Bi and trifilar suspension centering correction

Abstract

For accurate measurements of moments of inertia, or gyradii, about a vertical axis the offset of the center of mass of the suspended body from the symmetry axis of the suspension is required. A number of simple methods of determining this offset from yaw period measurements are described and experimentally shown to provide results of precision equal or better than suspension line tension measurements and thus eliminating the necessity for the use of strain gauges.

Appendix: Trifilar suspension with offset center of mass

When the center of mass is offset from the symmetry axis of a trifilar suspension by b = βd at an angle ψ, see Fig. 3b, the line tensions become:

$$\begin{aligned} T_{1} & = \frac{{M_{T} g}}{3}\left\{ {1 + 2\beta \text{Cos} \psi } \right\} \\ T_{2} & = \frac{{M_{T} g}}{3}\left\{ {1 - \beta \left( {\text{Cos} \psi - \sqrt 3 \text{Sin} \psi } \right)} \right\} \\ T_{3} & = \frac{{M_{T} g}}{3}\left\{ {1 - \beta \left( {\text{Cos} \psi + \sqrt 3 \text{Sin} \psi } \right)} \right\} \\ \end{aligned}$$

(23)

Then for a small yaw rotation θ about the symmetry axis the horizontal components and net horizontal force $\bar{F}_{\varSigma } = \sum\limits_{i} {\bar{F}_{i} }$ are:

$$\begin{aligned} \bar{F}_{1} & = T_{1} \frac{\theta d}{l}\left( \begin{aligned} \mathop {}\limits^{{}} 0 \hfill \\ - 1 \hfill \\ \end{aligned} \right) = \frac{{M_{T} gd}}{3l}\theta \left\{ {1 + 2\beta \text{Cos} \psi } \right\}\left( \begin{aligned} \mathop {}\limits^{{}} 0 \hfill \\ - 1 \hfill \\ \end{aligned} \right) \\ \bar{F}_{2} & = T_{2} \frac{\theta d}{l}\left( \begin{aligned} {{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 2}} \right. \kern-0pt} 2} \hfill \\ {{\mathop {}\limits^{{}} 1} \mathord{\left/ {\vphantom {{\mathop {}\limits^{{}} 1} 2}} \right. \kern-0pt} 2} \hfill \\ \end{aligned} \right) = \frac{{M_{T} gd}}{3l}\theta \left\{ {1 - \beta \left( {\text{Cos} \psi - \sqrt 3 \text{Sin} \psi } \right)} \right\}\left( \begin{aligned} {{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 2}} \right. \kern-0pt} 2} \hfill \\ {{\mathop {}\limits^{{}} 1} \mathord{\left/ {\vphantom {{\mathop {}\limits^{{}} 1} 2}} \right. \kern-0pt} 2} \hfill \\ \end{aligned} \right) \\ \bar{F}_{3} & = T_{3} \frac{\theta d}{l}\left( \begin{aligned} {{ - \sqrt 3 } \mathord{\left/ {\vphantom {{ - \sqrt 3 } 2}} \right. \kern-0pt} 2} \hfill \\ {{\mathop {}\limits^{{}} \mathop {}\limits^{{}} 1} \mathord{\left/ {\vphantom {{\mathop {}\limits^{{}} \mathop {}\limits^{{}} 1} 2}} \right. \kern-0pt} 2} \hfill \\ \end{aligned} \right) = \frac{{M_{T} gd}}{3l}\theta \left\{ {1 - \beta \left( {\text{Cos} \psi + \sqrt 3 \text{Sin} \psi } \right)} \right\}\left( \begin{aligned} {{ - \sqrt 3 } \mathord{\left/ {\vphantom {{ - \sqrt 3 } 2}} \right. \kern-0pt} 2} \hfill \\ {{\mathop {}\limits^{{}} \mathop {}\limits^{{}} 1} \mathord{\left/ {\vphantom {{\mathop {}\limits^{{}} \mathop {}\limits^{{}} 1} 2}} \right. \kern-0pt} 2} \hfill \\ \end{aligned} \right) \\ \bar{F}_{\varSigma } & = \frac{{M_{T} gb}}{l}\left( \begin{aligned} + \text{Sin} \psi \hfill \\ - \text{Cos} \psi \hfill \\ \end{aligned} \right)\theta \\ \end{aligned}$$

(24)

Thus $\bar{F}_{\varSigma }$ is perpendicular to the center of mass offset $\bar{b}$ and of magnitude $F_{\varSigma } = \left( {{{M_{T} gb} \mathord{\left/ {\vphantom {{M_{T} gb} l}} \right. \kern-0pt} l}} \right)\theta$ i.e. proportional to the yaw angle θ and thus will oscillate with the yaw frequency and excite sway-surge oscillations. If the suspension spacing d is close to the yaw gyradius k _yT then the sway-surge frequency is close to the yaw frequency which will be resonantly excited. The development of sway-surge oscillation, after initial yaw rotation about the symmetry axis, is an indication of an offset center of mass, with the offset perpendicular to the sway-surge oscillation.

The torque $\bar{\varGamma }_{CM}$ about the offset center of mass can be expressed as that about the symmetry axis plus the vector product of $\bar{b}$ and $\bar{F}_{\varSigma }$ as:

$$\begin{aligned} \bar{\varGamma }_{CM} & = \bar{d}_{11} \times \bar{F}_{1} + \bar{d}_{22} \times \bar{F}_{2} + \bar{d}_{33} \times \bar{F}_{3} - \bar{b} \times \bar{F}_{\varSigma } \\ \bar{\varGamma }_{CM} & = \left\{ {bF_{\varSigma } - d\left( {F_{1} + F_{2} + F_{3} } \right)} \right\}\hat{k} \\ \end{aligned}$$

(25)

Then substituting Eq. (24) leads to a torque about the center of mass of magnitude $\varGamma_{CM} = {{M_{T} g\left( {d^{2} - b^{2} } \right)\theta } \mathord{\left/ {\vphantom {{M_{T} g\left( {d^{2} - b^{2} } \right)\theta } l}} \right. \kern-0pt} l}$ and so a period of small angle yaw oscillation of $T_{yo} = 2\pi k_{yT} \sqrt {{l \mathord{\left/ {\vphantom {l {g\left( {d^{2} - b^{2} } \right)}}} \right. \kern-0pt} {g\left( {d^{2} - b^{2} } \right)}}}$.

For rotations about any axis other than the symmetry axis the horizontal displacements of the suspension points are not equal, so the angular displacements ϕ _i of the suspension lines, and hence the vertical displacements of the suspension points, are not equal, as becomes immediately obvious if the rotation is about one of the suspension lines. Thus the motion includes roll and pitch and the gyradii k _r , and k _p , as well as the center of mass height are required for a full description of the motion. Nonparallel suspensions exacerbate this problem. However, the roll-pitch angle is $\varphi \approx {{ - b\theta^{2} } \mathord{\left/ {\vphantom {{ - b\theta^{2} } l}} \right. \kern-0pt} l}$ and the vertical motion, which is the source of the potential energy, has an acceleration $\ddot{z} = {{\left( {d^{2} - b^{2} } \right)\left( {\dot{\theta }^{2} + \theta \ddot{\theta }} \right)} \mathord{\left/ {\vphantom {{\left( {d^{2} - b^{2} } \right)\left( {\dot{\theta }^{2} + \theta \ddot{\theta }} \right)} l}} \right. \kern-0pt} l}$, which are both small for l > d and of second order in θ. Thus the tensions can be assumed to have approximately their static values and second order effects due to the vertical, sway, surge, roll and pitch motions can to first approximation be ignored [29].