The design and construction of a mechanical Duffing oscillator suitable for use in an undergraduate laboratory is described. The oscillator provides a straightforward introduction to nonlinear vibrations and chaotic behavior that is both pedagogically appealing and easily adapted to students at various levels of sophistication. Good agreement is observed between the motion of the oscillator and computer simulations, which provides students with an introduction to the power of mathematical modeling. Period-doubling routes to chaos can be observed both experimentally and numerically.