The influence of suspension point stiffness on galloping motion was studied by means of computer simulation with the objective of reducing galloping magnitude and its harmful consequences. A dynamic simulation procedure was developed based upon a high order finite element method for modeling a galloping catenary and tower support structure. In order to test the results produced by the simulations several additional analytical models were employed. Two of these auxiliary models consist of linearizing the equations of motion about both the static equilibrium positions and a position the line would assume under the circumstances of a steady, blowing wind and the absence of galloping motion. Another model used to verify the simulation results consists of a unique variational, finite element, limit cycle analysis technique which determines both the fundamental frequency of galloping and the shape of each harmonic. The limit cycle method and the wind loaded linearization process are both derived in this work. The results of the investigation produce three different strategies for reducing galloping amplitudes through support stiffness, two of which are dependent upon the cable aerodynamics. The strategy independent of the line aerodynamics consists of making the vertical support stiffness as large as possible. The aerodynamic dependent solutions entail the tuning of the line modes to produce a particular trajectory beneficial to amplitude reduction and choosing a stiffness configuration which essentially renders the first symmetrical mode of oscillation stable, thus constraining galloping to occur in a higher frequency mode. This work also demonstrates that the fundamental frequency of galloping depends not only on the cable properties and support stiffness but also heavily upon the aerodynamic characteristics and wind speed. Also presented is a unique derivation of the equations of motion through Lagrangian mechanics which is well suited for variational finite element techniques