28 Innovative Solutions in the Field of Engineering Sciences

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1 Applied Mechanics and Materials ubitted: IN: , Vol. 59, pp 7-1 Accepted: doi:1.48/ Online: Trans Tech Publications, witerland The Vibration Based on Non-Zero Point orce Moent Elasticity Theory Wen-ba HAN 1, a*, huang-hua HUANG, b, Jie LIU, c and Jin-un UN 4, d 1 chool of Civil Engineering and Architecture,Panhihua University, Panhihua, China chool of Civil Engineering and Architecture,Panhihua University, Panhihua, China chool of Civil Engineering and Architecture,Panhihua University, Panhihua, China 4 chool of Civil Engineering and Architecture,Panhihua University, Panhihua, China a @qq.co, b @qq.co, c liujiewell@aliyun.co, d 66988@qq.co *Corresponding author Keywords: Point oent, Vibration, Resonance, Resonant frequency. Abstract. The traditional elastic theory believes that there exists noral stress in pure bending body (PBB) and shear stress in pure torsion body (PTB). However, the author proved that there is no noral stress but Bent Point Moent (BPM) in PBB. And it also concluded that there is no shear stress but hear Point Moent (PM) in PTB. This article overturns the preliinary theores of the Elasticity Theory, which believes that the value of the oent (Bending oent & Torsion oent) on a unit area converges to ero. Just as the copletely different natural frequencies of the forced vibration can lead to copletely different resonant conditions. Besides, this theory has also been validated in the Daage Mechanics National Key Laboratory of Tsinghua University. Therefore, it is significant to avoid destruction produced by resonance. Introduction Resonance occurs widely in nature when the frequency of driving force close to the syste's natural frequency. It ay cause structure failures then lead to treendous loss for huan life and properties. Deeply analying its causation, the proposed flaw in current theory of elasticity is no ore than ajor causation which ust be revised by the non-ero point force oent elasticity theory [1], for siplicity, it is naed by Point Moent Theory.This theory was first established by Wenba Han and huanghua Huang in 1.In their theory a new physical quantity naed Point Moent of orce was first introduced. In this paper, both vibration and resonance were resolved by the new elasticity theory. Analysis The orced Vibration Displaceent Based on Non-Zero Point orce Moent Elasticity Theory [1] and Current Elasticity Theory The new theory proved that no noral stress exists in pure bending body but exists bent point oent w, which deduced by the proposed new theory []: W M ( x) y (1) A y da where y, the distance between any point of cross section and neutral axis, (), absolute static oent,, the oent acting on cross section of the bea, respectively. What s ore, is All rights reserved. No part of contents of this paper ay be reproduced or transitted in any for or by any eans without the written perission of Trans Tech Publications, (ID: , Pennsylvania tate University, University Par, UA-5//16,1:49:55)

2 8 Innovative olutions in the ield of Engineering ciences non-negative and non-ero through the centroid of the section, in particular, as for rectangular bh cross-section,. 4 In the case external force P (the gravity of electrootor) loads at the iddle center point C of the bea with length of l. ig.1. External force P loads on iddle center C based on new theory, the deflection [1] due to P acting on C is Pl 48G P W () where is equivalent stiffness factor of vibration based on new theory, and G W is bending elasticity odulus. The above two paraeters are first defined and introduced. New theoretical results obtained fro treendous experients prove that the relationship between bending point oent and bending longitudinal strain is, W W G W (4) and Eq.(4) is derived fro Point Moent Theory. 9 As for carbon steel, G W 1 N / is obtained fro experients. Under the sae conditions, through current elasticity theory, axiu deflection [4-5] at the iddle center of bea is where Pl 48EI P (5) is equivalent to the stiffness factor of vibrations based on stress theory, E is the elastic 11 stretching odule, with regard to carbon steel, E 1 N /, I is oent of inertia relative to neutral axis, for rectangular cross-section, bh I 1 Copared Eq.() with Eq.(5), it is obvious that and are quite different since the deflections of vibrations at the iddle center of bea are distinct. orced Vibration Differential Equations and Motion Equations Based on tress Elasticity Theory [ ] The diension of the rectangular bea is l in length, B in width and h in height as shown in ig.1(a),where a electrootor with angular velocity ω is installed at the iddle center of the bea. The centrifugal inertial force induced by centrifugal otion of rotor is a periodical varying force which has two coponents in horiontal and vertical directions. The horiontal force is equal to cos t, which can lead to longitudinal vibration but is ignored in this condition. iilarly, the vertical force is equal to sin t, which can lead to lateral vibration. Thus, it is a vibration syste

3 Applied Mechanics and Materials Vol with one degree of freedo, scheeticly shown in ig.1(b). uppose velocity of the oveent is v, syste dap is, according to newton's second law, the differential equation [] of forced vibration is d x dx x cos t dt dt (6) g (7) where is natural angular frequency and β is daping coefficient. If the vibration is wealy daping, thus, the solutions of differential Eq.(6 ) [4,5] is, (8) t x Ae cos( t ) Acos( t ) (9) In which the first ter of right hand is daping vibration and second ter is siple haronic vibration. urtherly, Eq.(9) also deonstrates that the forced vibration is superposition by daping vibration and siple haronic vibration. The aplitude of forced vibration A and initial phase φ are, A ( ) 4 (1) arctan (11) orced Vibration Differential Equations and Motion Equations Based on Point Moent Theory Under the sae conditions in the previous section, the differential equations and otion equations of wealy daping vibration based on Point Moent Theory is d x dx x cost dt dt g (1) where is the natural angular frequency of forced vibration based on Point Moent Theory. The solution of Eq.(1 ) is X A e cos( t ) A cos( t ) t (14) when the vibration reaches a stable state, the second ter of the above equation equals to ero, and the otion equation of forced vibration becoes (1) x A cos( t ) The aplitude of forced vibration according to Point Moent Theory is (15) A ( ) 4 (16)

4 Innovative olutions in the ield of Engineering ciences and the initial phase is arctan (17) Copares With Cobining Eq.(7) and Eq.(1) the ratio of the natural frequencies based on Point Moent Theory becoe based on tress Theory and / / 48EI / l 48G / l EI G W W (18) or exaple, a steel bea with rectangular section is h in height and B in width, which eeps upright. ro Eq.(18) the ratio of and can be obtained as 11 1 bh /1 h bh / 4 (19) Eq.(19) reveals that the ratio of natural angular frequencies based on two theories are deterined by height of the bea. et h=1, it follows that et h, it follows that thus, the two results indicate that the frequencies ratio based on two theories are equal if h, therefore we define h* as the critical sie of rectangular bea under forced vibration. ince > and fro Eq.(1) and Eq (16),we can find the aplitude based on point oent theory is larger than that based on stress theory. in other words,it s ore dangerous. Resonance Based on Two Theories Resonance Based on tress Theory According to the tress Theory, we can achieve resonant frequency of wealy daping vibration,, resonance aplitude,, phase difference between resonance displaceent and driving force, as follows: r () A r (1) arctan r Resonance Based on Point Moent Theory ubstituting by () in Eq.(), the resonant angular frequency, aplitude of wea daping resonance and phase difference between resonance displaceent and driving force based on Point Moent Theory follow.

5 Applied Mechanics and Materials Vol r () A r (4) arctan r (5) Eq.() to Eq.(5) indicate that the resonant frequencies are distinct under the two theories, in addition, the aplitude and phase difference are also different. According to Eq.(4), if the height h of the steel bea equals to h*, then, and A >A It deonstrates that the higher bea suffers higher destruction ris due to resonance on account of Point Moent Theory. uary The results of this paper indicate that Non-Zero Point orce Moent Elasticity Theory can solve vibration and resonance probles. However, further investigations are needed to better understand the new theory and quantity experients would carry out to certify the advantages of Non-Zero Point orce Moent Elasticity Theory. References [1] Wenba Han, huanghua Huang.Non-ero point force oent elasticity theory [M].Chongqing: Chongqing University Press. [] Guoheng Yang,Yangyi, University physics,beijing, China Machine Press,1.8. [] A.D. Nashif, D.I.G. Jones,and J.P. Henderson.Vibration daping.newyor:john Wiley & ons, [4] A. A. habana. Theory of vibration, V. Ⅰand Ⅱ. NewYor:pringer-Verlag, [5] J. Argyris, H.P. Mlejen. Dynaicsof structures. NewYor: Elsevier cience Publishers, [6] Dianyuan Wang, Weijun Xie. General Physics [M].hang Hai: TongJi University Press,1. [7] Hongru Wang,ZhaoWang.Physics[M]. Beijing: Peing University Medical Press,1995. [8] Weihe Zhang,Guyue. Physics[M]. Beijing: Cheical Industry Press,1985. [9] Guano Xie. echanics of vibration[m]. Beijing: National Defence Industry Press,1. [1] Xijun Liu,Qinfen Jia. Engineering vibration theory and testing techniques.beijing: Higher Education Press,4. [11] J.M. Mechanics of Materials[M]. Beijing: China Machine Press,.

6 Innovative olutions in the ield of Engineering ciences 1.48/ The Vibration Based on Non-Zero Point orce Moent Elasticity Theory 1.48/