This sort of derivation is done by differentiating the coupled equations and then substituting back in the original, undifferentiated equations to eliminate the undesired variables.

In this case, if I was asked to reproduce the last equation, I would:

• Isolate $q$ from the first equation.
• Differentiate this equation twice to find expressions for $\partial^2 q/\partial t^2$ and $\partial^2 q/\partial x^2$ in terms of the derivatives of $w$ and $\phi$.
• Write down the right-hand side of the desired equation in terms of the derivatives of $w$ and $\phi$.
• Cancel out as many terms as I can and hope that the I don't make a sign error somewhere.

Note that for a homogenous and isotropic beam, all of the beam properties are constant with respect to $x$, which makes the derivatives simply somewhat.

This sort of derivation is done by differentiating the coupled equations and then substituting the results back into the original, undifferentiated equations to eliminate the undesired variables.

In this case, if I were asked to reproduce the last equation, I would:

• Isolate $q$ from the first equation.
• Differentiate this equation twice to find expressions for $\partial^2 q/\partial t^2$ and $\partial^2 q/\partial x^2$ in terms of the derivatives of $w$ and $\phi$.
• Write down the right-hand side of the desired equation in terms of the derivatives of $w$ and $\phi$.
• Cancel out as many terms as I can and hope that I don't make a sign error somewhere.

Note that for a homogenous and isotropic beam, all of the beam properties are constant with respect to $x$, which simplifies the differentiation somewhat.