Complex Dynamic Stiffness Research Articles

Measurement, Modeling, and Analysis of the Dynamic Properties of Resilient Elements Used for Vibration Isolation

Abstract Resilient elements are widely applied for vibration and noise control in many areas of engineering. Their complex dynamic stiffness gives fundamental information to describe their dynamic performance and is required for predicting structure-borne sound and vibration using dynamic modeling. Many laboratory measurement methods have been developed to determine the dynamic properties of resilient elements. This paper presents a review of recent developments in the measurement methods from the perspective of force–displacement relations of the resilient element assembly rather than of their material properties. To provide context, the review begins with an introduction to modeling methods for resilient elements, especially for rubber and rubber-like isolators, and three standardized measurement methods are introduced. Recent developments are then discussed including methods to extend the frequency range, which are mainly developments of the indirect method. Mobility methods, modal-based methods, recent active frequency-based substructuring (FBS), and inverse substructuring (IS) methods to study the dynamic properties of resilient elements are also described. Laboratory test rigs and the corresponding identification methods are outlined. Methods to evaluate nonlinear dynamic properties of resilient elements by laboratory measurements are also discussed. Finally, the review is concluded by discussing the advantages and limitations of the existing methods and giving suggestions for future research.

Machine foundation response subjected to vertical vibration: experimental and analytical using cone model

The fundamental aim in the proper design of machine foundations is to limit the response amplitude by preventing resonance in all vibration modes. The effects of static and dynamic forces and embedment depths for foundation-induced vertical vibration have been examined, experimentally and analytically using the Cone model. Therefore, various vibration indicators such as displacement amplitude, resonance frequency and complex dynamic stiffness were evaluated. Finally, the statistical properties, Quantile-Quantile plots, and Root-Mean-Square Error values were used to gain confidence in applying the Cone model for predicting the soil bed dynamic behaviour. Per the results, it was observed that embedment increases resonance frequency, natural frequency ratio and damping ratio while the maximum resonant amplitude reduces as the embedment depth increases. Given the findings, due to the increasing dynamic force, a drop in resonance frequency and a rise in peak displacement amplitude of the foundation bed were observed. On the other hand, resonance frequency and amplitude are attenuated with an increase in static force level. Further, the complex dynamic stiffnesses predicted by the Cone model shows good agreement with that obtained experimentally. Thus, whenever feasible, the convenient and relatively accurate Cone model can be performed for assessing foundation vibration in practical engineering projects.

Dynamic pile-head stiffness of laterally loaded end-bearing pile in linear viscoelastic soil − A comparative study

An approximate analysis following the Vlasov-Leontiev foundation model is developed to evaluate the pile-head stiffness (static and dynamic) of laterally loaded end-bearing pile, embedded in a homogeneous, isotropic, and linear viscoelastic soil with frequency-independent hysteretic material damping. In the analysis, the pile is modeled – a Euler-Bernoulli beam and the soil – a continuum. The horizontal soil displacement fields because of the pile motion is mathematically expressed as a product of separable functions while the vertical soil displacement is ignored for the sake of mathematical simplicity. The contribution of vertical soil displacement on the lateral response is indirectly included in the analysis by a modification of the soil modulus. The Extended Hamilton’s principle is applied to obtain the differential equations governing pile and soil motions, which are solved analytically following an iterative algorithm. The degree of accuracy, and the capability of the analysis to capture the essential mechanics (particularly because of dynamic loading) is assessed by comparing static and complex dynamic pile-head stiffness against other popular analyses developed in the literature. Based on the comparisons, appropriate discussions on the present, and analyses available in literature are provided; further, recommendations on the advantages of this analysis for use in practice are also outlined.

Numerical Study of Leakage and Rotordynamic Performance of Staggered Labyrinth Seals Working with Supercritical Carbon Dioxide

The numerical model of a staggered labyrinth seal working with supercritical carbon dioxide (S–CO2) is established. The dynamic and static characteristics of the staggered labyrinth seal for different axial shifting distances of the rotor, various cavity geometries (heights/widths of the rotor convex plate, heights of the seal cavity), and seal clearances were investigated and compared with the conventional see-through labyrinth seal. The results show that the effective damping coefficient (Ceff) with positive axial shifting distance is higher than that with negative axial shifting distance. When the rotor with convex plate operates without axial shifting, the cross-coupled complex dynamic stiffness (hR) of the staggered labyrinth seal shows little effect on the Ceff, and the average direct damping (Cavg) has a dominant influence on the Ceff. As the whirling frequency (Ω) is lower than 60 Hz, the Ceff decreases with increasing height of the rotor convex plate. For Ω < 140 Hz, the damping coefficient generally increases with the decreasing height of the seal cavity. For Ω < 160 Hz, the Ceff of the see-through labyrinth seal is about 107%–649% of the staggered labyrinth seal. Otherwise, the Ceff of the staggered labyrinth seal is about 105%–113% of the see-through labyrinth seal. The Ceff of the seal with the rotor convex plate width of 5.13 mm is relatively high, which is conducive to the stability of the system. The Ceff increases with the decreasing seal clearance. The Ceff of the seal with 0.4 mm clearance is about 116%–148% the seal with 0.6 mm. The leakage flow rate of the staggered labyrinth seal of the see-through labyrinth seal is increased by about 45.5%. The leakage flow rate of the staggered labyrinth seal decreases with the increasing convex plate height, the seal cavity height, and the decreasing seal clearance.

The optimal design of the mounting rubber system for reducing vibration of the air compressor focusing on complex dynamic stiffness

Urban rapid railway is one of the most popular transportation methods these days. It is equipped with large-capacity air compressors since it uses pneumatic pressure to supply power for braking and for opening and closing doors. Passengers tend to complain about vibration and noise from the air compressor. In this study, vibration reduction of air compressor was achieved by acquiring exact complex dynamic stiffness of the mounting rubber and optimizing the shape of it. Target stiffness values of the rubber was obtained from multi-body dynamics simulation of the air compressor-mounting system. Complex elastic modulus of the rubber mount was derived through EMA and Gent’s method and verified by finite element method. Through parametric study of mounting rubber, an optimal shape of mounting rubber was derived and produced. Lastly, the proposed value was verified by experiments comparing with baseline value.

A Study on the Leakage and Rotordynamic Coefficients of a Long-Smooth Seal With Laminar-Two-Phase, Mainly-Oil Mixtures

Abstract This paper conducts a comprehensive study on the effects of the presence of air in the oil on the leakage and rotordynamic coefficients of a long-smooth seal (inner diameter D = 89.306 mm, radial clearance Cr = 0.140 mm, and length-diameter ratio L/D = 0.65) under laminar-two-phase flow conditions. The mixture consists of air and silicone oil with inlet gas volume fraction (GVF) up to 10%. Tests are performed at inlet temperature Ti = 39.4 °C, exit pressure Pe = 6.9 bars, pressure drop PD = 31, and 37.9 bars, and rotor speed ω = 5, 7.5, and 10 krpm. The test seal is always concentric with the rotor, and no intentional fluid prerotation is provided at the seal inlet. The complex dynamic stiffness coefficients Hij of the test seal are measured and fitted by the frequency-independent direct stiffness K, cross-coupled stiffness k, direct damping C, cross-coupled damping c, direct virtual-mass M, and cross-coupled virtual-mass mq coefficients. Under laminar flow conditions, increasing inlet GVF has negligible effects on K, k, C, and effective damping Ceff, while it decreases c and M. These trends are correctly predicted by San Andrés's bulk-flow model with laminar flow friction formula. As inlet GVF increases, measured leakage flow rate m˙ increases slightly. In general, the predictions of K, k, C, c, Ceff, and m˙ are reasonably close to measurements.

Clearance Effects on Rotordynamic Performance of a Long Smooth Seal With Two-Phase, Mainly-Air Mixtures

This paper experimentally studies the effects of changing radial clearance Cr on the performance of a long (length-to-diameter ratio L/D = 0.65) smooth seal under mainly-air (wet-gas) conditions. The test fluid is a mixture of air and silicone oil. Tests are conducted with Cr = 0.188, 0.163, and 0.140 mm, inlet pressure Pi = 62.1 bars, exit pressure Pe = 31 bars, inlet liquid volume fraction LVF = 0%, 2%, 5%, and 8%, and shaft speed ω = 10, 15, and 20 krpm. The seal's complex dynamic stiffness coefficients Hij are measured. The real parts of Hij cannot be fitted by frequency-independent stiffness and virtual-mass coefficients. Therefore, frequency-dependent direct KΩ and cross-coupled kΩ stiffness coefficients are used. The imaginary parts of direct Hij produce frequency-independent direct damping C. Test results show that, for all pure- and mainly-air conditions, decreasing Cr decreases (as expected) the leakage mass flow rate m˙. Under mainly-air conditions, decreasing Cr decreases KΩ. This outcome is contrary to the test results at pure-air conditions, where KΩ increases as Cr decreases. Since an unstable centrifugal compressor rotor may precess at approximately 0.5ω, the effective damping Ceff at about 0.5ω is used as an indicator of the impact a seal would have on its associated compressor. For pure-air conditions, when Ω ≈ 0.5ω, decreasing Cr increases Ceff and makes the seal more stabilizing. This trend continues after the oil is added. A bulk-flow model developed by San Andrés (2011, “Rotordynamic Force Coefficients of Bubbly Mixture Annular Pressure Seals,” ASME J. Eng. Gas Turbines Power, 134(2), p. 022503) produces predictions to compare with test results. m˙ predictions correlate with measurements. Under pure-air conditions, the model correctly predicts the effects of changing Cr on KΩ and the Ceff value near 0.5ω. After the oil is added, as Cr decreases, predicted KΩ increases while measured KΩ decreases. Also, for mainly-air cases and Ω ≈ 0.5ω, decreasing Cr does not discernibly change predicted Ceff but increases the measured value.

On the Predicted Effect of Angular Misalignment on the Performance of Oil Lubricated Thrust Collars in Integrally Geared Compressors

Multiple-stage integrally geared compressors (IGCs) offer improved thermal efficiency and easier access for maintenance and overhaul than single-shaft centrifugal compressors. In an IGC, a main bull shaft drives pinion shafts, each having an impeller at its ends. The compression of process gas in the compressor stages induces axial loads along the pinion shafts that are transmitted via thrust collars (TCs) to the main bull gear (BG) shaft and balanced by a single thrust bearing. Manufacturing inaccuracies and a poor assembly process can lead to static angular misalignments of the TC and BG surfaces that affect the operating film thickness as well as the force and reaction moments of the lubricated mechanical element. In a follow-up to San Andrés et al. (2015, “On the Predicted Performance of Oil Lubricated Thrust Collars in Integrally Geared Compressors,” ASME J. Eng. Gas Turbines Power, 137(5), pp. 1–9), this paper presents an investigation of the performance of a single thrust collar configuration operating with increasing static angular misalignment of either the TC or BG. The flow model solves the Reynolds equation of hydrodynamic lubrication coupled to a thermal energy transport equation to determine the film pressure and bulk temperature fields, respectively. The model predicts performance parameters such as power loss and lubricant flow rate, and force and moment stiffness and damping coefficients. Predictions show that misaligning of either the thrust collar or bull gear alters the load-carrying area in the lubricated zone, shifts the pressure field with peak magnitudes doubling or more depending on the degree and direction of TC or BG misalignment. Static angular misalignment does not significantly affect the power loss, temperature rise, etc., but does have an effect on the dynamic coefficients (both axial and angular). Finally, a reduced complex dynamic stiffness matrix for the lubricated TC shows that some cross-coupled stiffness and moment coefficients are nonzero, indicating hydrodynamic coupling between axial and angular motions for the pinion and bull gear shafts. The coupling could affect the placement of the system natural frequencies and associated mode shapes as well as the system stability.

Measured Static and Rotordynamic Coefficient Results for a Rocker-Pivot, Tilting-Pad Bearing With 50 and 60% Offsets

Static and rotordynamic coefficients are measured for a rocker-pivot, tilting-pad journal bearing (TPJB) with 50 and 60% offset pads in a load-between-pad (LBP) configuration. The bearing uses leading-edge-groove direct lubrication and has the following characteristics: 5-pads, 101.6 mm (4.0 in) nominal diameter,0.0814 -0.0837 mm (0.0032–0.0033 in) radial bearing clearance, 0.25 to 0.27 preload, and 60.325 mm (2.375 in) axial pad length. Tests were performed on a floating bearing test rig with unit loads from 0 to 3101 kPa (450 psi) and speeds from 7 to 16 krpm. Dynamic tests were conducted over a range of frequencies (20 to 320 Hz) to obtain complex dynamic stiffness coefficients as functions of excitation frequency. For most test conditions, the real dynamic stiffness functions were well fitted with a quadratic function with respect to frequency. This curve fit allowed for the stiffness frequency dependency to be captured by including an added mass matrix [M] to a conventional [K][C] model, yielding a frequency independent [K][C][M] model. The imaginary dynamic stiffness coefficients increased linearly with frequency, producing frequency-independent direct damping coefficients. Direct stiffness coefficients were larger for the 60% offset bearing at light unit loads. At high loads, the 50% offset configuration had a larger stiffness in the loaded direction, while the unloaded direct stiffness was approximately the same for both pivot offsets. Cross-coupled stiffness coefficients were positive and significantly smaller than direct stiffness coefficients. Negative direct added-mass coefficients were obtained for both offsets, especially in the unloaded direction. Cross-coupled added-mass coefficients are generally positive and of the same sign. Direct damping coefficients were mostly independent of load and speed, showing no appreciable difference between pivot offsets. Cross-coupled damping coefficients had the same sign and were much smaller than direct coefficients. Measured static eccentricities suggested cross coupling stiffness exists for both pivot offsets, agreeing with dynamic measurements. Static stiffness measurements showed good agreement with the loaded, direct dynamic stiffness coefficients.

The effect of inertia and angular momentum of a fluid annulus on lateral transversal rotor vibrations

An extensive amount of work exists on experimental and theoretical analysis of unsteady flow phenomena in hydraulic turbines. Still, resonance phenomena and self-excited vibrations of the rotor of hydropower machines are not considered as a major problem during normal operation conditions. Nevertheless, in development and research it is not sufficient to rely on earlier experience. An accurate predictive rotor model is crucial in risk assessment of rotor vibrations of hydraulic generator units. This paper discusses the effects of inertia and the rotational energy of the fluid in the turbine on lateral transversal shaft vibrations of hydraulic generator units. There is a lack of agreement among engineers upon how fluid inertia of the turbine should be included in rotor models. The rotational energy of the fluid has a potential risk of feeding self-excited vibrations. A fluid-rotor model is presented that captures the effect of inertia and angular momentum of a fluid annulus on vibrations of an inner rigid cylinder. The purpose of the model is to gain physical understanding of the phenomena at work and it is not applicable to specific turbines. The linearized equation of motion of the cylinder surrounded by a fluid annulus is solved for by one single complex equation. The constrained cylinder has two degrees of freedom in the plane perpendicular to its axis. By the assumption of irrotational cyclic flow, the fluid motion is described by a complex potential function. The motion of the cylinder is described by three parameters. Two surfaces are defined that splits the parameter space into regions with different qualitative behaviour. One surface defines the limit of stability whereas the other defines a limit when the eigenvalues have opposite signs or are both positive. The response to an external periodic rotating force is visualized by the magnitude of the inverse of the complex dynamic stiffness.

Canonical description of the control system in the shimmy problem for the landing gear wheels of an aircraft

A control system for the landing gear is described in terms of complex dynamic stiffness. An analysis of the shimmy boundaries using values of complex dynamic stiffness and taking into account the maximum possible perturbations permits finding the necessary parameters of the control system (such as dry friction) for the prevention of shimmy at low amplitudes. The universal description of the control system in terms of complex dynamic stiffness, easy and convenient analysis, and the possibility of analytic estimation and experimental determination of complex dynamic stiffness are indicative of practicability of this canonical description

Identification and Prediction of Force Coefficients in a Five-Pad and Four-Pad Tilting Pad Bearing for Load-on-Pad and Load-Between-Pad Configurations

This paper presents the identification of the rotordynamic force coefficients for direct lubrication five-pad and four-pad tilting pad bearings. The bearing is 110 mm in diameter with a L/D of 0.4 pad axial length (44 mm). The experiments include load-on-pad and load-between-pad configurations, with 0.5 and 0.6 pivot offsets, for rotor speeds ranging from 7500 rpm to 15,000 rpm. The bearing force coefficients are identified from multiple frequency excitations (20–300 Hz) exerted on the bearing housing by a pair of hydraulic shakers and are presented as a function of the excitation frequency and rotor speed for a 300 kPa unit load. The experimental results also include temperatures at the trailing edge of three pads. The direct force coefficients, identified from curve-fits of the complex dynamic stiffness, are frequency independent if considering an added mass term much smaller than the test device modal mass. The force coefficients from the four-pad bearing load-between-pad configuration show similar coefficients in the loaded and orthogonal directions. On the other hand, as expected, the five-pad bearing load-on-pad shows larger coefficients (∼25%) in the loaded direction. The maximum pad temperature recorded for the 0.5 pivot offset configurations is up to 20°C higher than those associated to the 0.6 offset configuration. Results from a predictive code are within 50% of the experimental results for the direct stiffness coefficients and within 30% for the direct damping coefficients.

The effectiveness of wave barriers on the dynamic stiffness coefficients of foundations using boundary element method

A comprehensive numerical investigation has been carried out to indicate the influence of wave barriers on the complex dynamic stiffness coefficients of the surface supported foundations under dynamic loads. In the considered problems the underground is regarded as an elastic homogeneous half-space and as a layer on the top of a rigid bedrock. The results of the present numerical simulation are obtained by using the substructure approach in the frequency domain which is formulated on basis of the boundary element method (BEM). The validations of the numerical results are shown through comparison with published numerical data. The results point out that using an open trench barrier with appropriate distance surrounding the vibration source, a notable reduction level can be achieved.

Rotordynamic Coefficients Measurements Versus Predictions for a High-Speed Flexure-Pivot Tilting-Pad Bearing (Load-Between-Pad Configuration)

Experimental dynamic force coefficients are presented for a four pad flexure-pivot tilting-pad bearing in load-between-pad configuration for a range of rotor speeds and bearing unit loadings. Measured dynamic coefficients have been compared to theoretical predictions using an isothermal analysis for a bulk-flow Navier-Stokes (NS) model. Predictions from two models—the Reynolds equation and a bulk-flow NS equation models are compared to experimental, complex dynamic stiffness coefficients (direct and cross-coupled) and show the following results: (i) The real part of the direct dynamic-stiffness coefficients is strongly frequency dependent because of pad inertia, support flexibility, and the effect of fluid inertia. This frequency dependency can be accurately modeled for by adding a direct added-mass term to the conventional stiffness/damping matrix model. (ii) Both models underpredict the identified added-mass coefficient (∼32kg), but the bulk-flow NS equation predictions are modestly closer. (iii) The imaginary part of the direct dynamic-stiffness coefficient (leading to direct damping) is a largely linear function of excitation frequency, leading to a constant (frequency-independent) direct damping model. (iv) The real part of the cross-coupled dynamic-stiffness coefficients shows larger destabilizing forces than predicted by either model. The frequency dependency that is accounted for by the added mass coefficient is predicted by the models and arises (in the models) primarily because of the reduction in degrees of freedom from the initial 12 degrees (four pads times three degrees of freedom) to the two-rotor degrees of freedom. For the bearing and condition tested, pad and fluid inertia are secondary considerations out to running speed. The direct stiffness and damping coefficients increase with load, while increasing and decreasing with rotor speed, respectively. As expected, a small whirl frequency ratio (WFR) was found of about 0.15, and it decreases with increasing load and increases with increasing speed. The two model predictions for WFR are comparable and both underpredict the measured WFR values. Rotors supported by either conventional tilting-pad bearings or flexure-pivot tilting-pad (FPTP) bearings are customarily modeled by frequency-dependent stiffness and damping matrices, necessitating an iterative calculation for rotordynamic stability. For the bearing tested and the load conditions examined, the present results show that adding a constant mass matrix to the FPTP bearing model produces an accurate frequency-independent model that eliminates the need for iterative rotordynamic stability calculations. Different results may be obtained for conventional tilting-pad bearings (or this bearing at higher load conditions).

Frequency Response Function for Prediction of Planar Cellular Plastic Foam Acoustic Behavior

In this paper, Biot's dynamic poroelasticity theory is applied for deriving theoretical complex dynamic stiffness solutions for fixed lower surface, infinite planar cellular plastic foams with permeable and impermeable upper surfaces. A model using impulse pressure excitation to derive and acquire the acoustic response functions of foams is performed. The derived theoretical dynamic stiffness equations can also be used for predicting the dynamic stiffness of composite foams. The dynamic stiffness of a composite foam can be seen as a series combination of the dynamic stiffness of single planar foams according to their stacking order. Using the derived complex dynamic stiffness equations, frequency response functions of acoustic impedance, reflection coefficient, and sound absorption coefficient are obtained. Experimental results from earlier studies are used to validate the derived frequency response function. The experimental acoustic properties of planar polyurethane foams with the same cellular skeleton at various thicknesses, are compared to the predicted properties. The influences of thickness on the acoustic properties are discussed.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e., a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modelled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8×8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4×4 real dynamic stiffness matrix of the axisymmetric shaft.

Dynamic analysis of steel frames with flexible connections

This paper deals with the effects of flexibility and damping in the nodal connections on the dynamic behavior of plane steel frames. A flexible eccentric connection is idealized by nonlinear rotational spring and dashpot in parallel. Thus, the effects of viscous and hysteretic damping on dynamic response of frame structures are taken into consideration. A numerical model that includes both nonlinear connection behavior and geometric nonlinearity of the structure is developed. The complex dynamic stiffness matrix for the beam with flexible connections and linear viscous dampers at its ends is obtained. Several examples are included to illustrate the efficiency and accuracy of the present model.

THE PREDICITON OF SEAT TRANSMISSIBILITY FROM MEASURES OF SEAT IMPEDANCE

A method of predicting seat transmissibility from mathematical models of the seat and the human body is described. The complex dynamic stiffness of a seat is determined by measurement using an indenter rig, and its stiffness and damping subsequently determined by curve-fitting. By using the fitted stiffness and damping of the seat model, and a previously determined dynamic model of the human body, the seat transmissibility is predicted mathematically. The method is illustrated with data obtained with a car seat and also a rectangular sample of foam. The seat and foam transmissibilities were predicted over the frequency range 1·25–25 Hz using two alternative models of the human body (a one-degree-of-freedom model and a two-degree-of-freedom model). The predicted seat transmissibilities were close to those measured in a group of eight subjects over the entire frequency range. The two-degree-of-freedom model of the human body provided better predictions where the seat and foam showed a second resonance around 8 Hz. The need for a non-linear mathematical model of the human body and a non-linear seat model is discussed.

The Dynamic Stiffness Method for Linear Rotor-Bearing Systems

The behavior of linear rotor-bearing systems is investigated by using the exact approach of the dynamic stiffness method, which entails the use of continuous rather than lumped models. In particular, the theoretical formulation for rotor systems with anisotropic bearings is developed by utilizing the complex representation of all the involved variables. The proposed formulation eventually leads to the 8 × 8 complex dynamic stiffness matrix of the rotating Timoshenko beam; this matrix proves to be related, by a simple rule, to the 4 × 4 dynamic stiffness matrix, which describes rotor systems with isotropic bearings. The method is first applied to the critical speeds evaluation of a simple rotor system with rigid supports; for this case, the exact results of the dynamic stiffness approach are compared to the usual convergence procedure of the finite element method. Successively, the steady-state unbalance response of two rotor systems with anisotropic supports is analyzed; for these examples, the dynamic stiffness results compare favorably with the results of the finite element and the transfer matrix analysis performed by other authors.

Some comments on: Damped second‐order Rayleigh‐Timoshenko beam vibration in space‐an exact complex dynamic member stiffness matrix

Some comments on: Damped second‐order Rayleigh‐Timoshenko beam vibration in space‐an exact complex dynamic member stiffness matrix