Now that Michigan has become a right-to-work state, unions in this stronghold of organized labor confront a new and urgent problem: convincing members to continue paying for their services instead of taking them for free.

Brushing aside protests from thousands of labor supporters, the Republican-controlled state House approved measures Tuesday making it illegal to require that nonunion workers pay fees to unions for negotiating wage contracts and other services. The Senate did likewise last week, and Gov. Rick Snyder swiftly signed the bills into law.

The laws take effect 90 days after the Legislature adjourns this month, giving unions little time to devise a strategy for keeping members on board and convincing nonmembers to continue their financial support.

Union leaders said it was too soon to predict how the laws would affect their membership and recruiting, partly because workers covered by existing labor contracts won’t be able to stop paying union fees until those deals lapse — which in some cases will take several years. Contracts between unions and Detroit automakers, for example, are effective until September 2015.