In designed longitudinal studies, information from the same set of subjects are collected repeatedly over time. The longitudinal measurements are often subject to missing data which impose an analytic challenge. We propose a functional multiple imputation approach modeling longitudinal response profiles as smooth curves of time under a functional mixed effects model. We develop a Gibbs sampling algorithm to draw model parameters and imputations for missing values, using a blocking technique for an increased computational efficiency. In an illustrative example, we apply a multiple imputation analysis to data from the Panel Study of Income Dynamics and the Child Development Supplement to investigate the gradient effect of family income on children's health status. Our simulation study demonstrates that this approach performs well under varying modeling assumptions on the time trajectory functions and missingness patterns.