Physics based basis function for vibration analysis of high speed rotating beams

R. Ganesh,Ranjan Ganguli 국제구조공학회 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.39 No.1

The natural frequencies of continuous systems depend on the governing partial differential equation and can be numerically estimated using the finite element method. The accuracy and convergence of the finite element method depends on the choice of basis functions. A basis function will generally perform better if it is closely linked to the problem physics. The stiffness matrix is the same for either static or dynamic loading, hence the basis function can be chosen such that it satisfies the static part of the governing differential equation. However, in the case of a rotating beam, an exact closed form solution for the static part of the governing differential equation is not known. In this paper, we try to find an approximate solution for the static part of the governing differential equation for an uniform rotating beam. The error resulting from the approximation is minimized to generate relations between the constants assumed in the solution. This new function is used as a basis function which gives rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam. The results of finite element analysis with the new basis functions are verified with published literature for uniform and tapered rotating beams under different boundary conditions. Numerical results clearly show the advantage of the current approach at high rotation speeds with a reduction of 10 to 33% in the degrees of freedom required for convergence of the first five modes to four decimal places for an uniform rotating cantilever beam.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

Sarkar, Korak,Ganguli, Ranjan Techno-Press 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.51 No.5

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

Korak Sarkar,Ranjan Ganguli 국제구조공학회 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.51 No.5

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using ananalytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions canalso be used to check optimization algorithms proposed for modal tailoring.

A dragonfly inspired flapping wing actuated by electro active polymers

Sujoy Mukherjee,Ranjan Ganguli 국제구조공학회 2010 Smart Structures and Systems, An International Jou Vol.6 No.7

An energy-based variational approach is used for structural dynamic modeling of the IPMC (Ionic Polymer Metal Composites) flapping wing. Dynamic characteristics of the wing are analyzed using numerical simulations. Starting with the initial design, critical parameters which have influence on the performance of the wing are identified through parametric studies. An optimization study is performed to obtain improved flapping actuation of the IPMC wing. It is shown that the optimization algorithm leads to a flapping wing with dimensions similar to the dragonfly Aeshna Multicolor wing. An unsteady aerodynamic model based on modified strip theory is used to obtain the aerodynamic forces. It is found that the IPMC wing generates sufficient lift to support its own weight and carry a small payload. It is therefore a potential candidate for flapping wing of micro air vehicles.

Physics based basis function for vibration analysis of high speed rotating beams

Ganesh, R.,Ganguli, Ranjan Techno-Press 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.39 No.1

The natural frequencies of continuous systems depend on the governing partial differential equation and can be numerically estimated using the finite element method. The accuracy and convergence of the finite element method depends on the choice of basis functions. A basis function will generally perform better if it is closely linked to the problem physics. The stiffness matrix is the same for either static or dynamic loading, hence the basis function can be chosen such that it satisfies the static part of the governing differential equation. However, in the case of a rotating beam, an exact closed form solution for the static part of the governing differential equation is not known. In this paper, we try to find an approximate solution for the static part of the governing differential equation for an uniform rotating beam. The error resulting from the approximation is minimized to generate relations between the constants assumed in the solution. This new function is used as a basis function which gives rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam. The results of finite element analysis with the new basis functions are verified with published literature for uniform and tapered rotating beams under different boundary conditions. Numerical results clearly show the advantage of the current approach at high rotation speeds with a reduction of 10 to 33% in the degrees of freedom required for convergence of the first five modes to four decimal places for an uniform rotating cantilever beam.

Mechatronic Approaches to Synthesize Biomimetic Flapping-Wing Mechanisms: A Review

Nilanjan Chattaraj,Ranjan Ganguli 한국항공우주학회 2023 International Journal of Aeronautical and Space Sc Vol.24 No.1

Conventional electromechanical actuators cannot independently produce flapping-wing motion and typically require complimentary mechanical transmission mechanisms to achieve that motion. Hence, the selection and design of electromechanical actuators need to be considered in parallel with the selection and design of mechanical transmission mechanisms. The article presents a review on the mechatronics-based flapping-wing mechanisms applicable to micro air vehicles, which have been reported so far in the literature to the best of authors’ knowledge. The contribution of this review explicitly illustrates a design-map showing all the possible mechatronic methods to synthesize flapping-wing mechanisms, highlighting both attempted approaches in literature and unattempted approaches, which can be investigated in the upcoming time. The comparative discussion highlights both the capabilities and design-trade-offs of all the approaches to produce flapping-wing motion in their own way. The research gap recognized by the design-map presents the scope of future investigation in this domain.

Closed-form solutions for non-uniform axially loaded Rayleigh cantilever beams

Korak Sarkar,Ranjan Ganguli,Isaac Elishakoff 국제구조공학회 2016 Structural Engineering and Mechanics, An Int'l Jou Vol.60 No.3

In this paper, we investigate the free vibration of axially loaded non-uniform Rayleigh cantilever beams. The Rayleigh beams account for the rotary inertia effect which is ignored in Euler-Bernoulli beam theory. Using an inverse problem approach we show, that for certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation for Rayleigh beams. The derived property variation can serve as test functions for numerical methods. For the rotating beam case, the results have been compared with those derived using the Euler-Bernoulli beam theory.

Quadratic B-spline finite element method for a rotating non-uniform Rayleigh beam

Vijay Panchore,Ranjan Ganguli 국제구조공학회 2017 Structural Engineering and Mechanics, An Int'l Jou Vol.61 No.6

The quadratic B-spline finite element method yields mass and stiffness matrices which are half the size of matrices obtained by the conventional finite element method. We solve the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Rayleigh beam theory includes the rotary inertia effects in addition to the Euler- Bernoulli theory assumptions and presents a good mathematical model for rotating beams. Galerkin's approach is used to obtain the weak form which yields a system of symmetric matrices. Results obtained for the natural frequencies at different rotating speeds show an accurate match with the published results. A comparison with Euler-Bernoulli beam is done to decipher the variations in higher modes of the Rayleigh beam due to the slenderness ratio. The results are obtained for different values of non-uniform parameter ( n ).

A comparative study of dragonfly inspired flapping wings actuated by single crystal piezoceramic

Sujoy Mukherjee,Ranjan Ganguli 국제구조공학회 2012 Smart Structures and Systems, An International Jou Vol.10 No.1

A dragonfly inspired flapping wing is investigated in this paper. The flapping wing is actuated from the root by a PZT-5H and PZN-7%PT single crystal unimorph in the piezofan configuration. The nonlinear governing equations of motion of the smart flapping wing are obtained using the Hamilton’s principle. These equations are then discretized using the Galerkin method and solved using the method of multiple scales. Dynamic characteristics of smart flapping wings having the same size as the actual wings of three different dragonfly species Aeshna Multicolor, Anax Parthenope Julius and Sympetrum Frequens are analyzed using numerical simulations. An unsteady aerodynamic model is used to obtain the aerodynamic forces. Finally, a comparative study of performances of three piezoelectrically actuated flapping wings is performed. The numerical results in this paper show that use of PZN-7%PT single crystal piezoceramic can lead to considerable amount of wing weight reduction and increase of lift and thrust force compared to PZT-5H material. It is also shown that dragonfly inspired smart flapping wings actuated by single crystal piezoceramic are a viable contender for insect scale flapping wing micro air vehicles.

A dragonfly inspired flapping wing actuated by electro active polymers

Mukherjee, Sujoy,Ganguli, Ranjan Techno-Press 2010 Smart Structures and Systems, An International Jou Vol.6 No.7

An energy-based variational approach is used for structural dynamic modeling of the IPMC (Ionic Polymer Metal Composites) flapping wing. Dynamic characteristics of the wing are analyzed using numerical simulations. Starting with the initial design, critical parameters which have influence on the performance of the wing are identified through parametric studies. An optimization study is performed to obtain improved flapping actuation of the IPMC wing. It is shown that the optimization algorithm leads to a flapping wing with dimensions similar to the dragonfly Aeshna Multicolor wing. An unsteady aerodynamic model based on modified strip theory is used to obtain the aerodynamic forces. It is found that the IPMC wing generates sufficient lift to support its own weight and carry a small payload. It is therefore a potential candidate for flapping wing of micro air vehicles.