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jamorga37

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Please help -

I am trying to figure out the stress in a hyperbolic disk rotating at some RPM. The best equations I could find (and example) is from the book "advanced Strength of Materials" by J. P. Den Hartog. Google books shows page 62 with these formulas:

Thickness (t) = t

q = 1 shows a "ordinary hyperabola"

formula 48:

n

formula 49 after dividing t*r out is:

radial stress (s

tangential stress (s

I put these equations in EES (engineering equation solver) and matched the books numbers for the example (more or less since the author did the calculations without a calculator).

My problem is that as q goes to 0 (disk evens out to a flat disk), max stress: s

I've attached the EES plots. The bold lines show the q=1 solution for radial and tangential stresses (both divided by rho*omega

Using the equations for flat disks:

s

which gives a value of 190 at the bore (again divided by rho * omega

I've attached my EES code for reference.

Am I mis-interpreting the equations or what??

Thanks for any help in advance,

-Jeff

I am trying to figure out the stress in a hyperbolic disk rotating at some RPM. The best equations I could find (and example) is from the book "advanced Strength of Materials" by J. P. Den Hartog. Google books shows page 62 with these formulas:

Thickness (t) = t

_{i}/(r[tex]^{q}[/tex]) where r is radius.q = 1 shows a "ordinary hyperabola"

formula 48:

n

_{1},n_{2}= -0.5* q +/- sqrt(0.25*q^{2}+ mu*q + 1) where mu is poissons ratioformula 49 after dividing t*r out is:

radial stress (s

_{r}) = c_{1}*r^{n1}+ c_{2}*r^{n2}- [(3+mu)*rho*omega^{2}*r^{2}/(8-(3+mu)*q)] where omega is rotational velocity in radians per unit time, and rho is disk densitytangential stress (s

_{t}) = r * s_{r}' + rho * omega^{2}* r^{2}I put these equations in EES (engineering equation solver) and matched the books numbers for the example (more or less since the author did the calculations without a calculator).

My problem is that as q goes to 0 (disk evens out to a flat disk), max stress: s

_{t}(evaluated at the bore of the disk) becomes less and less. The opposite happens as q increases past 1. I've looked over the equations and I can't find a mistake.I've attached the EES plots. The bold lines show the q=1 solution for radial and tangential stresses (both divided by rho*omega

^{2}) which look just like the ones in the textbook (tangential larger then radial at the bore - r = 3in). The other lines are tangential and radial stresses for q = 0.5 and q = 0. The tangential stress at the bore drops as q drops. I would expect the tangential stress at the center to increase as q drops.Using the equations for flat disks:

s

_{t}=rho*omega^{2}* [(3+v)/(8)] * [(r_{i}^{2}+r_{o}^{2}+[(r_{i}^{2}*r_{o}^{2})/r^{2}]-[(1+3*mu) / (3+mu)] * r^{2})]which gives a value of 190 at the bore (again divided by rho * omega

^{2}) - far above the value of 36 shown in the plot for q = 0.I've attached my EES code for reference.

Code:

```
{start}
function s_r(r )
$Common mu, omega, rho,q, n1, n2, c1 , c2
s_r = c1*r^n1 + c2*r^n2 - (r^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
end
function s_t(r )
$Common mu, omega, rho,q, n1, n2, c1 , c2
s_r_prime = c1*n1*r^(n1-1) + c2*n2*r^(n2-1) - (2* r*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
s_t = s_r_prime* r + r^2 * rho * omega^2/ (386.4 [lbm-in/lbf-s^2])
end
function u(r )
$Common mu, omega, rho,q, n1, n2, c1 , c2, E
u = r * (s_t(r) - mu * s_r(r)) / E
end
n1 = -0.5 * q + (0.25*q^2 + mu * q + 1)^0.5
n2 = -0.5 * q - (0.25*q^2 + mu * q + 1)^0.5
Factor_Safety = Strength_Yield / s_t(r_i)
{book example - hyperbolic profile steel disk}
Strength_Yield = 11500 [lbf / in^2]
E = 30e6 [lbf / in^2]
rho = 0.28 [lbm / in^3]
r_i = 3 [in]
r_o = 15 [in]
mu = 0.3
omega =632 [1/sec]
q=1
0 = c1*r_i^n1 + c2*r_i^n2 - (r_i^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q) * 386.4 [lbm-in/lbf-s^2])
0 = c1*r_o^n1 + c2*r_o^n2 - (r_o^2*(3+mu)*rho * omega^2)/((8-(3+mu)*q)* 386.4 [lbm-in/lbf-s^2])
{max tangential stress @ r_i and max hoop stress since s_r_r_i is 0}
s_t_i = s_t(r_i)
s_t_o = s_t(r_o)
s_r_i = s_r(r_i)
s_r_o = s_r(r_o)
{s_t_r = s_t(r) * 386.4 [lbm-in/lbf-s^2]/ (r_o^2 * rho * omega^2)
s_r_r = s_r(r) * 386.4 [lbm-in/lbf-s^2]/ (r_o^2 * rho * omega^2)
t = t_i/(r^q)
t_i = 15}
bore_displacement = u(r_i)
tip_displacement = u(r_o)
{bore_displacementa = u(r_i) * E / (rho * omega^2)}
```

Am I mis-interpreting the equations or what??

Thanks for any help in advance,

-Jeff