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

1. May 2, 2010

### ashapi

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$$\ddot{x}$$ + b$$\dot{x}$$ + kx = 0

Where:
m$$\ddot{x}$$ = Facc = m$\frac{d^{2}x}{dt^{2}}$
b$$\dot{x}$$ = Fdamp (viscous friction) = C$\frac{dx}{dt}$
kx = Fpotential (gradient of some potential) = $$\nabla$$P(x)

So the equation becomes:

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

I then modeled the total energy as follows:

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

and

(4) $\frac{dE}{dt}$= $$\nabla$$P(x)$$\dot{x}$$ + m$$\dot{x}$$$$\ddot{x}$$

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

$\frac{dE}{dt}$= $$\nabla$$P(x)$$\dot{x}$$ + $$\dot{x}$$(-$$\nabla$$P(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:

2. May 2, 2010

### ashapi

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

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

PEtot = $\frac{1}{2}$(m11$${x}$$12 + m22$${x}$$22 + m12$${x}$$1$${x}$$2)

I took the generalized forms to be:

KEtot,nD = $\sum_{i=1}^{n}$ (mii)$$\dot{x}$$i2 + $\sum_{j=1}^{n-1}$ $\sum_{k=j+1}^n$ (mjk)$$\dot{x}$$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

Last edited: May 2, 2010