Strength Analysis of Composite Grinding Wheels
Abstract Generated abstract
This paper develops a strength calculation for rotating composite grinding wheels, modeled as thick walled rings subjected to inertial stresses. Starting from standard elastic solutions for radial and tangential stresses in rotating disks, it first reviews the case of a homogeneous ceramic bonded wheel, then extends the formulation to a wheel consisting of a metal inner ring and an abrasive outer ring joined by a thin adhesive layer. By neglecting the adhesive thickness and imposing equality of radial joint stresses together with deformation compatibility at the interface, the paper derives expressions for the interface radial stress and for radial and tangential stresses in both rings. The resulting formulas are presented as a corrected version of earlier similar calculations, with the author noting that a prior derivation contained an error capable of significantly affecting computed stress magnitudes.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV,
ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR
Volume 263 1975
STRENGTH CALCULATION OF COMPOSITE GRINDING WHEELS
V. A. Govorukhin
(Presented by the scientific seminar of the Department of Machine Tools and Metal Cutting
and the Department of Mechanical-Engineering Technology)
Any grinding wheel is a ring with a large wall thickness; therefore it may be regarded as a thick-walled ring. In the book [4], calculation equations are given for determining the radial \(\sigma_r\) and tangential \(\sigma_t\) stresses arising in a disk under the action of inertial forces during rotation of the disk:
\[ \sigma_r = B - \frac{A}{2r^2} - \frac{3+\mu}{8g}\,\gamma\omega^2 r^2, \tag{1} \]
\[ \sigma_t = B + \frac{A}{2r^2} - \frac{1+3\mu}{8g}\,\gamma\omega^2 r^2, \tag{2} \]
where
\(B\) and \(A\) are integration constants determined by substituting the boundary conditions into equations (1) and (2);
\(r\) is the current radius;
\(\mu\) is Poisson’s ratio of the disk material;
\(g\) is the acceleration due to gravity;
\(\gamma\) is the specific weight of the disk material;
\(\omega\) is the angular velocity of rotation of the disk.
Equations (1) and (2) are valid only for materials obeying Hooke’s law. In works [1, 2] it is shown that for grinding wheels on a ceramic bond there exists a linear dependence between load and deformation up to the destruction of the wheels. Consequently, for the strength calculation of grinding wheels on a ceramic bond, it is legitimate to use the indicated equations.
If we consider a grinding wheel without the action on it of an external load from cutting forces, then on its outer surface of radius \(r_2\) and inner surface of radius \(r_1\) the radial stresses \(\sigma_{r1}\) and \(\sigma_{r2}\) must be equal. If, taking this circumstance into account, the integration constants \(A\) and \(B\) in equations (1) and (2) are determined, then the latter take the form
\[ \sigma_r = C\left(1+a_2^2-\frac{a_2^2}{a^2}-a^2\right), \tag{3} \]
\[ \sigma_t = C\left(1+a_2^2+\frac{a_2^2}{a^2}-D\cdot a^2\right), \tag{4} \]
where
\[ x=\frac{r}{r_2}, \qquad x_1=\frac{r_1}{r_2}, \]
\[ C=\frac{3+\mu}{8g}\,\gamma\,(r_2\cdot\omega)^2, \qquad D=\frac{1+3\mu}{3+\mu}. \]
Equations (3) and (4) make it possible to calculate the radial and tangential stresses at any point of a rotating wheel. The calculation shows that the tangential stress on the inner surface of the wheel reaches a maximum value and greatly exceeds in magnitude the radial stress, which reaches its maximum value at
\[ x_2=\sqrt{\frac{r_1}{r_2}}. \]
Failure of the wheel occurs at the moment when the tangential stresses, as the greatest ones, at some point of it, according to the first theory of strength, reach the strength limit of the wheel material.
A composite grinding wheel consists of a metal ring on which an abrasive ring is fastened by means of an adhesive composition up to 1 mm thick.
Let us denote
\[ \begin{aligned} r_1 &\text{— inner radius of the metal ring,}\\ r_{1\mathrm{H}} &\text{— outer radius of the metal ring,}\\ r_{2\mathrm{B}} &\text{— inner radius of the abrasive ring,}\\ r_2 &\text{— outer radius of the abrasive ring.} \end{aligned} \]
Equations (1) and (2) also make it possible, for a composite grinding wheel, to determine the constants of integration and, consequently, the radial and tangential stresses, if the boundary conditions are substituted into them.
For the inner ring,
if
\[ r=r_1, \qquad \text{then } \sigma_{r1}=0, \]
\[ r=r_{1\mathrm{H}}, \qquad \text{then } \sigma_{r1\mathrm{H}}=\sigma_1. \]
For the outer ring,
if
\[ r=r_{2\mathrm{B}}, \qquad \text{then } r_{r2\mathrm{B}}=\sigma_2, \]
\[ r=r_2, \qquad \text{then } \sigma_{r2}=0. \]
Here \(\sigma_1\) and \(\sigma_2\) represent the radial stresses at the adhesive joint from the side of the inner and outer rings.
For the inner ring,
\[ A_1=2r_{1\mathrm{H}}^2\cdot r_1^2\left(-\frac{\sigma_1}{r_{1\mathrm{H}}^2-r_1^2}+C_1\cdot\omega^2\right), \]
\[ B_1=\sigma_1\cdot\frac{r_{1\mathrm{H}}^2}{r_{1\mathrm{H}}^2-r_1^2} +C_1\,(r_{1\mathrm{H}}^2+r_1^2)\cdot\omega^2. \]
For the outer ring,
\[ A_2=2r_2^2\cdot r_{2\mathrm{B}}^2\left(-\frac{\sigma_2}{r_2^2-r_{2\mathrm{B}}^2}+C_2\cdot\omega^2\right), \]
\[ B_2=-\sigma_2\cdot\frac{r_{2\mathrm{B}}^2}{r_2^2-r_{2\mathrm{B}}^2} +C_2\,(r_2^2+r_{2\mathrm{B}}^2)\cdot\omega^2; \]
where
\[ C_1=\frac{3+\mu_1}{8g}\,\gamma_1,\qquad D_1=\frac{1+3\mu_1}{8g}\,\gamma_1, \]
\[ C_2=\frac{3+\mu_2}{8g}\,\gamma_2,\qquad D_2=\frac{1+3\mu_2}{8g}\,\gamma_2, \]
\(\mu_1, \mu_2\) are the Poisson’s ratios for the inner and outer rings, respectively,
\(\gamma_1, \gamma_2\) are the specific weights of the material of the inner and outer rings, respectively.
For a given wheel design, all the quantities entering equations (1) and (2) are known, except for \(\sigma_1\) and \(\sigma_2\), which depend on the thickness of the adhesive layer and the modulus of elasticity of the adhesive composition. However, if the effect of the thin adhesive layer is neglected, then the stresses \(\sigma_1\) and \(\sigma_2\) may be set equal:
\[ \sigma_1=\sigma_2=\sigma. \]
But even after this, three unknown quantities \((\sigma_r, \sigma_t, \sigma)\) remain in two equations.
The third equation is obtained from the compatibility of deformations [3].
The relative tangential elongation of the material on the outer surface of the inner ring is
\[ \varepsilon_{t1\mathrm{H}}=\frac{\Delta r_{1\mathrm{H}}}{r_{1\mathrm{H}}} =\frac{1}{E_1}\left(\sigma_{t1\mathrm{H}}-\mu_1\cdot\sigma_1\right), \]
and on the inner surface of the outer ring
\[ \varepsilon_{t2\mathrm{B}}=\frac{\Delta r_{2\mathrm{B}}}{r_{2\mathrm{B}}} =\frac{1}{E_2}\left(\sigma_{t2\mathrm{B}}-\mu_2\cdot\sigma_2\right), \]
where \(E_1\) and \(E_2\) are the moduli of elasticity of the material of the inner and outer rings.
Assuming equality \(\varepsilon_{t1\mathrm{H}}=\varepsilon_{t2\mathrm{B}}\) in view of \(r_{1\mathrm{H}}\simeq r_{2\mathrm{B}}\), we obtain
\[ \sigma=\frac{\sigma_{t2\mathrm{B}}\cdot E_1-\sigma_{t1\mathrm{H}}\cdot E_2} {E_1\cdot\mu_2-E_2\cdot\mu_1}. \tag{5} \]
Let us calculate the stresses \(\sigma_{t2\mathrm{B}}\) and \(\sigma_{t1\mathrm{H}}\) from equation (2):
\[ \sigma_{t1\mathrm{H}}=\sigma\left(\frac{r_{1\mathrm{H}}^2+r_1^2}{r_{1\mathrm{H}}^2-r_1^2}\right) +2C_1\cdot r_1^2+r_{1\mathrm{H}}^2(C_1-D_1), \]
\[ \sigma_{t2\mathrm{B}}=-\sigma\left(\frac{r_2^2+r_{2\mathrm{B}}^2}{r_2^2-r_{2\mathrm{B}}^2}\right) +2C_2\cdot r_2^2+r_{2\mathrm{B}}^2(C_2-D_2) \]
and substitute them into equation (5). Then
\[ \sigma= \frac{ V^2\cdot E_1\cdot\gamma_2\cdot\left[(3+\mu_2)+\alpha_2^2(1-\mu_2)\right] - E_2\cdot\gamma_1\cdot\left[(3+\mu_1)\alpha^2+\alpha_2^2(1-\mu_1)\right] }{ 4gE_1\left(\mu_2+\frac{1+\alpha_2^2}{1-\alpha_2^2}\right) + 4\cdot g\cdot E_2\cdot\left(\frac{1+\alpha_1^2}{1-\alpha_1^2}-\mu_1\right) }, \]
where
\[ \alpha_1=\frac{r_1}{r_{1\mathrm{H}}},\qquad \alpha_2=\frac{r_{2\mathrm{B}}}{r_2},\qquad \alpha=\frac{r_1}{r_2}. \]
Introduce the notation:
\[ M_1=E_2\cdot\gamma_1\cdot\left[(3+\mu_1)\cdot\alpha^2+\alpha_2^2(1+\mu_1)\right], \]
\[ M_2=E_1\cdot\gamma_2\cdot\left[(3+\mu_2)+\alpha_2^2(1-\mu_2)\right]; \]
\[ N_1=4g\cdot E_2\left(\frac{1+\alpha_1^2}{1-\alpha_1^2}-\mu_1\right), \]
\[ N_2=4g\cdot E_1\left(\frac{1+\alpha_2^2}{1-\alpha_2^2}+\mu_2\right). \]
Then the radial stress in the adhesive joint will be equal to
\[ \sigma = V^2\frac{M_2-M_1}{N_2+N_1}. \tag{6} \]
Finally, the radial and tangential stresses in the inner and outer rings are determined from the equations:
\[ \sigma_{t1}=V^2\cdot\left\{ \frac{M_2-M_1}{N_2+N_1}\cdot \left[ \frac{1}{1-\alpha_1^2}+ \frac{r_1^2}{r^2(1-\alpha_1^2)} \right]+ C_1\left(\alpha_2^2+\alpha^2+\frac{r_1^2}{r^2}\alpha_2^2\right) -D_1\frac{r^2}{r_2^2} \right\}, \tag{7} \]
\[ \sigma_{r1}=V^2\cdot\left\{ \frac{M_2-M_1}{N_2+N_1}\cdot \left[ \frac{1}{1-\alpha_1^2}- \frac{r_1^2}{r^2(1-\alpha_1^2)} \right]+ C_1\left(\alpha_2^2+\alpha^2-\frac{r^2}{r_2^2}-\frac{r_1^2}{r^2}\alpha_2^2\right) \right\}, \tag{8} \]
\[ \sigma_{t2}=V^2\left\{ -\frac{M_2-M_1}{N_2+N_1}\cdot \left[ \frac{\alpha_2^2}{1-\alpha_2^2}+ \frac{r_{2B}^2}{r^2(1-\alpha_2^2)} \right]+ C_2\left(1+\alpha_2^2+\frac{r_{2B}^2}{r^2}\right) -D_2\cdot\frac{r^2}{r_2^2} \right\}, \tag{9} \]
\[ \sigma_{r2}=V^2\left\{ \frac{M_2-M_1}{N_2+N_1}\cdot \left[ \frac{r_{2B}^2}{r^2(1-\alpha_2^2)}- \frac{\alpha_2^2}{1-\alpha_2^2} \right]+ C_2\left(1+\alpha_2^2-\frac{r^2}{r_2^2}-\frac{r_{2B}^2}{r^2}\right) \right\}. \tag{10} \]
The equations obtained for calculating the stresses are not fundamentally new. In a similar form they were obtained in work [2]. However, in deriving them the authors made an error, which in some cases significantly affects the absolute magnitude of the stresses.
REFERENCES
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G. M. Ippolitov. Abrasive-Diamond Machining. Moscow, “Mashinostroenie,” 1969.
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E. O. Kienzle, H. I. Grasemann, K. Grüning. Wege zur Erhöhung der Umfangsgeschwindigkeit von Schleifscheiben, “VDI Zeitschrift,” No. 26, 1963.
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Machine-Builder’s Handbook. Vol. 3, Moscow, Mashgiz, 1962.
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N. M. Belyaev. Strength of Materials. Moscow, Gostekhizdat, 1958.