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In an underdamped system $\left(c<c_{c}\right),$ the period of vibration is commonly defined as the time interval $\tau_{d}=2 \pi / \omega_{d}$ corresponding to two successive points, where the displacement-time curve touches one of the limiting curves shown in Fig. $19.11 .$ Show that the interval of time $(a)$ between a maximum positive displacement and the following maximum negative displacement is $\frac{1}{2} \tau_{d},(b)$ between two successive zero displacements is $\frac{1}{2} \tau_{d},(c)$ between a maximum positive displacement and the following zero displacement is greater than $\frac{1}{4} \tau_{d}$

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Mechanical Vibrations

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Everybody so here we have a b and f to ask us and ask in different displacements here and because of this, we have this equation, which is equals x, not e, minus c, 2 m t times sine omega d, t plus pi and the derivative here or X, not is gonna, be x, not omega, b, minus c divided by 2 m t cosine, omega d plus 5 here that forgot plus x, not minus t 2 m p, minus c 2 m t will be the sign 100 here, so omega d, t plus 5. It and now we're going to set this equals 0 and we're gonna put this on 1 side and this on 1 side. So we have rated cosine omega t plus 5 equals x, not minus c 2 m p, minus t, 2 m p and then we wrote sine negative is t plus 5 really do not like it when to have tea and pluses that together wore or symbols here. So we're going to bring this here and this here and we know that is this, so we're left with q m neat c equals and w or negative, plus 5, and now we can actually write this as t initial, as the t initial is going to be, The base differences a little at the time when maximus height occurs in initial time, so for that we have minus 2 minus 5 divided by so they fly what we'll do in the 25. So now, when it's 0 or the time when the maximum height, we have the net divided by c, plus pi minus 5, and then they can be in this part, will probably put you up if you roland, that's the positive afraid. That'S because you're looking at a sign of pain, so you have this type of oscillation, so each mite! Less of for that and that's okay take the time so now we're gonna minus these together tete, we batoned, plus pi minus pi, divide us same. We can find it be like it out here all okay worse, so you can also cancel out your initial, not my initial, and you can obviously tell that this becomes positive. So you get this so we end up with. We can actually plug in up here to 5 in the caltha. They gave us for this problem and when we do that and up with that's the answer for a and now for, we have pretty much 20 displacing. So we have the initial plus 5 equal 5 and initial here equals minus 5 ay, and from this we have your guy. We also have not minus plus 5 equals to 5, and you get to be right. There we go in this table 3 pi minus 5. Divided by all that we have in this, we can have them minus each other same thing that we did above and eatethere we go. If i can crack myself guys and bolt, so we have 2 pi minus pi, and then we look like that rule answer. Maga minus pi, minus 5- and here you end up with similar to above part a and because of this were also going to put in what they gave us with was the pi of 2 pi oso, because there's yet to hoind. That'S the answer. For and now we can do and see what the max positive distribute of progenitors grantor max occurs, join your negative, the initial plus 5 and your 0 displacement. So this is your max. Displacement is elated. Okay, and here we have the hero, displacement and that's a d 0, and so because of this, what we're going to do is have place lent me. I plus 5 minus your hear replacementand here now now for this we're gonna take away omega out and we have not worried about. The 5 of the fives are gonna console each other. So here we have, and now from this we end up with taking the w over. So here we go initial and for us they gave us that we have above sohere. We have a tell you this at, as this has been multiplied here again. You have a world apart because we're plugging in here you piaand. This is your answerr c guy's right,

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