(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

1D spring-mass damper equation

n-D spring-mass damper equation

3. The attempt at a solution

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

(1) m[tex]\ddot{x}[/tex] + b[tex]\dot{x}[/tex] + kx = 0

Where:

m[tex]\ddot{x}[/tex] = F_{acc}= m[itex]\frac{d^{2}x}{dt^{2}}[/itex]

b[tex]\dot{x}[/tex] = F_{damp}(viscous friction) = C[itex]\frac{dx}{dt}[/itex]

kx = F_{potential}(gradient of some potential) = [tex]\nabla[/tex]P(x)

So the equation becomes:

(2) -[tex]\nabla[/tex]P(x) - C[itex]\frac{dx}{dt}[/itex] = m[itex]\frac{d^{2}x}{dt^{2}}[/itex]

I then modeled the total energy as follows:

(3) E_{tot}= P(x) + [itex]\frac{1}{2}[/itex]m[tex]\dot{x}[/tex]^{2}

and

(4) [itex]\frac{dE}{dt}[/itex]= [tex]\nabla[/tex]P(x)[tex]\dot{x}[/tex] + m[tex]\dot{x}[/tex][tex]\ddot{x}[/tex]

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

[itex]\frac{dE}{dt}[/itex]= [tex]\nabla[/tex]P(x)[tex]\dot{x}[/tex] + [tex]\dot{x}[/tex](-[tex]\nabla[/tex]P(x) - C[tex]\dot{x}[/tex])

=> -C[tex]\dot{x}[/tex]^{2}

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

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# Homework Help: N-D spring-mass damper total energy

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