Vibration Localization in Rotating Shafts: Part II — Experiment

This paper presents an experimental study of vibration localization in single-span, flexible, rotating shafts. It was shown in a companion paper (Part I) that a non-circular cross-section of the rotating shaft, leading to dissimilar lateral moments of inertia, can introduce disorder. Internal coupling between the principal directions of vibration is provided by the rotational speed through the gyroscopic moments. It is experimentally demonstrated here that directional vibration localization can occur for certain appropriate combinations of disorder and coupling. The steady state response, due to mass unbalance, of a simply supported rotating shaft is considered. It is shown that disorder and gyroscopic coupling lead to directional vibration localization; i.e., larger vibration amplitudes in one of the two orthogonal principal directions of the shaft cross section.

Active Vibration Control of a Triple Flexible Structures Combined With Actuators

In order to realize living comfort of tall buildings by reducing the vibration of higher floors by strong winds, this paper proposes a new method of vibration control for flexible structures with a large scale. The higher a tall building the lower its natural frequency. Since obtaining sufficient force to control the lower frequency vibrations of tall buildings is a difficult task, controlling the vibration of ultra-tall buildings using active dynamic absorbers is nearly impossible. This problem can be overcome by placing actuators between a pair of two or three ultra-tall buildings and using the vibrational force of each building to offset the vibrational movement of its paired mate. Therefore, it is able to obtain enough control force under the low frequency when the proposed method is used. In this paper, a reduced-order model expressed by 2DOF system under taking into consideration for preventing spillover instability is applied to control each flexible structure. The LQ control theory is applied to the design of such a control system. The effectiveness of this method is demonstrated theoretically as well as experimentally.

A Weighted Rearranged Spectral Equation Method for Structural Model Updating

A new structural model updating method based on the dynamic force balance is presented. The method consists of rearranging the spectral equation so that measured modes and natural frequencies can be used to compute directly updated stiffness coefficients. The proposed method preserves both the structural connectivity and reciprocity, which translate into sparsity and symmetry of the stiffness matrix, respectively. Large changes in small-valued stiffness coefficients are avoided using parameter weighting in the rearranged spectral equation solution. It is shown that the proposed method produces results which are similar to the results obtained using Alvar Kabe’s method, with the advantages of simpler formulation and smaller computational cost. A simple example of an 8 degrees-of-freedom mass-spring system, originally used by Kabe to present his method, is used here to evaluate the proposed method.

Identification of a Significantly Non-Proportionally Damped Structure Using Force Appropriation

Dynamic tests were carried out on an aluminium plate with significant non-proportional damping applied via two oil filled dampers. Normal mode force appropriation (phase resonance) methods were used to measure the undamped normal modes of the plate and the results compared with corresponding complex modes obtained using a standard curve fitting (phase separation) approach. It is demonstrated that, as long as suitable excitation positions are chosen, high quality undamped normal modes can be identified while the curve fitted modes are highly complex. A Finite Element analysis of the plate was used to show how the results of normal mode force appropriation are directly comparable, particularly when damping is non-proportional.

Prediction of Rotating Machinery Vibration Using Balance Simulation With Measurement Uncertainty

Vibration predictions for rotating machinery with high-speed flexible rotors must account for the methods and limitations of the balance test process which determine the residual rotor unbalance. Vibration predictions based on finite element analysis (FEA) methods are highly dependent upon the assumed rotor unbalance amplitude and phase. The actual residual unbalance distribution depends upon the measured influence coefficients and the least-mean-square (LMS) algorithm used to calculate balance correction weights. Repeatability of the vibration measurements is a key factor in successful balancing. The vibration predictions described in this paper use estimates of final residual unbalance obtained by simulating the balance test process. The simulation uses FEA based influence coefficients, a test based measurement uncertainty (repeatability) model, and LMS balance weight calculations including the specified vibration target levels. The simulations can be used to predict the limit of balance performance of the machinery and to evaluate design options for impact on residual unbalance levels.

Parameter Estimation for an Imperfectly Clamped Plate: Numerical Examples

The problems associated with maintaining truly fixed (zero displacement and slope) or simple (zero displacement and moment) boundary conditions in applications involving vibrating structures have led to the development of models which admit slight rotation and displacement at the boundaries. In this paper, numerical examples demonstrating the dynamics of a model for a circular plate with imperfectly clamped boundary conditions are presented. The latitude gained when using the model for estimating parameters through fit-to-data techniques is also demonstrated. Through these examples, the manner in which the model accounts for the physical manifestation of imperfectly clamped edges is illustrated, and issues regarding the use of the model in physical experiments are defined.

Development of V-Shaped Hybrid Mass Damper and its Application to High-Rise Buildings

A hybrid mass damper system has been developed with a view to counteracting wind- and earthquake-excited vibrations of large high-rise building structures. In order to eliminate the large space needed to accommodate a pendulum-type mass damper adapted to the long period of high-rise building, mechanism has been devised for suspending the auxiliary mass on a V-shaped rail sliding on rollers. The base angle of the V-shaped rail is varied for adjusting the natural period of the mass damper system. A suboptimal algorithm based on the minimum norm method has been adopted for designing the auxiliary mass driving system. Three units of this damper system, each equipped with auxiliary mass weighing 110 tons, have been installed on a 52-story building. Satisfactory performance conforming in all practical aspects with design has been verified from vibration test on actual building after installation. As sequel, the functioning of the system during the first year of service is also reported.

Hydraulic Actuator Tuning in the Control of a Rotating Flexible Beam Mechanism

The end point position and vibration control of a rotating flexible beam mechanism driven by a hydraulic cylinder actuator is considered. An integrated nonlinear system model comprised of beam dynamics, hydraulic actuator, control valves, and control scheme is presented. Control based on simple position feedback along with a hydraulic actuation system tuned to suppress beam vibration over a wide range of angular motion is investigated. For positioning to small to moderate mechanism angles, a linear system model with the actuator tuned for good open loop performance is developed. Actuator tuning is accomplished by varying the system hydraulic resistance according to a dimensionless parameter defining the interaction between the actuator and flexible beam. Simulation results for a closed loop system indicate that this simple tuned control provides comparable performance and requires less control effort than an untuned system with a more complex state feedback optimal controller. To compensate for geometric nonlinearities that cause instability when positioning to large mechanism angles, an active actuator tuning scheme based on continuous variation of hydraulic resistance is proposed. The active variable resistance controller is combined with simple position feedback and designed to provide a constant dimensionless actuator-flexible beam interaction parameter throughout the motion. Simulation results are presented to show the stabilizing effect of this control strategy.

Estimating the Interaction Kernel in a Mathematical Model for a Beam With Internal Damping Mechanism

Beams formed by long fiber composite materials have certain internal damping torque. A mathematical model for the displacement of this type of beams in cantilever configuration is the following initial-boundary value problem of an integro-differential equation [1, 14]: (1) ρ ( x ) w t t ( x , t ) − 2 ( ∫ 0 L h ( x , y ) [ w t x ( x , t ) − w t x ( y , t ) ] d y ) x + ( E I w x x ( x , t ) ) x x = f ( x , t ) , (2) w ( 0 , t ) = 0 , w x ( 0 , t ) = 0 , (3) w x x ( L , t ) = b l 1 ( t ) , (4) − ( E I w x x ( x , t ) ) x | x = L + 2 ∫ 0 L h ( L , y ) [ w t x ( L , t ) − w t x ( y , t ) ] d y = b l 2 ( t ) , (5) w ( x , 0 ) = w 0 ( x ) , w t ( x , 0 ) = w 1 ( x ) , where L is length of the beam, w(x, t) is the transverse displacement of the beam at time t and position x, ρ(x) is the mass density, EI is the stiffness parameter. The interaction integral kernel h(x, ξ) is introduced in this model by considering a restoring torque which comes from spatially variable bending of the beam. This kernel h(x, ξ) depends on the material properties of the beam. Choosing a different material (different h(x, ξ)) can realize a different damping effect for the beam.

An Efficient Approximated Algorithm for Balancing and Order Reduction of Lightly Damped Systems

A method of approximating the controllability gramian, observability gramian and the balancing transformation for lightly damped mechanical systems is presented, the approximation uses the special structure of the system and the fact that the damping is small to reduce the amount of computation considerably. Furthermore, one can avoid the calculation of the entire balancing transformation matrix and calculate only the parts that are required for order reduction. In cases where the reduced order is much smaller than the original that leads to another substantial reduction of computation effort.