Mass Imperfections Research Articles

Separation of mass and gap errors of hemispherical resonator gyroscopes by varying bias voltage

For hemispherical resonator gyroscopes (HRGs), mass imperfections and capacitive gap nonuniformity jointly influence the frequency mismatch and degrade the gyroscope performance. However, the coupling mechanism and separation of these two errors are rarely discussed. This paper proposes, for the first time, a novel method of separating mass imperfections and gap nonuniformity of HRGs, based on a dynamics model considering both the two errors. By measuring the frequency mismatches and principal axes of stiffness at different bias voltages, the mass imperfections and gap nonuniformity can be separately obtained. The method is investigated in detail by simulation, and validated by experiment on three different HRGs. The experimental results agree well with the theoretical analysis, demonstrating the effectiveness of the model and the separation method. Our method may contribute to high-precision HRG manufacture, and can be conveniently applied to other Coriolis vibrating gyroscopes (CVGs) with electrostatic actuation.

Dynamical behaviour of CNT reinforced plates with mass imperfections

Carbon nanotubes (CNTs) from their discovery have received great attention due to their remarkable structural, electrical, thermal and mechanical properties. Many notable applications of CNTs were explored such as transistors, sensors, membranes, energy storage, drug delivery etc. Further to that, incorporating the CNTs properties into existing materials forms a new form of material- CNTs reinforced structures. The understanding of mechanical behaviour of these structures has a pivotal role in ensuring the system functions appropriately. Several attempts have been made to help further the understanding of dynamics of CNT reinforced structures, including beams, shells, and plates. Mass imperfections are a common manufacturing error and should be taken into account when modelling CNT reinforced structures. Thus, to contribute to the research of the dynamics of CNT reinforced plates, the specific objective of this study is to explore the vibrational behaviour of a CNTs reinforced plate with mass imperfection. The plate is first modelled by using the Kirchhoff theory, and the potential and kinetic energy, incorporating the mass imperfections effects are formulated and inserted in the Hamilton principle leading to the equations of motions. The equations are then solved using the modal decomposition method to obtain the natural frequencies of the system. The effects of changing the plate dimensions, CNT volume fractions, magnitude and location of the point mass are explored in details in this study.

Porosity, mass and geometric imperfection sensitivity in coupled vibration characteristics of CNT-strengthened beams with different boundary conditions

Structures face different types of imperfections and defects during the fabrication process, installation and working environment. In this paper, the imperfection effects in the coupled vibration behaviour of axially functionally graded carbon nanotube (CNT)-strengthened beam structures with different boundary conditions are analysed considering porosity as well as geometric and mass imperfections in the structure. Porosity is modelled using different types of formulations for simple-cell, open-cell and closed-cell porous structures. The porosity is assumed to be either uniform or by varying through the thickness of the hollow beam using different functions. Mass imperfection effect is added to the system by considering a concentrated mass in the system affecting the mass homogeneity of the structure. Geometry imperfection is also considered by having an initial deformation in the structure which could be caused by an improper fabrication process. Coupled axial and transverse equations of motion are obtained using Hamilton’s principle and the von Karman geometrical nonlinearity. Governing equations are solved for different types of boundary conditions using a semi-analytical modal decomposition technique. It is shown that strengthening the base matrix with CNT fibres can improve the vibration behaviour of imperfect structures and the influence of CNT volume fraction and distribution through the length of the beam is discussed. The results provided in this paper may be used as a benchmark to validate future experimental results to prevent imperfection, delamination and stress singularities in the structures.

Mass imperfections in a toroidal micro-ring model with thermoelastic damping

• Thermoelastic dissipation is discussed for a toroidal micro ring with mass distributions. • A three-dimensional heat conduction equation is used for more accurate estimation. • The temperature profile is obtained in terms of Bessel functions in the cylindrical coordinate system . • The modified quality factor is estimated by use of the eigenfrequency of an imperfect ring. • Deviations of the temperature profile due to the imperfections are graphically observed as a function of the radial component . This work estimates the difference in quality factors (Q-factors) resulting from the effect of thermoelastic damping on toroidal solid micro rings with point mass distributions. The imperfections are irregularities in the angular locations with concentrated masses. On the basis of the modeling, eigenfrequencies are obtained as the in-plane vibration of the ring with an inextensional assumption. To evaluate the thermal expansion in the strains of the structure, a three-dimensional heat conduction equation is used for more accurate estimation. The temperature profile is obtained in terms of Bessel functions in the cylindrical coordinate system. Finally, the modified Q-factors are evaluated as the the ratio of the maximum strain energy of the model to the energy loss due to the thermoelastic damping. To check the validity of the study, the results obtained are compared with results in the literature for a uniform micro-ring model. The Q-factors for an imperfect toroidal micro ring are discussed on the basis of the sizes of the imperfections and the eigenmodes of the ring. Also, deviations of the temperature profile due to the imperfections are graphically observed as a function of the radial component.

Splitting of quality factors for micro-ring with arbitrary point masses

A ring resonator model with mass imperfections can be represented by a circular ring with irregularly distributed point masses. In this paper, in-plane vibration of the micro-ring is to be investigated. Also, the ThermoElastic Damping (TED) effect is considered in the structures. In order to get the temperature profile for the thermal flow, one-dimensional heat conduction equation is employed. Using the assumed mode shapes for the inextensional vibration of the imperfect micro-ring with TED, analytical expressions for the natural frequencies with concentrated masses are determined by the equations of motion. Then, the split of the natural frequencies are determined according to the effect of distributed point masses. Finally, imperfect micro-ring with TED is presented with the variations of Quality-factors (Q-factors) for lower and higher modes. Based on the verification of the present work, more parametric studies are performed for the ring with mass imperfections.

Using Fourier series to analyse mass imperfections in vibratory gyroscopes

When a vibrating structure is subjected to a rotation, the vibrating pattern rotates at a rate (called the precession rate) proportional to the inertial angular rate. This is known as Bryanʼs effect and it is employed to calibrate the resonator gyroscopes that are used to navigate in outer space, the stratosphere and under the polar cap. We study Bryanʼs effect for a non-ideal resonator gyroscope, using the computer algebra system (CAS) Mathematica to do the analysis involved, rendering this work accessible to undergraduate students with a working knowledge of college calculus and basic physics or mechanics (such as senior Engineering Mathematics students). In this paper the density of a slowly rotating vibrating annular disc is assumed to have small variations circumferentially, enabling a Fourier series representation of the density function. Using a CAS, the Lagrangian of the system of vibrating particles in the disc is calculated and, employing the CAS on the Euler–Lagrange equations, the equations of motion of the vibrating, rotating system are calculated in terms of “fast” variables, enabling us to demonstrate that the mass anisotropy induces a frequency splitting (beats). Unfortunately the fast variables are difficult to analyse (even with the aid of a CAS) and consequently a transformation from fast to slow variables is achieved. These slow variables are the principal and quadrature vibration amplitudes, precession rate and a phase angle. The transformation yields a system of four nonlinear ordinary differential equations (ODEs). This system of ODE demonstrates that the Fourier coefficients of the density function influence the precession rate and consequently a gyroscope manufactured from such a disc cannot use Bryanʼs effect for calibration purposes. Indeed, the CAS visualises that a capture effect occurs with the precession angle that appears to vary periodically and not increase linearly (Bryanʼs effect) as it would for a perfect structure. Keeping in mind that manufacturing imperfections will always be present in the real-world, the analysis shows how such density variations may be minimised. Using a symbolic manipulator such as Mathematica to do the “book-keeping” eliminates the plethora of technical detail that arises during calculations of a highly technical nature. This allows the aforementioned students to focus on the salient parts of the analysis, producing results that might have been beyond their capabilities without the aid of a CAS.

Dual-feed rotor spinning of cotton fiber: trash separation and yarn properties

In all industrial rotor spinning machines, opening of the fibers is performed by an opening roller, in which sliver is fed from a single point. Increasing number of feed rollers from one to two may improve fiber opening and trash ejection by two-step loading in opening zone of the opening roller. This may improve fiber orientation and blending in the produced yarn and yarn properties. In earlier two sliver feed laboratory systems (dual-feed), it was not possible to work with cotton fiber, because the trash removal zone was used for feeding a second sliver. The aim of this study was to design a system that can take advantage of dual-feed and also extract the trash. The experimental rig was a modified RU04 rotor spinning unit of Rieter in which two separate sliver feed systems were utilized. Raw material used and yarn count were cotton and 29 tex, respectively. Extracted trash and yarn properties produced with dual-feed system were compared with that of the original unit. Yarn properties tested were tenacity, extension, work of rupture, mass irregularity and imperfections, hairiness, and abrasion resistance. Test results were analyzed by ANOVA for any difference between the means and Duncan for ranking. Tenacity, strain at peak, work of rupture, and yarn abrasion of the dual-feed yarn increased, and their mass irregularity and imperfections decreased in comparison to that of the conventional yarn. According to the test results, it was concluded that, increasing number of feed rollers on the opening roller from one to two has improved the yarn properties and trash separation of cotton fiber is possible and comparable to the original single feed.

The influence of mass imperfections on the evolution of standing waves in slowly rotating spherical bodies

Standing waves can exist as stable vibrating patterns in perfect structures such as spherical bodies, and inertial rotation of the body causes precession (Bryan's effect). However, an imperfection such as light mass anisotropy destroys the standing waves. In this paper, an imperfection is introduced in the form of light mass anisotropy for a vibrating, slowly rotating spherical body. Assuming this light mass imperfection throughout this paper, the effects of slow rotation and light isotropic viscous damping are considered in a system of variables consisting of the amplitudes of principal and quadrature vibrating patterns, the angle of the rotation of the vibrating pattern (called the precession angle) and the phase shift of the vibrating pattern. We demonstrate how a combination of both qualitative and quantitative analysis (using, inter alia, the method of averaging) predicts that the inertial angular rate does not influence changes with time in the amplitudes of the principal and quadrature vibrations or the phase shift. The light mass imperfection causes changes with time which appear to be of a damped oscillatory nature for both the quadrature component as well as the principal component. The precession angular rate appears to depend on the inertial angular rate as well as the quadrature component of the vibration but is not influenced by the damping factor. It is not directly proportional to the inertial angular rate as is the case for a perfect isotropically damped structure. If the quadrature component is not suppressed, then a “capture effect” appears to occur, namely that the precession angle will not grow at a constant rate but is “captured” and shows periodic behaviour. It is evident that the damping factor does not influence changes with time in the phase shift and that the mass imperfection substantially influences these changes. The phase shift appears to be negative, strictly decreasing and unbounded.

On the statistics of natural frequency splitting for rings with random mass imperfections

This paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection. The analysis used to derive analytical expressions for the natural frequency splits is based on a Rayleigh–Ritz approach, in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring. The types of random mass imperfection investigated are: (i) random harmonic variations in the mass per unit length around the circumference of the ring; (ii) the attachment of random point masses at random locations on the ring; and (iii) the attachment of random point masses at uniformly spaced positions on the ring. For case (i) it is found that the frequency splits always have a Rayleigh distribution. For case (ii) an expression for the statistical distribution is deduced which tends to a Rayleigh distribution as the number of attached masses increases. For case (iii) it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses, and that in some situations the frequency splits have a “half Gaussian” (i.e., non-Rayleigh) distribution.

A Coupled Electromechanical Model of an Imperfect Piezoelectric Vibrating Cylinder Gyroscope

Coupled electromechanical equations of motion, describing the dynamics of a vibrating cylinder gyroscope, are derived using Hamilton's principle and the Rayleigh-Ritz method. The vibrating cylinder gyroscope comprises a thin walled steel cylinder which is closed at one end with discrete piezoceramic actuation and sensing elements bonded close to the open end. The operation of the gyroscope and the effect of imperfections are briefly described. The model allows direct comparison with experimental measurements in the form of electrical frequency response functions. The effects of ceramic location errors and mass imperfections were investigated. Comparisons with experimental measurements showed that the model could be used to predict the mass modifications required to reduce the effect of these typical imperfections in practical devices.