Physics Based Basis Functions
We have seen from the previous chapters that the convergence of finite element methods can be improved if we assume basis functions which will closely resemble the displacement variation in the physical problem (Chakraborty et al. Int J Mech Sci 45(3):519
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Physics Based Basis Functions
We have seen from the previous chapters that the convergence of finite element methods can be improved if we assume basis functions which will closely resemble the displacement variation in the physical problem [3, 4, 10]. The stiffness matrix for dynamic analysis of any structural system is the same as that in static analysis. Hence a very good choice for basis functions is one which satisfies the static part of the governing partial differential equation. Several researchers [1, 5–7] have tried to simplify the governing equation such that an exact analytical solution can be found for the equation. It has been observed that these modified basis functions show improved convergence for the fundamental mode but converge slower than the cubic functions for the higher modes. In order to obtain improved convergence for all the modes, the basis function must be able to capture the effects of the flexural stiffening as well as include the centrifugal stiffening effect. The centrifugal force varies along the length of the beam (maximum at the root and zero at the tip), hence the stiffness of the system varies along the length of the beam too. We try to split the static part of the differential equation into the flexural and the centrifugal part. This indirectly represents the analysis of a nonrotating beam and a rotating string separately. We assume a solution to the differential equation to be a linear combination of the two separate solutions. A method for solving differential equations is the collocation method [13]. In this method, we assume an approximate solution which is either defined in the global domain or in a piecewise manner and then use the residual in the differential equation to solve for the constants in the solution. We define an approximation in a given number of subintervals and then obtain a relation for the solution using the collocation method. This approximated solution is then used as a basis function in the finite element method. Conventional finite element methods do not directly take the error into account. Gunda and Ganguli [6] tried to use the local form of differential equation for obtaining the shape functions and showed that improved convergence was not obtained in the © Springer Science+Business Media Singapore 2017 R. Ganguli, Finite Element Analysis of Rotating Beams, Foundations of Engineering Mechanics, DOI 10.1007/978-981-10-1902-9_5
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5 Physics Based Basis Functions
higher modes. The combination of the collocation and the finite element method is what gives the method discussed in this chapter an advantage over conventional finite element methods.
5.1 Basis Function The governing differential equation of a Euler–Bernoulli rotating beam is given by [9]. ∂w ∂ 2w ∂ 2w ∂ ∂2 (T (x) ) + m(x) (E I (x) ) − =0 ∂x2 ∂x2 ∂x ∂x ∂t 2
(5.1)
where E I (x) is the flexural stiffness, T (x) is the axial force due to centrifugal stiffening, m(x) is the mass per unit length of the beam, w is the bending displacement and is the rotation speed. The axial force
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