A washing machine is provided herein. The washing machine appliance may include a cabinet, a tub disposed within the cabinet, and a passive pneumatic damper. The passive pneumatic damper may be attached to the tub within the cabinet. The pneumatic damper may include a rod, a casing, and a piston. The rod may extend between a first end portion and a second end portion. The casing may include a body that defines a damping chamber and an axis of motion. The body may include a first end cap and an oppositely-positioned second end cap, and a sidewall that extends between the first end cap and the second end cap. The piston may be slidable along the axis of motion within the damping chamber. The body may define a passive air aperture that extends in fluid communication between the damping chamber and an ambient atmosphere.