Shaft Cross-section Research Articles

Design and simulation of internal planetary wheel plunger-type ring molding machine for biomass pellets

Key components of the existing external meshing dorsal spine plunger-type molding machine were modeled in three dimensions, and the fatigue life analysis of the molding machine spindle was carried out by using Ansys software. Due to the nonlinear mechanical behavior of the material ring mold, and pressure roller in the granulation process, there are a lot of contacts and collisions. Using the linear mechanics model is difficult to analyze. To achieve more accurate and realistic results, an Edem-Ansys joint coupled simulation was carried out for the pressure roller and ring mold engagement process. The results showed that the stress concentration point and fatigue weak region of the spindle occurred at the shaft cross-section, where the stress value should be less than 0.75 F. The maximum stresses and strains in the engagement process of the pressure roller and the ring die body occurred at the engagement point. The maximum values of deformation, stress, and strain were 0.039 mm, 412 MPa, and 0.002 mm/mm, respectively, which are all within the reasonable range and meet the design requirements.

STATIC DEFORMATION ANALYSIS OF A MOBILE CUTTING MACHINE

A study of the special requirements and criteria to portable cutting machines was carried out. Based on the existing approaches to the deformation analysis of spindle units the calculation schemes and mathematical model for the static analysis of the problem were developed. The implemented approach makes use of the Timoshenko beam model. The resulting differential equation of the bent beam axis is solved by the method of initial parameters in the matrix form. The nonlinear equation system for the chosen calculation scheme is formed automatically and is solved iteratively. A special feature of the proposed approach is the nonlinear relation between the stiffness of the bearings and the forces acting upon them. As a model application example, the results of the static deformation analysis of portable boring machine XT7809-0524 are presented. An estimate of the bending stiffness and the corresponding distributions of bending deflections and rotations were obtained for several calculation schemes of the machine. The obtained solution quantities are presented in form of graphs along the spindle axis. It has been found, that the calculated equivalent stiffness of XT7809-0524 machine spindle shaft is lower than the normative value used in the industry. A strength assessment of the spindle shaft was also made and the distributions of internal forces and moments acting on shaft cross sections were obtained. It has been concluded, that under work loads the maximum stress in the shaft is acceptable to provide for the required factor of safety.

Full-Endoscopic Technique for Posterior Fossa Decompression in Chiari Malformation.

Full-endoscopic techniques are well-described for spinal procedures. Although endoscopic-assisted techniques are reported for posterior fossa decompression (PFD) in Chiari malformation (CM), a full-endoscopic technique is yet to be reported in these patients. The aim of this study was to present and describe a full-endoscopic technique for PFD in patients with CM. Two patients diagnosed with CM were operated on by the full-endoscopic PFD technique. The patients consented to the procedure and to the publication of their image. An endoscope with an oval shaft cross-section with a diameter of 9.3 mm, a working length of 177 mm, a viewing angle of 20°, and a working channel of 5.6 diameters were used. Operative videos were recorded. The surgical steps were easily applied after the clear anatomic landmarks, such as the C1 posterior tubercle and the rectus capitis posterior minor muscles. The patients were followed up for 6 months. Both patients were symptom-free with a significant decrease in Visual Analog Scale score and a good functional outcome assessed by Chicago Chiari Outcome Scale after surgery without any complications. All the steps of the full-endoscopic technique for PFD described by the authors in their previous human cadaveric study were also feasible on patients with CM.

Puberty in a Mesozoic reptile.

The histology of bone can be preserved virtually unaltered for hundreds of millions of years in fossils from all environments and all vertebrate taxa, giving rise to the flourishing field of paleohistology.1 The shafts of long bones are formed by the apposition of periosteal bone tissue, similar to the growth of wood, and preserve, an often cyclical, record of the growth of the individual and events in its life history. One such event is sexual maturation or puberty, during which hormonal changes transform the juvenile into a sexually mature adult. Puberty has been well studied in humans and some other living vertebrates. Here, we describe puberty in Keichousaurus, a small sexually dimorphic and live-bearing marine reptile from Middle Triassic rocks of SW China, about 240 million years old. Using a combination of bone histology and morphology, we detected puberty2 as one of the four life stages (the others being fetus, juvenile, and adult). Adult Keichousaurus males have a more robust humerus than females, with pronounced muscle attachment sites and a triangular shaft cross section. Midshaft sections of the humeri of the males show the transition from the rounded juvenile cross section to the triangular adult cross section, as reflected in the contour of the growth marks. This shape change is produced by differential bone apposition of the periosteum, presumably triggered by sex hormones, as in humans,3 and influenced by changes in loading regime during puberty. This is the first report of puberty in a fossil amniote.

Full-endoscopic technique for posterior fossa decompression in Chiari malformation type I: An anatomical feasibility study in human cadavers.

Although endoscope-assisted techniques have been described, a full-endoscopic approach is yet to be performed for posterior fossa decompression (PFD) in Chiari malformation type I (CM-I). This study aims to describe the full-endoscopic PFD technique and evaluate its feasibility. Five fresh-frozen anonymized adult human cadavers were operated on using an endoscope with an oval shaft cross-section with a diameter of 9.3 mm, a working length of 177 mm, and a viewing angle of 20°. It also had an eccentric working channel with a diameter of 5.6 mm, a light guide, a sheath for continuous irrigation, and a rod lens system. The instruments were introduced from the working channel. Posterior craniocervical structures were dissected, and PFD was achieved. The planned steps were performed in all five cadavers. The endoscope was introduced to the posterior craniocervical region, dissecting the structures to easily expose the suboccipital bone and C1 posterior arch. Important structures, such as the C1 posterior tubercle, rectus capitis posterior minor muscles, and posterior atlantooccipital membrane, were used as landmarks. PFD was feasible even with the dural opening. Using the full-endoscopic approach, posterior craniocervical structures can be reached, and PFD can be performed successfully. The instruments used are well-defined for spinal usage; thus, this full-endoscopic technique can be widely used in the surgical treatment of patients with CM-I.

Developing and testing a novel pressure-controlled hydraulic profile for siphon-shaft spillways

The operating heads of the siphon-shaft spillways are limited by cavitational pressure due to low siphonic pressure inside the spillway shaft. Therefore, these low pressures inside the shaft must be eliminated to control the cavitation. An innovative idea is to enlarge the shaft cross-section upwards where siphonic pressure decreases. For this purpose, in this study, a new pressure-controlled siphon-shaft profile was developed analytically based on Bernoulli principle and cavitational pressure. The hydraulic performance of the profile is numerically tested by means of computational fluid dynamics (CFD). The numerical model used was validated with the results of a benchmark model, and the discretization errors were estimated based on a procedure recommended by American Society of Mechanical Engineers (ASME). The verified models used to analyze the novel hydraulic design together with the classical Wagner profile originally applied to shaft/morning glory spillways. The analysis results indicated that the pressure-controlled siphon-shaft profile is quite effective in terms of cavitation protection and hydraulics performance for siphon-shaft spillways.

Pullout behavior of helical soil nails in cohesionless soil: a laboratory investigation

The soil reinforcing method of incorporating helical soil nails is a technique involving the introduction of passive elements into the soil. The combined action of helical bearing and shaft skin friction from helical soil nails gives better reinforcement against failure of the composite soil mass. The current study is aimed to explore the laboratory pullout response of helical soil nails in cohesionless soil (river sand) under varying shaft cross-section (Circular & Square), shaft type (Hollow & Solid), transition in the position of the third helix, and overburden pressure to optimize the most suitable soil nail. In this study, pullout tests are conducted on the soil nails in the laboratory using the physical model setup. The pullout capacity of circular nails is higher than the corresponding square nail. Moreover, a solid shaft nail depicts a higher pullout load capacity than a hollow shaft soil nail. Maximum pullout load is achieved at third helix spacing of about 3.8 Dh and gives a 7.40% increase in pullout load as compared to the reference soil nail. SSCN1. Further, as the overburden pressure is increased, the pullout capacity also increases.

Femoral neck anteversion in Xenarthra

Degree of femoral neck anteversion distinguishes extinct sloth genera from each other, particularly when analyzed in combination with femoral neck angle. These data have been previously analyzed using a comparative sample including only extant sloths, which limits functional interpretations. Here, we analyze both femoral neck anteversion and femoral neck angle broadly across Xenarthrans to reconstruct the locomotor behavior of extinct Caribbean sloths.Femora of four extinct sloth genera (n=49; Acratocnus; Megalocnus; Neocnus; Parocnus) and nine extant Xenarthra genera (n=63; Bradypus; Choloepus; Cyclopes; Dasypus; Euphractus; Myrmecophaga; Pridontes; Tamandua; Tolypeutes) were surface scanned. Morphological axes representing the femoral neck, shaft, geometric axis of the femoral condyles (GA), and maximum dimension of the femoral shaft cross section at 30% of whole bone length were extracted. Degree of femoral neck anteversion was calculated relative to both the GA and cross section axes. Femoral neck angle was additionally calculated as the angle between the femoral neck and shaft axes. Non‐parametric MANOVAs with post hoc multiple comparison tests were used to test how the genera differed when the angle of femoral neck anteversion and femoral neck angle were analyzed in conjunction.Parocnusdoes not significantly differ from any of the armadillo taxa when femoral neck angle is analyzed with either of the two anteversion angles (p>0.171 for all pairwise comparisons), supporting previous suggestions that Parocnuswas largely terrestrial. This finding also indicates that Parocnus may have had similar hindlimb biomechanics to armadillos, and possibly utilized digging behaviors. Megalocnusand Acratocnusdiffer significantly from all extant Xenarthrans (p<0.043 for all pairwise comparisons), but do not exhibit significantly different morphologies from each other (p=0.911). This suggests that the two genera may have utilized similar hip mechanics; however, their locomotor repertoires cannot be accommodated within the range of those exhibited by extant Xenarthrans.

Research and development of structures of single-row rope-profile guides for movement of vehicles in bar reinforcement

The paper studies the horizontal loads arising from the movement of vehicles under the action of dissipative forces (torsion of ropes, aerodynamic forces, loading cages with eccentricity in the frontal and lateral planes), which act on single-row rope-profile guides (hereinafter referred to as RPGs or guides). In the paper, for the case of two moving cages of one lifting unit, in accordance with the current regulatory techniques and Safety Rules, shaft cross-sections for various types of shaft reinforcement were designed, their advantages and disadvantages were analyzed, and their comparative analysis was carried out. On the basis of the research, the structures of single-row RPGs have been developed that have a smaller number of ropes and the structure of an enclosing profile in the form of an elongated hexagon, which makes it possible to increase the reliability of the kinematic link in the "vehicle-guide" system and ensure an increase of safety gaps between the vehicles and the shaft lining. The development of various structural solutions for single-row RPGs for cages makes it possible to achieve: - high reliability of a single-row RPG operation; - increased service life due to a significant decrease in the level of dynamic effects; - reduced labor intensity of maintenance and operation compared to rigid reinforcement; - reduced metal consumption compared to rigid reinforcement; - reduced level of dynamic oscillations at the meeting point of vehicles in the middle part of the shaft by reducing the RPG deflection; - increased reliability of the kinematic link in the "vehicle–RPG" system due to the use of specially designed guide legs; - reduced safety gaps per side (- 500 mm per side) for cage hoists along the entire depth of the shaft compared to flexible reinforcement; - reduced operating costs due to the high service life of a single-row RPG; - elimination of the use of fender ropes; - increased service life of rope-profile guides to the level inherent to guides of rigid shaft reinforcement. The result of the implementation of the developed structures of single-row RPGs with reinforcement is the extension of the service life of the reinforcement, and the reduction in the cost of capital construction and operation in the extraction of minerals from deep horizons.

Life-Extension Project Applies Assessment of Reinforced Concrete to Nonjacket Structures

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper OTC 31250, “Wandoo B: Application of Advanced Reinforced Concrete Assessment for Life Extension for Non-Jacket Structures,” by Robert Sheppard, Spire Engineering; Colin O’Brien, Vermilion Oil and Gas; and Yashar Moslehy, Spire Engineering, et al., prepared for the 2021 Offshore Technology Conference, originally scheduled to be held in Houston, 4–7 May. The paper has not been peer reviewed. Copyright 2021 Offshore Technology Conference. Reproduced by permission. Wandoo B is a concrete gravity-based structure (GBS) and is the main production facility for the Wandoo field offshore northwest Australia. It was installed in 1997 with a design life of 20 years. The structural assessments discussed in this paper are part of a comprehensive life-extension project encompassing wells, subsea systems, marine and safety systems, and topsides facilities and structures to demonstrate fitness for service through the end of field life. Background The GBS serves as the support structure for the Wandoo B facility and provides oil storage for the Wandoo field. The structure has four shafts approximately 11 m in diameter that support the top-sides facilities and a base structure with permanent ballast and oil storage cells (Fig. 1). It was originally developed as an ExxonMobil-led project and now is owned and operated wholly by Vermilion Oil and Gas Australia. The reinforced concrete (RC) shafts and the base top slab are pretensioned. In the shafts, tendons are enclosed in 20 ducts distributed around the circumference. The top of the shafts provides a mating point with the steel topsides structure with the connection formed by embedded anchor bolts in a bulge in the shaft cross section. The topsides structure is a three-level braced steel frame system supporting production operations for 12 well conductors contained within the northeast shaft and three outboard well conductors. Life-Extension Project The facility was designed with a target life of 20 years. The life-extension project was intended not only to satisfy the operator’s responsibility to continue safe operations and adhere to their safety case but also to meet the expectations of the regulator. The structural aspects of the project included four phases, the first two of which are detailed in this synopsis: - Design assessments per latest standards and modifications where required - Ultimate capacity assessments with retrofit modifications where required - Risk studies and workshops to demonstrate that risk is as low as reasonably practicable (ALARP) - Integrity-management manual and inspection plan The first two phases were addressed using the latest condition-assessment, weight, and environmental data available. The phased approach allowed the assessment team to use basic linear approaches to demonstrate code compliance and only use the more-advanced analysis techniques to evaluate the critical components that did not satisfy code or were needed to provide input to the ALARP assessment and establish target reliability for the facility.

Study of the Convective Stratification of Airflows in a Mine Shaft

The experimental study of heat and mass transfer in the atmosphere of a skip shaft of a Gypsum mine was carried out during the summer and winter seasons. The air flow overturning was observed in the winter season due to the ingress of cold heavy air from the surface into the shaft space. To analyze this phenomenon, a computational fluid dynamics (CFD) model of heat and mass transfer processes in a mine shaft was developed, considering the vertical temperature gradient, roughness of the shaft walls, heat exchange with the shaft lining. Based on numerical simulation, it was found that at relatively low air velocities in the shaft and with a relatively large difference between warm air in the shaft and cold air on the surface, the latter begins to penetrate the shaft and descend down it, gradually filling its space. The interaction of cold air from the surface with the warm air rising up the shaft, as well as with warm walls of the shaft leads to the formation of an unsteady convective vortex cellular structure along the shaft. The depth of the vortex depends on the velocity of the air in the shaft, as well as the difference between the temperatures of the warm and cold air. Based on the obtained numerical simulation data, it was possible to calculate the minimum allowable air velocity at which partial return air flows appear in the shaft cross-section.

A Vision Measurement Method for the Gear Shaft Radial Runout With Line Structured Light

The gear shaft needs to measure its radial runout for straightening before finishing, while it is difficult to measure the gear shaft radial runout without rotating the gear shaft by the existing measurement technology, which results the measurement inefficient. Present work puts forward a method of measuring a gear shaft radial runout with line structured light vision. This method can measure the radial runout value and direction online for straightening the bending of the gear shaft. The center position of the gear shaft cross section is obtained by the ellipse geometric fitting on the laser plane to eliminate the influence of projection distortion. The measurement method is tested by a specialized experiment, and the experimental results show that measurements of the method meet the requirements of straightening the gear shaft. The method can realize online measurement of the gear shaft radial runout just by a single image without rotating the gear shaft, so it features fast measurement compared with the existing measurement technology.

Smoke flow in full-scale urban road tunnel fires with large cross-sectional vertical shafts

In this study, a series of full-scale fire tests are performed in a tunnel with large cross-sectional vertical shafts. The heat release rates and fire locations are considered, and the general smoke development, smoke flow pattern in the shaft, smoke temperature distribution, and smoke front velocity are analysed. It is found that complete plug-holing occurs in the tunnel with large cross-sectional vertical shafts. The smoke flows into the shaft from the bottom sides and spreads out along the shaft sidewalls. The smoke cannot fill the shaft cross-section, even at the shaft outlet. In the early and full-development stages of a fire, a well-stratified smoke layer exists, whereas in the decay stage and after burning out, the visibility significantly reduces. The smoke can be controlled within 80–125 m on each side of the fire source, and it takes approximately 10 min for the visibility to self-recover naturally for common car fires after burning out. The results of the full-scale experiment indicate that the shaft has acceptable performance in smoke control for the test scenario investigated. According to an analysis of the tunnel environment during a common car fire, as well as the evacuation movement, it can be concluded that the occupants can escape safely. The experimental results are used to verify the smoke control performance of the vertical shaft and provide guidance for the design of naturally ventilated tunnels with vertical shafts.

Factors influencing the airflow rate of kitchens in cooking exhaust shaft system of high-rise residential buildings

Insufficient shaft cross-sectional area of cooking exhaust shaft system in high-rise buildings influences the ventilation capacity of range hoods in kitchens, which cause damage to human health or increase the energy consumption of the building. There are numerous factors that affect ventilation capacity of the exhaust system, such as exhaust shaft cross section, range hood performance, operating rates and distribution of range hoods. The purpose of this research is to conduct a quantitative study on the effects of these parameters on the airflow rate of kitchens. A theoretical model of the cooking exhaust shaft system was established according to the Fluid Network Theory, it was further validated by a full-scale model experiment and used to calculate the airflow rate of kitchens for a 33-storey residential building. The results show that when the exhaust shaft cross section is designed to be 500 mm × 400 mm according to standard, regardless of the performance of the range hood, the airflow rate of kitchens of some users will be either insufficient or excessive, and cannot meet 300~500 m3/h required by Chinese standards. But when the shaft cross section is 750 mm × 600 mm and the standard range hoods are used, the airflow rate of kitchens can reach 300~500 m3/h at any operating rate. The established model can be applied in other countries to study the ventilation effect of the similar cooking exhaust shaft system in residential or other buildings. The research results can be useful for improving the design of the cooking exhaust shaft system in residential buildings.

Ultrasonic velocity tomography for inspecting the condition of a bridge pylon

A two-phase ultrasonic velocity tomography technique was introduced to inspect the concrete quality of bridge pylons in Taiwan. In phase I, the modified cross-hole sonic logging method was applied to rapidly identify the aberrant sections of pylon shafts and horizontal beams. Most of the concrete quality was classified as good (G), ie the apparent P-wave velocity ranged from 3600 m/s to 4300 m/s. In phase II, the cross-sectional velocity image method examined the sections with possible anomalies and, in contrast, the uniform sections of the pylon shafts. The cross-sectional velocity image approach effectively demonstrated the spatial position and range of anomalies in the pylon shaft cross-sections. It was suggested that the relatively low-velocity zones, confirmed with surface concrete spalling, be listed as future repair targets for the bridge agency.

Longitudinal Fracture Analysis of Nonlinear Elastic Circular Shafts Loaded in Torsion

The longitudinal fracture of circular shafts loaded in torsion is analytically investigated. It is assumed that a longitudinal crack with a circular crack front is arbitrarily located to the radial direction of the shaft cross section. The shafts are made of nonlinear elastic material, which exhibits continuous inhomogeneity in the radial direction. The material nonlinearity is described by the Ramberg–Osgood stress–strain relation. The longitudinal fracture is studied in terms of the strain energy release rate. The developed analysis is applied to evaluate the longitudinal fracture behavior of a clamped nonlinear elastic inhomogeneous shaft loaded in torsion by a torque applied to a free end of internal crack arm. For the method verification, the strain energy release rate is also assessed by analyzing the energy balance in the clamped shaft. A parametric study is performed to assess the effects of material inhomogeneity, its nonlinearity, and crack orientation to the shaft radial direction on the longitudinal fracture behavior of the clamped shaft.

Analysis of the periodic breathing of a transverse annular crack propagated in a real rotating machine

This paper shows the numerical results of a model-based analysis carried out to investigate the experimental dynamic behaviour of the shaft of a real steam turbine in which a deep annular – or circumferential – crack propagated. The shape of this transverse crack was caused by the generation of multiple initiation points, equally spaced over the whole circumferential surface of the shaft cross-section located at the first high pressure stage of the steam turbine.The periodic breathing of the crack caused changes in the area moments of inertia of the damaged cross-section and fault symptoms. Conversely, the shaft heating that affected the turbine during the load rises performed at the end of the runups carried out from a cold state pointed out an abnormal sensitivity of the synchronous vibrations to the machine thermal state. As a result, the fault symptoms caused by this crack were quite different from those induced by common shaft transverse cracks. The breathing of this crack, caused by gravity, has been studied by means of a mathematical model in order to investigate the influence of the crack opening and closure on the harmonic content of the turbine vibration.A new approach is used in this paper, considering a more appropriate 3D FE model that allowed the non-linear distribution of the stresses in the cracked section to be estimated. The method used for the present study has proved that the non-linear distribution of the axial stresses in the area close to the crack tip gives only minor effects on the breathing mechanism. This is an original contribution, which could have not been observed in previous authors' papers that employed simplified models for the periodic breathing.The good accordance between numerical results and experimental data confirms the adequacy of the investigation method used for this study. With reference to diagnostic purposes, this involves that the early on-line detection of annular cracks propagated in the shafts of rotating machinery needs to use more accurate techniques, based on the analysis of vibration data, which can point out fault symptoms, even in the case of very deep transverse cracks.

Numerical Calculation and Stability of the Yield and Enhanced Support Technology for Shaft

This work study the auxiliary shaft in Jinchuan No. 3 mining area. A numerical calculation method is adopted in this study to optimize retention space to yield for the support of the shaft and to analyze the stability of the surrounding rock mass to solve the large deformation problem occurring in shaft excavation. Firstly, three different sections are investigated, including the shaft wellhead section, main body and cross-cut under the shaft. Support and reinforcement solutions based on the retention space to yield support technology of shaft are proposed twice to resolve the corresponding problems appearing during construction. The relationship between the thickness of the lining and the convergence of the lining, the thickness of the plastic zone and the maximum principal stress of the lining is then identified through a planar model analysis on the shaft cross-section with the depth of 250 m, 400 m, 500 m, and 650 m. Lastly, a three-dimensional numerical simulation and safety evaluation of the first support design (50 mm bolt–shotcrete mesh + 450 mm plain concrete), as well as a stability analysis on the adjusted support solution (50 mm bolt–shotcrete mesh + 750 mm plain concrete) for a stratum of poor conditions are carried out using software FLAC3D.

The full-endoscopic uniportal technique for decompression of the anterior craniocervical junction using the retropharyngeal approach: an anatomical feasibility study in human cadavers and review of the literature.

Acute or progressive myelopathy may necessitate direct anterior decompression of the craniocervical junction and odontoidectomy. Different techniques with individual advantages and disadvantages can be used. In addition to the gold standard-the transoral approach-there is also increasing experience with the endoscopic transnasal technique. Other alternative methods are also being developed to reduce technical and perioperative problems. The aim of this anatomical study was to investigate the feasibility of the full-endoscopic uniportal technique with a retropharyngeal approach for decompression of the craniocervical junction, taking into consideration the specific advantages and disadvantages compared with conventional methods and the currently available data in the literature. Five fresh adult cadavers were operated on. The endoscope used has a shaft cross-section of 6.9 × 5.9 mm and a 25° viewing angle. It contains an eccentric intraendoscopic working channel with a diameter of 4.1 mm. An anterior retropharyngeal approach was used. The anatomical structures of the anterior craniocervical junction were dissected and the bulbomedullary junction was decompressed. The planned steps of the operation were performed in all cadavers. The retropharyngeal approach allowed the target region to be accessed easily. The anatomical structures of the anterior craniocervical junction could be identified and dissected. The bulbomedullary junction could be adequately decompressed. No resections of the anterior arch of the atlas were necessary in the odontoidectomy. Using the full-endoscopic uniportal technique with an anterior retropharyngeal approach, the craniocervical region can be adequately reached, dissected, and decompressed. This is a minimally invasive technique with the known advantages of an endoscopic procedure under continuous irrigation. The retropharyngeal approach allows direct, sterile access. The instruments are available for clinical use and have been established for years in other operations of the entire spine.

Refined Beam Elements for the Free Vibration Analysis of Stepped Shafts With and Without Transition Area

Investigation of the free vibration of stepped shafts with and without transition area is presented in this paper. The refined one-dimensional beam theories are chosen to be used based on the Carrera Unified Formulation (CUF) in the present work. According to CUF, Taylor-type expansion is employed to describe the displacement field over cross section of shafts. With the implementation of Hamilton’s principle, the governing differential equations and the related mass matrix, the stiffness matrix and the load vector are obtained. To obtain the numerical solutions, the finite method is adopted in this paper. The accuracy and reliability of the present model are demonstrated by comparing with classical beam model (Timoshenko beam model) and a solid model generated in the commercial software ANSYS. Meanwhile, the influences of step ratio, step location, boundary conditions and the slope of generatrix attached to the transition area on vibration characteristics are studied.