N-D spring-mass damper total energy

[ashapi]

Homework Statement

I am trying to derive the total energy for an n-dimensional linear (Hookeian) spring-mass damper system and show that the total energy is either decreasing or constant.

Homework Equations

1D spring-mass damper equation
n-D spring-mass damper equation

The Attempt at a Solution

I started with the following force equation in 1D to describe the system:

(1) m\ddot{x} + b\dot{x} + kx = 0

Where:
m\ddot{x} = F_acc = m\frac{d^{2}x}{dt^{2}}
b\dot{x} = F_damp (viscous friction) = C\frac{dx}{dt}
kx = F_potential (gradient of some potential) = \nablaP(x)

So the equation becomes:

(2) -\nablaP(x) - C\frac{dx}{dt} = m\frac{d^{2}x}{dt^{2}}

I then modeled the total energy as follows:

(3) E_tot = P(x) + \frac{1}{2}m\dot{x}²

and

(4) \frac{dE}{dt}= \nablaP(x)\dot{x} + m\dot{x}\ddot{x}

In (4) I substituted m\ddot{x} from (1) and got:

\frac{dE}{dt}= \nablaP(x)\dot{x} + \dot{x}(-\nablaP(x) - C\dot{x})

=> -C\dot{x}²

...which shows that total energy is either decreasing (|\dot{x}| > 0) or constant (\dot{x} = 0)

I then tried to derive the total energy for the n-dimensional case and relate it to an n-D force equation as I did above. To do this I looked at the 2D case and then generalized the formulas for kinetic and potential energy to n-D but I'm not sure if this was correct. Here are my kinetic and potential energy equations for 2D:

 

[ashapi]

My latex seems to bug out sometimes so if it looks weird please click on the source but let me continue...

KE_tot,2D = \frac{1}{2}(m₁₁\dot{x}₁² + m₂₂\dot{x}₂² + m₁₂\dot{x}₁\dot{x}₂)PE_tot = \frac{1}{2}(m₁₁{x}₁² + m₂₂{x}₂² + m₁₂{x}₁{x}₂)

I took the generalized forms to be:

KE_tot,nD = \sum_{i=1}^{n} (m_ii)\dot{x}_i² + \sum_{j=1}^{n-1} \sum_{k=j+1}^n (m_jk)\dot{x}_j\dot{x}_k

...and the same for PE with {x} replacing \dot{x}.

My plan was to take the derivatives of these and replace the m\ddot{x} terms with the force equivalences (as above). Is this right? I also can't think of how to make a generalized form of the force equation for an n-D vector.

Any help would be greatly appreciated.

Thanks,

ashapi