On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations

2014 ◽  
Vol 228 (14) ◽  
pp. 2532-2544 ◽  
Author(s):  
Dragi Radomirovic ◽  
Ivana Kovacic
Keyword(s):  
Potential Energy ◽  
Stable Equilibrium ◽  
Dissipative Function ◽  
Equivalent System ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Out Of Plane ◽  
Stable Equilibrium Position ◽  
Equivalent Systems ◽  
Plane Oscillations

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.

On the equivalent systems for concurrent springs and dampers – Part 1: Small in-plane oscillations

2014 ◽  
Vol 228 (14) ◽  
pp. 2520-2531 ◽  
Author(s):  
Dragi Radomirovic ◽  
Ivana Kovacic
Keyword(s):  
Equilibrium Position ◽  
Stable Equilibrium ◽  
Damping Coefficient ◽  
Viscous Dampers ◽  
Stiffness Coefficients ◽  
Stable Equilibrium Position ◽  
Equivalent Systems ◽  
Stiffness And Damping ◽  
Plane Oscillations

In this work, concurrent linear springs placed in the system that performs small in-plane oscillations around the stable equilibrium position are considered. New theorems defining how they can be replaced by two mutually orthogonal springs are provided. The same concept is applied to find two mutually orthogonal linear viscous dampers that can replace a system of concurrent linear viscous dampers. The directions of such springs and dampers correspond to the principal stiffness and damping axes, respectively. So far unknown invariants related to the sum of stiffness coefficients and damping coefficient of the original and equivalent systems are presented. A few examples are given to illustrate the use and benefits of this approach. In addition, it is shown how the concept of two mutually orthogonal springs can be beneficially used for analysing problems concerned with oscillations of a particle on elastic frames.

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

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Chris D. Kulhanek ◽  
Dara W. Childs
Keyword(s):  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Excitation Frequency ◽  
Quadratic Function ◽  
Added Mass ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Tilting Pad ◽  
Frequency Independent ◽  
Direct Stiffness

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.

On the equivalence of systems of different sizes, with applications to system comparisons

2016 ◽  
Vol 48 (2) ◽  
pp. 332-348 ◽  
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Arne B. Huseby
Keyword(s):  
Optimization Problem ◽  
Coherent System ◽  
Sufficient Condition ◽  
Equivalent System ◽  
Coherent Systems ◽  
Two Systems ◽  
Equivalent Systems ◽  
Engineered Systems ◽  
Signature Vector ◽  
Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

scholarly journals Annular Honeycomb Seals: Test Results for Leakage and Rotordynamic Coefficients; Comparisons to Labyrinth and Smooth Configurations

1989 ◽  
Vol 111 (2) ◽  
pp. 293-300 ◽  
Author(s):  
D. Childs ◽  
D. Elrod ◽  
K. Hale
Keyword(s):  
Labyrinth Seal ◽  
Test Results ◽  
Rotordynamic Coefficients ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Shaft Rotation ◽  
Honeycomb Seal ◽  
Cell Dimensions ◽  
Honeycomb Seals ◽  
Rotordynamic Coefficient

Test results are presented for leakage and rotordynamic coefficients for seven honeycomb seals. All seals have the same radius, length, and clearance; however, the cell depths and diameters are varied. Rotordynamic data, which are presented, consist of the direct and cross-coupled stiffness coefficients and the direct damping coefficients. The rotordynamic-coefficient data show a considerable sensitivity to changes in cell dimensions; however, no clear trends are identifiable. Comparisons of test data for the honeycomb seals with labyrinth and smooth annular seals shows the honeycomb seal had the best sealing (minimum leakage) performance, followed in order by the labyrinth and smooth seals. For prerotated fluids entering the seal, in the direction of shaft rotation, the honeycomb seal has the best rotordynamic stability followed in order by the labyrinth and smooth. For no prerotation, or fluid prerotation against shaft rotation, the labyrinth seal has the best rotordynamic stability followed in order by the smooth and honeycomb seals.

Floating Body Equilibrium by Potential-Energy Minimization

1991 ◽  
Vol 28 (03) ◽  
pp. 153-162
Author(s):  
Antonio Campanile
Keyword(s):  
Potential Energy ◽  
Energy Minimization ◽  
Stable Equilibrium ◽  
Mathematical Formulation ◽  
Floating Body ◽  
Offshore Platforms ◽  
Equilibrium Equations ◽  
Hull Form ◽  
Computer Based ◽  
Marine Vehicle

The paper presents a computer-based approach aiming to solve the equilibrium equations of a floating body by means of potential-energy minimization. After a brief and general discussion, the principle that the potential energy of the system must be at a minimum is adopted as the condition for identifying the stable equilibrium positions of a marine vehicle. A consistent mathematical formulation is then developed. The problem is solved, therefore, by searching for the minimum of a function of three variables using a simple and efficient iterative method. This makes it possible for the equilibrium positions to be determined directly, unlike the classic methods-that is, without any previously constructed table of hydrostatic data and regardless of the magnitude of waterplane rotation compared with the initial orientation. No restriction is stipulated on hull form. Some study cases relating to a prismatic barge and a jacket-type platform are presented and analyzed. Relevant numerical results allow the procedure to be optimized so as to improve convergence. Finally, peculiar features of the proposed method are discussed, with particular reference to jacket-type offshore platforms, and further valuable applications are indicated.

An Edge Dislocation in a Three-Phase Composite Cylinder Model

1991 ◽  
Vol 58 (1) ◽  
pp. 75-86 ◽  
Author(s):  
H. A. Luo ◽  
Y. Chen
Keyword(s):  
Edge Dislocation ◽  
Stable Equilibrium ◽  
Phase Model ◽  
Composite Cylinder ◽  
Two Phase ◽  
Trapping Mechanism ◽  
Three Phase ◽  
Cylinder Model ◽  
Stable Equilibrium Position ◽  
The Stability

An exact solution is given for the stress field due to an edge dislocation embedded in a three-phase composite cylinder. The force on the dislocation is then derived, from which a set of simple approximate formulae is also suggested. It is shown that, in comparison with the two-phase model adopted by Dundurs and Mura (1964), the three-phase model allows the dislocation to have a stable equilibrium position under much less stringent combinations of the material constants. As a result, the so-called trapping mechanism of dislocations is more likely to take place in the three-phase model. Also, the analysis and calculation show that in the three-phase model the orientation of Burgers vector has only limited influence on the stability of dislocation. This behavior is pronouncedly different from that predicted by the two-phase model.

The Stress Field Due to an Edge Dislocation Interacting With Two Circular Inclusions

2016 ◽  
Vol 33 (2) ◽  
pp. 161-172 ◽  
Author(s):  
C.-K. Chao ◽  
F.-M. Chen ◽  
T.-H. Lin
Keyword(s):  
Stable Equilibrium ◽  
Complex Stress ◽  
Explicit Dependence ◽  
Closed Form Solutions ◽  
General Series ◽  
Stable Equilibrium Position ◽  
Arbitrary Plane ◽  
Plane Elastostatics ◽  
Combination Of Material ◽  
Extended Matrix

AbstractA general series solution to the problem of interacting circular inclusions in plane elastostatics is presented in this paper. The analysis is based on the use of the complex stress potentials of Muskhelishvili and the theorem of analytical continuation. The general forms of the complex potentials are derived explicitly for the circular inhomogeneities under arbitrary plane loading. Using the alternation technique, these general expressions were subsequently employed to treat the problem of an infinitely extended matrix containing two arbitrarily located inhomogeneities. The major contribution of the present proposed method is shown to be capable of yielding approximate closed-form solutions for multiple inclusions, thus providing the explicit dependence of the solution on the pertinent parameters. The result shows that the dislocation has a stable equilibrium position at a certain combination of material constants. The case of an inhomogeneity interacting with a circular hole under a remote uniform load is also investigated.

Characterization of the Dynamical Response of a Micromachined G-Sensor to Mechanical Shock Loading

2007 ◽  
Author(s):  
Daniel E. Jordy ◽  
Mohammad I. Younis
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Element Model ◽  
Squeeze Film ◽  
Dynamical Response ◽  
Mems Devices ◽  
Damping Coefficients ◽  
Squeeze Film Damping ◽  
Out Of Plane ◽  
Gap Width

Squeeze film damping has a significant effect on the dynamic response of MEMS devices that employ perforated microstructures with large planar areas and small gap widths separating them from the substrate. Perforations can alter the effect of squeeze film damping by allowing the gas underneath the device to easily escape, thereby lowering the damping. By decreasing the size of the holes, the damping increases and the squeeze film damping effect increases. This can be used to minimize the out-of-plane motion of the microstructures toward the substrate, thereby minimizing the possibility of contact and stiction. This paper aims to explore the use of the squeeze-film damping phenomenon as a way to mitigate shock and minimize the possibility of stiction and failure in this class of MEMS devices. As a case study, we consider a G-sensor, which is a sort of a threshold accelerometer, employed in an arming and fusing chip. We study the effect of changing the size of the perforation holes and the gap width separating the microstructure from the substrate. We use a multi-physics finite-element model built using the software ANSYS. First, a modal analysis is conducted to calculate the out-of-plane natural frequency of the G-sensor. Then, a squeeze-film damping finite-element model, for both the air underneath the structure and the flow of the air through the perforations, is developed and utilized to estimate the damping coefficients for several hole sizes. Results are shown for various models of squeeze-film damping assuming no holes, large holes, and assuming a finite pressure drop across the holes, which is the most accurate way of modeling. The extracted damping coefficients are then used in a transient structural-shock analysis. Finally, the transient shock analysis is used to determine the shock loads that induce contacts between the G-sensor and the underlying substrate. It is found that the threshold of shock to contact the substrate has increased significantly when decreasing the holes size or the gap width, which is very promising to help mitigate stiction in this class of devices, thereby improving their reliability.

Measurement of Microwave Induced Forces

1992 ◽  
Vol 269 ◽  
Author(s):  
John L. Watkins ◽  
H. W. Jackson ◽  
M. Barmatz
Keyword(s):  
Electric Field ◽  
Stable Equilibrium ◽  
Resonant Cavity ◽  
Single Mode ◽  
Microgravity Environment ◽  
Force Measurements ◽  
Stable Equilibrium Position ◽  
Infinite Set ◽  
Frequency Radiation ◽  
Made In

ABSTRACTElectrically polarizable materials in an inhomogeneous electric field experience a dielectrophoretic force which is proportional to the gradient of the square of the electric field. For high frequency radiation, the time averaged electromagnetic field provides a dc force. The force exerted on a sapphire and a metal sphere by the microwave field in a single mode resonant cavity has been measured. The force measurements have been made at equally spaced points along three orthogonal axes centered in a cylindrical cavity operating in the TE111 mode using 20 watts of microwave power at about 4.84 GHz. These data are compared to the simple theory of dielectrophoresis on a small sphere and verify that, for this mode and in the absence of other forces, there is a unique and stable equilibrium position at the center of the cavity. Measurements of the quality factor of the cavity provide a means of making an absolute check to the theory. The TE111 mode is one of an infinite set of modes predicted to have stable equilibrium positions that exist away from the cavity walls. It is proposed that dielectrophoresis in a microwave resonant cavity can be used to position samples for material processing applications in a microgravity environment. Knowledge of these forces may also be important for interpreting thermogravimetric measurements made in a microwave oven.

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