This paper presents a methodology for synthesizing planarlinkages to approximate any prescribed periodic function. Themechanisms selected for this task are the slider-crank and thegeared five-bar with connecting rod and sliding output (GFBS),where any number of drag-link (or double crank) four-bars areused as drivers. A slider-crank mechanism, when comparing theinput crank rotation to the output slider displacement, producesa sinusoid-like function. Instead of directly driving the inputcrank, a drag-link four-bar may be added that drives the crankfrom its output via a rigid connection between the two. Drivingthe input of the added four-bar results in a function that is lesssinusoid-like. This process can be continued through the additionof more drag-link mechanisms to the device, slowly alteringthe curve toward any periodic function with a single maximum.For periodic functions with multiple maxima, a GFBS is usedas the terminal linkage added to the chain of drag-link mechanisms.The synthesis process starts by analyzing one period ofthe function to design either the terminal slider-crank or terminalGFBS. A randomized local search is then conducted as thefour-bars are added to minimize the structural error between thedesired function and the input-output function of the mechanism.Mechanisms have been “grown” in this fashion to dozens of linksthat are capable of closely producing functions with a variety ofintriguing features.