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PhMichael
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I need to find the equations of motions via Lagrange's formulation when the generalized coordinates are:
\vec{q}=[x,y,z]^T2. 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!
Homework Statement
I need to find the equations of motions via Lagrange's formulation when the generalized coordinates are:
\vec{q}=[x,y,z]^T2. 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!
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