# Scleronomic or Rheonomic Mechanical System?

• sams
In summary, the conversation discusses the distinction between scleronomic and rheonomic mechanical systems, and how to identify them based on their constraint equations. A scleronomic system has constraint equations that only relate the positions of the masses in the system, while a rheonomic system has constraint equations that also involve velocities. The conversation also mentions the concept of Pffafian form for constraint equations and provides an example of a rheonomic system. The conversation also clarifies that this is not a schoolwork question and emphasizes the importance of not confusing rheonomic and nonholonomic constraints.
sams
Gold Member
I would really appreciate if someone could advise me whether the system below is a scleronomic or a rheonomic mechanical system, or a mix of both. If we consider the first pendulum, the constraint is fixed which leads to a scleronomous case while the constraint of the second pendulum is not fixed (varies with time) which leads to a rheonomous case.

Any advice is much appreciated.

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• Double Pendulum.PNG
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Is this a schoolwork question?

Present equations of constraints as ##f_i(x_1,y_1,x_2,y_2)=0,\quad i=1,2##

berkeman said:
Is this a schoolwork question?
No

berkeman
No, the constraints here are not the pendulums but the rod that connects the pendulum. Now, try to find the equations that describe each constraint (i.e. the constraint equations) in the inertial frame of reference. Put all terms on the LHS (i.e. set RHS equals 0). This is then called the Pffafian form of the constraint. The definition of a scleronomic system is that the constraint equations of the system relate only the positions of the masses in the system, can be arranged into the Pffafian form. Since you can find a Pffafian form of the constraints, you have a scleronomic system. An example of a rheonomic system would be a rolling disk in 3 dimensions, in which the constraint equation relates the positions as well as velocities of the masses of the system, and in particular it cannot be integrated to a constraint equation that only relates the positions of the masses in the system.

aldo sebastian said:
An example of a rheonomic system would be a rolling disk in 3 dimensions, in which the constraint equation relates the positions as well as velocities of the masses of the system, and in particular it cannot be integrated to a constraint equation that only relates the positions of the masses in the syst
do not confuse with rheonomic and nonholonomic constraints

## What is a Scleronomic or Rheonomic Mechanical System?

A Scleronomic or Rheonomic Mechanical System is a type of mechanical system in which the relationships between the system's components and their movements or displacements are predetermined and do not change. This means that the system's behavior is solely determined by its initial configuration and external forces acting on it.

## What are the key differences between Scleronomic and Rheonomic Mechanical Systems?

The main difference between these two types of mechanical systems is the presence of constraints. In Scleronomic systems, the constraints are rigid and do not change, while in Rheonomic systems, the constraints can vary. This means that Rheonomic systems can adapt to different environments, while Scleronomic systems have a fixed behavior.

## What are some real-world examples of Scleronomic Mechanical Systems?

Some common examples of Scleronomic Mechanical Systems include simple machines like levers, pulleys, and gears. Other examples include bridges, buildings, and other structures that have a fixed shape and do not change their behavior based on external forces.

## What are some real-world examples of Rheonomic Mechanical Systems?

Rheonomic Mechanical Systems are often found in more complex systems such as robots, vehicles, and machines that need to adapt to different environments or tasks. For example, a car's suspension system is a Rheonomic Mechanical System that adjusts to different road conditions to provide a smooth ride.

## What are the advantages and disadvantages of Scleronomic and Rheonomic Mechanical Systems?

The advantages of Scleronomic systems include their simplicity and predictability, making them easier to design and analyze. However, they lack flexibility and cannot adapt to changing environments. On the other hand, Rheonomic systems can adapt to different situations, but they are more complex and can be harder to design and analyze.

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