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Question: Analysis and Design of Machine Foundation

Question:Analysis and Design of Machine Foundation

moses 119 Maple 17
2dmath programming assignment
May 03 2016
1

 

ANALYSIS AND DESIGN OF MACHINE FOUNDATION

 


restart

Loading Optimization

 

Loading LinearAlgebra  

 

Loading plots  

with(ScientificConstants):

Loading DynamicSystems  

with(Units:-Standard)

with(Units:-Natural)

with(StringTools)

FormatTime("%m-%d-%Y, %H:%M")

FormatTime("%m-%d-%Y, %H:%M")

 (1)

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Introduction
 

This document deals with vibration analysis and design of machine foundations subjected to dynamic load.

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GetConstant(g);

standard_acceleration_of_gravity, symbol = g, value = 9.80665, uncertainty = 0, units = m/s^2

 (2)

g__SI := evalf(Constant(g, system = SI, units))

9.80665*Units:-Unit(('m')/('s')^2)

 (3)

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Richart and Lysmer's Model
 

Richart et al. (1970) idealised the foundation as a lumped mass supported on soil which is idealised as frequency independent springs which he described in term of soil parameter

dynamic shear modulus or shear wave velocity of the soil for circular footing when footings having equivalent circular radius. The Tables below shows the different values of spring and damping vlaues as per Richart and Lysmer.

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In which, G = dynamic shar modulus of he soil and is given G = `ρ__s`*V__s^2 ; ν = Piosson's ratio of the soil; ρs = mass density of the soil; Vs = shear wave velocity of the soil obtained

from soil testing; g = acceleration due to gravity; m = mass of the machine and foundation; J = mass moment of inertia of the machine and foundation about the appropriate axes; K = equivalent spring stiffness of the soil; C = damping value of the soil; B = interia factor contributing to the damping factor; D = damping ratio of the soil; r = equivalent radius of a circular foundation; L = length of foundation, and B = width of the foundation.

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Sketch
 

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nu := .25

Table : Values of soil springs as per Richart and Lysmer (1970) model
 

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SI No.

Direction

Spring value

Equivalent radius

Remarks

1

Vertical

K__z = 4*G*r__z/(1-nu)"(->)"

r__z = sqrt(L*B/Pi)"(->)"

This is in vertical Z direction

2

Horizontal

K__x = (32*(1-nu))*G*r__x/(7-8*nu)
"(->)"

r__x = sqrt(L*B/Pi)"(->)"

This induce sliding in horizontal X

2.1

Horizontal

K__y = (32*(1-nu))*G*r__y/(7-8*nu)
"(->)"

r__y = sqrt(L*B/Pi)"(->)"

This induce sliding in horizontal Y

3

Rocking

`K__φx` = 8*G*`r__φx`^3/(3*(1-nu))"(->)"

`r__φx` = (L*B^3/(3*Pi))^(1/4)"(->)"

This produces roxking about Y axis

3.1

Rocking

`K__φy` = 8*G*`r__φy`^3/(3*(1-nu))"(->)"

`r__φy` = (L^3*B/(3*Pi))^(1/4)"(->)"

This produces roxking about X axis

4

Twisting

`K__ψ` = 16*G*`r__ψ`^3*(1/3)"(->)"

`r__ψ` = ((B^3*L+B*L^3)/(6*Pi))^(1/4)
"(->)"

This produces twisting about vertical Z axis
 

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Table : Values of soil damping as per Richart and Lysmer (1970) model
 

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SI No.

Direction

Mass ratio (B)

Damping ratio and Damping values

Remarks

1

Vertical

B__z = .25*m__U*(1-nu)*g__SI/(`ρ__s`*r__z^3)
"(->)"

`ζ__z` = .425/sqrt(B__z)"(->)"C__z = 2*`ζ__z`*sqrt(K__z*m__U)"(->)"

This damping value is in vertical Z direction

2

Horizontal

B__x = (7-8*nu)*m__U*g__SI/((32*(1-nu))*`ρ__s`*r__x^3)
"(->)"

`ζ__x` = .288/sqrt(B__x)"(->)"

C__x = 2*`ζ__x`*sqrt(K__x*m__U)"(->)"

This damping value is in lateral X direction

2.1

Horizontal

B__y = (7-8*nu)*m__U*g__SI/((32*(1-nu))*`ρ__s`*r__y^3)
"(->)"

`ζ__y` = .288/sqrt(B__y)"(->)"

`ζ__y` = .288/((2.145204688-2.451662500*nu)*m__U*Units:-Unit(('m')/('s')^2)/((1-nu)*`ρ__s`*(L*B/Pi)^(3/2)))^(1/2)

 (5.1)

NULLError, invalid left hand side in assignmentError, invalid left hand side in assignment

`ζ__ψ` = .5/(1+117.6798000*`J__ψ`*Units:-Unit(('m')/('s')^2)*6^(1/4)/(`ρ__s`*((B^3*L+B*L^3)/Pi)^(5/4)))

 (5.2)

C__y = 2*`ζ__y`*sqrt(K__y*m__U)"(->)"

This damping value is in lateral Y direction

3

Rocking

`B__φx` = (.375*(1-nu))*`J__φx`*g__SI/(`ρ__s`*`r__φx`^5)
"(->)"

`ζ__φx` = .15/((1+`B__φx`)*sqrt(`B__φx`))
"(->)"

`ζ__φx` = .15/((1+11.03248125*(1-nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))*((11.03248125-11.03248125*nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))^(1/2))

 (5.3)

Error, invalid left hand side in assignmentError, invalid left hand side in assignmentNULLError, invalid left hand side in assignment

.15/((1+11.03248125*(1-nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))*((11.03248125-11.03248125*nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))^(1/2)) = .15/((1+11.03248125*(1-nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))*((11.03248125-11.03248125*nu)*`J__φx`*Units:-Unit(('m')/('s')^2)*3^(1/4)/(`ρ__s`*(L*B^3/Pi)^(5/4)))^(1/2))

 (5.4)

`C__φx` = 2*`ζ__φx`*sqrt(`K__φx`*`J__φx`)"(->)"

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This damping value is for rocking about Y direction

3.1

Rocking

`B__φy` = (.375*(1-nu))*`J__φy`*g__SI/(`ρ__s`*`r__φy`^5)
"(->)"

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`ζ__φy` = .15/((1+`B__φy`)*sqrt(`B__φy`))
"(->)"

.15/((1+`B__φy`)*`B__φy`^(1/2)) = .15/((1+`B__φy`)*`B__φy`^(1/2))

 (5.5)

`C__φy` = 2*`ζ__φy`*sqrt(`K__φy`*`J__φy`)"(->)"

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This damping value is for rocking about X direction

4

Twisting

`B__ψ` = `J__ψ`*g__SI/(`ρ__s`*`r__ψ`^5)"(->)"

`ζ__ψ` = .5/(1+2*`B__ψ`)"(->)"NULLError, invalid left hand side in assignmentError, invalid left hand side in assignmentNULLError, invalid left hand side in assignmentNULLError, invalid left hand side in assignment

.5/(1+117.6798000*`J__ψ`*Units:-Unit(('m')/('s')^2)*6^(1/4)/(`ρ__s`*((B^3*L+B*L^3)/Pi)^(5/4))) = .5/(1+117.6798000*`J__ψ`*Units:-Unit(('m')/('s')^2)*6^(1/4)/(`ρ__s`*((B^3*L+B*L^3)/Pi)^(5/4)))

 (5.6)

`C__ψ` = 2*`ζ__ψ`*sqrt(`K__ψ`*`J__ψ`)"(->)"NULL

``

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This damping value is valid for twisting about vertical Z axis
 

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 (4)

``

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Vertical Motion Considering damping of the Soil
 

For vertical direction the equation becomes that of a lumped mass having single degree of freedom when

deq := m__U*(diff(z(t), t, t))+'C__z'*(diff(z(t), t))+'K__z'*`#mi("z")` = P__0*sin(`ω__m`*t)

m__U*(diff(diff(z(t), t), t))+C__z*(diff(z(t), t))+K__z*`#mi("z")` = P__0*sin(`ω__m`*t)

 (6.1)

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t1 := subs(P__0*sin(`ω__m`*t)/m__U = F, expand(deq/m__U));

diff(diff(z(t), t), t)+1.085721853*(G*(L*B/Pi)^(1/2)*m__U/(1-nu))^(1/2)*(diff(z(t), t))/(m__U*(m__U*Units:-Unit(('m')/('s')^2)/(`ρ__s`*(L*B/Pi)^(3/2))-m__U*Units:-Unit(('m')/('s')^2)*nu/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2))+4*G*(L*B/Pi)^(1/2)*`#mi("z")`/(m__U*(1-nu)) = F

 (6.2)

NULL

 (6.3)

By algebraically manipulating the expression, the form traditionally used by engineers is derived:

t2 := algsubs('C__z'/m__U = 2*zeta*omega, t1)

diff(diff(z(t), t), t)-(-1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))*nu+1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))+4.*G*(L*B/Pi)^(1/2)*`#mi("z")`*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2))/((nu-1.)*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2)*m__U) = F

 (6.4)

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t3 := algsubs('K__z'/m__U = omega^2, t2)

diff(diff(z(t), t), t)-(-1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))*nu+1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))+4.*G*(L*B/Pi)^(1/2)*`#mi("z")`*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2))/((nu-1.)*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2)*m__U) = F

 (6.5)

This form includes the damping ratio , the natural frequency , and the external forcing term .  Consider only free vibration by setting

gen3 := subs(F = 0, t3)

diff(diff(z(t), t), t)-(-1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))*nu+1.085721853*(-G*(L*B/Pi)^(1/2)*m__U/(nu-1))^(1/2)*(diff(z(t), t))+4.*G*(L*B/Pi)^(1/2)*`#mi("z")`*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2))/((nu-1.)*(-Units:-Unit(('m')/('s')^2)*(nu-1)*m__U/(`ρ__s`*(L*B/Pi)^(3/2)))^(1/2)*m__U) = 0

 (6.6)

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sol1 := dsolve({gen3, z(0) = P, (D(z))(0) = V}, z(t))

z(t) = -(1000000000/1178791942081753609)*Pi*m__U*exp(-(1085721853/1000000000)*Units:-Unit(('s')/('m')^(1/2))*t/(Pi*((nu-1)^2/(`ρ__s`*L^2*B^2*G))^(1/2)*m__U))*(1085721853*V*Units:-Unit(('s')/('m')^(1/2))*((nu-1)^2/(`ρ__s`*L^2*B^2*G))^(1/2)*`ρ__s`*L^2*B^2-4000000000*(L*B/Pi)^(1/2)*Pi*nu*`#mi("z")`+4000000000*`#mi("z")`*Pi*(L*B/Pi)^(1/2))/(Units:-Unit(('s')/('m')^(1/2))^2*B^2*L^2*`ρ__s`)+(4000000000/1085721853)*`#mi("z")`*G*Pi*(L*B/Pi)^(1/2)*((nu-1)^2/(`ρ__s`*L^2*B^2*G))^(1/2)*t/((nu-1)*Units:-Unit(('s')/('m')^(1/2)))+(1/1178791942081753609)*(1085721853000000000*V*Units:-Unit(('s')/('m')^(1/2))*Pi*((nu-1)^2/(`ρ__s`*L^2*B^2*G))^(1/2)*m__U*`ρ__s`*L^2*B^2+1178791942081753609*Units:-Unit(('s')/('m')^(1/2))^2*P*`ρ__s`*L^2*B^2-4000000000000000000*(L*B/Pi)^(1/2)*Pi^2*m__U*nu*`#mi("z")`+4000000000000000000*`#mi("z")`*Pi^2*(L*B/Pi)^(1/2)*m__U)/(B^2*L^2*`ρ__s`*Units:-Unit(('s')/('m')^(1/2))^2)

 (6.7)

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Download Analysis_and_Design_of_Machine_Foundations_1.mw

Good Morning Mapleprime Community,

Would anybody please help in the attached worksheet. I'm trying to use the new function in Maple that is the clicable method, but I was having problem in some of my output such as zeta_y and zeta_phi as this two equations are generating an error message.

 

Regards,

Moses
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