 Hi everyone. In this video I want to talk about coil springs. Coil springs being extremely common, you know, coil, squish it, provides a bounce back. Again elastic energy storage device, useful in all sorts of situations need, you know, damping elastic recovery, things like that. So when we talk about coil springs and our ability to analyze them, suppose we have, okay that's a bad coil spring, I don't know how to draw a good coil spring though, so we have a coil spring and let's have a little retaining retainer here and our force is applied to that and we'll define this outer diameter as d and a wire diameter as little d and if we assume that this force is uniformly distributed on this retainer so that it's applied evenly, you know, there's no offset, it's right at the middle. We can look at, you know, this spring and we'll make a cut and just look at a portion of it. So let's say right here and we have a cross section that looks something like that. Now we have to translate this force that we applied to the spring to this new free body diagram so we can kind of figure out what's going on internal to the spring. So for this to be balanced we would expect there to be, you know, a force and I'm just going to use the same notation here of f equivalently applied so like a shear stress or like a shear force over here and then I need a moment right, but because of the orientation of our coil in that, you know, I'm looking at it and it's into or out of the board in this case, this gets represented as more like a torque and it's going to be equal to f times half of the distance so one radius of the outer, you know, overall diameter of the spring. This shear force ends up being, you know, relatively minor in terms of what's actually happening so really we only need to be thinking about this torque and taking this torque and turning it into a torsional stress using again our standard equation and if I make some substitutions because I have a round cross section I get something like that. Now it's in terms of torque and diameter and I've taken j out by substituting in the equation for a round cross section and I can also substitute in what I just have for torque up here in terms of force and I can get an equation in terms of the force applied and this is just a general equation for the shear stress we would find in a torsion bar based on the force applied to the spring in this case. Now one place that we do have to be a little bit careful is that our equation slightly doesn't really apply completely to this scenario because the assumption with this with a torsion equation like this is that I have a straight a straight beam a straight long axis of my part and in reality so if I just kind of you know draw a neutral neutral axis this way in a cross section that I'm going to look at and in general I represent the torsional stress like this right I have uh stress applied oops internal to this part and it's it's zero at the middle and goes out to some maximum at the outer edge and that's great right but in reality now it's a little bit I'm obviously going to exaggerate this in my drawing I have a curved member because the coil spring is is circular in shape so I have a curved bar so in reality when I when I look at how my stress is distributed it's a little bit different right because it's got to take into account that geometry so what that means is is out here somewhere my stress is greater than tr by j and down here you know my negative stress or my opposite direction stress is greater than tr by j as well so that's it's not quite a perfect system to say we can just use that that tr by j equation so what we do instead is we make some adjustments and basically we we factor in a correction so first under static but also under fatigue we have corrections and we'll look at both at the same time so I start with my base basic equation that I've already written and that's the same in each case of course and now I'm going to multiply by a static correction factor or a fatigue correction factor I'm just using the notation in the book here and those factors look like this in case of w where c equals something we call the spring index which is a relationship between the outer diameter and the wire diameter so a ratio there so this is the spring index which oftentimes you'll find springs defined by this the spring index great so now we have these correction factors and what they're intending to do is correct for the fact that our shape of our rod that we're looking at in torsion is not straight as the normal tr by j equation suggests or assumes but has a slight curvature to it and it factors in how tight that curvature is too right by the fact that we're including this spring index which accounts for the diameter the outer diameter and the wire diameter so both things are going to have an influence when when it comes to determining the stress so this allows us to calculate the torsional stress in our springs you know static or fatigue you know whatever our situation may be one other concern that we have when we talk about springs is that they can take on a set and you may have heard of this before generally or basically what we're talking about is that you know they can they can have plastic deformation so deformation that isn't recovered which means i've you know compressed them a bunch of times maybe in fatigue or i've overloaded them you know drastically one time and and cause them to take on a set and that's something we generally we want to avoid so what we do is we can set a limit we set a design limit and we can set it such that the the the stress in the spring and i'll define this in just a second is less than or equal to 0.35 somewhere in the range of 0.35 to 0.65 of the ultimate strength of the material depending on what the material is and if there's any pre-setting so sometimes in the design of our spring we could we could have the manufacturing process include a pre-setting which makes it less likely that we would have problems with this down the road this tau sub s is the stress at spring solid so spring solid is is basically exactly how it sounds if i have a spring and i compress it all the way till all the coils touch you know and it effectively becomes a solid cylinder that's as far as you can really deform it right i mean obviously you can you can plastically deform it when it's a solid cylinder as well but while it's still a spring we can compress it until those coils touch and that's what we would consider to be spring solid in that case so that's a situation we want to avoid so more definitions that can be useful um to talk about or to know our spring deflection or is spring deflection and deflection we've defined you know deflection of things before but basically just taking the equations that we have and then substituting all of the stuff that we've talked about in i get 8 fd to the third n over d to the fourth g and here n is the number number of active turns so what we mean by active is it's coils of the spring that are actually um deforming deflecting when we compress it um usually our springs will have a coil or two at the top and at the bottom that are passive they they don't do anything they're just you know making up the end of the spring um and they might you know have a different um different spacing because uh of that's what because that's what they're doing so n uh is the total number of active turns we can then define n sub t to be the total number of turns so including active and passive and generally speaking n sub t is equal to n plus two all right one more thing to define here is spring rate or spring constant relation of force to deflection and again taking all the stuff we we now know i can substitute in or and rearrange this equation to get spring rate so last thing um i really want to mention in this video is uh one one we can define the solid length length of the spring so this is again when it's compressed to solid uh we have the solid length and what that is specifically somewhat depends on what the end of our spring looks like so if i have plane ends which would be my spring comes over and then it's just cut you know perpendicular oops drew that the wrong way it's cut perpendicular to the axis of the shaft that's just called plane ends and then my spring or my solid length is equal to n t plus one times d if i have plane ground ends oops well pretty close so the end is ground to make it flat we have l s equals n sub td if i have square ends that is where it's cut square rather than perpendicular to the axis of the spring and in that case l so that's equals again n t plus one times d and finally square ground something like that l s equals n t times d okay so depending on which um type of end i have on the spring i would have a different calculation for what my solid length is and and that's fine now we can use that um you know when we're specifying what we're doing um generally when we're designing a spring um there's there's kind of a lot of variables that we might need to consider so one of the first things we might do is design the outer diameter the the overall size d large d and the wire diameter small d for whatever stress we're setting as our limit you know relation to um trying to avoid a set and uh usually more than one combination of those two variables can be used to achieve the same thing so we might need to limit it based on uh the geometry of the application you know what our limitations are um we can then design the number of turns based on the spring rate for what we're trying to do uh design the length of the spring to allow for um some clash allowance so usually we're trying to avoid that spring solid scenario right because it's it's where the spring stops being a spring um and we usually build in some sort of clash allowance there which is effectively a safety factor and after doing all that if the design doesn't work then you know we can um if the design doesn't work we can go back we can go back we can change uh d um the outer diameter and the wire diameter uh to something else and try again and kind of iterate through that until we've come up with a a good solution thanks