An apparatus for damping vibrations in an internal combustion engine, the apparatus comprising: a crankshaft comprising a lobe, the lobe comprising: a body; a flange coupled to the body, the flange having an opening (364, 366) therein, the flange opening (364, 366) having an inner surface (465, 467); a pendulum having an opening (354, 356) therein, the pendulum opening (354, 356) having an inner surface (455, 457); and a pin (374, 376) extending through the pendulum opening (354, 356) and the flange opening (364, 366) to pivotally couple the pendulum to the flange, characterized in that the pendulum opening inner surface (455, 457) and the flange opening inner surface (465, 467) have identical epicycloidal configurations so that the pendulum follows an epicycloidal path defined by ρ 2 = ρ 0 2 -A 2 S 2 that lies between, and does not include, a tautochronic path where λ equals λ e and a cycloidal path where λ equals 1, an epicycloidal configuration of the inner surface (465, 467) of the flange opening (364, 366) being rotated 180 degrees with respect to an epicycloidal configuration of the inner surface (455, 457) of the pendulum opening (354, 356), where po is a radius of curvature of a path corresponding to the pendulum apex, S is an arc length distance along the pendulum path from the apex, and λ e is the square root of the quantity n 2 over the sum n 2 +1, where n is a target order to be corrected. a crankshaft comprising a lobe, the lobe comprising: a body; a flange coupled to the body, the flange having an opening (364, 366) therein, the flange opening (364, 366) having an inner surface (465, 467); a pendulum having an opening (354, 356) therein, the pendulum opening (354, 356) having an inner surface (455, 457); and a pin (374, 376) extending through the pendulum opening (354, 356) and the flange opening (364, 366) to pivotally couple the pendulum to the flange, characterized in that the pendulum opening inner surface (455, 457) and the flange opening inner surface (465, 467) have identical epicycloidal configurations so that the pendulum follows an epicycloidal path defined by ρ 2 = ρ 0 2 -A 2 S 2 that lies between, and does not include, a tautochronic path where λ equals λ e and a cycloidal path where λ equals 1, an epicycloidal configuration of the inner surface (465, 467) of the flange opening (364, 366) being rotated 180 degrees with respect to an epicycloidal configuration of the inner surface (455, 457) of the pendulum opening (354, 356), where po is a radius of curvature of a path corresponding to the pendulum apex, S is an arc length distance along the pendulum path from the apex, and λ e is the square root of the quantity n 2 over the sum n 2 +1, where n is a target order to be corrected. a body; a flange coupled to the body, the flange having an opening (364, 366) therein, the flange opening (364, 366) having an inner surface (465, 467); a pendulum having an opening (354, 356) therein, the pendulum opening (354, 356) having an inner surface (455, 457); and a pin (374, 376) extending through the pendulum opening (354, 356) and the flange opening (364, 366) to pivotally couple the pendulum to the flange, characterized in that the pendulum opening inner surface (455, 457) and the flange opening inner surface (465, 467) have identical epicycloidal configurations so that the pendulum follows an epicycloidal path defined by ρ 2 = ρ 0 2 -A 2 S 2 that lies between, and does not include, a tautochronic path where λ equals λ e and a cycloidal path where λ equals 1, an epicycloidal configuration of the inner surface (465, 467) of the flange opening (364, 366) being rotated 180 degrees with respect to an epicycloidal configuration of the inner surface (455, 457) of the pendulum opening (354, 356), where po is a radius of curvature of a path corresponding to the pendulum apex, S is an arc length distance along the pendulum path from the apex, and λ e is the square root of the quantity n 2 over the sum n 2 +1, where n is a target order to be corrected.