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This study discusses model reduction techniques for second-order index 3 descriptor systems using the balanced truncation methods; in particular, linearised equations of motion with holonomic constraints are considered which arise in mechanics and multibody dynamics. It is shown that the index 3 system can be converted into an equivalent form of index 0 system by projecting it onto the hidden manifold. When model reduction is applied to the projected system, explicit formulation of the projected system is not required. The low-rank alternating direction implicit iteration is also discussed for solving the projected Lyapunov equations of the underlying descriptor system efficiently in an implicit way. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]