The objective of this study is to perform a pioneering research about a viscoelastic hyperboloidal helical rod having a standard type of distortional behavior and a Kelvin type of bulk compressibility. Field equations are based on the Timoshenko beam theory, and the exact curvatures of the hyperboloidal geometry are considered through the formulation. The numerical analysis is carried out by the mixed finite element method, considering the rotary inertia, in the Laplace space, and the results are transformed back to time space numerically using the modified Durbin's algorithm. A cantilevered hyperboloidal helical rod having solid circular, hollow circular, and thin-walled hollow circular cross sections is handled, and the rod is loaded by rectangular and triangular impulsive types of point load at the tip. Through the analysis, different values of retardation time, three different relaxation functions associated with shear modulus, and three different creep functions associated with bulk modulus are handled. Finally, a benchmark example is presented, and the influence of the loading and the material parameters on the helix geometry is discussed.