# Equations of motion for lever balance [duplicate]

What I am trying to do is to derive the equations of motion of a lever balance like the one in the picture

As can be seen the lever balance has achieved an static balance position, nevertheless it is not horizontal since its mass distribution will not allow it. Diagram of the system can be observed in next figure:

Where:

$$T_{1} = m_{1}*g*r_{1}*cos(\theta)$$

$$T_{2} = m_{2}*g*r_{2}*cos(\theta)$$

$$T_{d} = d\dot{\theta}$$

I defined the differential equation as:

$$J\ddot{\theta} = \frac{1}{J}*(m_{1}*r_{1}*g*cos(\theta) - m_{2}*r_{2}*g*cos(\theta) - d*\dot{\theta})$$

And implemented it on Matlab by using ODE45:

clc;clear all
tspan = 0:0.01:200;  %Time span

r1 = 0.03;                       % distance from pivot to m1 (m)
r2 = 0.02;                       % distance from pivot to m2 (m)
m1 = 0.02;                       % mass of m1(Kg)
m2 = 0.02;                       % mass of m2(Kg)
J = (m1*(r1^2)) + (m2*(r2^2));   % Moment of innertia
d = 0.01;                        % damping coeficcient

[t,x,U] = ode45(@(t,x)diffEq_simple2(x,J,r1,r2,m1,m2,d),tspan,x0);

theta = xd(:,1);
plot(t,theta);
axis([0 max(t) -180 180])
grid on;

function dx = diffEq_simple2(x,J,L1,L2,m1,m2,d)
g = 9.8;
dx(1)  =  x(2);
dx(2)  =  (1/J)*(m1*L1*g*cos(x(1)) - m2*L2*g*cos(x(1)) - d*x(2));
dx = dx';
end


After plotting the results of angle $$\theta$$ I get this:

It doesnt matter which value I assign to r2 or r1, the angle $$\theta$$ will rotate to the correct direction but will always converge to 90 degrees, which it is not correct, I was hopping the model to oscillate a bit but converge to a different value each time I change the value or r1,r2,m1 or m2, but it always converges to 90 degrees. Is my ode wrong derived ? which is the correct method to derive the equation of motion for a lever balance?

If an ideal frictionless lever as in the picture isn't balanced when it is flat, it won't be balanced at some angle $$\theta<\pi/2$$ to the horizontal either. This is because when the lever tilts, the torques exerted by the masses on both sides are both scaled by $$\cos{\theta}$$, so that any torque imbalance persists upon rotation of the lever.
The text passage describes a physics problem involving the derivation of the equations of motion for a lever balance and includes a discussion of the results obtained from a MATLAB simulation. The author questions whether the derived ordinary differential equation (ODE) is correct and seeks an explanation for why the angle always converges to 90 degrees.
single_question

the passage contains a single mathematical question regarding the correctness of the derived ode and the behavior of the lever balance system. no answer to this specific question is provided within the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Equations of motion for lever balance [duplicate]

What I am trying to do is to derive the equations of motion of a lever balance like the one in the picture

As can be seen the lever balance has achieved an static balance position, nevertheless it is not horizontal since its mass distribution will not allow it. Diagram of the system can be observed in next figure:

Where:

$$T_{1} = m_{1}*g*r_{1}*cos(\theta)$$

$$T_{2} = m_{2}*g*r_{2}*cos(\theta)$$

$$T_{d} = d\dot{\theta}$$

I defined the differential equation as:

$$J\ddot{\theta} = \frac{1}{J}*(m_{1}*r_{1}*g*cos(\theta) - m_{2}*r_{2}*g*cos(\theta) - d*\dot{\theta})$$

And implemented it on Matlab by using ODE45:

clc;clear all
tspan = 0:0.01:200;  %Time span

r1 = 0.03;                       % distance from pivot to m1 (m)
r2 = 0.02;                       % distance from pivot to m2 (m)
m1 = 0.02;                       % mass of m1(Kg)
m2 = 0.02;                       % mass of m2(Kg)
J = (m1*(r1^2)) + (m2*(r2^2));   % Moment of innertia
d = 0.01;                        % damping coeficcient

[t,x,U] = ode45(@(t,x)diffEq_simple2(x,J,r1,r2,m1,m2,d),tspan,x0);

theta = xd(:,1);
plot(t,theta);
axis([0 max(t) -180 180])
grid on;

function dx = diffEq_simple2(x,J,L1,L2,m1,m2,d)
g = 9.8;
dx(1)  =  x(2);
dx(2)  =  (1/J)*(m1*L1*g*cos(x(1)) - m2*L2*g*cos(x(1)) - d*x(2));
dx = dx';
end


After plotting the results of angle $$\theta$$ I get this:

It doesnt matter which value I assign to r2 or r1, the angle $$\theta$$ will rotate to the correct direction but will always converge to 90 degrees, which it is not correct, I was hopping the model to oscillate a bit but converge to a different value each time I change the value or r1,r2,m1 or m2, but it always converges to 90 degrees. Is my ode wrong derived ? which is the correct method to derive the equation of motion for a lever balance?

If an ideal frictionless lever as in the picture isn't balanced when it is flat, it won't be balanced at some angle $$\theta<\pi/2$$ to the horizontal either. This is because when the lever tilts, the torques exerted by the masses on both sides are both scaled by $$\cos{\theta}$$, so that any torque imbalance persists upon rotation of the lever.
user
The text passage describes a physics problem involving the derivation of the equations of motion for a lever balance and includes a discussion of the results obtained from a MATLAB simulation. The author questions whether the derived ordinary differential equation (ODE) is correct and seeks an explanation for why the angle always converges to 90 degrees.

TYPE: SINGLE_QUESTION

The passage contains a single mathematical question regarding the correctness of the derived ODE and the behavior of the lever balance system. No answer to this specific question is provided within the text.