# Mechanical system with two degree of freedom

• ohmaley
In summary, we are tasked with determining the horizontal and rotational movements of two cylinders connected by a spring on a horizontal plane. We can do this by creating free body diagrams and using equations of motion and angular momentum. Solving the resulting differential equation system will give us the frequency and associated modes of the system.
ohmaley

## Homework Statement

1) Two cylinder can roll without sliding on a horizontal plan. they are connected throught the center by a spring of rigidity "k".

They both share the same radius "R" and mass "m". The left cylinder has its mass around the perimeter and the right cylinder's mass is spread uniformly.

http://tinypic.com/r/16kpevc/7

a)Determine the free body diagram. Figure out the horizontal and rotational mouvements of each rolling bodies.

b) adding the cinematic equations (the Xi in function of the rotation angles θi), determine the differential equation system in x1 and x2.

c) Calculate both w and their associated modes.

## Homework Equations

sum Fx = m*agx

sum Mg = Jg*α (alpha)

Inertia momentum of the two cyclinders:

Jg=m*R^2 for mass around perimeter

## The Attempt at a Solution

I figured since there is no sliding -> X1=R* θ1 and X2=R" θ2.

m1*ag=-k(x1-x2)

m2*ag=k(x1-x2)

with x(t) = Rcos(wt), "R" being the amplitude

Last edited:
Any help would be greatly appreciated. For the free body diagram, the forces acting on the cylinders are the gravitational force, the spring force due to the connection between the two cylinder, and the normal force. The gravitational force will act downwards, the spring force will be equal in magnitude but opposite in direction between the two cylinders, and the normal force will act perpendicular to the surface. For the equations of motion, we can use the equations of Newton's second law:sum Fx = m1*agx1 = -k(x1 - x2)sum Fx = m2*agx2 = k(x1 - x2)For the rotational equations, we can use the equation of angular momentum:sum Mg = Jg1*α1 = -k(θ1 - θ2)sum Mg = Jg2*α2 = k(θ1 - θ2)where Jg1 and Jg2 are the inertial moments of the cylinders.Then, by combining the equations of motion and angular momentum, we can find the differential equation system in x1 and x2.Finally, solving the differential equation system will give us the frequency and associated modes of the system.

## What is a mechanical system with two degrees of freedom?

A mechanical system with two degrees of freedom refers to a system that has two independent variables that affect its motion or behavior. These variables are typically position and rotation, and they can vary independently from each other.

## What are some examples of mechanical systems with two degrees of freedom?

Some common examples of mechanical systems with two degrees of freedom include a pendulum, a seesaw, a bike with two wheels, and a double pendulum. These systems have two independent variables that control their motion and behavior.

## What is the significance of studying mechanical systems with two degrees of freedom?

Studying mechanical systems with two degrees of freedom is important because it allows us to understand and analyze the behavior and motion of complex systems. It also helps us design and improve machines and structures that have multiple moving parts.

## What are the equations used to describe a mechanical system with two degrees of freedom?

The equations used to describe a mechanical system with two degrees of freedom are typically derived from Newton's laws of motion. These equations help us understand the relationship between the system's variables and predict its behavior under different conditions.

## How can we apply the principles of mechanical systems with two degrees of freedom in real-life scenarios?

The principles of mechanical systems with two degrees of freedom can be applied in various fields, such as engineering, robotics, and physics. For example, engineers use these principles to design efficient and stable structures, while robotics engineers use them to create more advanced and versatile robots.

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