Vibrations of Rotating Ring Under Electric Field

旋轉環於靜電場下的振動

于涛(Tao Yu) ； 寇建阁(Jiange Kou) ； 胡毓忠(Yuh-Chung Hu)

《中國機械工程學刊》 42卷1期 (2021/02) Pp. 51-62

摘要

The vibration of rotating ring has got attention due to its wide application in engineering, such as vibratory ring gyroscope. This paper proposes an analytical model for the vibration of a rotating ring subjected to uniform electric field. The static pull-in voltage of a stationary ring is derived ahead the vibration analysis of the rotating ring. Then, a linearized electric-field model is derived and agrees well with the non-linear model when voltage lower than the static pull-in voltage. It is then applied on the analysis of the vibrations and dynamic stabilities of the rotating ring before static pull-in. The electric field will introduce an equivalent negative stiffness, which is proportional to the square of the voltage, into the rotating ring and thereby soften the ring and probably makes the vibration of the rotating ring unstable. The most interesting is that once the voltage enters the unstable region, one can further raise the rotational speed higher than the so-called critical speed to escape the unstable region. This is due to the initial stress induced by centrifugal force, which is proportional to the square of the rotational speed, makes the ring more rigid.

關鍵字

Electrostatics ； initial stress ； MEMS ； vibratory ring gyroscope ； rotating ring ； rotor dynamics ； vibration

並列摘要

因其應用廣泛，旋轉環頗受矚目，例如：振動環陀螺儀。本文提出一個旋轉環在均勻電場作用下之振動的解析模型。在分析旋轉環的振動之前，先推導出靜止環在靜電場作用下的靜態吸附電壓。然後推導出線性化的靜電場模型。當電壓小於靜態吸附電壓時，該線性化的靜電場模型與非線性模型相當吻合。因此，在分析旋轉環於靜態吸附之前的振動和穩定性時，則沿用該線性化的靜電場模型。靜電場會對旋轉環產生一個正比於電壓平方的等效負勁度，該等效負勁度會軟化環結構進而可能造成旋轉環的振動呈不穩定。最特別的是，當電壓值達到產生不穩定振動的範圍時，藉由增加環的轉速至超過所謂的臨界轉速，可使的旋轉環離開不穩區。此乃因離心力對環所造成的初始應力正比於轉速的平方，而該初始應力則可強化環的剛性。