3D pendulum-spring-damper (Lagrange)

[PhMichael]

[PLAIN]http://img337.imageshack.us/img337/3623/pensp.jpg

Homework Statement

I need to find the equations of motions via Lagrange's formulation when the generalized coordinates are:
$$\vec{q}=[x,y,z]^T$$2. The attempt at a solution

I need to verify whether what I obtained so far is true or not.

The position vector if the bob (from the support) is:$$\vec{r}=x\hat{e}_{1}+y\hat{e}_2+(L-z)\hat{e}_{3}$$

And the velocity vector is: $$\vec{v}=\dot{x}\hat{e}_{1}+\dot{y}\hat{e}_{2}-\dot{z}\hat{e}_{3}$$

The Kinetic energy is: $$T=\frac{1}{2}mv^{2}=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})$$

The dissipation function is: $$D=\frac{1}{2}C(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})$$

The potential energy (gravitational energy is taken with respect to the support) is: $$V=-mg(L-z)+\frac{1}{2}k\left ( \sqrt{x^{2}+y^{2}+(L-z)^{2}}-l_{0} \right )^{2}$$

The generalized forces (given in the original question): $$Q_{x}=F_{x}$$ , $$Q_{y}=F_{y}$$ , $$Q_{z}=0$$

Each of these expression will be substituted into Lagrange's equation of motion:

$$\frac{d}{dt}\left ( \frac{\partial T}{\partial \dot{q_j}} \right )-\frac{\partial T}{\partial q_j}+\frac{\partial V}{\partial q_j}+\frac{\partial D}{\partial \dot{q_j}}=Q_j$$

Are those expressions, especially that one of the potential energy, correct?

Thanks!

 

[PhMichael]

anyone? =/