From[Polynomial Cam Function (Derivation of Fifth-degree function) - Part 2], we get the equations for displacement, velocity and acceleration of cam follower using

**fifth-degree (3-4-5) polynomial**. All of these functions can be plotted in Excel spreadsheet as shown in the picture. This is for the case of zero velocity at both ends i.e v_{0}= 0 and v_{1}= 0. We can see that it looks like cycloid cam profile which has zero starting and ending velocity. But fifth-degree polynomial has ability to change the starting and ending velocity, while cycloid can't do that. So we have at least 4 parameters to configure the cam curve of fifth-degree polynomial i.e. the total displacement (h_{m}), total angle (b_{m}), starting velocity (v_{0}) and ending velocity (v_{1}). We also have to make sure that the connection between curves must not have any discontinuity. The functions such as cycloid and fifth-degree polynomial has continuity in its displacement, velocity and acceleration. But when connecting it with another curves, it may create discontinuity if we don't properly connect them together.
Now let's see what happen when the starting velocity and ending velocity are both greater than zero (v

_{0}= 1 and v_{1}= 3). From the second graph, we can see that the displacement of this fifth-degree (3-4-5) polynomial function is the same as the first case (v_{0}= 0 and v_{1}= 0). This time the starting velocity is not zero and it has to decelerate and accelerate so that it reaches the desire ending velocity. So we can see the different in the velocity as well as acceleration.*. For other cam function such as cycloid, modified sine, etc, it will move from point A to point B without moving back (just move forward). But we can see that this fifth-degree cam function does the different thing. It starts moving forward from point A until certain degree and it moves backward beyond point A! then move forward again until it reaches point B where it's the end of sector.***Very important remark for fifth-degree polynomial cam function**: though the displacement is still the same for this case (same hm), but the total traveling "distance" for this case is much more than the first case**Therefore if you don't properly check the displacement graph you'll be surprised to see your mechanism moving back and forth**. In most cases, we don't need that, unless it's planned motion from the design.
The third graph is for the same displacement but we change ending velocity to be less than the second case (v

_{0}= 1 and v_{1}= 0.5). Now it keeps moving forward without moving back. But it depends on the constraints of your design whether you allow to change the velocity or displacement, etc. Just make sure that it moves in the way that you expect. On the other hand, moving back and forth is also good for movement if we design it properly. I think only moving back and forth is fine, but moving forward then back then forward again using only one function of fifth-degree polynomial like the second case is maybe too much and difficult to control. In some situations we may need the mechanism to move back and forth without any stop. For this case we can use fifth-degree polynomial cam function to control the movement as shown in the last picture of this post.
The last graph shows the movement when v

_{0}= 1 and v_{1}= -1. The total lift for this movement is more than what specified in the value of h_{m}. But we can calculate or even use Excel spreadsheet to adjust the total lift to be the same value as what we need. That means the h_{m}for fifth-degree cam function may be different from the total lift that we need. It doesn't matter as long as it satisfies all the timing diagram and moves without discontinuity in the displacement, velocity and acceleration.
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