- #1
PhMichael
- 134
- 0
[PLAIN]http://img337.imageshack.us/img337/3623/pensp.jpg
I need to find the equations of motions via Lagrange's formulation when the generalized coordinates are:
[tex]\vec{q}=[x,y,z]^T[/tex]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:[tex] \vec{r}=x\hat{e}_{1}+y\hat{e}_2+(L-z)\hat{e}_{3} [/tex]
And the velocity vector is: [tex]\vec{v}=\dot{x}\hat{e}_{1}+\dot{y}\hat{e}_{2}-\dot{z}\hat{e}_{3}[/tex]
The Kinetic energy is: [tex]T=\frac{1}{2}mv^{2}=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})[/tex]
The dissipation function is: [tex]D=\frac{1}{2}C(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})[/tex]
The potential energy (gravitational energy is taken with respect to the support) is: [tex]V=-mg(L-z)+\frac{1}{2}k\left ( \sqrt{x^{2}+y^{2}+(L-z)^{2}}-l_{0} \right )^{2}[/tex]
The generalized forces (given in the original question): [tex]Q_{x}=F_{x}[/tex] , [tex]Q_{y}=F_{y}[/tex] , [tex]Q_{z}=0[/tex]
Each of these expression will be substituted into Lagrange's equation of motion:
[tex]\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[/tex]
Are those expressions, especially that one of the potential energy, correct?
Thanks!
Homework Statement
I need to find the equations of motions via Lagrange's formulation when the generalized coordinates are:
[tex]\vec{q}=[x,y,z]^T[/tex]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:[tex] \vec{r}=x\hat{e}_{1}+y\hat{e}_2+(L-z)\hat{e}_{3} [/tex]
And the velocity vector is: [tex]\vec{v}=\dot{x}\hat{e}_{1}+\dot{y}\hat{e}_{2}-\dot{z}\hat{e}_{3}[/tex]
The Kinetic energy is: [tex]T=\frac{1}{2}mv^{2}=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})[/tex]
The dissipation function is: [tex]D=\frac{1}{2}C(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})[/tex]
The potential energy (gravitational energy is taken with respect to the support) is: [tex]V=-mg(L-z)+\frac{1}{2}k\left ( \sqrt{x^{2}+y^{2}+(L-z)^{2}}-l_{0} \right )^{2}[/tex]
The generalized forces (given in the original question): [tex]Q_{x}=F_{x}[/tex] , [tex]Q_{y}=F_{y}[/tex] , [tex]Q_{z}=0[/tex]
Each of these expression will be substituted into Lagrange's equation of motion:
[tex]\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[/tex]
Are those expressions, especially that one of the potential energy, correct?
Thanks!
Last edited by a moderator: