Journal bearings mounted on viscoelastic supports are often used to improve the stability. However, proper support conditions have not been established especially for herringbone-grooved aerodynamic journal bearings. This study aims to make clear the effect of support stiffness and damping on the stability characteristics of the bearings. For that purpose, the linear perturbation analysis and the nonlinear transient analysis are performed for various viscoelastic support conditions. This investigation reveals that the groove configuration that gives the highest threshold speed of whirl instability changes with the viscoelastic support conditions. This investigation also reveals that it is possible to stabilize the bearings by viscoelastic supports. However, the threshold speed for lower damping support conditions decreases rather than that for rigid support condition. (C) 2016 Elsevier Ltd. All rights reserved.

Herringbone-grooved aerodynamic journal bearings supported by external stiffness and damping elements such as O-rings offer the higher stability. In the previous paper the authors presented the stability charts of the bearing systems using the liner Voigt model simulating external stiffness and damping elements. However, the effect of bearing-support-elements on the stability of the bearing systems is not fully clarified because of their frequency dependence. In addition, the nonlinearity has prevented comparisons between theoretical and experimental results. This study, therefore, investigates the effect of the frequency dependence of bearing-support-elements (O-rings) on the stability threshold of the herringbone-grooved aerodynamic journal bearings. This paper, first of all, clarifies the dynamic properties of O-rings under bearing supporting up to 8 kHz and proposes new empirical formulae for their stiffness and damping coefficients. Subsequently, the stability threshold of the bearing systems is calculated using the empirical formulae and compared with the experimental one. Results show that the stiffness coefficients increase exponentially with the frequency while the damping coefficients decrease and then converge on constant. The predicted stability threshold has good agreement with the experiments.

The purpose of this study is to find out the effects of hematocrit of the sealed blood on the leakage characteristics of blood cell components and plasma components in a mechanical seal, which separates blood from cooling water. The hematocrit was controlled to 0, 10, 20, 30 and 40%. Each component leakage was obtained by measuring the increase in the potassium ion concentration and the sodium ion concentration in the cooling water. It was revealed that each component leakage became larger with the increase of the hematocrit in the three-hour measurement. When changing the hematocrit from 20% to 30%, the volume percentage of the blood cell component leakage remarkably increased by 7.7 times. The friction characteristics of the mechanical seal under blood sealing and the hemolytic properties of the sealed blood were also studied. There is a typical trend under the mixed lubrication condition where the friction torque decreased with each of the sealed blood as the rotational speed increased. The hemolytic index of the sealed blood after the measurement experiments of the blood leakage increased with the increase of the hematocrit.

This paper discusses the stability of herringbone-grooved aerodynamic journal bearings supported by rubber O-rings. In this research, the dynamic stiffness and the damping properties of the O-rings are identified in a variety of operating and design conditions. The identification results are fitted with the empirical formulae previously proposed, and the distributions of the dynamic properties in a variety of support conditions are obtained. It is theoretically investigated how the stability of the bearing system changes with the dynamic properties. The bearing tests are also performed to verify the analytical results with the empirical formulae. The empirical formulae used here was reasonable for simulating the dynamic properties of the O-rings in a variety of bearing support conditions. The comparisons between the analysis and experiments verified the effectively of analyzing the stability of the bearing system by using the empirical formulae.

Hydrodynamic journal bearings induced serious shaft vibrations called oil whirl. Since they often give severe damages to the mechanical systems, it is important to fully understand the phenomenon to avoid the occurrence by the proper bearing design. This study theoretically and experimentally investigated both cylindrical and conical whirl instabilities, which can be appeared in journal bearings, and their relation was discussed. As the result, the span length of the bearings had a significant effect on the vibration mode. The transitional state in which the cylindrical and conical whirl instabilities alternately appeared was observed.

It is absolutely important for ultra-compact rotational machineries to achieve sable shaft rotation at ultra-high-speed. This paper discussed herringbone-grooved aerodynamic journal bearing systems developed for the purpose. In this system, the bearings are supported by rubber-O-rings for accurate and stable operations. To grasp the possibility for stabilization, two types of O-rings with different stiffness and damping properties under bearing supporting were tested in the experiment. As the results, the bearing system demonstrated the maximum rotational speed over 460,000 rpm without unstable phenomenon called whirl. However, the difference in rubber O-rings definitely affected the stability of the bearing system.

A miniature mechanical seal with the cooling water circulating system has been developed as a solution of the shaft seal problems in rotary blood pumps. The present paper describes the effect of the surface roughness in the sealing gap on the shaft seal problems under blood sealing. The sealing surface roughness of tested three seat rings were designed to be Ra=0.009, 0.088 and 0.170 mu m. The frictional loss torque in the sealing gap and the leakage of the blood and the cooling water were measured in this work. Results show that the maximum frictional loss torque measured in the seat ring of Ra=0.009 mu m, and the frictional loss torque decreased as the surface roughness increased. The leakage rate of the cooling water is larger than that of the blood in all seat rings, even though the blood chamber has higher pressure. The sealing characteristics between under the blood and water sealing were different. Moreover, the best surface roughness in the torque and the leakage characteristics were also different. Thus, it is important to study under blood sealing in the design of the surface roughness of the mechanical seals for rotary blood pumps.

Herringbone grooved aerodynamic journal bearings have better stability characteristics than plane aerodynamic journal bearings. In addition, the use of flexibly supported bearings is one of the efficient way to improve the bearing stability. Therefore, the combination bearing systems have prospects of realization of ultra-high-speed operations. The purpose of this work is to evaluate the effect of external stiffness and damping elements on the bearing stability. The Voigt rheological model was used for the external elements, in which the stiffness and the damping were treated as the linear elements. It was clarified that the stability boundary of the herringbone grooved aerodynamic journal bearings supported by suitable stiffness and damping elements can be heightened more than three times compared with the rigidly supported bearings.

The surface roughness of sealing face has effects on the characteristics of mechanical seals. An average flow model has been applied to the lubrication problems considering those effects. However, that model is limited to a type of normally-distributed surface roughness despite of a number of lapped surfaces which have non-normal distribution. Therefore, this study focuses on the skewness of surface roughness and investigates the effects of that on the correction factors in the average flow model. As a result, the smallest pressure flow factors were obtained under negative skewness. The pressure flow factor has effects on the leakage rate of mechanical seals, so that the surface roughness has negative skewness results in small leakage.

A miniature mechanical seal with cooling water circulating system has been developed as a shaft seal for rotary blood pumps. The present paper describes the establishment of the experimental methods under blood sealing and the estimation of the effect of the surface roughness on the shaft seal problems, especially in the increase of the friction and the leakage. Micro-leakage rates of the blood and the cooling water were measured by an ion chromatography. In this method, the ion concentration was converted into the leakage rate. The frictional loss torque was measured by a torque meter. Results are provided with three types of surface roughness, Ra = 0.009 mu m, Ra = 0.088 mu m and Ra = 0.170 mu m. It is shown that as the surface roughness increases the leakage decreases while the frictional loss torque increases under blood sealing. The best surface roughness in the torque and the leakage characteristics were different. Copyright (C) 2010 John Wiley & Sons, Ltd.

A miniature mechanical seal with the cooling water circulating system have been developed as a solution of the shaft seal problems in rotary blood pumps. The present paper describes the effect of the surface roughness in the sealing gap on the shaft seal problems under blood sealing. The sealing surface roughness of tested three seat rings were designed to be Ra=0.009μm, Ra=0.088μm and Ra=0.170μm. The frictional loss torque in the sealing gap and the leakage of the blood and the cooling water were measured in this work. Results show that the maximum frictional loss torque measured in the seat ring of Ra=0.009μm, and the frictional loss torque decreased as the surface roughness increased. The leakage rate of the cooling water is larger than that of the blood in all seat rings, even though the blood chamber has higher pressure. The sealing characteristics between under blood and water sealing were different. Moreover, the best surface roughness in the torque and the leakage characteristics were also different. Thus, it is important to study under blood sealing in the design of the surface roughness of the mechanical seals for rotary blood pumps.

Herringbone grooved aerodynamic journal bearings have better stability characteristics than plane aerodynamic journal bearings. In addition, the use of the flexibly supported bearings is one of the efficient way to improve the bearing stability. Therefore, the combination bearing systems have prospects of realization of ultra-high-speed operations. The purpose of this work is to evaluate the effect of the external stiffness and damping elements on the bearing stability. The Voigt rheological model was used for the external elements, in which the stiffness and the damping were treated as the linear elements. It was cleared that the stability boundary of the herringbone grooved aerodynamic journal bearings supported by the suitable stiffness and damping elements can be heightened more than three times compared with the rigidly supported bearings.

The purpose of this study is to establish an internationally-acceptable methodology of fatigue testing for prosthetic heart valves, based on the improvement process of a Chinese-made accelerated fatigue test apparatus. This test system had four parallel water cylinders, in which valves were incorporated. Open and closing motion of each valve was controlled at a repetition speed of 10-20 Hz with a displacement of 1-2mm. The Jerryfish (JF) valve was used in this study. It had the longest implantation record in animal trials. In this case, the polyurethane membrane with the thickness of 50-70 mum was ruptured due to a creep effect observed at the middle portion between two spokes after 300days in goat.
First of all, in vitro fatigue test machine has been innovated in order to reproduce the above mentioned in-vivo fatigue test results. Dynamic conditions which enabled the identical forces between accelerated and normal cycles were determined : maximum load during valve closure was 5.2N (increased by 20% to the normal), and impulse was 2.5*10^<-2>N・s. Under this condition, valves were ruptured for 1.0*10^7 cycles, but no distinct creep was noticed. Secondary, a temperature of the test-fluid was changed from 20ﾟC to 50ﾟC,expecting a creep effect to the membrane shape. Change in shape of the membrane was observed after 3.2*10^6 cycles, however, its distortion value was only 50% of the animal experimental results. For the next step, a material of the membrane was changed from polyurethane to polystirene to achieve bigger change in shape. As a result, the identical distortion level was successfully reproduced, but a location of the ruptured area was different between in vitro and in vivo results. In order to clarify the above mentioned difference, FEM analysis was performed. The calculated result exhibited that there are three major areas of the maximum elastic strain between two spokes. Moreover, it was confirmed that the ruptured areas observed by both in vitro/in vivo results could be fully superimposed to the areas of the calculated maximum elastic strain.
Further investigation should by carried out to determine more reliable methodology for fatigue testing, based on the in vitro and FEM analysis data.

潤滑剤の諸特性を改善するために多量の高分子重合物を添加した潤滑油は,粘弾性挙動を示すことが知られている.この粘弾性挙動が軸の安定性に及ぼす影響を,理論的に明らかにすることが本研究の目的である.研究方法としては,まず粘弾性モデルとして線形2要素モデルを用い,基礎方程式を流体潤滑理論に適用し潤滑方程式を得て,さらにこれを無限長幅ジャーナル軸受に適用し,非線形偏微分方程式を得る.圧力の線形近似と,軸の微少振動を正弦波と仮定した「近似解法1」,圧力の線形近似を高次の項まで考慮した「近似解法2」の2通りの解き方によりこの方程式を解いて圧力,油膜力,油膜係数を求める.ここで中央にロータを持ち両端で軸受に支えられている弾性軸の系を考え,その振動方程式から導かれた特性方程式に油膜係数を代入して安定限界線図を求める.その結果,「近似解法1」から油膜係数について,弾性係数は粘弾性流体の方がニュートン流体よりも高くなり,粘性係数については逆に粘弾性流体の方が低くなった.また安定限界線図から潤滑剤として粘弾性流体を用いると中偏心率では不安定領域が広がり,低偏心率及び高偏心率では逆に安定領域が広がった.「近似解法2」では近似の次数が低いときには「近似解法1」と同じような結果を得ることができたが,高くなると結果が不安定になった

The purpose of this study was to fabricate viscoelastic tubes, which was anatomically identical and functionally equivalent to the natural aorta. A rotational molding workstation which was originally developed at the National University of Singapore was used for this research. First of all, we tried to fabricate aortic arch-shaped tubes by a conventional method ; a single rotational axis was employed after silicone liquid was painted onto the mold by a brush. However, it was hard to make an even thickness tube, and it was also hard to control the quality. For the next step, several trials and errors were conducted using a rotational molding workstation under various combination of rotational speeds between two axis with different painting volume, interval and times. At last, the optimal condition to develop 1mm-thickness tube was determined as follows : rotational speeds of major and minor axis were 0.02rpm and 8rpm, respectively, a weight of silicone was 20g each for three times, and painting interval was 3hours. Then, thickness deviation which was defined as a ratio between standard deviation and average thickness culd be reduced from 30% to 20%.It was well-known that natural aorta had specific mechanical properties ; increase in pressure is small for the range of less incremental volume, whereas it becomes more for the range of higher incremental volume. We have tried to fabricated multi-layr tube, however, we haven't get a favorable result. So, other trials that contained different properties from silicone was carried out. A sponge layr was sandwiched with soft and stiff silicone layrs. The compliance values of this 3-layr tube was 0.41 1/mmHg in the range of small incremental volume up tp 40%, but it was decreased to 0.36 1/mmHg for over 40%, due to a shift of major elasctic effects from outer stiff to inner soft silicon layr.Above findings were worthwhile for the further improvements of the rotational molding machine

In order to predict a durability of heart valve prostheses, two types of commercially available (ISU-Standard) accerelated fatigue tests were employed, while improving a life time of Jellyfish polymer valve.A chronic animal experiment of Jellyfish valves incorporated into conventional sac-type blood pumps for over 300days exhibited that there were a fracture and calcification on this flexible membrane. Therefore, a long durability (approx. 1 year guarantee) of Jellyfish valve has been developing. while clearing a cause of the problems.First of all, design improvement of Jellyfish valve was attempted. Results of finite element analysis during closed phase of Jellyfish valve indicated that mechanical strain generated on valve membrane fracture and calcification in the animal experiment. Moreover further analysis suggested that an additional circular rim located at 7.0mm in radius was quite effective to avoid strain concentration. In vitro steady and pulsatile tests demonstrated that a modified valve with 0.5mm width rim located at 70mm in radius had a comparable hydrodynamic performance to the original Jellyfish valve.For the next step, commercially available two types of accelerated fatigue tests were employed. Fatigue tests successfully revealed that a durability of the modified valve was much more extended than the original Jellyfish valve by 7 times.Thus, chronic animal experiment with conventional sac-type blood pumps in which the modified valves were incorporated was performed for months. Hemolysis related to the application of the modified valve was not observed. This would be owing to the tapered configuration of the additional rim with 0.1mm width of inflow side.From the data above, the modified polymer valve showed great promise for clinical artificial heart application, and it was also clarified that our prevent fatigue tester could provide comparative durability data of polymer valves

An originally-designed mock circulatory system (MCS) has been developed in order to evaluate functions of several types of artificial organs for the surgical treatment of cardiovascular deseases. The features of the present MCS has a capability of exhibiting a practical simulation of hemodynamics in aortic vascular prostheses, and of having a mechanical feasibility of the natural left ventricular contraction by the computer-based control.As mechanical properties of arterial vascular grafts are different from those of elastic natural arteries, it can be speculated that the left ventricular afterload might be elevated after the graft implantation due to its low compliance. By switching blood flow to a bypass graft which was inserted in parallel with the natural descending aorta of dog, the characteristic impedance was elevated by 2.6 times, furthermore, the cardiac output was decreased by 14%, and the cardiac work calculated from the pressure-volume relationships in the left ventricle, was increased by 1.5 times. MCS was also employed to represent hemodynamic characteristics as well as to compare the data with those in animal experiments. When the flow was set to 4L/min in MCS by the mechanical driven pump, either characteristic impedance of the arterial system or external work of the left ventricular model was increased by 2.4 or 1.5 times, respectively. Thus, it was anticipated that the external cardiac work would be elevated by the increase of the arterial impedance after the graft implantation

A centrifugal blood pump has been developed as an implantable left ventricular assist device. A newly designed mechanical seal with a recirculating cooling water system is used for the shaft seal. The mechanical seal separate cooling water and blood in the sealing face. The objective of this study is to establish the measuring method of bidirectional leakage of the sealed fluid and the cooling-water through the sealing face, especially when blood is as the sealed fluid. The leakage of the blood to the cooling-water side was measured by the atomic emission spectroscopy. The leakage of the cooling-water to the blood side was measured by using the atomic absorption spectroscopy, by using the standard addition method, by the correction of hemolysis caused by leakage of the cooling water. As the results of the experiments, the leakage rate of both the cooling water and the blood were below 0.4[ml/6h]. Therefore, the mechanical seal performed acceptably well to support its use in the centrifugal blood pump

The objective of this work is to study the effects of the surface roughness of the sealing face in the mechanical seal for rotary blood pumps on the characteristics of the shaft seal. This mechanical seal is incorporated with a circulating cooling water system for the lubrication, cooling and washing in the sealing face. In this paper, the several types of mechanical seals which have the different surface roughness of Ra=0.009μm, Ra=0.088μm, Ra=0.170μm and Ra=0.250μm were used, and the leakage rate, the torque and the temperature change were measured. The leakage was measured by ion chromatography. The torque was measured by a torque meter and the temperature change was measured by a thermometer which was set in the seat ring. As the results, the torque and the temperature change have the minimum value by using the seat ring of Ra=0.088μm, and the sealed fluid leakage rate has the minimum value of less than 0.1 ml/3h by using the seat ring of Ra=0.009μm

　冷却システムを備えた血液ポンプ用メカニカルシールの潤滑特性を検討するために，インペラを取り外したシールしゅう動面のみを検討できる実験装置を開発した．それを用いて，しゅう動面での摩擦損失トルク，温度変化，漏れの観点から，人工心臓用メカニカルシールのしゅう動面の表面粗さが軸シール特性に及ぼす影響について検討した．実験には，表面粗さRa=0.009&micro;m，Ra=0.088&micro;mおよびRa=0.170&micro;mの3種類を用いた．得られた結論を以下に示す．（１）Ra=0.009&micro;mのシートリングでは，しゅう動面間が非常に狭くなるため流体摩擦の影響が大きく，摩擦損失トルクおよび温度上昇が大きかった．一方，Ra=0.170&micro;mのシートリングでは，局所的な混合潤滑も影響により，摩擦損失トルクおよび温度上昇が大きくなった．Ra=0.088&micro;mのシートリングでは，すべての実験条件において摩擦トルクが最も小さくなり，トルク特性の観点からみれば現在使用されているRa=0.088&micro;mのシートリングが最も優れていた．（２）シールしゅう動面の温度変化を計測する手法を確立し，しゅう動面の温度特性を検討した結果，温度特性はトルク特性とよく似た傾向を示し，温度変化はRa=0.088&micro;mのシートリングで最も低くなった．また，そのときの温度変化は0.3K以下に抑えられた．よって，温度特性の観点からみれば現在使用されているRa=0.088&micro;mのシートリングが最も優れていた．（３）Ra=0.009&micro;mのシートリングを使用した場合，しゅう動面間が非常に狭くなり，シール面に対して両サイドの圧力差の影響を受けにくくなり，血液漏れ量および血液中へのクーリングウォータの漏れ量をともに少なく抑えることができた．漏れ量は，密封溶液漏れ量，クーリングウォータ漏れ量ともに0.03 ml/3hであった．そのため，漏れ量の観点からは現在使用されているRa=0.088&micro;mのシートリングよりも，Ra=0.009&micro;mのシートリングのほうが優れていた．

軸シール部の冷却・洗浄を目的に開発された「Cool-Seal-System」を搭載した定常流型補助人工心臓をさらに小型化・長寿命・高信頼とするためには，血液による摩擦損失トルクの上昇，密封流体およびC.W.(クーリングウォータ)の漏洩の問題を解決する必要がある．そこで，本研究では，定常流型補助人工心臓用メカニカルシールの摺動面粗さに注目し，Ra=0.009，Ra=0.088，Ra＝0.170のシートリング，およびRa＝0.093のシールリングを用いて，摺動面の表面粗さが人工心臓用メカニカルシールの潤滑特性に及ぼす影響を，トルク特性，密封特性の両観点から調査することを目的とした． メカニカルシールの漏れ量に関しては，血中たんぱく質の影響により血液をそのまま用いての測定が困難なため，血液相当液としてMgCl2水溶液を用い，チャンバ内圧力100[mmHg]，回転数3000[rpm]，マグネット反発力3.0[N]とし，C.W.側(純水中)に漏れ出したMg元素濃度をイオンクロマトグラフィ－により測定した．トルク特性に関しては，チャンバ内にHt値30%のブタ血液，C.W.に純水を用いて，回転数を1000～8000[rpm]，マグネット反発力3.0～5.0[N]に変化させ，摺動面にかかる摩擦損失トルクの大きさを測定した． 実験結果より，シールの漏れ量は摺動面粗さに依存し，粗さの増加とともに漏れ量も増加し，Ra=0.170での漏れ量は約0.16[ml/3ｈ]であった．摩擦損失トルクの最低値は表面粗さが最も大きいRa=0.170で最も低くなり，その大きさは約2.5[mN･m]であった．これにより，定常流型補助人工心臓用メカニカルシールの潤滑特性向上には，摺動面の表面粗さを考慮した設計が重要であることがわかった．

潤滑剤の諸特性を改善するために多量の高分子重合物を添加した潤滑油は、非線形の粘弾性挙動を示すことが知られている。しかし、ジャーナル軸受によって支持された軸の安定性に関する研究は、潤滑剤としてニュートン流体を用いた場合のみで、そのレオロジ方程式の複雑さから粘弾性流体に対する研究はほとんどなされていない。そこで本研究では、粘弾性流体に対する潤滑方程式の解法として微小振動の過程を用いた線形化手法による解析を提案し、潤滑剤の粘弾性特性がジャーナル軸受の運転安定限界特性に及ぼす影響を明らかにすることを目的とする。　本研究では、粘弾性モデルとしてばねとダッシュポットが直列に結合されたマックスウェルモデルを用いる。マックスウェルモデルに対するレオロジ方程式と、連続の条件式、および力の釣り合い条件式を連立させることにより、粘弾性流体に対する潤滑方程式を得る。この潤滑方程式を解くために、ジャーナルの微小振動を仮定し、油膜の弾性係数および粘性係数に対応する微分方程式の誘導を行い、これより油膜係数を算出する。この結果をもとに微小振動時の油膜力を求め、ジャーナルの運動方程式と連立させることによってその特性方程式を求め、これを解くことによりジャーナルの運転安定限界特性を求める。　その結果、潤滑剤の粘弾性特性がジャーナル軸受の安定限界に及ぼす影響として、粘弾性流体ではニュートン流体に比較して、低偏心率では不安定領域が広がり、高偏心率では逆に安定領域が広がるという結論を得た。研究成果の発表1998年５月富岡淳、柴崎健一、折橋健治、林洋次、粘弾性流体で潤滑された有限幅ジャーナル軸受の動特性、平成10年度日本設計工学会春季発表講演会

　近年、高速回転や高荷重など過酷な条件下で使用されている滑り軸受では、潤滑剤の諸特性を改善するために多量の高分子重合物を添加した潤滑油が使用されている。この種の潤滑油は非線形の粘弾性挙動を示すことが知られている。しかし、この非線形粘弾性挙動が、ジャーナル軸受などの滑り軸受において特に問題となる自励振動に及ぼす影響は明らかではない。　そこで本研究では、潤滑剤の非線形粘弾性挙動が、ジャーナル軸受の安定限界曲線およびオイルホワール発生時におけるジャーナルの軸心軌跡に及ぼす影響を明らかにする事を目的とする。　解析において、微小振動の仮定を用いる場合と用いない場合の2とおりを検討し比較した。　まず、微小振動の仮定を用いる場合は、油膜の弾性係数および粘性係数に対応する微分方程式の誘導を行い、これより油膜係数を算出する。この結果をもとに微小振動時の油膜力を求め、ジャーナルの運動方程式と連立させることによって特性方程式を導き、これを解くことにより軸受の安定限界曲線を求めた。　次に、微小振動の仮定を用いない場合は、非線形粘弾性流体に対するレイノルズ方程式を直接解くことにより油膜力を求め、これとジャーナルの運動方程式とを連立させ、時間に関する高次差分法を導入することにより、ジャーナルの軸心軌跡を求め、軸受の安定特性を求めた。　その結果、いずれの解析においても、低偏心率の場合は潤滑剤として粘弾性流体を用いた方がニュートン流体より安定域が増大し、高偏心率の場合は逆の傾向を示すという結論が得られた。

潤滑剤の諸特性を改善するために多量の高分子重合物を添加した潤滑油は，粘弾性挙動を示すことが知られている。この粘弾性挙動が軸の安定性に及ぼす影響を，理論的に明らかにすることが本研究の目的である。 研究方法としては，粘弾性モデルとして非線形マックスウェルモデルを用い，基礎方程式を流体潤滑理論に適用し潤滑方程式を得て，さらにこれを無限長幅ジャーナル軸受に適用し，非線型偏微分方程式を得る。圧力の線形近似と，軸の微少振動を正弦波と仮定した「近似解法1」，圧力の線形近似を高次の項まで考慮した「近似解法2」の2通りの解き方によりこの方程式を解いて，圧力，油膜力，油膜係数を求める。ここで中央にロータを持ち両端で軸受に支えらている弾性軸の系を考え，その振動方程式から導かれた特性方程式に油膜係数を代入して安定限界線図を求める。 その結果，「近時解法1」から油膜係数について，弾性係数は粘弾性流体の方がニュートン流体よりも高くなり，粘性係数については逆に粘弾性流体の方が低くなった。また安定限界線図から潤滑剤として粘弾性流体を用いると中偏心率では不安定領域が広がり，低偏心率及び高偏心率では逆に安定領域が広がった。「近似解法2」では，近似に次数が低いときには「近似解法1」と同じ結果を得ることが出来たが，高くなると結果が不安定になった。