Equation of motion of a system

I am studying the problem in Vibration effects. I would like to obtain the simplified equation of motion in (2.7).

\[Lambda] =Sqrt[l^2 + a^2 (3 Cos[\[Phi]] - Sqrt Sin[\[Phi]])];\[Lambda]ang = Simplify[\[Lambda] /. \[Phi] -> \[Theta] + 5/6 Pi]


and

h = Sqrt[ l^2 - 2 a^2 (1 - Cos[\[Phi]])]; hang = Simplify[h /. \[Phi] -> \[Theta] + 5/6 Pi]


Elastic energy in (2.3) can be written as:

U = 3/2 k (\[Lambda]ang - \[Lambda]n)^2


How can I write the Kinetic energy in (2.4)? Moreover, How can I write the equations in (2.5) and (2.6)?

You can do as follows: Here are the definitions:

  \[Lambda][x_] := Sqrt[L^2 - 2 Sqrt a^2*Cos[x]];
\[CapitalPhi][\[Theta]_] :=
3/2 \[Kappa] (\[Lambda][\[Theta][t]] - \[Lambda]N)
\[CapitalDelta][t_] := 3*\[Gamma]*D[\[Lambda][\[Theta][t]], t]^2
h[x_] := Sqrt[L^2 - a^2*(2 + Sqrt*Cos[x] + Sin[x])];


Now - the definition of the energy:

k[t_] := 3/2*m*a^2*\[Theta][t]^2 + 3/2*m*D[h[\[Theta][t]], t]^2;
k[t]


yielding: Now let us just substitute the above definitions into the formula (2.6):

(\[Theta]''[t]*D[k[t], \[Theta]'[t]] + \[Theta]'[t]*
D[k[t], \[Theta][t]]) + \[Theta]'[t]*
D[\[CapitalPhi][\[Theta][t]], \[Theta][t]] + \[CapitalDelta][
t] == 0 // Simplify


this gives the following heavy formula: Have fun!

• In your definitions, I found some discrepancies: [CapitalPhi][[Theta]_] := 3/2 [Kappa] ([Lambda][[Theta][t]] - [Lambda]N)^2...is it correct? – Gae P Oct 16 '18 at 14:46
• Considering (2.4), the definition of the energy should be: k[t_] := 3/2*ma^2*D[[Theta][t], t]^2 + 3/2*mD[h[[Theta][t]], t]^2...is it correct? – Gae P Oct 16 '18 at 15:51
• If you find any errors - correct them. I gave you an idea. The rest is up to you. – Alexei Boulbitch Oct 16 '18 at 18:45