralized vibro-acoustic control

Direct velocity feedback control of structures is well known to increase structural damping and thus reduce vibration. In multi-channel systems the way in which the velocity signals are used to inform the actuators ranges from decentralized control, through distributed or clustered control to fully centralized control. The objective of distributed controllers is to exploit the anticipated performance advantage of the centralized control while maintaining the scalability, ease of implementation, and robustness of decentralized control. However, and in seeming contradiction, some investigations have concluded that decentralized control performs as well as distributed and centralized control, while other results have indicated that distributed control has significant performance advantages over decentralized control. The purpose of this work is to explain this seeming contradiction in results, to explore the effectiveness of decentralized, distributed, and centralized vibro-acoustic control, and to expand the concept of distributed control to include the distribution of the optimization process and the cost function employed.     

Non-stationary response of a beam to a moving random force

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.     

Revisiting the moving force problem

The problem of the vibration of a beam subject to a travelling force is considered. The purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force. It is shown that, for given boundary conditions, there exists a unique response-velocity dependence function. A technique to determine this function is suggested, which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode. To illustrate this, the maximum response function is calculated analytically for a simply supported (SS) beam and constructed numerically for a clamped-clamped beam. The effect of the higher modes on the maximum response is investigated, and the relative error of the one-mode approximation for a SS beam is constructed. The estimates obtained substantiate the assumption about adequacy of the one-mode approximation in a wide range of velocities; in particular, the relative error in the neighborhood of the velocity that results in the largest response is less than one percent.     

Characteristics of an active feedback system for the control of plate vibrations

In the interests of improving airborne insulation of panels and of controlling room reverberation, a technique is studied for establishing control of the transverse vibrations of a thin plate by the application of active energy feedback. A localized point control force is derived from the sensed motion of some point on the plate surface. The superposition principle is applied in the form of a mobility analysis which shows that open loop gain conditions cannot result in a specific motion, including that of complete damping, of any arbitrary point on the plate surface but can be effective for particular points and for control of resonant modal motions under conditions of light damping. With velocity sensing, the characteristics for stable operation under the convenient condition of constant gain depend on maintenance of like symmetry, in the sense of an identity of velocity magnitude and sign, in the relative motion of sensing and control-force points. Bandwidth limitations are avoidable only by closure of the loop between these points. Two varieties of control force generator are involved: namely, where the generator mass is rigidly mounted and again where a spring mounting on the plate provides a self-supporting role.This is the first of two companion papers on active control of plate vibrations. Systems in which an array of multiple control units is used will be described in the second paper.     

Optimal Damping of Perturbations of Moving Thermoelastic Panel

The translational motion of a thermoelastic web subject to transverse vibrations caused by initial perturbations is considered. It is assumed that a web moving with a constant translational velocity is described by the model of a thermoelastic panel simply supported at its ends. The problem of optimal damping of vibrations when applying active transverse actions is formulated. For solving the optimization problem, modern methods developed in control theory for systems with distributed parameters described by partial differential equations are used.     

Passive vehicle suspensions employing inerters with multiple performance requirements

This paper investigates passive vehicle suspensions with inerters by considering multiple performance requirements including ride comfort, suspension deflection and tyre grip, where suspension deflection performance is novelly considered which is formulated as a part of objective functions and a constraint separately. Six suspension configurations are analyzed and the analytical solutions for each performance measure are derived. The conditions for each configuration to be strictly better than the simpler ones are obtained by presenting the analytical solutions of each configuration based on those of the simpler ones. Then, two stages of comparisons are given to show the performance limitations of suspension deflection for passive suspensions with inerters. In the first stage, it is shown that although the configurations with inerters can improve the mixed performance of ride comfort and tyre grip, the suspension deflection performance is significantly decreased simultaneously. In the second stage, it is shown that for passive suspensions with inerters, suspension deflection is the more basic limitation for both ride comfort and tyre grip performance by doing comparisons among mixed ride comfort and suspension deflection optimization, mixed ride comfort and tyre grip optimization, and mixed suspension deflection and tyre grip optimization. Finally, the problem of mixed ride comfort and tyre grip performance optimization with equal suspension deflection is investigated. The limitations of suspension deflection for each configuration are further highlighted.     

Optimum vibration absorber (tuned mass damper) design for linear damped systems subjected to random loads

Optimum design of dynamic vibration absorbers (DVAs) installed on linear damped systems that are subjected to random loads is studied and closed-form design formulas are provided. Three cases are considered in the optimization process: Minimizing the variance of the displacement, velocity and acceleration of the main mass. Exact optimum design parameters for the velocity case, which to the best knowledge of the author do not exist in the literature, are derived for the first time. Exact solutions are found to be directly applicable for practical use with no simplification needed. For displacement and acceleration cases, a solution for the optimum absorber frequency ratio is obtained as a function of optimum absorber damping ratio. Numerical simulations indicate that optimum absorber damping ratio is not significantly related to the structural damping, especially when the displacement variance is minimized. Therefore, optimum damping ratio derived for undamped systems is proposed for damped systems for the displacement case. When acceleration variance is minimized, however, the optimum damping ratio derived for undamped systems is found not as accurate for damped systems. Therefore, a more accurate approximate expression is derived. Numerical comparisons with published approximate expressions at the same level of complexity indicated that proposed design formula yield more accurate estimates. Another important finding of the paper is that for specific applications where all of the response parameters are desired to be minimized simultaneously, DVAs designed per velocity criteria provide the best overall performance with the least complexity in the design equations.     

Effect of viscous damping on power transmissibility for the vibration isolation of building services equipment

Vibration isolation plays an important role in both the vibration and noise control of building services equipment. To evaluate vibration isolation performance, the force transmissibility method is commonly adopted. However, increasing the damping effect in the force transmissibility method reduces both the resonance peak value and the isolation performance in the “isolation region”. The limitation of the method is that the transmitted displacement of a floor structure and the interaction of mounting points are neglected. To include the floor displacement and the interaction of mounting points, Mak and Su recently proposed the power transmissibility method to assess the performance of vibration isolation. In this paper, the effect of viscous damping on power transmissibility is investigated. A practical procedure for experimentally determining the damping ratio is also given.     

Tracking Consensus for Second-Order Multi-Agent Systems with Nonlinear Dynamics in Noisy Environments

This paper investigates the leader-following tracking consensus problem for second-order multi-agent systems with time delays and nonlinear dynamics in noisy environments on the conditions of fixed and switching directed topologies. Based on a novel velocity decomposition technique and stochastic analysis, a measurement-based distributed tracking control protocol is proposed, under which all agents can track the leader in mean square. Simulation results are also given to illustrate the effectiveness of the proposed protocol.     

An alternative to modal analysis for material stiffness and damping identification from vibrating plates

This paper presents an alternative to modal analysis to extract stiffness and damping parameters from thin vibrating plates. Full-field slope measurements are performed through a deflectometry technique on a plate vibrating at a given frequency. Images are recorded in phase and at π/2 lag from the excitation. From this information, deflection fields are computed by integration and curvature fields are obtained by differentiation. This information is then input into the principle of virtual work to extract both stiffness and damping parameters. This procedure, known as the Virtual Fields Method, is detailed in the paper and the notion of special optimized virtual fields is extended to the present problem. Validation on simulated data is performed before moving to experimental data. One of the main advantages of this technique is that it is completely insensitive to the damping coming from the boundary conditions. This is illustrated experimentally on two tests where a viscoelastic layer and rubber washers are added in the experimental set up.     

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate is presented. The active modal control of the vibro-acoustic response is implemented using piezoelectric actuators/sensors. The active modal damping is added to the coupled system via negative velocity feedback. The feedback gain between the piezoelectric actuators/sensors for the modal control is obtained using the in-vacuo modal matrix and the incompressible fluid-loaded modal matrix. The modal control performance of structural vibration and acoustic radiation of a baffled plate is numerically studied. It is shown that the proposed method increases the modal damping ratio and achieves reduction in the mean square velocity and the sound power for given modes of the fluid-loaded plate.     

Active control of cylindrical shell with magnetostrictive layer

This paper analyses the damping characteristics of a titanium shell with a magnetostrictive layer bonded to it. The magnetostrictive layer produces an actuating force required to control vibration in the shell, based on a negative velocity feedback control law. The control input is the current to the solenoid surrounding the shell. In the present study, a finite element formulation, physically consistent with the problem has been developed. Vibration reduction in the shell by changing the position of the magnetostrictive layer and its current carrying actuating coil pair along the shell is investigated.     

Transverse vibrations of beams with variable moment of inertia and an intermediate support

The above problem is tackled using the Ritz method and approximating the deflection function by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental frequency coefficients are determined in the case of a rather complex structure taking into account different types of boundary conditions.     

Elastic plate as floating wave breaker in a beach zone

Wave propagation and damping mechanism due to elastic coating of the sea surface is considered. The hydrodynamic performance of an elastic plate is analyzed for various conditions in terms of wave reflection and transmission, plate deflection, and surface strain. Rigidity and geometrical scales of the coating plate essentially affect the wave transmission characteristics. The model of wave propagation and scattering is constructed in the long-wave approximation. The case of elastic plate with fixed edges is considered. It is shown that optimally designed horizontal flexible membrane can be a very effective wave barrier in a beach zone.     

Reduction of bridge dynamic amplification through adjustment of vehicle suspension damping

This paper presents a novel approach to the reduction of short-span bridge dynamic responses to heavy vehicle crossing events. The reductions are achieved through adjustment of the vehicle suspension damping coefficient just before the crossing. Given pre-calculations of the response of a vehicle-bridge system to a set of ‘unit’ road disturbances, it is shown that a single optimum damping coefficient may be determined for a given velocity and any specified road profile. This approach can facilitate implementation since the optimum damping is selected prior to the bridge and there is no need to continuously vary the damping coefficient during the crossing. The concept is numerically validated using a bridge-vehicle interaction model with several road profiles, both measured and artificially generated. The bridge-friendly damping control strategy is shown to reduce bridge dynamics across a typical range of vehicle velocities, proving most effective for road profiles that induce large vibrations in the vehicle-bridge system.     

A study on radial directional natural frequency and damping ratio in a vehicle tire

The measurement of radial directional natural frequency and damping ratio in a vehicle tire has been studied. Natural frequencies and damping ratios in the radial direction of various tires, from passenger car tires to truck bus tires, are reported. The radial direction modal parameters of tires subjected to different levels of inflation pressure, have been determined by using a frequency response function method. To obtain the theoretical natural frequency and mode shape, the plane vibration of a tire has been modeled as though it were that of a circular beam. By using the Tielking method that is based on Hamilton’s principle, theoretical results have been determined by considering the rotational velocity, tangential and radial stiffness, radial directional velocity and tension force which is due to tire inflation pressure. The results show that experimental conditions can be considered as the parameters that shift the natural frequency and damping ratio.