1. **Stating the problem:** We want to derive the equation of motion for a Timoshenko cantilever beam using Newton-Euler principles. 2. **Background:** The Timoshenko beam theory accounts for both bending and shear deformations, unlike Euler-Bernoulli beam theory which neglects shear deformation. 3. **Key variables:** - $w(x,t)$: transverse displacement - $\theta(x,t)$: rotation of the cross-section - $\rho$: density - $A$: cross-sectional area - $I$: second moment of area - $k$: shear correction factor - $G$: shear modulus - $E$: Young's modulus 4. **Newton-Euler principles:** - Apply Newton's second law for linear momentum: $$\rho A \frac{\partial^2 w}{\partial t^2} = \frac{\partial Q}{\partial x} + q(x,t)$$ where $Q$ is shear force and $q$ is distributed load. - Apply Euler's equation for rotational motion: $$\rho I \frac{\partial^2 \theta}{\partial t^2} = \frac{\partial M}{\partial x} - Q$$ where $M$ is bending moment. 5. **Constitutive relations:** - Shear force: $$Q = kGA \left(\theta - \frac{\partial w}{\partial x}\right)$$ - Bending moment: $$M = EI \frac{\partial \theta}{\partial x}$$ 6. **Substitute $Q$ and $M$ into equations:** - Linear momentum: $$\rho A \frac{\partial^2 w}{\partial t^2} = \frac{\partial}{\partial x} \left[kGA \left(\theta - \frac{\partial w}{\partial x}\right)\right] + q(x,t)$$ - Rotational motion: $$\rho I \frac{\partial^2 \theta}{\partial t^2} = \frac{\partial}{\partial x} \left(EI \frac{\partial \theta}{\partial x}\right) - kGA \left(\theta - \frac{\partial w}{\partial x}\right)$$ 7. **Final coupled equations of motion for Timoshenko beam:** $$\boxed{\begin{cases} \rho A \frac{\partial^2 w}{\partial t^2} = \frac{\partial}{\partial x} \left[kGA \left(\theta - \frac{\partial w}{\partial x}\right)\right] + q(x,t) \\ \rho I \frac{\partial^2 \theta}{\partial t^2} = \frac{\partial}{\partial x} \left(EI \frac{\partial \theta}{\partial x}\right) - kGA \left(\theta - \frac{\partial w}{\partial x}\right) \end{cases}}$$ These equations describe the dynamic behavior of a Timoshenko cantilever beam derived using Newton-Euler principles.