The detailed solution to determine the basic type of hinged four-bar mechanism by using the hinged f

Expanding on the analysis of the relationship between the relative motion and relative size and position of the connecting rods in a four-bar kinematic chain, we can further examine different scenarios to draw conclusive observations. By considering the relative size and position of each rod, we can determine solutions for various basic types of mechanisms within the assumed framework.

The relative size relationship can be divided into three cases. In the first case, when "ad < bc" and a and b are adjacent bars, we can take the shortest bar (a) as the reference member and analyze its relative motion with the two adjacent bars. Since "a < b" and "c < d", we can conclude that rod a can be straightened and collinear with the two adjacent rods. In other words, a can overlap and be aligned with the neighboring rods.

To establish this overlap condition, the following inequality must hold: "d + c > b + a". This implies that "d > c + a", which is consistent with the given conditions. Additionally, it can be proven that a can also be overlapped and collinear with the opposite adjacent rod (c), using a similar reasoning. Hence, the opposite pole (b or c) of the shortest bar (a) can only swing relative to the two adjacent poles at an angle less than 180°. This means that the angle of swing is limited to less than 180°, but greater than 50° (as per the condition).

In the second case, when a and b are opposite poles, the same method as the first case can be applied to demonstrate that the shortest bar (a) can still rotate relative to the two adjacent bars. Therefore, we can omit the discussion of this scenario.

In the third case, we consider the relative motion between the longest bar (d) and the two adjacent bars. From the known conditions, we can deduce that "d > b + ac" and "d > e + ab". This implies that the rod d cannot be straightened and collinear with the two adjacent rods. Additionally, if d could be overlapped and collinear with the neighboring rods, we would have "d + C > b + a" and "d + a > b + c" (contradicting the given conditions). Therefore, we conclude that the rod d cannot overlap and be collinear with the neighboring rods. In this scenario, both angles of swing are limited to less than 180°.

Combining the conclusions from the first, second, and third cases, we can summarize that, under the condition of "ad < bc", regardless of the various positions, the shortest bar (a) and the second adjacent bar (b) can rotate relative to each other for a whole circle. However, the opposite pole (b or c) and the two adjacent poles of the shortest bar can only swing relative to each other, with the relative swing angle being less than 180°.

By clarifying the relative motion between the two hinged rods in this type of four-bar kinematic chain, we can easily determine the type of mechanism by selecting any rod as the frame of reference. For example, if the shortest rod (a) with the adjacent side is used as the frame, a crank-rocker mechanism is obtained. On the other hand, if the opposite side of the shortest rod is used as the frame, a double-rocker mechanism is obtained. It is important to note that the rocker, in the latter case, cannot swing to a position collinear with the rack or the extension line of the rack, and the swing angle must be less than 180°.

In conclusion, through the analysis of the relationship between the relative motion, relative size, and relative position of the connecting rods in a hinged four-bar kinematic chain, we can determine the type of mechanism and its limitations. By understanding these principles, engineers and designers can develop and optimize various types of mechanisms for specific applications.