
#1
Feb2212, 03:44 PM

P: 3

1. The problem statement, all variables and given/known data
Essentially the problem that I am trying to solve is the same as in this topic except that it is for 3 springs and 3 masses http://www.physicsforums.com/showthread.php?t=299905 2. Relevant equations I have found similar equations as in the topic but I face a problem in describing the potential energy due to gravity for the system. The eulerlagrange equations aren't supposed to have any constants but due to the PE I find that I have constants in my eulerlagrange equations. How do you define the gravitational potential energy for a vertical two spring system? Thank you all in advance 



#2
Feb2212, 04:31 PM

PF Gold
P: 3,173





#3
Feb2212, 04:36 PM

P: 3

mg(l+x_1 )+mg(2l+x_2 )+mg(3l+x_3 ) where x_1 is the closest to the ceiling mass and l is the length of each spring at rest. We consider all springs to have the same length. Somehow I feel that the potential I found is not quite right 



#4
Feb2212, 04:59 PM

PF Gold
P: 3,173

Lagrangian Mechanics for two springs (revisited)Notice that in your Lagrangian must also appear the potential energy of each spring (that you must express in terms of your generalized coordinates x_1, x_2 and x_3). 



#5
Feb2212, 06:59 PM

PF Gold
P: 3,173

Hey guys, so I'm stealing this problem because I study the same course. I put my work into a spoiler so that the OP does not cheat if he doesn't want to. But I need help.
Spoiler
My Lagrangian is [itex]\frac{m}{2}(\dot x_1 ^2 + \dot x_2 ^2 + \dot x _3 ^2)+\frac{k}{2}[2x_2^2+x_1^2+x_3^22(x_1x_2+x_2x_3)][/itex].
Using EulerLagrange equations, I reach the equations of motion: (1)[itex]m\ddot x_1 +k (x_2x_1)=0[/itex]. (2)[itex]m\ddot x_3 +k (x_2x_3)=0[/itex]. (3)[itex]m\ddot x_2+k (x_3+x_12x_2)=0[/itex]. At first glance they look reasonable in my opinion. Now I must find the normal modes and frequencies (and the equilibrium positions too). I do not know how to do any of these, which I know is very important. Hmm to get the equilibrium position I guess I must solve the system of DE's and find the stationary solutions, but this doesn't look like a piece of cake. No idea about the others. I'd appreciate any help. 



#6
Feb2312, 10:58 AM

P: 3

Can I ask you something?I find the same Lagrangian but the only difference is that the coefficient of x_1^2 is 3 and instead of the product x_1x_2 I found the product x_3x_1



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