# Hyperbolic disk centrifugal stress calculation

• jamorga37
In summary, a hyperbolic disk centrifugal stress calculation is a scientific method used to determine the maximum stress on a spinning disk based on its diameter and rotational speed. It is important for designing safe and durable spinning disks and involves factors such as material properties and external forces. This calculation is performed using mathematical equations and has applications in industries such as aerospace, automotive, and manufacturing, as well as in the study of material strength and development of new technologies.
jamorga37

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) = ti/(r$$^{q}$$) where r is radius.
q = 1 shows a "ordinary hyperabola"
formula 48:
n1,n2 = -0.5* q +/- sqrt(0.25*q2 + mu*q + 1) where mu is poissons ratio

formula 49 after dividing t*r out is:
radial stress (sr) = c1*rn1 + c2*rn2 - [(3+mu)*rho*omega2*r2/(8-(3+mu)*q)] where omega is rotational velocity in radians per unit time, and rho is disk density
tangential stress (st) = r * sr' + rho * omega2 * r2

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: st (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*omega2) 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:
st=rho*omega2 * [(3+v)/(8)] * [(ri2 +ro2+[(ri2*ro2)/r2]-[(1+3*mu) / (3+mu)] * r2)]

which gives a value of 190 at the bore (again divided by rho * omega2) - 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

#### Attachments

• hyperbolic disk stress profile.JPG
32.4 KB · Views: 548
Answer:It looks like the equations you are using are correct, but the results you are getting are not. It could be because of a mistake in the calculation or an issue with the parameters used. You should double-check your calculations and make sure all the parameters you used are correct. If that doesn't help, then you may need to look for other equations or methods to calculate the stress in a hyperbolic disk.

## 1. What is a hyperbolic disk centrifugal stress calculation?

A hyperbolic disk centrifugal stress calculation is a scientific method used to determine the amount of stress or force exerted on a spinning disk due to centrifugal force. This calculation takes into account the diameter and rotational speed of the disk to determine the maximum stress that it can withstand before failing.

## 2. Why is hyperbolic disk centrifugal stress calculation important?

Hyperbolic disk centrifugal stress calculation is important because it allows scientists and engineers to design and manufacture durable and safe spinning disks. By understanding the maximum stress that a disk can handle, they can ensure that it can withstand the forces it will experience during operation and avoid potential failures or accidents.

## 3. What factors are involved in a hyperbolic disk centrifugal stress calculation?

The main factors involved in a hyperbolic disk centrifugal stress calculation are the diameter and rotational speed of the disk. Other factors that may also be considered include the material properties of the disk, such as its density and strength, as well as any external forces acting on the disk.

## 4. How is a hyperbolic disk centrifugal stress calculation performed?

To perform a hyperbolic disk centrifugal stress calculation, a scientist or engineer will use mathematical equations and formulas, such as the stress-strain relationship and the centrifugal force equation, to determine the maximum stress on the disk. These calculations may be done by hand or using specialized software programs.

## 5. What are the applications of hyperbolic disk centrifugal stress calculation?

Hyperbolic disk centrifugal stress calculation has many applications in various fields, such as aerospace, automotive, and manufacturing industries. It is used to design and test spinning disks for various purposes, such as turbines, engines, and centrifuges. It is also important in the study of material strength and durability, as well as in the development of new materials and technologies.

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