In the name of GodIn the name of God

Application of the Application of the generalized generalized differential differential

quadrature rule toquadrature rule toinitial–boundary-initial–boundary-value problemsvalue problems

Key Words:Key Words:

Partial differential equations (PDEs)Partial differential equations (PDEs) Generalized differential quadrature rule Generalized differential quadrature rule

(GDQR)(GDQR) Differential quadrature method (DQM)Differential quadrature method (DQM) Ordinary differential equations(ODEs)Ordinary differential equations(ODEs)

AbstractAbstract (PDEs) for the forced vibration of structural beams are (PDEs) for the forced vibration of structural beams are

solved in this paper using the recently proposed(GDQR).solved in this paper using the recently proposed(GDQR). The GDQR techniques are first applied to both spatial The GDQR techniques are first applied to both spatial

and time dimensions simultaneously as a whole.and time dimensions simultaneously as a whole. The objective of this paper is to formularize the GDQR The objective of this paper is to formularize the GDQR

expressions and corresponding explicit weighting expressions and corresponding explicit weighting coefficients, while the derivation of explicit weighting coefficients, while the derivation of explicit weighting coefficients is one of the most important aspects in the coefficients is one of the most important aspects in the DQMs.DQMs.

An Euler beam and a Timoshenko beam are employed An Euler beam and a Timoshenko beam are employed as examples.as examples.

The proposed procedures can be applied to problems in The proposed procedures can be applied to problems in other disciplines of sciences and technology, where the other disciplines of sciences and technology, where the problems may be governed by other PDEs with different problems may be governed by other PDEs with different orders in the time or spatial dimension.orders in the time or spatial dimension.

IntroductIntroductionion The DQM is usually applied only in the spatial dimension.The DQM is usually applied only in the spatial dimension.

Classical methods, such as Runge–Kutta methods, are Classical methods, such as Runge–Kutta methods, are used in the time dimension.used in the time dimension.

The DQM has never been implemented in the time The DQM has never been implemented in the time dimension of PDEs when the temporal order is second dimension of PDEs when the temporal order is second order or higher.order or higher.

The present authors have proposed the (GDQR) to deal The present authors have proposed the (GDQR) to deal with initial-value (ODEs) of second through fourth orders with initial-value (ODEs) of second through fourth orders and boundary-value problems of fourth, sixth, and eighth and boundary-value problems of fourth, sixth, and eighth orders in solid mechanics and of third and sixth orders in orders in solid mechanics and of third and sixth orders in fluid mechanics.fluid mechanics.

The PDEs for the forced vibration of some structural The PDEs for the forced vibration of some structural beams are of second order in the time dimension and of beams are of second order in the time dimension and of fourth order in the spatial dimension.fourth order in the spatial dimension.

The explicit weighting coefficients in the conventional The explicit weighting coefficients in the conventional DQM have been obtained using the Lagrange interpolation DQM have been obtained using the Lagrange interpolation functions in Refs.functions in Refs.

Reference Reference problemsproblems

The governing equation for the forced vibration of an Euler The governing equation for the forced vibration of an Euler beam is expressed as:beam is expressed as:

(1)(1)

EI =4.7726*10^7 is the stiffness coefficientEI =4.7726*10^7 is the stiffness coefficient

ρ=420 the mass density per unit lengthρ=420 the mass density per unit length

Q=10^7 the maximum forceQ=10^7 the maximum force

L=10 the beam lengthL=10 the beam length

P=2π/0.28335 the frequency of the dynamic forceP=2π/0.28335 the frequency of the dynamic force

Both the spatial domain [0, L] and time domain [0, T] are transformed to [0, 1], using X=x/L and τ =t/T.Eq. (1) is then non-dimensionlized as:

(2)

If the beam is simply supported at both ends, the boundary conditions are:

(3)(3)

The initial conditions are as follows:

(4)

The analytic solutions for the displacement and bending moment are obtained as:

(5)

(6)

(7)

Eq. (2) does not contain terms of mixed derivatives, while PDEs for classical rectangular plates do.To illustrate the application of the GDQR to initial –boundary–value problems, a Timoshenko beam with a mixed derivative term is used as the other example. The PDE for a Timoshenko beam considering the effect of rotary inertia or shear deformation can be expressed as

(8)(8)

where ξ is a constant related to the effect of rotary inertia or shear deformation. The corresponding frequency is:

(9)(9)

If the boundary and initial conditions also adopt Eqs. (3) and (4), its corresponding analytical solutions are written as:

(10)

(11)

The data in the later numerical analysis are as follows:

(12)(12)

FormulationThe conventional DQM’s expression for ODEs is written as, if the Lagrange interpolation shape functions are used as trial functions

(13)

N : N : is the number of all the discrete sampling points : are the weighting coefficients of the rth order derivative of the function y(x) associated with points : is the rth order derivative : were obtained

ix)()(i

rj xl

)(xl j

)(rijA

)()()(i

rj

rij xlA

The GDQR expression for a two-point boundary value fourth order ODE is expressed as:

(14)

are the corresponding Hermite–Fejer interpolation functions. :are the GDQR’s weighting coefficients of the rth order derivative of the function y(x) at point :have been applied to Euler beam analysis

)(}.,,,...,,,{}{ )1(132

)1(1 xhyyyyyyW jNNN

T

)()()(i

rj

rij xhE

)(rijE

ix

The GDQR expression for an initialvalue second order ODE is expressed as:

(15)

: : are the corresponding Hermite–Fejer interpolation functions :are the GDQR’s weighting coefficients of the rth order derivative of the function y(x) at point have been applied to one degree of freedom dynamic problems are initial conditions

)(}.,,...,,{}{ )1(21 xPyyyyQ jNN

T

)(rijF

)t( i)()( r

jr

ij PF

(1)Ny andNy

The key point for choosing independent variables for PDEs is that the number of independent variables at a point is equal to the number of equations/conditions to be satisfied at the same point.

Eqs. (2) and (8) are similar in the eye of the GDQR, since both of them are of second order in the time dimension and of fourth order in the spatial dimension.

Fig. 1. The GDQR’s grids for both the spatial and time domains.

Next, the independent variables for each point are chosen according to the GDQR’s definitions, as shown in Table 1(a).

The symbol Uij in Table 1(a) is used as the replacement of independent variables just for convenient formulations.

The Hermite-Fejer interpolation functions for each independent variable are shown in Table 1(b).

Therefore, the interpolation expression of the displacement function are summarized as:

(16)

Using Eq. (16), the GDQR’s expression for the rth order X-partial derivative at points X=Xi along any line parallel to the X-axis may be written as:

(17)

for the sth order t-partial derivative at points along any line X=Xi parallel to the t-axis may be written as:

(18)

The GDQR’s expression for a mixed derivative at point is in the form of:

(19)

Using the equations derived, governing equation (8) is discretized as:

(20)

Boundary Eq. (3) can be transformed to:

(21)

Initial condition equation (4) may be written as:(22)

The formed algebraic equations from (20)–(22) can be solved to obtain the required independent variables.

Using Eq. (17), the bending moment at any point can be obtained.

The velocity and acceleration are calculated using Eq. (18).

If ξ is taken as zero in Eq. (20), the results for the Euler problem can be obtained.

Using only three points in the spatial domain and six points in the time domain, the GDQR’s relative errors for the Euler beam are shown in Tables 2 and 3, and those for the Timoshenko beam in Tables 4 and 5.

Results and discussion It is clearly shown from this work that the DQ

techniques can directly transform PDEs to discrete algebraic equations.

The procedures here can be applied to the forced vibration of circular plates if the time dimension is added. since their governing equations can all be expressed as the PDEs with the fourth order in spatial dimension and the second order in temporal dimension.

The GDQR is demonstrated here to solve high order PDEs without using the three conventional techniques such as:

building the boundary conditions into weighting coefficients

dropping equations at points closest to the domain ends

For the four corner points as shown in Fig. 1, the two points at the initial line have three equations and three independent variables, while the two points at the time-domain end have only two equations and two independent variables

The principle about the choice of independent variables is formularized here

From the formulation of the GDQR proposed for classical rectangular plates, an application of the GDQR to other high order initial–boundary value problems can be expected

Prepared by:Prepared by:

S-khajeh hassaniS-khajeh hassani

Thanks for your Thanks for your attentionattention

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