Animals LOVE when you sing to them! You don't have to be a 'singer' to lull your pet to sleep, or help them relax. They don't care if you're monotone!
When you sing to them, include compliments and praises in your song, just make it up as you go. I always sing to my cat about how much I love him, and how handsome he is and what an amazing shaman healer cat he is, he just closes his eyes and relaxes his body. :) If you have a dog and know that he or she is a real 'protector' for you and your family, thank them about that in your song, and whatever else you think they think their 'job' is. If you're not sure what your animals' job is, just praise and compliment them.
Go ahead, don't be shy, sing to your animal partner, they will love it!!