 In this video I want to talk about the critical speeds of shafts under rotation. So I've previously mentioned that shafts when they rotate at high speeds can have a natural frequency like basically anything and when the center of rotation does not coincide with the center of mass with a shaft and anything that may be mounted to it. We can have problems with this critical speed in that we may get large displacements wobbling and things like that that can cause problems for our machine or for our device. So I want to talk about how we look at these critical speeds and then know to avoid them in our design. So the first example here is just a simple shaft with a single point load mounted to it and the single point load represented here as a mass is causing a deflection delta st in that shaft and we're kind of neglecting the mass of the shaft itself here and just kind of assuming that the the mass of whatever is mounted to it is is much larger. So you know as we expect each shaft would have a some stiffness and we've previously seen the relationship between a weight applied to a shaft and its deflection and that stiffness by this equation. So then without going into details of derivation we can find the natural frequency natural rotational frequency using an equation that looks like this where it's equal to the square root of that stiffness over the mass and if I do some substitutions of different things so weight is related to mass and gravity I can express that equation like this and therefore I can also express it like this as the square root of g acceleration of gravity over delta st. Now oftentimes we would want to or usually would express our critical speed in rpm rather than you know radians per second so we can do that conversion just as a unit conversion and therefore this is expressed in rpm by multiplying that natural frequency by 30 over pi. Now what if we have different variations on this we get slightly different equations but but still we're generally the same principle. So suppose we have several objects mounted on this shaft and I'm just going to draw three potential masses here m1 m2 and m3 and then each of those causes some deflection measured at the location of that mass delta 1 draw that a little better delta 1 delta 2 and delta 3 and I'm leaving off the the sts here just to make it a little easier to write so now if we want to express this and determine its critical speed we have our same unit conversion in here and now we need to treat this as a discrete like summation of the number of number of parts so we can write this as g times the summation of w delta so this is all the little components of delta and w divided by the summation of w delta squared and really what these two summations are doing is getting like a weighted average for these various components so it's it's taking all the little components and waiting them over the over the hole and finding their contribution then to that critical speed now in these previous two we've more or less neglected the the shaft itself but of course a shaft itself is likely to have deflection of its own with its own mass so if we have just the shaft mass by itself considered we can determine again some deflection where it's at its maximum value and if we were to do this we can find an equation for natural frequency which looks like this 5g over 4 delta st and of course then we know already how to convert that to to critical speed but basically this is just giving us a relationship based on the mass of the shaft by itself we also have an expression which we can use to find delta st in this scenario which is 5 w l to the fourth over 384 EI and this might look familiar from a from beam bending and when you talk about beam bending problems and things like that in units of inches here so this gives us the ability to to calculate what's going on with a given shaft