Buckling of a twisted and compressed rod supported by Cardan joints

Glavardanov, V.B.; Maretic, Ratko; Grahovac, Nenad

doi:10.1016/j.euromechsol.2008.04.004

Cited by 8 publications

(4 citation statements)

References 15 publications

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“…Dynamic angle changes between co-working, connected shafts strongly argue for the application of cardan joints, which are used in ship drives, rolling mills, or automobiles [22,23]. For this reason, modelling and analysis of transient states in drive systems that include susceptible motion transmissions including cardan joints is very topical [9][10][11]. In some cases of rotational systems operating in difficult conditions, a mathematical model of motion transmission should be analysed addressing a variety of latent motions, both internal and external [4].…”

Section: Drive Systems Including Cardan Jointsmentioning

confidence: 99%

“…Only the susceptibility of the middle shaft needs to be addressed in our system, since it is several times longer than the first or third shaft. Introduction of the second (middle) shaft susceptibility causes the appearance of a mutual spatial shift of planes of fork shaft endings; as a result, the rotational speeds of the first and third shafts will not be the same [10]. This is an adverse effect, especially in precise control systems where signals from a rotational speed sensor are supplied to drive motors [27][28][29][30], for instance, in PMSM and BLDC motors.…”

Section: Drive Systems Including Cardan Jointsmentioning

confidence: 99%

“…Drive systems including cardan joints are free from this shortcoming of non-aligned shafts [9][10][11]. A cardan joint is a mechanical element serving to transmit torque and normally used to connect driving and driven shafts that cannot be connected directly due to the distance or the need to allow some relative motion between them.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

et al. 2021

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Beginning with the classic methods, a mathematical model of an electromechanical system is developed that consists of a deep bar cage induction motor that, via a complex motion transmission with distributed mechanical parameters, drives a working machine, loading the drive system with a constant torque. The electromagnetic field theory serves to create the motor model, which allows addressing the displacement of current in the rotor cage bars. Ordinary and partial differential equations are used to describe the electromechanical processes of energy conversion in the motor. The complex transmission of the drive motion consists of a long shaft with variable geometry cardan joints mounted on its ends. Non-linear electromechanical differential equations are presented as a system of ordinary differential equations combined with a mixed problem of Dirichlet first-type and Poincaré third-type boundary conditions. This system of equations is integrated by discretising partial derivatives by means of the straight-line methods and successive integration as a function of time using the Runge–Kutta fourth-order method. Starting from there, complicated transient processes in the drive system are analysed. Results of computer simulations are presented in the graphic form, which is analysed.

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Section: Drive Systems Including Cardan Jointsmentioning

confidence: 99%

Section: Drive Systems Including Cardan Jointsmentioning

confidence: 99%

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Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

et al. 2021

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“…The regions of stability follow from the bifurcation points of an equation, which results from the first integrals of the constitutive equations. Glavardanov et al (2009) examined the regions of stability for an elastic beam supported by Cardan joints at both ends. The beam is loaded by a compressive force and a couple.…”

Section: Introductionmentioning

confidence: 99%

Stability optimization for a simultaneously twisted and compressed rod

Kobelev

2022

MMMS

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PurposeThe authors search the optimal distribution of bending flexure along the axis of the rod. For the solution of the actual problem, the stability equations take into account all possible convex, simply connected shapes of the cross-section. The authors study the cross-sections with equal principal moments of inertia. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The cross-section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The optimal form of the cross-section is known to be an equilateral triangle. The distribution of material along the length of a twisted and compressed rod is optimized so that the rod must support the maximal moment without spatial buckling, presuming its volume remains constant among all admissible rods. The static Euler's approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force.Design/methodology/approachThe optimization problems for stability of twisted and compressed rods are studied in this manuscript. The complement for Euler's buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under simultaneous torsion and compression (Greenhill, 1883).FindingsFor determining the optimal solution, the authors directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form.Research limitations/implications(1) The linear stability equations are applied. (2) No nonlinear or postbuckling effects were accounted. (3) The moment-free, ideal boundary conditions on both ends of the rod assumed.Practical implicationsOne of the most common design cases in mechanical engineering is the concurrent compression and twisting of the straight members. The closed-form solution allows the immediate estimation of the optimization effect for axes and rotors in industrial and automotive engineering.Social implicationsThe application of lighter and material-saving structural elements allow the saving fabrication resources, reducing the mass of vehicles and industry machines. The systematic usage of material optimized structural elements assists the stabilization of global energy balance of Earth.Originality/valueAlbeit the governing ordinary differential equations are linear, the application of the optimality conditions leads to the nonlinearity of the final optimization equations. The search of closed form solution of the nonlinear differential equations is one of the mathematically hardest tasks in engineering mathematics. The closed-form solution presents in terms of higher transcendental functions.

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Optimization of Concurrently Compressed and Torqued Rod

Kobelev

2023

Mathematical Engineering

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The applications of this method for stability problems in the context of twisted and compressed rods are demonstrated in this manuscript. The complement for Euler's buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under simultaneous torsion and compression (Greenhill, 1883). We search the optimal distribution of bending flexure along the axis of the rod. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. We study the cross-sections with equal principle moments of inertia. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The optimal form of the cross-section is known to be an equilateral triangle. The distribution of material along the length of a twisted and compressed rod is optimized so that the rod must support the maximal moment without spatial buckling, presuming its volume 𝑉 remains constant among all admissible rods. The static Euler's approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force. For determining the optimal solution, we directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form in terms of the higher transcendental functions.

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Buckling of a twisted and compressed rod supported by Cardan joints

Cited by 8 publications

References 15 publications

Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

Stability optimization for a simultaneously twisted and compressed rod

Optimization of Concurrently Compressed and Torqued Rod

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