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Anyone please help me to find out the distance in following case.

Refer to the attached image. Consider an arbitrary point P on the circumference of a circle of radius r (mm). The point makes an angle Ɵ with the vertical axis when subtended to circle center at that particular instant. The circle is simply rotating about its center at a fixed speed. The center of the circle (O) once again has a linear velocity of V (mm/rev). This V is related to the rotational speed in such as a way that the center (O) advances by V mm in every complete rotation (360 degree). Assume 2D motion only. Moreover, V < r.

I think the point P will trace a trochoid (not sure). Whatever be the case, assume P made two complete rotations (i and i+1), and at the same time the center moved linearly by some distance. In (i+1)th rotation, if we join the arbitrary point P with its instantaneous center, then this straight line will also intersect the trochoid curve traced in i-th pass (previous pass). This intersection point (say, Q) lies in between point P and its corresponding center. I need the linear distance between P and Q (may be in terms of Ɵ and other known parameters). Please provide little bit calculation and relevant diagram also, if required. Note: Center of the circle must be considered on instantaneous basis. Remember Q will not become P after 360 degree rotation as they make different angles with corresponding instantaneous centers. See image for more details. Schematic of the problem

Thanks in advance.

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