In this study, optimum design of viscous dampers under different mode behaviors are investigated for structures exposed to earthquakes. The structure is modeled as a shear frame and considered as linear. The upper and lower limit values of the dampers are defined as passive constraints, while the damping coefficients of the dampers placed in each storey are considered as the design variables. The use of these technological tools in construction has caused a serious cost increase. It is therefore important to use minimum of these elements. For this reason, the sum of the damping coefficients of the dampers is regarded as the objective function and is minimized as an indication of the capacities of the dampers placed in the building stories and thus their costs. It is well known that the addition of dampers to the structure increases the damping ratio of the structure. A new active constraint is included in the optimization problem as the target damping ratio. The equation required for the calculation of the value of the target damping ratio corresponding to any mode is also derived. The simple optimization problem is solved using three different optimization algorithms: Simulated Annealing, Nelder Mead and Differential Evolution algorithms. Optimum designs are found to minimize the cost function and provide all constraints. In the shown numerical example, the effect of the variation of the building period and the changes of the target damping ratio on the optimization is investigated. Furthermore, earthquake behavior of the structure corresponding to these optimum designs is investigated using El Centro Earthquake (NS) record and examined in terms of period and additional damping ratios of the maximum displacements to the floors. In addition, the proposed method finds the optimum damper distribution considering the first two modes. The proposed optimum damper design method is very simple and it is a method which reaches the optimum designs in different mode behaviors. As a result, it has been shown in the numerical examples that the optimum damper design can be changed according to the variations of the designer's constraints. (C) 2018 Elsevier Ltd. All rights reserved.