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- Determining coefficients from an equation with 3 variables

Hi,

Some of the background related to this question is in this thread, but I've got another question as I'm looking at another problem that has come up with the same application which I'm trying to solve using the equation of meshing for a worm gear and the cutting/grinding tool that creates it. This is taken from the paper "Contact Characteristics of Recess Action Worm Gear Drives With Double-Depth Teeth", which you can view here, specifically Equation 16:

$$f(l_1,\theta_1,\phi_1(\phi_2))=\omega_1{[(m_21 cos \gamma_1-1) Y_1+m_21(cos \phi_1 sin \gamma_1 Z_1 + sin \phi_1 cos \gamma_1 C_1)]N_x1+[-(m_21 sin \gamma_1-1) X_1+m_21(-sin \phi_1 sin \gamma_1 Z_1 + cos \phi_1 cos \gamma_1 C_1)]N_y1+[m_21 sin \gamma_1(-cos \phi_1 X_1 + sin \phi_1 Y_1 + C_1)]N_z1}=0$$

This has 3 variables in it - ##l_1##, ##\theta_1##, ##\phi_1(\phi_2)## - now, some of the notation is not familiar to me since it's been about 20 years since I last studied math(s), but I've found that I may be able to use Cramer's Rule to solve for the 3 variables if I can construct a 3x3 matrix of the coefficients and a 3x1 vector for the results.

What I also find strange is there's the ##f(l_1,\theta_1,\phi_1(\phi_2))## part but the main equation only references ##\theta_1##, but since this equation is created using other equations, that could cause the loss of ##l_1## and ##\theta_1## due to the substitutions and expansions I suppose?

So, how would I solve this equation, and how would I get the coefficients from it to use with Cramer's Rule? It looks to me like a matrix system of linear equations, but is it really?

Some of the background related to this question is in this thread, but I've got another question as I'm looking at another problem that has come up with the same application which I'm trying to solve using the equation of meshing for a worm gear and the cutting/grinding tool that creates it. This is taken from the paper "Contact Characteristics of Recess Action Worm Gear Drives With Double-Depth Teeth", which you can view here, specifically Equation 16:

$$f(l_1,\theta_1,\phi_1(\phi_2))=\omega_1{[(m_21 cos \gamma_1-1) Y_1+m_21(cos \phi_1 sin \gamma_1 Z_1 + sin \phi_1 cos \gamma_1 C_1)]N_x1+[-(m_21 sin \gamma_1-1) X_1+m_21(-sin \phi_1 sin \gamma_1 Z_1 + cos \phi_1 cos \gamma_1 C_1)]N_y1+[m_21 sin \gamma_1(-cos \phi_1 X_1 + sin \phi_1 Y_1 + C_1)]N_z1}=0$$

This has 3 variables in it - ##l_1##, ##\theta_1##, ##\phi_1(\phi_2)## - now, some of the notation is not familiar to me since it's been about 20 years since I last studied math(s), but I've found that I may be able to use Cramer's Rule to solve for the 3 variables if I can construct a 3x3 matrix of the coefficients and a 3x1 vector for the results.

What I also find strange is there's the ##f(l_1,\theta_1,\phi_1(\phi_2))## part but the main equation only references ##\theta_1##, but since this equation is created using other equations, that could cause the loss of ##l_1## and ##\theta_1## due to the substitutions and expansions I suppose?

So, how would I solve this equation, and how would I get the coefficients from it to use with Cramer's Rule? It looks to me like a matrix system of linear equations, but is it really?