Point Receptances Research Articles

Influence of guideway friction on the cutting point receptance in machine tools

Motivated by the fact that the tool cutting point (TCP) receptance demonstrates significant variation between the idle and in-motion conditions of a machine tool, this paper studies the relationship between guideway friction and TCP dynamics. The fundamental effect is modelled and demonstrated on a laboratory testbench considering structural mechanics, servo dynamics, and guideway friction. The findings are then extended to an industrial-scale machine, for which improved measurement of the TCP receptance is obtained using a dedicated inertial linear actuator. Finally, practical in-motion impact hammer testing guidelines are proposed for more accurate characterization of the TCP receptance and chatter stability predictions.

Controlling the vibration and noise of a ballasted track using a dynamic vibration absorber with negative stiffness

In this study, a rail dynamic vibration absorber with negative stiffness is developed to reduce the vibration transmission and radiated noise from the rail components of a ballasted track. The compound models of the ballasted track system with and without the proposed dynamic vibration absorber and a traditional dynamic vibration absorber are constructed. A parametric study is performed to evaluate the effects of the design parameters of the proposed dynamic vibration absorber on the vibration and noise reduction of the track system in terms of the point receptance, the decay rate of rail vibration along the track, and the vibration energy level of the rail. Compared with the traditional dynamic vibration absorber, the proposed counterpart can work effectively over a broad frequency range around resonance. The efficiency of the dynamic vibration absorber can be improved by adjusting the design values of the active mass and damping coefficient. A comparison with the traditional dynamic vibration absorber shows that the vibration and noise suppression capability of the proposed one can be enhanced by increasing the value of the stiffness ratio. However, different from the traditional dynamic vibration absorber, the design parameters of the proposed one can also affect the decay rate and vibration energy at low-frequency regions. A discrete track with the proposed dynamic vibration absorber, which is arranged in continuous or discrete distribution along the rail, is illustrated to study the influences of the rail components on the decay rate and vibration energy level of rails. These calculated results could provide a theoretical basis for the design of the proposed dynamic vibration absorber in controlling the vibration and radiated noise from rails.

Multi-point coupling for tool point receptance prediction

This paper describes a multi-point receptance coupling substructure analysis (RCSA) technique for tool point receptance prediction. The portion of the tool inside the holder is coupled to the portion of the holder that clamps the tool at multiple points along the insertion length. The coupling can be rigid or flexible-damped. The procedure is described analytically and then experimental results are presented. The tool point receptances, or frequency response functions, for four carbide tool blank diameters are tested at multiple extension lengths and compared to predictions for an ER32 collet connection. Stiffness matrices are identified for each of the tool blank diameters and reported. Given the predicted tool point receptances, milling process models can be used to select optimized operating parameters at the process planning stage.

An examination of the concept of driving point receptance

In the field of vibration, driving point receptance is a well-established and widely applied concept. However, as demonstrated in this paper, when a driving point receptance is calculated using the finite element (FE) method with solid elements, it does not converge as the FE mesh becomes finer, suggesting that there is a singularity. Hence, the concept of driving point receptance deserves a rigorous examination. In this paper, it is firstly shown that, for a point harmonic force applied on the surface of an elastic half-space, the Boussinesq formula can be applied to calculate the displacement amplitude of the surface if the response point is sufficiently close to the load. Secondly, by applying the Betti reciprocal theorem, it is shown that the displacement of an elastic body near a point harmonic force can be decomposed into two parts, with the first one being the displacement of an elastic half-space. This decomposition is useful, since it provides a solid basis for the introduction of a contact spring between a wheel and a rail in interaction. However, according to the Boussinesq formula, this decomposition also leads to the conclusion that a driving point receptance is infinite (singular), and would be undefinable. Nevertheless, driving point receptances have been calculated using different methods. Since the singularity identified in this paper was not appreciated, no account was given to the singularity in these calculations. Thus, the validity of these calculation methods must be examined. This constructs the third part of the paper. As the final development of the paper, the above decomposition is utilised to define and determine driving point receptances required for dealing with wheel/rail interactions.

Tool Point Analysis for Bending, Torsional and Axial Receptances of tool-holder-spindle Assembly

A generalized method for the analysis of the tool point receptances related to all axes (X, Y and Z) is presented in this paper. In order to facilitate modeling, the tool-holder-spindle assembly is divided into four substructures, i.e., spindle-holder subassembly, shank of tool, fluted part of tool and tool-holder joint interface. The fluted part of tool is modeled using three-dimensional Timoshenko theory. The tool-holder joint interface is regarded as a zero-thickness distributed layer. A set of independent spring-damper elements is employed to simulate the dynamic properties of the joint interface. The dynamic responses of all substructures is assembled to calculate the tool point receptances. Finally the proposed method is experimentally verified.

Generalized method for the analysis of bending, torsional and axial receptances of tool–holder–spindle assembly

Existing researches for the prediction of tool point receptances have focused on developing dedicated methods for cutting tools used in each single cutting operation such as milling and drilling processes. This paper presents a generalized method for the analysis of the tool point receptances of cutting tools suitable for being mounted on a rotating spindle. Translational and rotational dynamic responses related to all axes (X, Y and Z) are simultaneously modeled in a unified way to predict the tool point bending, torsional and axial receptances of all kinds of rotating tools, such as milling, drilling and boring cutters. To facilitate modeling, the tool–holder–spindle assembly is divided into four substructures, i.e., spindle–holder subassembly, shank of tool, fluted part of tool and tool–holder joint interface. The fluted part of tool is modeled as a three-dimensional Timoshenko beam with varying cross-section, while the tool–holder joint interface is regarded as a zero-thickness distributed layer and modeled as a joint substructure composed of a set of independent spring–damper elements. Assembling criterion is derived to couple the dynamic responses of all substructures to calculate the tool point receptances. Meanwhile, compared with past experimental means, a measurement procedure to eliminate the adapter's mass effect on torsional and axial receptances is designed. The proposed method is experimentally proven for two kinds of rotating tools, i.e., mills and drills.

A frequency-based substructuring approach to efficiently model position-dependent dynamics in machine tools

Structural deformations of machine tool components that translate and/or rotate relative to each other to realize tool motion results in tool point dynamics varying along the tool path. These changing dynamics interact with the cutting process and the control loop of the drives to limit machine performance, making it necessary to virtually characterize these interactions such as to guide design decisions. To facilitate rapid evaluation of these varying dynamics, this paper describes a generalized frequency-based substructuring approach that combines the position-invariant component level receptances at the contacting interfaces between substructures to obtain the position-dependent tool point response. Receptances at the contacting interfaces are approximated by projecting them to a point to facilitate a multiple point receptance coupling formulation. Complete machine behavior is represented by just a few sets of receptances, making the model computationally more efficient than full-order finite element models and other dynamic substructuring methods. Position-dependent dynamic behavior for a representative three axis milling machine is simulated and numerically verified. Rapid investigations of the varying dynamics assist in virtually characterizing machine performance before eventual prototyping.

The effects of in-plane loading on vibration and buckling of the grooved plates

This paper presents an analytical study on the influence of in-plane forces on the dynamic characteristics of a plate with grooves in different sizes. A sub-structuring approach, the reverse receptance method, is applied to solve the eigenvalues of an in-plane forces loaded plate with two parallel grooves and two orthogonal grooves, respectively. On considering coupling and interaction between two orthogonally connected grooves, the point receptance arrays are employed. Accuracy is evaluated by comparing the analytical solutions with numerical results obtained from the finite element model.Parametric study reveals, the deeper the groove, the lower the natural frequencies and buckling loads of the grooved plate will be. With in-plane tensions applied and increased linearly, the grooved plate is stiffened gradually and its natural frequencies increase accordingly. With equal bi-axial tensions, the natural frequencies of the grooved plates increase twice as much as those with uni-axial tension do. With in-plane compressive forces on the plate, the situations are reversed. The lowest natural frequency of a grooved plate with sufficient tensions can be higher than that of a non-grooved unloaded plate. When the groove size and the loading condition are the same, the orthogonal grooves decrease the natural frequencies and the buckling loads of a plate more than the parallel grooves do. The in-plane forces do not affect the mode shapes of the grooved plate; but the grooves do.The analytical evaluation demonstrates the accuracy and applicability of the proposed approach.

A point receptance array approach for stiffened panel structures

A point receptance array is proposed to study behaviors of panel structures with stiffeners. Although the overall objective is intended to develop a general algorithm for large structural members with stiffeners of arbitrary arrangements, the present study focus on an analytical evaluation of the accuracy of the approximation. To simplify the formulations, a thin plate with two diagonally connected beam stiffeners is selected. By solving governing differential equations for the plate and beam, point receptances that represent the interaction and compatibility conditions at the plate–beam interfaces are analytically obtained. Buckling modes of the stiffened plate are obtained by solving an eigenvalue problem. Accuracy of the approach is verified by a comparison of the analytical solution with data obtained from a direct finite element model. A parametric study was also performed. Good agreement indicates that the point receptance is a viable approach for general applications.

Linear Vibrations of Frames Carrying a Concentrated Mass

The free and forced in-plane and out-of-plane vibrations of frames are investigated. The beam has a straight and a curved part. It has a circular cross-section. A concentrated mass is also located at different points of the frame with different mass ratios. FEM is used to analyze the problem. The in-plane and out-of-plane natural frequencies, point and transfer receptances of the system are obtained to determine the sensitive and non-sensitive frequency intervals depending on the location and direction of the force.

Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies

This paper presents a numerical study concerning the active vibration control of smart piezoelectric beams. A comparison between the classical control strategies, constant gain and amplitude velocity feedback, and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, is performed in order to investigate their effectiveness to suppress vibrations in beams with piezoelectric patches acting as sensors or actuators. A one-dimensional finite element of a three-layered smart beam with two piezoelectric surface layers and metallic core is utilized. A partial layerwise theory, with three discrete layers, and a fully coupled electro-mechanical theory is considered. The finite element model equations of motion and electric charge equilibrium are presented and recast into a state variable representation in terms of the physical modes of the beam. The analyzed case studies concern the vibration reduction of a cantilever aluminum beam with a collocated asymmetric piezoelectric sensor/actuator pair bonded on the surface. The transverse displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and closed-loop classical and optimal control systems. The case studies allow the comparison of their performances demonstrating some of their advantages and disadvantages.

Determining the response of infinite, one-dimensional, non-uniform periodic structures by substructuring using waveshape coordinates

A method is presented for determining the wavenumbers, waveshapes and point receptances for an infinite, one-dimensional, non-uniform periodic structure with distributed periodic attachments or supports. The approach is based on a general theory of harmonic wave propagation in one-dimensional periodic systems. Ill-conditioning was previously reported as an impediment to applying the theory to problems of practical importance. In this paper ill-conditioning problems are overcome and a method of substructuring using waveshape coordinates is presented that dramatically improves computational efficiency. The accuracy and generality of the new method are tested by comparing computed and measured receptances for a typical TGV railway track with UIC60 rail, rail pad, ballast and concrete sleepers. The computed results are found to correlate well with measured data.

Linear Forced In-Plane and Out-of Plane Vibrations of Frames Having a Curved Member

The forced, in-plane and out-of-plane vibrations of frames comprised of straight and curved members are investigated using Finite Element Methods. The straight and curved beams are assumed as Euler-Bernoulli type and they have circular cross-sections. The frame lies in a single plane. In the analysis, elongation, bending and rotary inertia effects are included. Four degrees of freedom for in-plane vibrations and three degrees of freedom for out-of-plane vibrations are assumed. The in-plane and out-of-plane point and transfer receptances are obtained in order to determine the sensitive and non-sensitive frequency interval of the frame system.

An integrated thermo-mechanical-dynamic model to characterize motorized machine tool spindles during very high speed rotation

High speed machining (HSM) is a promising technology for drastically increasing productivity and reducing production costs. Development of high-speed spindle technology is strategically critical to the implementation of HSM. Compared to conventional spindles, motorized spindles are equipped with built-in motors for better power transmission and balancing to achieve high-speed operation. However, the built-in motor introduces a great amount of heat into the spindle system as well as additional mass to the spindle shaft, thus complicating its thermo-mechanical-dynamic behaviors. This paper presents an integrated model with experimental validation and sensitivity analysis for studying various thermo-mechanical-dynamic spindle behaviors at high speeds. Specifically, the following effects are investigated: the bearing preload effects on bearing stiffness, and subsequently on overall spindle dynamics; high-speed rotational effects, including centrifugal forces and gyroscopic moments on the spindle shaft and, subsequently, on overall spindle dynamics; and the spindle dynamics on the cutting point receptance. The proposed integrated model is a useful tool for differentiating quantitatively different effects on the spindle behaviors. The results show that a motorized spindle softens at high speeds mainly due to the centrifugal effect on the spindle shaft.

Dynamic Analysis of a Rigid Body Mounting System with Flexible Foundation Subject to Fluid Loading

This paper presents an investigation of the force transmission from a rigid body mounting system to a flexible foundation with light and heavy fluid loading under the force and moment excitation. The analytical expression has been derived in which the flexible foundation effects are incorporated into a revised system stiffness matrix that is derived from the receptance matrix at mounting points. A typical case with a thin infinite plate as the foundation has been studied with the point and transfer receptances theoretically and numerically analysed in the case of light and heavy fluid loading. The results show that, compared with the rigid foundation, the force transmission is reduced and system natural frequencies are shifted. The detailed analysis demonstrates that the force reduction and frequency shifting are more obvious at low frequencies where the receptance value is significant. The study is also carried out to compare the transfer receptances from different waves in plate as it couples with water with the objective to simplify the calculation of receptance. It is found that, in the low frequency and after a short distance from driving point, the transfer receptance calculation for the heavy fluid loading can be simplified by only accounting the contribution from free wave which may easily be evaluated from the point receptance in air. It implies the plate response under heavy fluid loading could be directly derived from that with light fluid loading.

The vibration behavior of railway track at high frequencies under multiple preloads and wheel interactions

The track foundation is preloaded by multiple wheel loads due to the train weight and, as the pad and ballast are nonlinear, their stiffness depends upon the preload in them. Due to the influence of these resilient components of the track, the track vibration is affected by the wheel loads. It is also affected by the wheel/rail interactions. In this article the preloads in the pad and ballast are calculated by considering the nonlinear properties of the track foundation, and thus the preloaded pad and ballast stiffnesses are determined. The vibration properties are explored for the track under multiple wheel loads and multiple wheel/rail interactions by comparing the results from different track models with and without these effects. It is found that the point receptance of the track is reduced and the vibration decay rate is enhanced at low frequencies due to the wheel loads. The effects of the wheel/rail interactions are most significant for frequencies 400-2000 Hz. Because of the wheel/rail interactions, the point receptance fluctuates and the vibration decay is enhanced in the regions around the wheels.

A DOUBLE TIMOSHENKO BEAM MODEL FOR VERTICAL VIBRATION ANALYSIS OF RAILWAY TRACK AT HIGH FREQUENCIES

Models of track vibration representing the rail as a beam are insufficiently detailed for high frequencies where the rail cross-sectional deformation is significant. A double Timoshenko beam model is developed for vertical vibration analysis at high frequencies. Although simple, this model includes the essential cross-sectional deformation of a rail in vertical vibration at high frequencies. Based on this model, the dispersion relation of propagating waves in a free and a continuously supported rail is studied. Vertical vibration receptances of the rail are calculated using both continuously and discretely supported models. The results show good agreement with an FE model and measurement data in terms of frequency-wavenumber curves, point receptance and vibration decay rate.

THE EFFECTS OF LOCAL PRELOAD ON THE FOUNDATION STIFFNESS AND VERTICAL VIBRATION OF RAILWAY TRACK

Track vibration is one of the main sources of railway rolling noise. The resilient components in the railway foundation such as the pad and ballast have significant effects on the track vibration. Usually, their parameters vary with load and frequency. The wheel load of a train causes local deformation in the track and the foundation, so the railpads and the ballast are preloaded and thus their stiffness has different values at the supports near the wheel load from the other supports. In this paper the preloads in the pad and ballast caused by a single wheel load are studied by considering the non-linear properties of the track foundation, and thus the preload pad and ballast stiffnesses are determined. A model of the track dynamics based on an infinite discretely supported Timoshenko beam with preloaded pad and ballast dynamic stiffness is developed. Its vertical vibration behaviour under the influence of the local wheel load is investigated. It is shown that the point receptance for vertical vibration is strongly affected by the local wheel load at low frequencies, but the effect of the local wheel load on the average wave propagation decay rate along the rail is very limited.

Analysis of the Beating Response of Bell Type Structures

The impulse response of a ring-stiffened cylindrical shell, with an attached concentrated mass, is analyzed. It is a simplified model of a Korean bell structure. The objective is to study the beating response, which is a purposefully designed, unique feature of Korean bells. Based on the receptance method, modal characteristics of the total system are determined in terms of natural modes and frequencies of an unstiffened cylindrical shell and a ring. In this study, the receptance technique is employed repeatedly (in two steps), since it would have been difficult to simultaneously handle the line receptance of a ring and the point receptance of a concentrated mass. Numerical results show how the frequency levels and major modal characteristics of the original shell can be changed by the ring stiffening, and how the asymmetry due to the concentrated mass introduces beating when the structure is subjected to impact. The analysis can be utilized for the control of dominant frequencies and beat frequencies of Korean bell structures in the design stage, but has much more general application possibilities.