# Constraints of a mechanical system

• I
• Ahmed1029
In summary: Yes, that's correct. A continuous body usually has to be described by fields (e.g., density, velocity, pressure for a fluid). An exception is the rigid body, which has only 6 degrees of freedom (3 position-vector coordinates to any fixed point within the body and 3 Euler angles to describe the rotation of a body-fixed Cartesian coordinate system wrt. a space-fixed Cartesian one). It's a good exercise to derive this with continuum-mechanical methods.
Ahmed1029
I'm studying theoretical mechanics and I kind of find the notion of a "mechanical system" very slippery, especially when it comes to constraints. Take an example :

I know that when a system consists of N particles and p constraints, it has 3N-p degrees of freedom; this is the definition. Then I come across something like this example in the picture above, in which I have a wire which includes an infinite number of particles, and in the solution it's completely ignored. Here there are 2 constraints and the auther treated the whole setup as if there is only one particle while the wire is completely ignored, thus the system has one degree if freedom because 3(1)-2=1
In general, how do I know the number of degrees of freedom of a "mechanical system" that is not just made of ordinary accumilatios of discrete particles?

vanhees71 and Ahmed1029
Baluncore said:
The wire is a hypothetical perfect guideline for the sliding bead.
Only the position of the bead on the wire is being considered.
The model is applicable to this real situation.
https://en.wikipedia.org/wiki/Liquid-mirror_telescope
Oh! So it's considered a time dependent geometrical constraint. What about real wires though? Is it right to say that " When I can specify completely the state of a system just by the positions of the discrete particles, then I should ignore any other continuous body as a mere time dependent geometrical constraint"?

The constraint is a bit inconveniently defined. In cylindrical coordinates it's much simpler and valid for any time,
$$z=\alpha \rho^2.$$
Now first parametrize the position vector (wrt. the inertial frame of reference) of the particle on the parabolic wire and then you can simply put the entire problem into the Lagrangian machinery.

Baluncore
vanhees71 said:
The constraint is a bit inconveniently defined. In cylindrical coordinates it's much simpler and valid for any time,
$$z=\alpha \rho^2.$$
Now first parametrize the position vector (wrt. the inertial frame of reference) of the particle on the parabolic wire and then you can simply put the entire problem into the Lagrangian machinery.
Suppose I don't know which generalized coordinates I should choose to specity the system, but I nevertheless want to know the number of degrees of freedom of the system. In case of discrete particles it's easy, you just find independent equations relating the coordinates to each other until there is no more ( which is done by inspection), then subtract from 3N the number of constraints, where N is the number of particles. Suppose now I have not just discrete particles, but also continuous bodies: How do I know the number of degrees of freedom for sure? My guess is to find any number of coordinates that specify the whole system completely, find the maximum number of equations relating them to each other, and subtract the number of independent equations from the number of original arbitrary coordinates. Is this right.

vanhees71
That sounds right. A continuous body usually has to be described by fields (e.g., density, velocity, pressure for a fluid). An exception is the rigid body, which has only 6 degrees of freedom (3 position-vector coordinates to any fixed point within the body and 3 Euler angles to describe the rotation of a body-fixed Cartesian coordinate system wrt. a space-fixed Cartesian one). It's a good exercise to derive this with continuum-mechanical methods.

Orodruin and Ahmed1029

## 1. What are the main types of constraints in a mechanical system?

The main types of constraints in a mechanical system are kinematic constraints, force constraints, and geometric constraints. Kinematic constraints limit the motion of a system, force constraints limit the forces applied to a system, and geometric constraints limit the shape or orientation of a system.

## 2. How do constraints affect the behavior of a mechanical system?

Constraints play a crucial role in determining the behavior of a mechanical system. They restrict the degrees of freedom of a system and limit the possible motions and forces that can act on it. This can result in predictable and stable behavior, but can also lead to undesired limitations or failures if not properly designed.

## 3. What are some common methods for analyzing constraints in a mechanical system?

There are several methods for analyzing constraints in a mechanical system, including free body diagrams, kinematic equations, and force equations. These methods can be used to determine the forces and motions within a system, and to identify any potential issues or limitations caused by the constraints.

## 4. How can constraints be used to improve the performance of a mechanical system?

Constraints can be strategically designed and implemented to improve the performance of a mechanical system. For example, they can be used to increase stability, reduce vibrations, or control the motion of a system. Constraints can also be used to optimize the strength, efficiency, or accuracy of a mechanical system.

## 5. What are some challenges associated with constraints in a mechanical system?

Constraints can present several challenges in the design and operation of a mechanical system. These include balancing conflicting requirements, ensuring compatibility with other system components, and minimizing the impact on overall system performance. Additionally, constraints can introduce complexity and increase the risk of failure if not carefully considered and managed.

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