This study presents a theory that approximates steady forced oscillations up to the second-order with respect to vibration amplitude a. This is the extension of the first-order approximation (linear theory) that enable us to obtain the optimal forcing without any parametric studies (Ishida et al., 2012). The theory is constructed on an ordinary differential equation system (ODEs) with an arbitrary small time varying external forcing and is applicable to obtain the steady response including the contribution of the vibration to the direct-current component more accurately than the linear theory while a is relatively small. It also enables us to evaluate a sensitivity, i.e. the derivative of an objective measure, indicator, with respect to the amplitude a. Thereby the region of a where the linear theory is effective can be identified. The application of the second-order approximation to a thermal convection field in a square cavity subjected to a heat-flux vibration on the bottom wall shows that the second-order sensitivity (second derivative) well explains the parameter region where the nonlinearity appears. Outside the region we can utilize in a positive manner the useful linear theory. Moreover, the change in the time-mean skin friction on the wall surface is well evaluated by the second-order approximation while a is small enough. 2016 Elsevier Ltd. All rights reserved.