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Finite difference method for fourth order PDE 

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February 14, 2021, 12:26 
Finite difference method for fourth order PDE

#1 
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Box
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How can I solve this initial boundary problem?
WhatsApp Image 20210214 at 20.58.08.jpeg Are my approximations right? WhatsApp Image 20210214 at 23.18.58.jpeg it's the nonlinear mathematical model of the dynamics of a rotating drill string compressed at both ends by the longitudinal force N(x3, t) u2  displacement of the center of bending of the crosssection of the bar along the x2 axe due to bending E = 2.1e11 # Pa  Young's modulus rho = 7800 # kg/м3  density v = 0.28 # Poisson's ratio d1 = 0.12 # м  drill string inner diameter d2 = 0.2 # м  external diameter A = 2.01e2 # Column crosssectional area Ix1 = PI / 64. * (d2 ** 4  d1 ** 4) # Ring section moment of inertia Omega = 0.33 # drill string angular velocity C1 = 0.01 # constant determining the rate of deviation of the drill string axis from the initial position in the Ox2x3 plane at the initial moment time. Thank you in advance Last edited by BoxQwerty; February 15, 2021 at 23:37. 

February 14, 2021, 14:23 

#2 
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Filippo Maria Denaro
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I don't know the physics described by this PDE, the problem is not only in a consistent discretization but also on the mathematical character of the equation. A second order in time PDE could be hyperbolic but I see also fourth order spatial terms. Central discretization in time and space is consistent but I don't know if it is also stable.
Then, also the BCs for this problem should be analysed. Have you some specific reference? 

February 15, 2021, 05:48 

#3 
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Box
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Material I have is all in Russian.
In short, it's the nonlinear mathematical model of the dynamics of a rotating drill string compressed at both ends by the longitudinal force N(x3, t) u2  displacement of the center of bending of the crosssection of the bar along the x2 axe due to bending E = 2.1e11 # Pa  Young's modulus rho = 7800 # kg/м3  density v = 0.28 # Poisson's ratio d1 = 0.12 # м  drill string inner diameter d2 = 0.2 # м  external diameter A = 2.01e2 # Column crosssectional area Ix1 = PI / 64. * (d2 ** 4  d1 ** 4) # Ring section moment of inertia Omega = 0.33 # drill string angular velocity C1 = 0.01 # constant determining the rate of deviation of the drill string axis from the initial position in the Ox2x3 plane at the initial moment time. 

February 15, 2021, 05:49 

#4 
New Member
Box
Join Date: Feb 2021
Posts: 4
Rep Power: 4 
Material I have is all in Russian.
In short it's the nonlinear mathematical model of the dynamics of a rotating drill string compressed at both ends by the longitudinal force N(x3, t) u2  displacement of the center of bending of the crosssection of the bar along the x2 axe due to bending E = 2.1e11 # Pa  Young's modulus rho = 7800 # kg/м3  density v = 0.28 # Poisson's ratio d1 = 0.12 # м  drill string inner diameter d2 = 0.2 # м  external diameter A = 2.01e2 # Column crosssectional area Ix1 = PI / 64. * (d2 ** 4  d1 ** 4) # Ring section moment of inertia Omega = 0.33 # drill string angular velocity C1 = 0.01 # constant determining the rate of deviation of the drill string axis from the initial position in the Ox2x3 plane at the initial moment time. 

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finite difference, numerical solution, pde 
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