Finding the Lowest Position of a Kinematics Mechanism: A Mathematical Challenge

[Matik]

Hello community. I have a task from my college and no matter how hard i tried i failed to find a solution for it. The task itself is resolvable (by drawing) but i can't figure the mathematical way to solve it. And it bothers me so bad.

Homework Statement

So i have this kinematics mechanism. At the OA lever i have a known constant velocity, let's say - ω with direction counter clockwise around point O. I have given these dimensions: OA, AB, AD, DB1 and h aswell. My task is to determine the lowest position of element/point B1. Of course i plotted enough points and determined that (see the picture)... but there should be mathematical way to determine this. And i can't find it.

Initialy i thought... the lowest position of B1 is when OA is alongside to DB1. But i was wrong. Then i thought that if the velocity that occurs at point D is perpendicular to DB1 then the velocity at point B1 will be equal to 0. But this solution does not match with the position from the picture either.

 

[paisiello2]

Is point D a pinned joint?

 

[Matik]

No, its a mobile joint. My bad, i fixed it:

 

[paisiello2]

is length B1-D less than h?

 

[Matik]

.

 

[Matik]

No. Here are all dimensions and values i have:

OA = 0.45m
AB = 1.1m
BD = 1.6m
B1D=0.65m
h=0.5m
ωoa = 4 rad/s (counter clockwise)

I already plotted full cycle with 6 positions and by the lowest position of B1 there i plotted another 10-12 points near it, increasing the resolution. Thus i found (roughly) the lowest position of B1 shown on the picture up there. But i still think there has to be another - much more accurate and mathmatical way to solve this.

I think that if B1 is in the lowest position its velocity should be equal to 0. Therefore at point D velocity should be perpendicular to DB1:
ωoa =Voa . OA We have ωoa and OA so Voa is known. If we separate vector Voa by sinα.Voa and cosα.Voa we will have at point B - ωbd=(sinα.Voa)/AB and the point D this velocity will be vector Vda=ωbd.BD=(sinα.Voa).(BD/AB) plus the other vector cosα.Voa. But when i calculate and plot them it doesn't seems to be perpendicular at all. And i can't figure it out why is that.

 

[paisiello2]

I think your approach looks valid. However, might a simpler be simply geometrical? Can you express y in terms of θ? Then set dy/dθ=0.

 

[Matik]

I didn't understand you. What are y and θ?

 

[paisiello2]

y is the vertical position of point B1 and theta is the angle OA makes with the x-axis.