We present a set of nonlinear dynamic equations for bicycles and motorcycles. The model is based on Euler-Lagrange equation of 4-body nonholonomic system. The software package Maple is used to obtain the complex coefficients of the dynamic equations. The final analytic results were converted into C++ for fast calculation. 3 coupled second order nonlinear ordinary different equations are obtained for roll, steer and lateral dynamics. The coefficients of the equations are shown only depend on the roll and steer angles. We also show that under small roll and steer angle, the first two nonlinear ODEs yield the well-known linear model. The coefficients are computed and compared with published linear model. The nonlinear model can be used for design new bicycle and motorcycle, study their dynamic behavior for large roll and steer angles under high speed, and extreme dynamic conditions.