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Problem 129 Easy Difficulty

In the case of light damping $\left(c<c_{c}\right),$ the displacements $x_{1}, x_{2},$ and $x_{3}$ shown in Fig. 19.11 may be assumed equal to the maximum displacements. Show that the ratio of any two successive maximum displacements $x_{n}$ and $x_{n+1}$ is a constant and that the natural logarithm of this ratio, called the logarithmic decrement, is
$$\ln \frac{x_{n}}{x_{n+1}}=\frac{2 \pi\left(c / c_{c}\right)}{\sqrt{1-\left(c / c_{c}\right)^{2}}}$$

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Vector Mechanics for Engineers: Statics and Dynamics

Chapter 19

Mechanical Vibrations

Section 5

Damped Vibrations

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everybody. So we're looking at this figure right here, and I kind of uploaded of giant one. Um, those bring it Go. Not bad. Down a little bit, but we're looking on this figure right here, and we want to show the ratio any two successive maximum displacements of axe, not civil has me ex ny and looks like X and plus one okay, and is a constant is a constant in the natural log with him. With this ratio, call the log of interest. Placemat. Okay, so here we're trying to prove what they gave us, and we're also trying to find the exit An accent plus one. So from this, we have x not here and appear, if you can find it here. Okay. Done. Here we have this equation. X equals e minus. C Teoh. That's just be a to you, Teoh. See? See? One sign their U D T plus C to co sign w. Leave the damping a teeny noise. Tell that I always make my tease like this, so I don't could confuse sign. Um, that is one way to help you when you're Matt, we do. Math is trying. Make your letters as clear as possible. Especially your plus sign. So now we have we couldn't see, uh, damping for Quincy is this And it's the number of times it goes, and we have one of minus the damping factor squared up. Then these other equations were looking at and for the vibration of the amp. Here we have X equals x not e minus c to EM t And then we have sign I'm dancing factor almost got t times t plus by, and I'm just going to separate this drama will remind you guys see it better. Things from the sign is because this is a sine wave and you can see that if you see right here, we're getting this number by okay, you're just curious. But so here we have, we're gonna put in these boxes that your tea is gonna equal t end. That's the number of times your acts is gonna equal the number of times for sex in the displacements. Okay, so now we're gonna plug it into the boat, This one right here. So we have X and equals X not e minus C to, um t n times sign W d t plus by okay. And we have this equal to line and this is gonna be a And now I want to do the maxim displacements and wanted to the max this placement it is Here we did the vibration of diminishing so vibe of timidity. Okay, so when that's displacement, we're gonna have tea cools on. You can do this in greeting people kaido the same so t equals t in plus wine in absence. Be one. And here you're looking at Siri's, by the way. And this is Siri's. Okay, that's something that you will get used to doing your learning patterns and you'll have the Siri's. And finally to co sign for you guys. Okay, back to this So x and plus wine equals that's not e minus C divided right to writings, tea and plus one. And from that, we have a sign that you d see and plus why, Plus by this one in this year, Alright, school down. So from this, we're gonna set our give you see and plus wine people to utter beauty, saying a and B A and B pulled from the inside here and minus five part. And from this we do this we enough where? Get too high. And we're gonna bring this over here. I'm not going to. And plus one minus two my he and my equals two pi. So now we have this divided by each other, and this is make sure to get all this river now. Way of acts, not times e minus C to, um, and provided by That's not e minus c to, um, eating line. If you Yes, with me C two times he minus t in class one. And because of that, that from here we end up with e equals C two m highs too high. And, you know, from this we're gonna have this people to si Hai. And now we need to figure out Ow D and we already have that above. But I can rewrite this for you. Um, by let's wait for second with you want to do now is, uh this people see pie and this First we want to get the WG down. So to do that, we're going to you l n Besides, Okay, So we do you got and you end up. We have Poland, X and X and plus one, it's one pulls sleep high. Okay? I know it's not a w. I just like saying wgi. Um, so now from above, I gave a beginning. If you don't see here, so one my sees efficient, it's square. And here we're gonna have this equal C to m. So because I have because why, my c We're so because we're not well that into this, please. Here. All right, so this is gonna be see hi to M M C C. Wine divided by wine minus C C C square. Yeah, I know you can burn this. And because this we have to buy up, out if you have c c here divided by one minus. And so we have got and that group, But we were getting. And we also find the constraints that were poor with his A a and B. You have that right. Okay, guys. Thank you.

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Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek

Vector Mechanics for Engineers: Statics and Dynamics

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek

Vector Mechanics for Engineers: Statics and Dynamics