A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts

The aim of this research was to design a physically consistent model for the forced torsional vibrations of automotive driveshafts that considered aspects of the following phenomena: excitation due to the transmission of the combustion engine through the gearbox, excitation due to the road geometry, the quasi-isometry of the automotive driveshaft, the effect of nonuniformity of the inertial moment with respect to the longitudinal axis of the tulip–tripod joint and of the bowl–balls–inner race joint, the torsional rigidity, and the torsional damping of each joint. To resolve the equations of motion describing the forced torsional nonlinear parametric vibrations of automotive driveshafts, a variational approach that involves Hamilton’s principle was used, which considers the isometric nonuniformity, where it is known that the joints of automotive driveshafts are quasi-isometric in terms of the twist angle, even if, in general, they are considered CVJs (constant velocity joints). This effect realizes the link between the terms for the torsional vibrations between the elements of the driveshaft: tripode–tulip, midshaft, and bowl–balls–inner race joint elements. The induced torsional loads (as gearbox torsional moments that enter the driveshaft through the tulip axis) can be of harmonic type, while the reactive torsional loads (as reactive torsional moments that enter the driveshaft through the bowl axis) are impulsive. These effects induce the resulting nonlinear dynamic behavior. Also considered was the effect of nonuniformity on the axial moment of inertia of the tripod–tulip element as well as on the axial moment of inertia of the bowl–balls–inner race joint element, that vary with the twist angle of each element. This effect induces parametric dynamic behavior. Moreover, the torsional rigidity was taken into consideration, as was the torsional damping for each joint of the driveshaft: tripod–joint and bowl–balls–inner race joint. This approach was used to obtain a system of equations of nonlinear partial derivatives that describes the torsional vibrations of the driveshaft as nonlinear parametric dynamic behavior. This model was used to compute variation in the natural frequencies of torsion in the global tulip (a given imposed geometry) using the angle between the tulip–midshaft for an automotive driveshaft designed for heavy-duty SUVs as well as the characteristic amplitude frequency in the region of principal parametric resonance together the method of harmonic balance for the steady-state forced torsional nonlinear vibration of the driveshaft. This model of dynamic behavior for the driveshaft can be used during the early stages of design as well in predicting the durability of automotive driveshafts. In addition, it is important that this model be added in the design algorithm for predicting the comfort elements of the automotive environment to adequately account for this kind of dynamic behavior that induces excitations in the car structure.

Axial dipole moments of solar active regions in cycles 21-24

<p>The axial dipole moments of emerging active regions control the evolution of the axial dipole moment of the whole photospheric magnetic field and the strength of polar fields. Hale's and Joy's laws of polarity and tilt orientation affect the sign of the axial dipole moment of an active region, determining the normal sign for each solar cycle. If both laws are valid (or both violated), the sign of the axial moment is normal. However, for some active regions, only one of the two laws is violated, and the signs of these axial dipole moments are the opposite of normal. The opposite-sign axial dipole moments can potentially have a significant effect on the evolution of the photospheric magnetic field, including the polar fields.</p><p>We determine the axial dipole moments of active regions identified from magnetographic observations and study how the axial dipole moments of normal and opposite signs are distributed in time and latitude in solar cycles 21-24.We use active regions identified from the synoptic maps of the photospheric magnetic field measured at the National Solar Observatory (NSO) Kitt Peak (KP) observatory, the Synoptic Optical Long term Investigations of the Sun (SOLIS) vector spectromagnetograph (VSM), and the Helioseismic and Magnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO).</p><p>We find that, typically, some 30% of active regions have opposite-sign axial dipole moments in every cycle, often making more than 20% of the total axial dipole moment. Most opposite-signed moments are small, but occasional large moments, which can affect the evolution of polar fields on their own, are observed. Active regions with such a large opposite-sign moment may include only a moderate amount of total magnetic flux. We find that in cycles 21-23 the northern hemisphere activates first and shows emergence of magnetic flux over a wider latitude range, while the southern hemisphere activates later, and emergence is concentrated to lower latitudes. We also note that cycle 24 differs from cycles 21-23 in many ways. Cycle 24 is the only cycle where the northern butterfly wing includes more active regions than the southern wing, and where axial dipole moment of normal sign emerges on average later than opposite-signed axial dipole moment. The total axial dipole moment and even the average axial moment of active regions is smaller in cycle 24 than in previous cycles.</p>

Optimization of the cross-sectional shape of the belts of trihedral lattice supports

The article considers the problem of optimizing the geometric parameters of the cross section of the belts of a trihedral lattice support in the shape of a pentagon. The axial moment of inertia is taken as the objective function. Relations are found between the dimensions of the pentagonal cross section at which the objective function takes the maximum value. We introduce restrictions on the constancy of the consumption of material, as well as the condition of equal stability. The solution is performed using nonlinear optimization methods in the Matlab environment.

Reconstructing solar magnetic fields from historical observations

Context. The axial dipole moments of emerging active regions control the evolution of the axial dipole moment of the whole photospheric magnetic field and the strength of polar fields. Hale’s and Joy’s laws of polarity and tilt orientation affect the sign of the axial dipole moment of an active region. If both laws are valid (or both violated), the sign of the axial moment is normal. However, for some active regions, only one of the two laws is violated, and the signs of these axial dipole moments are the opposite of normal. Those opposite-sign active regions can have a significant effect, for example, on the development of polar fields. Aims. Our aim is to determine the axial dipole moments of active regions identified from magnetographic observations and study how the axial dipole moments of normal and opposite signs are distributed in time and latitude in solar cycles 21−24. Methods. We identified active regions in the synoptic maps of the photospheric magnetic field measured at the National Solar Observatory (NSO) Kitt Peak (KP) observatory, the Synoptic Optical Long term Investigations of the Sun (SOLIS) vector spectromagnetograph (VSM), and the Helioseismic and Magnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO), and determined their axial dipole moments. Results. We find that, typically, some 30% of active regions have opposite-sign axial moments in every cycle, often making more than 20% of the total axial dipole moment. Most opposite-signed moments are small, but occasional large moments, which can affect the evolution of polar fields on their own, are observed. Active regions with such a large opposite-sign moment may include only a moderate amount of total magnetic flux. We find that in cycles 21−23 the northern hemisphere activates first and shows emergence of magnetic flux over a wider latitude range, while the southern hemisphere activates later, and emergence is concentrated to lower latitudes. Cycle 24 differs from cycles 21−23 in many ways. Cycle 24 is the only cycle where the northern butterfly wing includes more active regions than the southern wing, and where axial dipole moment of normal sign emerges on average later than opposite-signed axial dipole moment. The total axial dipole moment and even the average axial moment of active regions is smaller in cycle 24 than in previous cycles.