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Sensitive Element Dynamics

  • Vladislav Apostolyuk
Chapter

Abstract

After CVG sensitive element motion equations were obtained in the previous chapter, the next step would be to find solutions of these equations. As it has been already mentioned, obtained equations are not quite suitable for analysis in a closed form in terms of the primary and secondary coordinates due to the presence of the unknown and variable angular rate as a coefficient

Keywords

External Rotation Excitation Frequency Sensitive Element Angular Rate Proof Mass 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

After CVG sensitive element motion equations were obtained in the previous chapter, the next step would be to find solutions of these equations. As it has been already mentioned, obtained equations are not quite suitable for analysis in a closed form in terms of the primary and secondary coordinates due to the presence of the unknown and variable angular rate as a coefficient. However, given certain assumptions and using the fact that useful information in vibratory gyroscopes is present in amplitudes and phases of oscillations, solutions suitable for analysis still can be obtained.

Analysis of the sensitive element dynamics on a fixed and rotating base will be presented in this chapter. Solutions in terms of amplitudes and phases of the primary and secondary oscillations will be obtained. Sensitive element motion trajectory analysis will be presented as well. Finally, methodology of realistic numerical simulation of CVG dynamics will be explained in detail.

3.1 Primary Motion of the Sensitive Element

Sometimes it is more convenient to view at the angular rate sensing by means of a Coriolis vibratory gyroscope as an amplitude modulation. Indeed, output secondary oscillations are the primary oscillations modulated by the external angular rate. This is apparent from the analysis of the generalised motion Eq. ( 2.29). First equation describes primary oscillations that are induced by the excitations system and is coupled with the second motion equation via Coriolis term, which is linearly related to the external angular rate.

If there is no external rotation (\({\Omega } = 0\)) the motion equations become independent set of two unconnected equations
$$\left[ {\begin{array}{*{20}l} {\mathop {x_{1} }\limits^{..} \,+\,2 {\upzeta}_{1} {\upomega }_{1} \dot{x}_{1} + {\upomega }_{1}^{2} x_{1} = q_{10} \sin ({\upomega } t),} \hfill \\ {\mathop {x_{2} }\limits^{..}\,+\,2 {\upzeta}_{2} {\upomega }_{2} \dot{x}_{2} + {\upomega }_{2}^{2} x_{2} = 0.} \hfill \\ \end{array} } \right.$$
(3.1)

Here there are no external forces are applied to the secondary motion, and \(q_{10}\) is the amplitude of the accelerations from the excitations system, \({\upomega }\) is the excitation frequency. From Eq. (3.1) it is apparent that without external rotation secondary motion is absent, and only the first equation needs to be solved.

Closed form solutions for Eq. (3.1) are
$$\begin{aligned} x_{1} (t) =&\,C_{1} e^{{ - {\upzeta}_{1} t}} \sin \left( {t{\upomega }_{1} \sqrt {1 - {\upzeta}_{1}^{2} } + {\upvarphi }_{1} } \right) \\ & + \frac{{q_{10} }}{{\sqrt {\left( {{\upomega }_{1}^{2} - {\upomega }^{2} } \right)^{2}\,+\,4{\upomega }_{1}^{2} {\upzeta}_{1}^{2} {\upomega }^{2} } }}\sin \left( {{\upomega } \, t + {\upgamma} } \right), \\ x_{2} (t)\,=\,& 0. \\ \end{aligned}$$
(3.2)
Phase shift \({\upgamma}\) of the resulting primary oscillations is given by
$$tg\left( {\upgamma} \right) = - \frac{{2 {\upzeta}_{1} {\upomega }_{1} {\upomega }}}{{{\upomega }_{1}^{2} - {\upomega }^{2} }},$$
and constants \(C_{1}\) and \({\upvarphi}_{1}\) are determined from the initial conditions.
Analysis of the obtained solutions (3.2) shows us that settled oscillations of the CVG sensitive element occur with an amplitude that is proportional to the excitation force after natural oscillations disappear due to the damping (see Fig. 3.1).
Fig. 3.1

Primary oscillations (ω1 = 1000 Hz, ω1 = ζ2 = 0.025, Ω = 0)

At the same time proof mass remains motionless, and output signal from the sensor will be zero.

Furthermore transient time of the natural oscillations will determine start-up time of CVG, when only settled oscillations of the sensitive element remain and occur with the constant amplitude.
$$A_{10} = \frac{{q_{10} }}{{\sqrt {\left( {{\upomega }_{1}^{2} - {\upomega }^{2} } \right)^{2}\,+\,4{\upomega }_{1}^{2} {\upzeta}_{1}^{2} {\upomega }^{2} } }}.$$
(3.3)

Since primary oscillations are the carrier that is modulated by the external angular rate, in order to perform reliable measurements, the carrier must be highly stable in terms of amplitude and frequency and its amplitude must be as high as possible.

The natural way to achieve the highest amplitude of the primary oscillations is to excite them in resonance with its eigenfrequency
$${\upomega } = {\upomega }_{1} \sqrt {1-2 {\upzeta}^{2} }.$$
(3.4)
In case of resonance primary amplitude (3.3) becomes
$$A_{10} = \frac{{q_{10} }}{{2 {\upzeta}_{1} {\upomega }_{1}^{2} \sqrt {1 - {\upzeta}_{1}^{2} } }}.$$
(3.5)

Analysis of the expression (3.5) shows that the primary amplitude is higher when damping \({\upzeta}_{1}\) is lower, as well as the natural frequency \({\upomega }_{1}\) of the primary oscillations.

3.2 Sensitive Element Motion on a Rotating Base

Having looked at the primary oscillations of the sensitive element and methods of their efficient excitation, let us move on to the secondary oscillations. Studying solutions (3.2), one could see that if external angular rate is absent then secondary oscillations are absent as well.

Let us study now behaviour of the CVG sensitive element on the base that rotates with constant angular rate, e.g. \(\dot{\Upomega } = 0\). Motion Eq. ( 2.29) takes slightly simpler form
$$\left\{ \begin{aligned} \mathop {x_{1} }\limits^{..} +\,2 {\upzeta}_{1} {\upomega }_{1} \dot{x}_{1} + ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} )x_{1} = q_{1} - g_{1} {\Upomega } \dot{x}_{2} , \hfill \\ \mathop {x_{2} }\limits^{..} +\,2 {\upzeta}_{2} {\upomega }_{2} \dot{x}_{2} + ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} )x_{2} = q_{2} + g_{2} {\Upomega } \dot{x}_{1} . \hfill \\ \end{aligned} \right.$$
(3.6)

Unlike the set of Eq. (3.1) here we have a system of equations that are cross-coupled by gyroscopic terms with the angular rate. Most importantly it becomes a system of ordinary differential equations with constant coefficients.

If no external forces affect secondary motion (\(q_{2} = 0\)) and assuming ideal elastic suspension with no coupling, secondary motion will depend only on the angular rate \({\Upomega }\).

We are also interested in the forced solution of the system (3.6) since it is responsible for the angular rate measurement, while natural solution describes transient processes.

If excitation of the primary oscillation is harmonic, acceleration from the excitation forces could be represented in a complex form as
$$q_{1} \left( t \right) = \text{Re} \left\{ {q_{10} e^{{i{\upomega }t}} } \right\} .$$
(3.7)
Here ω is the excitation frequency and phase is assumed zero. Primary and secondary oscillations of the proof mass and the decoupling frame we shall search as a particular solution of the system (3.6) in the following form
$$\begin{aligned} x_{1} (t) = \text{Re} \left\{ {A_{1} e^{{i{\upomega }t}} } \right\}, \hfill \\ x_{2} (t) = \text{Re} \left\{ {A_{2} e^{{i{\upomega }t}} } \right\}. \hfill \\ \end{aligned}$$
(3.8)
We search here for the forced oscillations occurring at the excitation frequency ω, and parameters of these oscillations are described by the constant complex primary and secondary amplitudes \(A_{1}\) and \(A_{2}\) as
$$\begin{aligned} A_{1} = A_{10} e^{{i {\upvarphi}_{10} }} , \hfill \\ A_{2} = A_{20} e^{{i {\upvarphi}_{20} }} . \hfill \\ \end{aligned}$$
(3.9)
Here \(A_{10}\), \(A_{20}\), \({\upvarphi}_{10}\), and \({\upvarphi}_{20}\) are the constant real amplitudes and phases of the primary and secondary oscillations, respectively. Substituting suggested solutions (3.8) into Eq. (3.6) we obtain the following system of algebraic equations in terms of complex amplitudes instead of differential equations:
$$\left\{ {\begin{array}{*{20}l} {({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} + 2 {\upzeta}_{1} {\upomega }_{1} i{\upomega })A_{1} + g_{1} i{\upomega }{\Upomega } A_{2} = q_{10} ,} \hfill \\ {({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} + 2 {\upzeta}_{2} {\upomega }_{2} i{\upomega })A_{2} - g_{2} i{\upomega }{\Upomega } A_{1} = 0.} \hfill \\ \end{array} } \right.$$
(3.10)
Solutions of the system (3.10) with respect to complex amplitudes are
$$\begin{aligned} A_{1} = & \frac{{q_{10} ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} + 2 {\upzeta}_{ 2} {\upomega }_{2} i{\upomega })}}{\Updelta }, \\ A_{2} = &\, \frac{{g_{2} q_{10} i{\upomega }}}{\Updelta }{\Upomega } , \\ {\Updelta } = &\, ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} )({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} ) \\ & - (g_{1} g_{2} {\upomega }^{2} {\Upomega }^{2} + 4 {\upzeta}_{ 1} {\upzeta}_{ 2} {\upomega }_{1} {\upomega }_{2} {\upomega }^{2} ) \\ & + 2i{\upomega }\left[ { {\upzeta}_{1} {\upomega }_{1} ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} ) + {\upzeta}_{2} {\upomega }_{2} ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} )} \right] \\ \end{aligned}$$
(3.11)

Using of the complex amplitude method, which is modification of the method of averaging, results in possibility to analyse amplitudes and phases of the sensitive element motion instead of analysing in terms of its displacements \(x_{1}\) and \(x_{2}\). The former are of great interest from the angular rate sensing point of view.

Looking at (3.11) one can see that the amplitude of the secondary oscillations is almost linear function of the unknown angular rate. However, amplitudes (3.11) are the complex valued quantities. Conversion from the complex amplitudes to the real amplitudes and phases is performed as follows:
$$\begin{aligned} & A_{i0} = \left| {A_{i} } \right| = \sqrt {\text{Re}^{2} A_{i} + \text{Im}^{2} A_{i} } , \\ & \tan {\upvarphi }_{i0} = \frac{{\text{Im}\, A_{i} }}{{\text{Re}\,A_{i} }}. \\ \end{aligned}$$
(3.12)

Here subscript \(i = 1, 2\) for the primary and secondary amplitude and phase.

Applying transformations (3.12) to the complex amplitudes (3.11) results in
$$\begin{aligned} & A_{10} = \frac{{q_{10} \sqrt {({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} )^{2} + 4 {\upzeta}_{2}^{2} {\upomega }_{2}^{2} {\upomega }^{2} } }}{{\left| {\Updelta } \right|}}, \\ & A_{20} = \frac{{g_{2} q_{10} {\upomega }}}{{\left| {\Updelta } \right|}}{\Upomega } , \\ \end{aligned}$$
(3.13)
where
$$\begin{aligned} \left| {\Updelta } \right|^{2}\,= &\, \left[ {({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} )({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} )} \right. \\ &\, \left. { - {\upomega }^{2} (g_{1} g_{2} {\Upomega }^{2} + 4 {\upzeta}_{1} {\upzeta}_{2} {\upomega }_{1} {\upomega }_{2} )} \right]^{2} \\ &\,+ 4{\upomega }^{2} \left[ { {\upzeta}_{1} {\upomega }_{1} ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} ) + {\upzeta}_{2} {\upomega }_{2} ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} )} \right]^{2} . \\ \end{aligned}$$
The real phases of the primary and secondary oscillations are given by the following expressions:
$$\begin{aligned} &{\text{tg}}\left( {\phi_{1} } \right) = \frac{{2\upomega \left[ {(\upomega_{2}^{2} - d_{2} \Upomega^{2} - \upomega^{2} )b_{1} + \upomega_{2} \upzeta_{2} b_{2} } \right]}}{{(\upomega_{2}^{2} - d_{2} \Upomega^{2} - \upomega^{2} )b_{2} - 4\upomega_{2} \upzeta_{2} \upomega^{2} b_{1} }}, \hfill \\ &{\text{tg}}\left( {\phi_{2} } \right) = \frac{{(\upomega_{1}^{2} - d_{1} \varOmega^{2} - \upomega^{2} )(\upomega_{2}^{2} - d_{2} \Upomega^{2} - \upomega^{2} ) - \upomega^{2} (4\upzeta_{1} \upzeta_{2} \upomega_{1} \upomega_{2} + g_{1} g_{2} \Upomega^{2} )}}{{2\upomega \left[ {\upomega_{1} \upzeta_{1} (\upomega_{2}^{2} - d_{2} \Upomega^{2} - \upomega^{2} ) + \upomega_{2} \upzeta_{2} (\upomega_{1}^{2} - d_{1} \varOmega^{2} - \upomega^{2} )} \right]}}, \hfill \\ \end{aligned}$$
(3.14)
where
$$\begin{aligned} & b_{1} = {\upomega }_{1} {\upzeta}_{1} ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} ) + {\upomega }_{2} {\upzeta}_{2} ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} ), \\ & b_{2} = ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} - {\upomega }^{2} )({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} - {\upomega }^{2} ) - {\upomega }^{2} (4 {\upzeta}_{1} {\upzeta}_{2} {\upomega }_{1} {\upomega }_{2} + g_{1} g_{2} {\Upomega }^{ 2} ). \\ \end{aligned}$$

Using formulae (3.13) and (3.14) we can now study how CVG sensitive element responds to the external angular rate and how it could be measured with the highest possible efficiency.

Amplitude of secondary oscillations \(A_{20}\) (or secondary amplitude), as a function of the excitation frequency \({\upomega }\), is shown in Fig. 3.2.
Fig. 3.2

Secondary amplitude as a function of the excitation frequency (ω1 = 3000, ω2 = 3200, ζ1 = ζ2 = 0.01, Ω = 1)

It is apparent that maximum response to the constant angular rate is achieved when resonance with the lower natural frequency occurs. At the same time, the lower is damping, the higher are peaks of the secondary amplitude.

As it becomes apparent from (3.13) and (3.14), amplitude-related performances of a CVG are strongly related to such parameters of its sensitive element as natural frequencies and damping factors, as well as type related coefficients \(d_{i}\). In order to make analysis of the CVG sensitive element motion more intuitive, let us introduce the following new variables: \(k = {\upomega }_{1}\) is the natural frequency of the primary motion, δk = ω21 is the relative natural frequency of the secondary motion, \({\updelta } {\upomega } = {\upomega }/k\) is the relative excitation frequency, δΩ = Ω/k is the relative angular rate, which is apparently small in comparison to primary natural frequency (Ω ≪ k, δΩ ≪ 1). In terms of these new variables, expressions (3.13) for the primary and secondary amplitudes can be rewritten as
$$\begin{aligned} & A_{10} = \frac{{q_{10} k^{2} \sqrt {({\updelta } k^{2} - d_{2} {\updelta } {\Upomega }^{2} - {\updelta } {\upomega }^{2} )^{2} + 4 {\upzeta}_{2}^{2} {\updelta } k^{2} {\updelta } {\upomega }^{2} } }}{{\left| {\Updelta } \right|}}, \\ & A_{20} = \frac{{g_{2} q_{10} {\updelta } {\upomega }}}{{\left| {\Updelta } \right|}}{\updelta } {\Upomega } , \\ \end{aligned}$$
(3.15)
and squared denominator is
$$\begin{aligned} \left| {\Updelta } \right|^{2}\, = &\, k^{8} \left[ {(1 - d_{1} {\updelta } {\Upomega }^{2} - {\updelta } {\upomega }^{2} )({\updelta } k^{2} - d_{2} {\updelta } {\Upomega }^{2} - {\updelta } {\upomega }^{2} )} \right. \\ &\, \left. { - {\updelta } {\upomega }^{2} (g_{1} g_{2} {\updelta } {\Upomega }^{2} +\,4\upzeta_{1}\upzeta_{2} {\updelta } k)} \right]^{2} \\ & +\,4k^{8} {\updelta } {\upomega }^{2} \left[ {\upzeta_{1} ({\updelta } k^{2} - d_{2} {\updelta } {\Upomega }^{2} - {\updelta } {\upomega }^{2} ) + {\upzeta}_{2} {\updelta } k(1 - d_{1} {\updelta } {\Upomega }^{2} - {\updelta } {\upomega }^{2} )} \right]^{2} . \\ \end{aligned}$$
Expressions (3.15) can be simplified even further, if small relative angular rate is assumed, e.g. δΩ ≪ 1 and δΩ2 ≈ 0:
$$\begin{aligned} A_{10} \approx &\, \frac{{q_{10} }}{{k^{2} \sqrt {(1 - {\updelta } {\upomega }^{2} )^{2} + 4 {\upzeta}_{1}^{2} {\updelta } {\upomega }^{2} } }}, \\ A_{20} \approx &\, \frac{{g_{2} q_{10} {\updelta } {\upomega }}}{{k^{2} \sqrt {({\updelta } k^{2} - {\updelta } {\upomega }^{2} )^{2} + 4\upzeta_{2}^{2} {\updelta } k^{2} {\updelta } {\upomega }^{2} } \sqrt {(1 - {\updelta } {\upomega }^{2} )^{2} + 4 {\upzeta}_{1}^{2} {\updelta } {\upomega }^{2} } }}{\updelta } {\Upomega } \\ \approx &\, \frac{{g_{2} A_{10} {\updelta } {\upomega } }}{{\sqrt {({\updelta } k^{2} - {\updelta } {\upomega }^{2} )^{2} + 4 {\upzeta}_{2}^{2} {\updelta } k^{2} {\updelta } {\upomega }^{2} } }}{\updelta } {\Upomega } . \\ \end{aligned}$$
(3.16)
Expressions for the phases (3.14) are also can be rewritten in terms of relative sensitive element parameters. In case of small relative angular rate they become
$$\begin{aligned} & {\text{tg}}\left( {\upvarphi }_{1} \right) = \frac{{2{\updelta } {\upomega } {\upzeta}_{1} }}{{1 - {\updelta } {\upomega }}}, \\ & {\text{tg}}\left( {\upvarphi }_{2} \right) = \frac{{{\updelta } k^{2} - (1 + 4 {\upzeta}_{1} {\upzeta}_{2} {\updelta } k + {\updelta } k^{2} ){\updelta } {\upomega }^{2} + {\updelta } {\upomega }^{4} }}{{2{\updelta } k{\updelta } {\upomega }( {\upzeta}_{2} + {\upzeta}_{1} {\updelta } k) - 2{\updelta } {\upomega }^{3} ( {\upzeta}_{1} + {\upzeta}_{2} {\updelta } k)}}. \\ \end{aligned}$$
(3.17)

Expressions (3.16) and (3.17) allow to analyse motion of the CVG sensitive element in the presence of external rotation using dimensionless parameters, which gives a certain level of generalisation of obtained results.

Another important aspect of motion equation analysis is the characteristic equation that gives eigenfrequencies of the sensitive element. In order to obtain characteristic equation, the differential operator \(s = d/dt\) is used to transform the motion Eq. (3.6) to the following form:
$$\left\{ \begin{aligned} s^{2} x_{1} + 2 {\upzeta}_{1} {\upomega }_{1} sx_{1} + ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} )x_{1} = q_{1} - g_{1} {\Upomega } sx_{2} , \hfill \\ s^{2} x_{2} + 2 {\upzeta}_{2} {\upomega }_{2} sx_{2} + ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} )x_{2} = q_{2} + g_{2} {\Upomega } sx_{1} . \hfill \\ \end{aligned} \right.$$
(3.18)
This transformed equation can be then rewritten in a matrix form as
$$A \cdot \left[ {\begin{array}{*{20}c} {x_{1} } \\ {x_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {q_{1} } \\ {q_{2} } \\ \end{array} } \right].$$
(3.19)
where A is the principal system matrix given defined by the Eq. (3.18) terms as
$$A = \left[ {\begin{array}{*{20}c} {s^{2} + 2 {\upzeta}_{1} {\upomega }_{1} s + {\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} } & {g_{1} {\Upomega } s} \\ { - g_{2} {\Upomega } s} & {s^{2} + 2 {\upzeta}_{2} {\upomega }_{2} s + {\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} } \\ \end{array} } \right] .$$
(3.20)
Characteristic equation is defined as
$$\det A = 0 .$$
(3.21)
Substituting expressions (3.20) into Eq. (3.21) yields the characteristic equation as
$$s^{4} + a_{3} s^{3} + a_{2} s^{2} + a_{1} s + a_{0} = 0.$$
(3.22)
where coefficients \(a_{j}\) are
$$\begin{aligned} & a_{3} = 2s^{3} ( {\upzeta}_{1} {\upomega }_{1} + {\upzeta}_{2} {\upomega }_{2} ), \\ & a_{2} = s^{2} ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} + {\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} + 4 {\upzeta}_{1} {\upomega }_{1} {\upzeta}_{2} {\upomega }_{2} + g_{1} g_{2} {\Upomega }^{2} ), \\ & a_{1} = 2s[ {\upzeta}_{1} {\upomega }_{1} ({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} ) + {\upzeta}_{2} {\upomega }_{2} ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} )], \\ & a_{0} = ({\upomega }_{1}^{2} - d_{1} {\Upomega }^{2} )({\upomega }_{2}^{2} - d_{2} {\Upomega }^{2} ). \\ \end{aligned}$$
Characteristic Eq. (3.22) allows analysis of stability of the sensitive element oscillations by using Routh criteria as
$$\begin{aligned} & a_{1} a_{2} a_{3} - a_{1}^{2} - a_{3}^{2} a_{0} > 0, \\ & a_{j} > 0. \\ \end{aligned}$$
(3.23)
Coefficients of the characteristic Eq. (3.22) are functions of the angular rate, damping factors and natural frequencies of the system, and only angular rate is unknown. All other parameters are subjects to design in accordance to the stability conditions (3.23). Analysing coefficients of (3.22) one can find that stable secondary oscillations occur when the external angular rate is less than natural frequency of primary oscillations:
$$- {\upomega }_{1}\,<\,{\Upomega }\,<\,{\upomega }_{1} .$$
(3.24)

Relationships (3.24) usually are satisfied, since external angular rate is much less than the natural frequency of primary oscillations.

Closed form solution of the complete fourth-order Eq. (3.22) is quite complicated and therefore useless for subsequent analysis. However, if damping is assumed to be negligibly small (\({\upzeta}_{1} = {\upzeta}_{2} = 0\)), then all odd order terms will disappear and characteristic Eq. (3.22) becomes bi-quadratic, which is being rewritten using dimensionless variable introduced earlier, is as follows:
$$\begin{aligned} & s^{4} + s^{2} k^{2} (1 - d_{1} {\updelta } {\Upomega }^{2} + {\updelta } k^{2} - d_{2} {\updelta } {\Upomega }^{2} + g_{1} g_{2} {\updelta } {\Upomega }^{2} ) \\ & +\,k^{4} (1 - d_{1} {\updelta } {\Upomega }^{2} )({\updelta } k^{2} - d_{2} {\updelta } {\Upomega }^{2} ) = 0. \\ \end{aligned}$$
(3.25)
Roots of Eq. (3.25) are relating to the relative eigenfrequencies \({\updelta } {\upomega }_{j0}\) of the primary and secondary oscillations as
$$\begin{aligned} & s_{1,2} = \pm ik{\updelta } {\upomega }_{10} , \\ & s_{3,4} = \pm ik{\updelta } {\upomega }_{20} . \\ \end{aligned}$$
(3.26)
Solving Eq. (3.25) results in the following expressions for the eigenfrequencies:
$$\begin{aligned} {\updelta } {\Omega }_{j0}^{2} = & \frac{1}{2}\left[ {1 + {\updelta } k^{2} - (d_{1} + d_{2} - g_{1} g_{2} ){\updelta } {\Omega }^{2} } \right] \\ & + \frac{{\left( { - 1} \right)^{j} }}{2}\left\{ {4({\updelta } k^{2} - d_{1} {\updelta } {\Omega }^{2} )(d_{2} {\updelta } {\Omega }^{2} - 1)} \right. \\ & + \left. {\left[ {1 + {\updelta } k^{2} - (d_{1} + d_{2} - g_{1} g_{2} ){\updelta } {\Omega }^{2} } \right]^{2} } \right\}^{1/2} \\ \end{aligned}$$
(3.27)
Graphic plot of the frequencies (3.27) as a functions of the external angular rate is shown in Fig. 3.3, where relative natural frequency is \({\updelta } k = 1.05\).
Fig. 3.3

Relative eigenfrequencies (dashed—primary, solid—secondary)

When there is no external rotation, eigenfrequencies become equal to corresponding natural frequencies. When external rotation is applied, these frequencies start to shift due to the angular rate. Although this dependence does not appear to be linear, it becomes very close to linear when natural frequencies are equal (\({\updelta } k = 1\)). Such a frequency modulation could be considered as complementary to amplitude modulation, demonstrated by (3.15), and used to improve angular rate measurements.

3.3 Modelling Proof Mass Motion Trajectory

It is possible to view the CVG sensitive element as a two-dimensional pendulum. In presence of the constant external angular rate, trajectory of its centre of gravity forms an ellipse, as shown in Fig. 3.4.
Fig. 3.4

Sensitive element motion trajectory

In this figure, a and b are the big and small half-axes of the ellipse, θ is the angle of the ellipse rotation relatively to the axes of primary \(x_{1}\) and secondary \(x_{2}\) oscillations. It is well known that these parameters (namely half-axes and angle of rotation) depend on amplitudes and phases of primary and secondary oscillations, which in turn depend on parameters of the sensitive element design and unknown angular rate.

The problem, which is to be addressed in this section, is to develop and analyse a mathematical model of the ellipse parameters as a function of the sensitive element design and its parameters.

Without loss of generality, primary and secondary coordinates of the CVG sensitive element in its steady motion can be represented as
$$\begin{aligned} & x_{1} (t) = A_{10} \cos {\upomega }t, \\ & x_{2} (t) = A_{20} \cos ({\upomega }t + \upvarphi ) = A_{20} [\cos {\upomega }t\cos \upvarphi - \sin {\upomega }t\sin \upvarphi ], \\ \end{aligned}$$
(3.28)
where \(A_{10}\) and \(A_{20}\) are the primary and secondary amplitudes, φ is the phase shift between the primary and secondary oscillations, ω is the oscillations circular frequency.
First of all, we have to exclude terms containing oscillation phase ωt from Eq. (3.28). In order to achieve this, we express sine and cosine of the ωt from the first equation and substitute them into the second one, yielding
$$\frac{{x_{1} }}{{A_{10} }}\cos \upvarphi - \frac{{x_{2} }}{{A_{20} }} = \pm \sqrt {1 - \frac{{x_{1}^{2} }}{{A_{10}^{2} }}} \sin \upvarphi .$$
(3.29)
Squaring both sides of the Eq. (3.29) results in
$$\frac{{x_{1}^{2} }}{{A_{10}^{2} }} + \frac{{x_{2}^{2} }}{{A_{20}^{2} }} - \frac{{2x_{1} x_{2} \cos \upvarphi }}{{A_{10} A_{20} }} = \sin \upvarphi^{2} .$$
(3.30)
Assuming \(X_{1}\) and \(X_{2}\) as the coordinates of the sensitive element centre of gravity in the coordinate system that is rotated by the angle θ, it is easy relating them to the original coordinates as
$$\begin{aligned} & x_{1} = X_{1} \cos \uptheta - X_{2} \sin \uptheta , \\ & x_{2} = X_{1} \sin \uptheta + X_{2} \cos \uptheta . \\ \end{aligned}$$
(3.31)
In terms of these coordinates ellipse equation has its conventional form
$$\frac{{X_{1}^{2} }}{{a^{2} }}\,+\,\frac{{X_{2}^{2} }}{{b^{2} }} = 1.$$
Substituting (3.31) into (3.30) results in
$$\begin{aligned} & X_{1}^{2} \frac{{A_{20}^{2} \cos^{2} \uptheta - A_{10} A_{20} \cos {\upvarphi} \sin 2\uptheta + A_{10}^{2} \sin^{2} \uptheta }}{{A_{10}^{2} A_{20}^{2} }} \\ & +\,X_{1} X_{2} \frac{{(A_{10}^{2} - A_{20}^{2} )\sin 2\uptheta - 2A_{10} A_{20} \cos {\upvarphi} \cos 2\uptheta }}{{A_{10}^{2} A_{20}^{2} }} \\ & +\,X_{2}^{2} \frac{{A_{10}^{2} \cos^{2} \uptheta + A_{10} A_{20} \cos {\upvarphi} \sin 2\uptheta + A_{20}^{2} \sin^{2} \uptheta }}{{A_{10}^{2} A_{20}^{2} }} = \sin^{2} {\upvarphi} . \\ \end{aligned}$$
(3.32)
Apparently, in order to transform Eq. (3.32) to the conventional form, the second term must disappear. This will occur if θ satisfies the following condition
$$\uptheta = \frac{1}{2}\tan^{ - 1} \frac{{2A_{10} A_{20} \cos {\upvarphi} }}{{A_{10}^{2} - A_{20}^{2} }} .$$
(3.33)
If angle θ is defined be (3.33), then half-axes of the ellipse will be given by the following expressions:
$$\begin{aligned} & a = \frac{{A_{10} A_{20} \sin {\upvarphi} }}{{\sqrt {A_{20}^{2} \cos^{2} \uptheta - A_{10} A_{20} \cos {\upvarphi} \sin 2\uptheta + A_{10}^{2} \sin^{2} \uptheta } }}, \\ & b = \frac{{A_{10} A_{20} \sin {\upvarphi} }}{{\sqrt {A_{10}^{2} \cos^{2} \uptheta + A_{10} A_{20} \cos {\upvarphi} \sin 2\uptheta + A_{20}^{2} \sin^{2} \uptheta } }}. \\ \end{aligned}$$
(3.34)

Now we have to express half-axes a and b of the ellipse, and its angle of rotation θ in terms of the sensitive element parameters and external angular rate.

In perfectly tuned CVG, where primary and secondary eigenfrequencies are perfectly matched, phase shift is zero (\({\upvarphi} = 0\)). This allows us to approximate expressions (3.33) and (3.34) with the linear terms of its Taylor series representation around zero phase shift:
$$\begin{aligned} a & \approx \frac{{A_{10} A_{20} {\upvarphi} }}{{A_{20} \cos \uptheta - A_{10} \sin \uptheta }}, \\ b & \approx \frac{{A_{10} A_{20} {\upvarphi} }}{{A_{10} \cos \uptheta + A_{20} \sin \uptheta }}. \\ \end{aligned}$$
(3.35)
One should note that in approximations (3.35) zero phase shift results in zero big half-axis a, which apparently is not acceptable in trajectory analysis applications. Let us have a close look at the dependencies (3.34) for the elliptical trajectory of the CVG sensitive element. Obviously, expression for the big half-axis has a peculiarity in vicinity of the zero phase shift, when the denominator of the formula may become zero. However, far from the zero phase, this function is quite smooth and has no such peculiarities. From the extensive analysis of this dependency the following simple approximation is suggested:
$$a \approx A_{10} + \frac{{A_{20}^{2} }}{4}(1 + \cos 2 {\upvarphi} ) .$$
(3.36)
Dimensionless relative error of the approximation (3.36) is shown in Fig. 3.5.
Fig. 3.5

Relative error of the big half-axis approximation

Observable singularities along the zero phase shift in Fig. 3.5 demonstrate that approximation is more relevant than the initial formula. It is easy to see that approximation (3.36) is certainly accurate enough for small phase shifts and small relative secondary amplitudes, which are typical for the CVG sensitive element motions.

Let us now analyse how elliptical trajectory parameters depend on the external angular rate and sensitive element characteristics.

One should note that the sensitive element trajectory parameters depend on the phase shift \({\upvarphi} = {\upvarphi}_{2} - {\upvarphi}_{1}\) between primary and secondary phases. Most importantly, based upon (3.17), phases do not depend on the angular rate.

In case of primary resonance (\({\updelta } {\upomega } = 1\)), sine and cosine of this phase shift can be calculated as
$$\begin{aligned} \sin {\upvarphi} = & \frac{{1 - {\updelta } k^{2} }}{R}, \\ \cos {\upvarphi} = & \frac{{2 {\upzeta}_{2} {\updelta } k}}{R}, \\ R = & \sqrt {{\updelta } k^{4} - 2(1-2 {\upzeta}_{2}^{2} ){\updelta } k^{2} + 1} . \\ \end{aligned}$$
(3.37)

Expressions (3.16) and (3.17) along with the phase shift representations (3.37) can now be used to analyse parameters of the actual trajectory of the CVG sensitive element.

Let us now substitute expressions (3.16) and (3.37) into the formula for the big half-axis approximation (3.36):
$$a = \frac{{q_{10} (4k^{2} R^{2} {\upzeta}_{1} + g_{2}^{2} q_{10} {\updelta } {\Upomega }^{2} )}}{{8k^{4} R^{2} {\upzeta}_{1}^{2} }}.$$
Remembering that the relative angular rate is small (\({\updelta } {\Upomega }^{2} \approx 0)\), big half-axis, as expected, becomes the primary amplitude:
$$a \approx \frac{{q_{10} }}{{2k^{2} {\upzeta}_{1} }} = \left. {A_{10} } \right|_{{{\updelta } {\Upomega } = 1}} .$$
(3.38)
Small half-axis of the sensitive element trajectory after substitution is
$$b = \frac{{q_{10} g_{2} (1 - {\updelta } k^{2} )}}{{\sqrt 2 k^{2} R {\upzeta}_{1} \sqrt {R^{2} + g_{2}^{2} {\updelta } {\Upomega }^{2} + \sqrt {R^{4} - 2g_{2}^{2} (R^{2} - 8 {\upzeta}_{2}^{2} {\updelta } k^{2} ){\updelta } {\Upomega }^{2} + g_{2}^{4} {\updelta } {\Upomega }^{4} } } }}{\updelta } {\Upomega } ,$$
and dropping higher powers of the angular rate results in
$$b = \frac{{q_{10} g_{2} (1 - {\updelta } k^{2} )}}{{2k^{2} {\upzeta}_{1} [1 - 2(1 - 2 {\upzeta}_{2}^{2} ){\updelta } k^{2} + {\updelta } k^{4} ]}}{\updelta } {\Upomega } .$$
(3.39)

Analysing formula (3.39) is apparent that small half-axis is absent in either of two cases: perfect match of the natural frequencies (\({\updelta } k = 1\)) or absence of the external angular rate (\({\updelta } {\Upomega } = 0\)).

Finally, the last but not the least, angle of the trajectory rotation θ can be calculated using the following expression:
$$\uptheta = \frac{1}{2}\arctan \left[ {\frac{{4g_{2} {\upzeta}_{2} {\updelta } k{\updelta } {\Upomega } }}{{1 - 2(1 - 2 {\upzeta}_{2}^{2} ){\updelta } k^{2} + {\updelta } k^{4} }}} \right].$$
(3.40)
Similarly to the previous case, we can linearly approximate formula (3.40) for small angular rates:
$$\uptheta \approx \frac{{2g_{2} {\upzeta}_{2} {\updelta } k}}{{1-2(1-2 {\upzeta}_{2}^{2} ){\updelta } k^{2} + {\updelta } k^{4} }}{\updelta } {\Upomega } .$$
(3.41)
Or, in case of perfectly matched primary and secondary natural frequencies (\({\updelta } k = 1\)), (3.41) becomes
$$\uptheta \approx \frac{{g_{2} }}{{2 {\upzeta}_{2} }}{\updelta } {\Upomega } .$$
(3.42)
Both accurate expression (3.40) and its linear approximation (3.42) are shown in Fig. 3.6.
Fig. 3.6

Trajectory rotation angle as a function of angular rate (solid—accurate, dashed—linear approximation)

From the graphs in Fig. 3.6 one can see that trajectory rotation angle is almost linear function of the unknown angular rate, which allows to use it to measure angular rate.

Figure 3.7 demonstrates how trajectory approximations (3.38), (3.39) and (3.42) related to the realistically simulated trajectory, based on the numerical solution of the original motion equations.
Fig. 3.7

CVG Trajectory simulation (solid—simulated, dashed—theoretical)

Here the solid line demonstrates steady motion trajectory of the simulated CVG sensitive element, and the dashed line corresponds to the trajectory, generated from the obtained trajectory parameters.

Comparison with the results of numerical simulation demonstrates high accuracy of the obtained mathematical model of the sensitive element motion trajectory. These dependencies allow not only further analysis of the sensitive element motion, but efficient synthesis of many different control loops improving overall CVG performance as well.

3.4 Numerical Simulation of CVG Dynamics Using Simulink®

Accurate numerical simulation of CVG is essential for proper verification of the developed mathematical models. Generalised Simulink model for CVG simulation in an open-loop operation mode is presented in Fig. 3.8.
Fig. 3.8

CVG simulation model

Here block “Angular Rate” provides angular rate as an input to the system, block “Excitation” provides sinusoidal signal to the primary mode input, “Process Noise” can be added to the secondary mode input, and the “Sensor Noise” can be added to the CVG output. Secondary coordinate \(x_{2}\) must be demodulated by the “Secondary Detector” to remove primary carrier signal.

To make the simulation results as realistic as possible, the following most generalised sensitive element motion Eq. ( 2.29) simplified for small angular rates (\({\Upomega }^{2} \approx 0\)) will be used:
$$\left\{ \begin{aligned} \mathop {x_{1} }\limits^{..} + 2 {\upzeta}_{1} {\upomega }_{1} \dot{x}_{1} + {\upomega }_{1}^{2} x_{1} = q_{1} - g_{1} {\Upomega } \dot{x}_{2} - d_{3} \dot{\Upomega }x_{2} , \hfill \\ \mathop {x_{2} }\limits^{..} + 2 {\upzeta}_{2} {\upomega }_{2} \dot{x}_{2} + {\upomega }_{2}^{2} x_{2} = q_{2} + g_{2} {\Upomega } \dot{x}_{1} + d_{4} \dot{\Upomega }x_{1} . \hfill \\ \end{aligned} \right.$$
(3.43)
Corresponding simulation model (contents of the “Sensitive Element Dynamics” subsystem block in Fig. 3.8) is shown in Fig. 3.9.
Fig. 3.9

Sensitive element dynamics model

Primary and secondary dynamics in this model is simulated using transfer function block “Transfer Fcn” from Simulink. Parameters of the sensitive element are replaced with the following variables: \({\text{k1}} = {\upomega }_{1}\), \({\text{k2}} = {\upomega }_{2}\), \({\text{h1}} = {\upzeta}_{1}\), \({\text{h2}} = {\upzeta}_{2}\), etc.

Demodulation of the secondary output is performed by the synchronous demodulator (“Secondary Detector” block in Fig. 3.8) as shown in Fig. 3.10.
Fig. 3.10

Secondary detector model

Secondary output is multiplied by the sinusoidal signal at the excitation frequency and the result is passed through the low-pass eighth-order Butterworth filter. Filter output is multiplied with the factor, which scales it to the rate-like output.

Resume

As demonstrated in this chapter, solutions of the CVG motion equations not only give useful insights into how the sensitive element reacts to the external rotation, but also allow us to calculate and optimise the main performances of these types of gyroscopes.

Attentive readers may notice that we did not actually solve the original motion equations, but obtained steady state solutions for amplitudes, phase, and eigenfrequencies related to the external angular rate. This was done due to the fact that closed formed solutions of the original equations in terms of primary and secondary motions, which are oscillatory, are complicated and therefore useless for further analysis. The good news is we do not need them, since all the important dependencies from angular rate express themselves in amplitudes and phases, rather than actual sensitive element motions. However, steady state solutions, related to constant angular rate, do not allow analysis of the sensitive element behaviour when angular rate varies in time. And this problem is yet to be solved in the next chapter.

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Aircraft Control Systems and InstrumentsNational Technical University of UkraineKievUkraine

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