Finding instantaneous rotation and translation from multiple forces

Question

I'm having trouble extrapolating torque, rotation, and such into the information I want.

I'm trying to set up an estimation of a sea ship's movement for a game. The system is 2D. The ship has multiple arbitrary forces acting upon it to represent propulsion. I am assuming the ship is always stationary and facing forward. I want to get the rotational and translation accelerations for this simple ship and use them to change the velocities of a moving ship.

1. Does this approach make sense?

If so the main question is: 2. How do I determine how fast it accelerating in x,y and θ?

And relatedly

3. If there are opposite rotational forces, say +80 and -60 units, you would get +20 units rotation, but would the 60 that cancels out be converted back to translational force and added back into x and y?

Note: Ignoring other forces such as drag, which will be calculated separately. I understand all of the math just not the physics. The biggest problem I have is splitting up the forces and vectors in the correct way.

Answer 1

Does this approach make sense?

Maybe. However outside of a spaceship simulation, it's rare that you would already know all the forces involved. Real engines don't produce constant forces. As you move faster, the force that is available from your car wheel or your boat propeller decreases.

It is often much easier to not bother with a full simulation of forces and simplify what's happening. But there's nothing inherently wrong with your approach.

How do I determine how fast it accelerating in x,y and θ?

For linear acceleration, sum all the forces (with vector addition) to create a net force. The object accelerates in that direction with inverse proportion to its mass.

For angular acceleration, you need to calculate the torque from the forces. The ship accelerates in that direction with inverse proportion to its moment of inertia.

If there are opposite rotational forces, say +80 and -60 units, you would get +20 units rotation, but would the 60 that cancels out be converted back to translational force and added back into x and y?

Not in the way you say. But adding the forces will give different figures. If the forces point toward each other, the linear acceleration will be reduced, but the angular acceleration may be increased. You'll do vector addition for the linear acceleration. You'll convert the forces to a torque and then sum the torques for angular acceleration.

Answer 2

A way of analyzing such a situation is to "convert" the original forces $F$ and $f$ into couples (black arrows) and forces whose line of action go through the centre of mass (red arrows).

You now have an object of mass $m$ which is under the action of forces $F$ and $f$ (red) which will cause a linear acceleration $a$ of the centre of mass $F+f=m\,a$ and black forces which constitute two couples whose torque is independent of any axis and if $I_{\rm c}$ is the moment of inertia of the object about the centre of mass the angular acceleration of the object $\alpha$ can be found using $Fx=fx=I_{\rm c} \alpha$.

As the object undergoes translation and rotation you will have to decide whether or not the directions of the forces relative to the object change.

Answer 3

Why can't you use the equations of motion in 2D? Assume you have $n$ forces applied to a rigid body. Each force $\boldsymbol{F}_i = (Fx_i,Fy_i)$ is located at $\boldsymbol{r}_i = (x_i,y_i)$. The motion of the body is described by the motion of the center of mass. Solve for the acceleration components $\ddot{x}_{com}$, $\ddot{y}_{com}$ and $\ddot{\theta}$ of the COM from the equations below:

$$ \begin{aligned} \sum_{i=1}^n( Fx_i) & = m\, \ddot{x}_{com} \\ \sum_{i=1}^n( Fy_i) & = m\, \ddot{y}_{com} \\ \sum_{i=1}^n( x_i Fy_i - y_i Fx_i + \tau_i) & = I_{com} \ddot{\theta} \end{aligned} $$

where $m$ is the mass, $I_{com}$ is the mass moment of inertia at the center of mass .