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Finite difference method for fourth order PDE

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Old   February 14, 2021, 13:26
Exclamation Finite difference method for fourth order PDE
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Box
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How can I solve this initial boundary problem?
WhatsApp Image 2021-02-14 at 20.58.08.jpeg

Are my approximations right?
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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 cross-section 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 = 2.01e-2 # Column cross-sectional 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 16, 2021 at 00:37.
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Old   February 14, 2021, 15:23
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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?
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Old   February 15, 2021, 06:48
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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 cross-section 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 = 2.01e-2 # Column cross-sectional 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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Old   February 15, 2021, 06:49
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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 cross-section 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 = 2.01e-2 # Column cross-sectional 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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