Sometimes I run/hike with my dog. I was planning to take my camera with me, however I'm afraid that running with a DSLR camera can damage it. Is it true? Are there any steps I could take to ensure my camera is safe? To be more clear, I assume that environmental conditions are not severe (temperature, pressure are okay, no sand or perspiration). So, the only factor I want to discuss is the vibration.
Having the camera on your body while running will subject it to relatively low frequency vibrations (1 Hz or thereabouts). That's way too low to hit the resonance of any part inside the camera. The issue is therefore just force due to accelleration. At such a low frequency, that will be small, probably not more than ±2g, which the camera should be able to take pretty much indefinitely.
The real threat to the camera is you falling and the camera hitting the ground suddenly, especially if it's a hard surface like asphalt, concrete, or rock. You should be able to run with your camera bouncing along with you all day long, but just a single 1 m drop to hard ground could cause serious damage.
Don't worry about it, your camera is probably tougher than you are.
I've dragged DSLRs all over Asia for nearly 10 years now and the only problem was the mould.
Motorbikes, planes, running, cycling, walking, climbing, they've survived them all and the time I hit myself in the face jumping over a fence I had to sit down for 10 minutes to recover and my camera didn't even blink.
PS. I have heard that the strong, high frequency vibration from helicopter will kill a camera in short order, but I've not had the chance to find out yet.
Some Canon and Nikon are known for hardy, robust build quality that will survive an array of physical punishment. Not all cameras are equal however. To avoid a long term problem consider switching to a mirrorless camera for your runs. I suspect that moving parts are most vulnerable to damage during vibration. The simplification of a mirrorless design would likely avoid such damage without compromising on photo quality. And when the helicopter picks you up for the ride home your camera will remain trouble free.