animation-name.js

// Attatch an event listener to your element for when the first animation 
//  finishes, but only when it's named for the first animation
$('.my-element').on('animationend', function() {
  
  if ($(this).css('animation-name') == "animation-1") {
    // Change the CSS animation-name property for the 2nd animation
    $(this).css('animation-name', 'animation-2');
  }
});