Singing well is as much about technique as it is enthusiasm and passion. You can really mean the words you are singing, but if you aren't able to properly control the pitch and volume of your voice, you aren't going to be able to convey what you're trying to get across. Just like playing any other instrument, it's necessary to practice the techniques required, so your muscles can learn the relevant movements and commit them to memory.
Practice moderating your breathing. Controlling the pressure of the air in your lungs as it passes over the vocal chords is the key to obtaining more volume without sacrificing control. Take a deep breath inward, then practice slowly releasing it at a steady rate. A fast breath in, followed by a slow, steady exhale.
Focus on your diaphragm. You should be breathing and singing from this point just at the base of your ribcage. It may feel alien at first but it is the technically correct way to sing, and once you master it, it will greatly improve your control and volume. When your singing voice is controlled by the muscular diaphragm, you'll be able to generate much more air pressure when you want to sing loudly, than you possibly could by just exhaling harder.
Feel the resonance inside your head. When you sing certain notes, you'll be able to feel the vibrations through your skull, it may even make the tip of your nose tingle. Practice singing different notes in a scale and feel the different points in your head where they resonate. Resonance and learning to properly control it is key to being able to push your voice without yelling.
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