Study on synthesis method of λ-formed 4-bar linkages approximating a straight line

This paper presents an analytical synthesis method of λ-formed 4-bar linkages approximating a straight line for given three pairs of coincident curve points. Through formulating the constraints (including two separate positions constraints and a velocity constraint) and solving the equations analytically, three 4-bar linkages, with two symmetrical coupler curves and one asymmetrical curve, will be synthesized simultaneously for given independent design parameter φ0 (or y01). Also, there are an infinite number of different λ-formed linkages with extraordinary good approximations of the straight line by changing the value of φ0 (or y01). The key advantages of this method are that it can be easily coded into programs suitable for processing by computers, it is more concise and clear than complex geometry method, and highly accessible. To demonstrate the utility of this displacement–matrix method, a few examples are illustrated for different values of the design parameter in this paper.