# Analysis of a Beam under Non-static Case

I am interested in the distribution of the beam in nonstatic cases.

Typically we discuss the stress, and moment distribution when a beam is in a static case. However, I am interested in this distortion of a robot arm. In such cases, the arm is not in static equilibrium (i.e. it has nonzero angular acceleration.)

I would like to know if such cases could be analyzed by adequate static models, or if angular acceleration itself affects the distortion and therefore could not be transformed into a static case.

Let's consider the case below.

If the moment exerted on the point A is not equal to net moment exerted on the point B, the beam will rotate with nonzero angular acceleration.

Regarding the stress and distortion analysis on section C,

1. Is there an equivalent static model? (does angular acceleration affects the stress and distortion on point C)
2. If there is no equivalence static model, but in case the angular acceleration is relatively small and could be disregarded then what would be the equivalent static model?
3. In dynamics cases, could the load on point B could be exchanged by an equivalent moment on point B as in static cases?

The most important part IMHO is:

1. If there is no equivalence static model, but in case the angular acceleration is relatively small and could be disregarded then what would be the equivalent static model?

In most robotic cases the angular acceleration (or angular velocity see below) should be so small that the robotic arm (if designed properly) should not suffer any additional structural deformations.

## angular acceleration and motor torque

The angular acceleration is a product of the imbalance of the moments (point A: motor torque, point C: load moment) that are applied according to the law:

$$\sum M = I \alpha \tag {eq.1}$$

So, if motor torque is not applied then there is no angular acceleration. The above equation is used mainly to determine the kinematics of the robotic arm (and its acceleration).

Of course, if you wanted to calculate a specific acceleration profile which is a common case, you'd use eq.1 to specify the required motor torque. However, again the motor torque would be responsible for loading the arm.

Bottom line is that the angular acceleration plays a role indirectly, through the motor torque (and the reaction forces).

## angular velocity

Angular velocity is another matter though. Angular velocity ($$\omega$$) is directly proportional to a the centifugal/centripetal force (proportional to $$\omega^2 r$$) that develops on the robotic arm.

So angular velocity plays (usually) a more active and direct part in the calculation, compared to the angular acceleration.