Assignment on Non-constant thickness

The elastic wedge for the two-dimensional unsteady problem is considered for the ideal and incompressible liquid. In the analysis the main concern is to determine liquid flow, deflections of wedge walls, pressure distribution, stress distribution of the wedge plating. The dimensions of wedge will be considered based on edge. The surface of wedge is free surface and ridge wedge. 

Initial condition: The initial condition for the wedge touching the liquid horizontal free surface is considered for the single point at t^'=0 and then it starts penetration in the liquid with a constant velocity V. Cartesian coordinates are used to calculate the dimension variables. The liquid free surface is defined as y^'=0 that correspond to the symmetric flow of liquid. Based on wedge side walls, the equation becomes y^'=  |x^' |  tan⁡〖y 〗and |x^' |=  L cos⁡y. The plane unsteady problem is used for the elastic wedge penetration. The length of walls is L and it can be used for both sides of the walls. In the Analysis the impact of wedge on the elastic side walls is dropped down with the liquid free surface. The non-dimensional thickness is considered to be measured. The undeformed wedge is used initially for symmetric and potential conditions. The side walls can be measured by modeling through the bending stresses. In order to measure the maximum bending stresses the penetration stages are different from each other. in the wedge surface process, the wetted wedge is considered as vibrating around the axis point. Different approximate approaches are used to evaluate numerical solution with different stages of penetration. It is observed that thin plating is observed for the maximum bending stresses that occur at the different points. The penetration point is different at each level due to thickness of the plat in the liquid surface. The approximate model is concerned with the simple case of thick plating and the wedge is treated as a rigid. In the analysis, it is demonstrated that model is simplified decoupled model. The beam inertia is considered for the hydrodynamic loads that is being calculated within the rigid wedge impact problem. The analysis provides reasonable estimate for the maximum bending stresses and for the thick wedge plating. The plane and potential flow is generated by using the wedge penetration and considering the velocity potential ϕ (x,y,t) and it satisfy following equation through the Wagner theory (Khabakhpasheva & Korobkin, 2003). Wagner theory is used for the two dimensional and potential flow that is generated through the elastic wedge impact for the velocity potential ϕ (x,y,t).

∇^2  ϕ = 0     ( y < 0)    …… … … … … .  .  .  .  ( 1 )

ϕ =  0     (  y =0    |x|   >  c (t)

ϕ_( y)  =  - 1  +   w_( t)  ( x,t )

ϕ→  0   ( x^( 2)   +  y^( 2)   →  ∞ )   …… … … … … .  .  .  .  ( 2 )

 Now consider the hydrodynamic pressure   p (x ,y  ,t )as a non-dimensional variables that is given by the linearized bernoulli’s equation as p ( x,y ,t )=  -   ϕ_( t)   (x  ,y ,t )  (  y ≤   0 ). The normal beam deflection in the non-dimensional variable becomes 

α  (∂^2  w)/(∂ t^( 2) )  +β  (∂^4  w)/(∂ x^( 4) )   =  p  ( x ,0 ,t )     ( 0 < x <   1 ,    t > 0 ) …… … … … … .  .  .  .  ( 3 )

w =  w_( x x)  =  0   (x = ±  0 ,t ≥ 0 )

    w =  w_( t)  =  0   (x = ±  1  ,t ≥  0 )

w_( xx)  =    0   w_( xxx)  =  sign (x)   k_( j)  k_( l)    w  (  x = ± 1,t ≥0)

The one dimensional variable in the equation gives for α =((p_b  h))/ρL   and β =  (E J c^( 2))/(ρ L^( 3)  V^( 2) ). Where b is high of thickness, E is young modulus, and J is moment of inertia (Khabakhpasheva & Korobkin, 2003).

Beam equation 

The deadrise angle is used and because of symmetry the ride side wall of the wedge can be considered. Normal deflection of the beam can be denoted by w^' (s^',t^' ) and in the equation s’ is used to define coordinates for side walls as s’ = 0 and it correspond to the wedge tip and beam end point is s’ = L. In the equation k_( l)  =  (K_( l)  〖L 〗^3)/(E J) is used for the spring rigidity. Different boundary conditions are used to represent the structural problems through the hydrodynamic pressure. The solution for the wetting part of the wedge surface is used. The boundary condition is used to evaluate vertical positions at the intersection point x = ±c ( t ) (Khabakhpasheva & Korobkin, 2003). The condition become different for the evaluation of the surface. In the defection process the Wagner condition is reduced to the following equation,

∫_( 0)^(( π)/( 2))▒〖 y_( b)  [c(t)  sin⁡θ,t]  〗 d θ = 0   …… … … … … .  .  .  .  ( 1 ) 

The selected equation in the beam deflected process depends on the actual position of the elastic wedge and the variables are dimensionless under the wagner conditions of  x =c  (t)  sin⁡θ

 c  =   π/2  t - ∫_( 0)^(( π)/( 2))▒〖 w [c(t)  sin⁡θ,t]  〗 d θ = 0   …… … … … … .  .  .  .  ( 2 )

  The bending stress in the present beam is along the upper side of the beam and the dimensionless variables are used for the measurement of stress scale. The hydrodynamic part of the equation is considered for the maximum strain condition. The strain module is considered as a part of the equation and geometrical part of the equation. The coupled problem is used to solve the issues without simplification (Khabakhpasheva & Korobkin, 2003). The normal mode method is used to describe the algorithm section for the structure of the beam and varying the thickness of the beam. The structure of the mode further represents the eigen modes and variables if the FEM based simulation code is used in the analysis of the beam deflection in the non-compressible liquid surface (Khabakhpasheva & Korobkin, 2003).

The equation 1 represent the structural part of the beam deflection equation that can be resolved by the hydrodynamic pressure p(x, 0, t) that is considered as the wetted part for the wedge surface. The J parameter in the equation shows inertia momentum for the beam cross section. The beam consists of constant thickness for the solution J=h^3/12. The boundary condition defined in equation 1 and 5 is considered for the elevation of the free surface before penetration of the beam. The vertical position is used to define the deformed wedge at the intersection points. The equation of Wagner Condition is further reduced as equation 2 (Khabakhpasheva & Korobkin, 2003). 

The bending stress distribution is considered on the upper side of the beam that is given as σ (s,t) for the dimensionless variables and it becomes σ (s,t)=   w_( xx)  ((x,t))/2along with 2∈  (E H)/( L) with the stress scale. The end solution is hydrodynamic part, structural part and geometrical part. The Wagner problem is closely related to the ridge wedge method under large number of beta parameters.  The structural part of the equation provides approximate value for the calculated parameters. The solution of the hydrodynamic problem is linked with the approximate method. The applicability is measured for the approximate solution and one need to solve the Wagner problem to solve the process accurately.  The equation becomes as follow 

α ( x )  α^' φ (x)   +  α ( B ψ )^'' = 0

(a^''  (t))/(a (t))   +   ( B ψ )^''/(a ψ)

Normal mode method 

The deflection method can be further extended to the normal mode method for deflection w( x,t ) at the wedge plating and equation becomes,

w (x,t)  =  ∑_(n =   1)^( ∞)▒〖a_( n)  (t)  ψ_( n)  (x)               ( -1 ≤  x≤  1 )      .  .  .  .  .  .  .  .  .  .  .  ( 1 ) 〗

In the equation ψ_( n)  (x) shows the shape of the free vibrating edges of the structure in the dry air. The function is also known as normal mode of the structure and becomes as 

ψ_( n)  (x)    =  sin⁡〖 (λ_( n)  ┤|  x |)〗   while  λ_n  =  ϕ n (n= 1,2,.  .  .  .  .)  

The simplified function is supported for the normal wedge plating. In case of the linear spring connection the wedge plating for the main structure analysis use elastic wedge impact. The corresponding dry modes are used for the linear spring correction of the wedge plating. These dry modes are mentioned below, 

ψ_( n)  (x)   =  A_( n)   sin⁡〖 (λ_( n)  ┤|  x |)〗  +  C_( n)  sin⁡〖h (λ_n   ┤|  x |〗)     c_( n)    =   sin⁡〖 (λ_( n))〗/sin⁡〖h (λ_( n) )〗     .  .  .  .  .  .  .  .  .( 1 )  

In this case, it is convenient to consider the principle coordinates as a_n ( t ) for the unknown functions of beam deflection. The infinite system is used for the ordinary differential equation under the principle coordinates a = (a_( 1),a_( 2),a_3,.  .  .)^( T) along with the auxiliary vector-function ~v = 〖 (〖 v〗_( 1),v_( 2),v_( 3),.  .  .)〗^( T). The corresponding dry read modes becomes, 

( 1)/(A_(  n)^( 2) )    =  1 -  C_( n)^( 2)  +   (C_( n)^( 2)  sin⁡〖h (2 λ_n  )  - sin⁡〖 (2 λ_n  )〗  〗)/(2  λ_( n) )  -  (2 C_( n))/λ_( n)   (cos⁡〖h  (λ_( n) )  sin⁡(λ_( n) ) 〗 -  sin⁡〖h (λ_( n) )  〖cos 〗⁡(λ_( n) ) 〗    .  .  .  .  .  .  .  .  .( 2 )

In this equation λ_n  are the positive roots so 

coth⁡〖(λ_n )= (2k_1)/(λ_n^3 )〗+cot⁡〖(λ_n)〗

The orthogonal condition for the solution becomes as follows, 

∫_(- 1)^( 1)▒  ψ_( n)  (x)  ψ_(  m)  (x)  dx  =  δ_( nm)  .  .  .  .  .  .  .  .  .( 3 )

The principle coordinates are used to determine the beam deflection under the new and unknown functions. The process is used under the defined interval of the liquid boundary - 1 <x<1 y =0. At this point the contact region is used for the end impact range measurement. The liquid boundary is considered as wedge surface is completely wetted. The pressure distribution and velocity potential become as 

φ(x,0,t)=∑_(m=1)^∞▒〖b_m (t)  ψ_( n)  (x),p(x,0,t)=-∑_(m=1)^∞▒〖(b_m ) ̇(t)  ψ_( n)  (x),〗〗

b_( m)  ( t )  =  ∫_(- c(t))^( c(t))▒  ψ_( n)  (x,0,1)  ψ_(  m)  (x)  dx   .  .  .  .  .  .  .  .  .( 4 )

Now new functions will be defined again from the boundary value problem solution 

(∂^2 φ_n)/(∂x^2 )+(∂^2 φ_n)/(∂y^2 )=0    (y<0)

φ_n=0    (y=0,|x|>c(t)),

(∂ φ_n)/∂y=ψ_n (x)       (y=0,|x|<c(t)),

φ_n=0    (x^2+y^2→∞)

Now this equation will become with integrable singularities 

φ(x,0,t)=-φ_0 (x,0,c)+∑_(n=1)^∞▒〖(a_n  ) ̇  〗 (t) φ_n (x,0,c)

The function of principle coordinate a_( n (t)) and derivate of the equation is used in the integration process. The hydrodynamic part of the equation 2 is further considered under the boundary value problem and the function is integrable at first derivative. The contact point of the solution is x  = ±c and n =  0,1,2,...... 

Differentiation of the equation with respect to time gives,

dt/dc   =  Q (a ⃗,( v) ⃗,c)

F ⃗(c,v ⃗ )=(al+S(c))^(-1) (βDv ⃗+ f ⃗(c))

K_(ln⁡(c))=2/π  ∫_0^(π/2)▒〖〖ψ_n^,〗_  [c sin⁡0 ]  sin⁡〖θ dθ,〗 〗  K_(0n(c))=2/π  ∫_0^(π/2)▒〖〖ψ_n^,〗_  [c sin⁡0 ]  ⁡〖 dθ,〗 〗

While the initial boundary condition of the ordinary differential equation is given as a=  0, v  =  0, c=  0 and t =0. The efficient numerical solutions for the hydroelastic behavior are used in case of elastic wedge entering the liquid. The algorithm is efficient to compute the function (Khabakhpasheva & Korobkin, 2003).

Reference of Formulation

Khabakhpasheva, T., & Korobkin, A. (2003). APPROXIMATE MODELS OF ELASTIC WEDGE IMPACT. Hydrondynamics, 01(03), 01-10.