  We show how to utilize magnetostriction to synchronize two mechanical
vibration modes in a cavity magnomechanical system. The dispersive
magnetostrictive interaction provides necessary nonlinearity required for
achieving synchronization. Strong phase correlation between two mechanical
oscillators can be established, leading to the synchronization robust against
thermal noise. We develop a theoretical framework to analyze the
synchronization by solving the constraint conditions of steady-state limit
cycles. We determine that the strong cavity-magnon linear coupling can enhance
and regulate the synchronization, which offers a new path to modulate
synchronization. The work reveals a new mechanism for achieving and modulating
synchronization and indicates that cavity magnomechanical systems can be an
ideal platform to explore rich synchronization phenomena.
