# Equations of motion for two-mass torsional oscillator with the gear train

This is my first topic and I continue work on that: Lagrangian of three-mass system with Mathematica

I found interesting problem here, and try reproduce results.

Assumption: $$d_1=0$$

Algorithm:

1. Write Lagrangian:

$$L=W_k-W_p=\frac{J_1 \omega_1^2}{2}+\frac{J_2 (\frac{r_2}{r_1}\omega_2)^2}{2}-\frac{c_1(\phi_1-\frac{r_2}{r_1}\phi_2)^2}{2}$$

where $$W_k$$ and $$W_n$$ - kinetic and potential energy.

1. Using formula:

find equation of motion for coordinates:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{\phi_1}}-\frac{\partial L}{\partial \phi_1}=0$$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{\phi_2}}-\frac{\partial L}{\partial \phi_2}=0$$

I implemented all this in Mathematica:

Clear["Derivative"];

ClearAll["Global*"];

Remove[c, J, r];

L = Sum[1/2 Subscript[J, i] D[Subscript[\[Phi], i][t], t]^2, {i, 2}] -
Sum[1/2 Subscript[c,
10 i + i +
1] (Subscript[\[Phi], i][t] -
Subscript[\[Phi], i + 1][t])^2, {i, 1}];

L = -(1/2) Subscript[c,
12] (Subscript[\[Phi], 1][
t] - ((Subscript[r, 2]/Subscript[r, 1]) Subscript[\[Phi], 2][
t]))^2 +
1/2 Subscript[J, 1] Derivative[1][Subscript[\[Phi], 1]][t]^2 +
1/2 Subscript[J,
2] ((Subscript[r, 2]/Subscript[r, 1]) Derivative[1][
Subscript[\[Phi], 2]][t])^2;

eq1 = D[D[L, Derivative[1][Subscript[\[Phi], 1]][t]], t] -
D[L, Subscript[\[Phi], 1][t]]
eq2 = D[D[L, Derivative[1][Subscript[\[Phi], 2]][t]], t] -
D[L, Subscript[\[Phi], 2][t]]


But the results didn't match the picture. Where did I go wrong?

EDIT:

There is my code for three-mass system:

Clear["Derivative"]

ClearAll["Global*"]

Remove[c, J]

(***Gear ratios)

gr = {Subscript[n, 1], Subscript[n, 2]};

L = Sum[1/2 Subscript[J, i] D[Subscript[\[Phi], i][t], t]^2, {i, 3}] -
Sum[1/2 Subscript[c,
10 i + i +
1] (Subscript[\[Phi], i][t]/gr[[i]] -
Subscript[\[Phi], i + 1][t])^2, {i, 2}];

eq1 = D[D[L, Derivative[1][Subscript[\[Phi], 1]][t]], t] -
D[L, Subscript[\[Phi], 1][t]] == Subscript[T, 1][t] // Simplify;
eq20 = D[D[L, Derivative[1][Subscript[\[Phi], 2]][t]], t] -
D[L, Subscript[\[Phi], 2][t]] == Subscript[T, 2][t] // Simplify;
eq3 = D[D[L, Derivative[1][Subscript[\[Phi], 3]][t]], t] -
D[L, Subscript[\[Phi], 3][t]] == Subscript[T, 3][t] // Simplify;

eq2 = ApplySides[Expand[Subscript[n, 1]^2*#1] &,
eq20 /. Subscript[n, 1] -> 1/Subscript[n, 1]];


gr =.; (*Subscript[r, 2]/Subscript[r, 1]*)
igr =.; (*Subscript[r, 1]/Subscript[r, 2]*)
twist = -gr*Subscript[ϕ, 2][t] - Subscript[ϕ, 1][t];

L = Sum[(1/2)*Subscript[J, i]*D[Subscript[ϕ, i][t], t]^2, {i, 2}] -
(1/2)*Subscript[c, 1]*twist^2;

eq1 = D[D[L, Derivative[1][Subscript[ϕ, 1]][t]], t] - D[L, Subscript[ϕ, 1][t]] ==
Subscript[T, 1][t] + Subscript[d, 1]*D[twist, t]//Expand
eq20 = D[D[L, Derivative[1][Subscript[ϕ, 2]][t]], t] - D[L, Subscript[ϕ, 2][t]] ==
Subscript[T, 2][t] + Subscript[d, 1]*D[twist, t]*gr;
eq2 = ApplySides[Expand[igr^2*#1] & , eq20 /. gr -> 1/igr]


$$c_1 \text{gr} \phi _2(t)+c_1 \phi _1(t)+J_1 \phi _1''(t)=-d_1 \text{gr} \phi _2'(t)-d_1 \phi _1'(t)+T_1(t)\\c_1 \text{igr} \phi _1(t)+c_1 \phi _2(t)+\text{igr}^2 J_2 \phi _2''(t)=-d_1 \text{igr} \phi _1'(t)-d_1 \phi _2'(t)+\text{igr}^2 T_2(t)$$

• How can this approach be extended to the case of three-mass systems?
– dtn
Jun 28, 2021 at 15:52
• just for example: i.sstatic.net/qPodJ.gif
– dtn
Jun 28, 2021 at 16:08
• and it is natural that the gear ratios $N_1=\frac{\omega_1}{\omega_2}$ and $N_2=\frac{\omega_2}{\omega_3}$
– dtn
Jun 28, 2021 at 16:13
• It doesn't matter if it's two, three or more masses. Just add the additional terms to the Lagrangian and the generalized forces, and use the gear ratios in terms of the radii to eliminate the intermediate velocities. Jun 28, 2021 at 16:38
• Please add a figure to show what you mean by 'after the gearboxes'. ApplySides has been in the WL since 11.3. What is the output you are getting and is it the same as eq20? And what version are you using and which OS? Jun 28, 2021 at 22:22