# Free-body diagrams for constrained rigid multibody dynamics

• Bastian1978
In summary: Ahmed Shabana explaining how to draw free body diagrams. In summary, this video explains how to determine the direction of the joint reaction forces using calculus.

#### Bastian1978

Hi all,
I have been reading the book "Computational Dynamics" by Ahmed Shabana, and have a few questions.
The book talks about creating "free-body diagrams" for constrained rigid multibody dynamics problems.
I have attached two examples:
The first is a "pendulum with a moving base":

which is described in the book as:
"Body 1 is a slider block that has a specified motion defined by the function z(t). Body 2 is a uniform slender rod that has mass m2, mass moment of inertia about its centre of mass J2, and length l. The rod that is subjected to the external moment M2 is connected to the sliding block by a pin joint at O."

The second,

is described in the book as:
"The system consists of the ground denoted as body 1, a rod OA denoted as body 2, and a disk denoted as body 3. The rod is connected to the ground by a pin joint at O, while the disk is connected to the rod by a pin joint at A. The rod is assumed to be uniform and its length is l."

I have also attached the book's free-body diagrams for these two examples.

I'm unclear about how the free-body diagrams are constructed (and it is assumed knowledge in the book).
In the first free-body diagram, I understand that m2g (directed down) is the force of gravity, and M2 (anticlockwise) is the externally applied torque. However I am unclear about F12x and F12y, which the book says are the "joint reaction forces at the pin joint".
(Similarly I am also unclear about the forces F12x, F12y, F23x and and F23y in the second free-body diagram).

I don't understand how the direction of the joint reaction forces is determined (eg why is F12x directed from left to right in both diagrams)?
Also, are these directions constant regardless of the orientation of the rod? Or will they change as the rod rotates?

Does anyone know of any references that deal with this topic (constructing free-body diagrams specifically for constrained rigid multibody dynamics -
particularly describing how to direct the joint reaction forces)? Any reference would be greatly appreciated, as I haven't been able to find an answer to this question so far online or in books.

Nick.

The reaction forces in these diagrams result from Newton's Third Law (ie. why doesn't a mass on a table sink into the table?).

Each reaction force only points in one direction, but the reason you see x and y components is because that is the same force resolved to the x and y-axis respectively.

To construct free body diagrams, you need to consider all the forces acting on a body. F23x and and F23y for example, would be due to the weight of mass $$M^3$$.

As the system evolves, all the pieces move, and hence, the directions and magnitude of some forces will change, resulting in the magnitude of the resolved components changing.

Last edited:
I do understand how to draw the free body diagram of a book resting on a table but I don't understand how the direction of the X components of the forces was determined in those diagrams either.

Any help would be greatly appreciated

Thanks

Prismaticus

## What is a free-body diagram for constrained rigid multibody dynamics?

A free-body diagram is a simplified representation of a physical system, where all external forces acting on the system are shown as arrows with the direction and magnitude. In constrained rigid multibody dynamics, these diagrams are used to analyze the motion of connected objects or bodies, taking into account the constraints that limit their motion.

## Why are free-body diagrams important in constrained rigid multibody dynamics?

Free-body diagrams help in understanding the forces and torques acting on a system, which is crucial in predicting the motion of constrained rigid multibody systems. They also help in identifying and visualizing the constraints that affect the motion of the bodies.

## How do you draw a free-body diagram for constrained rigid multibody dynamics?

To draw a free-body diagram for constrained rigid multibody dynamics, first identify the bodies involved and their points of contact. Next, draw the bodies as simplified shapes and add arrows to represent the external forces acting on each body. Finally, add the constraints as lines or other symbols to show their effects on the motion of the bodies.

## What is the difference between a free-body diagram and a kinematic diagram?

A free-body diagram shows the forces and constraints acting on a physical system, while a kinematic diagram illustrates the motion of the system's components. In constrained rigid multibody dynamics, both diagrams are used together to analyze the motion of connected objects.

## What are some common mistakes when drawing free-body diagrams for constrained rigid multibody dynamics?

Some common mistakes when drawing free-body diagrams for constrained rigid multibody dynamics include missing forces or constraints, incorrect direction or magnitude of forces, and neglecting frictional forces. It is also essential to ensure that the diagrams accurately represent the physical system and its motion.

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