Everybody Matt here for late in sports cards ripping two boxes of 20 22 tops chrome update baseball hobby These are for Samuel H. Good luck there's Abrams Tony and Kevin Smith O'Neill Cruz It's a rookie debut purple Martin Perez Nick Allen Josh Winder there's a wonder rookie debut Melendez and Garcia Shane Bos debut purple Ronald Acuna all-star Peña Full sands rookie purple Albert Pujols die-cut greats and winker Jayrod debut Juan Lewis Jack Sawinsky Brendan Donovan rookie purple Jason Crizzan Steven Kwan rookie Pedro die-cut greats and bra so so contra bra so Ohtani and La Vestita Michael fulmer Eleanor Nato This is rookie refractor Ryan Pepio That is the 299 42 of 299. There's uh, Joe Perez favorite die-cut Pepio Roberts Thomas and brash seager Donaldson There's Jayrod rookie stallings Bryce Harper die-cut greats and Kwan Well, let's see a good auto in here Nicoledolo rookie refractor purple refractor Steven Kwan generation now in Jack Sawinsky rookie This is rookie auto Adam Euler tribino Buxton and do little Carrie Sanchez Gary and Correa Miranda and Naquan Shane Bos generation now Rios Adore Jeremy Peña rookie Nick Allen purple Tony Gwyn die-cut greats and Wander Franko rookie debut purple Seya Suzuki rookie and Scherzer Seya Suzuki generation now in Joandron rookie purple Albert is up after this Albert Kevin Joseph Bobby Whit jr. rookie and Royce Lewis debut purple Autista Cole Thomas debut purple Meadows RT's and MJ Melendez rookie purple Please be sure to hit the like button on the live stream. It is appreciated Jack Peterson Yes, Edward. Yours was um last What's up, Yogi? How's it going? Kunya and Joe Ryan It's Freddy Freeman purple All right, that'll do it for your box of Samuel appreciate it. We'll get these right out to you


All right, I'm gonna jump straight into today's question from our buddy Imogen. She says hey I've been enjoying your new website and emails. Thank you They're very interesting and helpful. I'm wondering if you can help me with something specific I've not yet had a lucid dream and I've not been able to try the wake-back-to-bed strategy because I'm afraid I'll be tired the next day Excuse me Will this affect my ability to lose a dream? She says she's an insomniac and she takes melatonin at night so To start off with when you do the wake-back-to-bed technique It involves waking up earlier than you would normally wake up.  It's just it's a very natural and common thing But unfortunately the wake-back-to-bed technique does involve doing that So yes, if you have a job to get up for in the morning early and you need to be alert all day The next, you know, the following day, maybe try a different technique Try the mild technique or the what the waking juice lucid dream or even, you know, something else There are plenty of techniques on my website. How to lucid how to lucid.com So you can go ahead and check those out So I'm going to be starting to do a video a day Um, and I'd really appreciate if you could leave a comment on this video or whichever video you Watch that you want to reply to asking me a question And, you know, even if it's just to say hi just I'd like to get some feedback from you guys So I know where to go with the future of this youtube channel Do you want to see whiteboard videos, you know, whatever the case may be? So leave some feedback give this a like if you enjoyed it and Also, let me know if you'd prefer longer or shorter videos. Okay So, yeah, I hope that makes sense and I'll see you next time


Hey, this is Brett and this 2017 Ram 3500 Crew Cab Long Box Dual Rear Wheel Laramie is stock number 12439Z. We're here at Summit Automotive in Fond du Lac, Wisconsin, you're new and used heavy duty truck and ram headquarters. This 2017 Ram 3500 has the 6.7 liter Cummins diesel engine. This truck has been fully safety and inspected by our service shop per the state of Wisconsin inspection process, has a fresh oil and filter change all the fluids have been checked and topped off and this truck is 100% ready to go. We're going to go inside underneath, start it up, look under the hood in this video. White pearl is the color and we shoot all of our videos in 1080p, 60 frames per second so if you have HD capabilities on your computer, tablet, smartphone or television I highly recommend turning them on right now because it is your best way to check out the quality, condition and options of the truck before seeing it in person. And if you like the video and how we do things here at Summit Auto in the upper right hand part of the screen is a subscribe button to our YouTube channel. Click that and then click the bell notifications and you'll get updates every time we do videos here at Summit Automotive. This one comes with the polished aluminum rims.  Nitto Ridge Grappler LT28575R17 tires and these tires look like they have probably about 80-90% of the tread left on them, frame and underbody is an exceptionally clean condition. Front fender I didn't see any major dents or dings on there, it does have the projector lamp headlamps and LED running lights, factory fog lights, front bumper, excellent condition, no dents or dings on there, it does have the front bumper parking sensors. The hood is in really nice condition as well and it does have the cab lights or clearance lights up top. Passenger side front fender looks really good and the passenger side rim, no major scuffs or scrapes on that. White pearl has just a little bit of metal flake too, it's kind of hard to see because it's white, there we go, it's got just a little bit of metal flake in it when the sun hits it just right and as you go down this side of the truck take note of how clean the body is, how reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and just cannot make the trip down but you're still interested in purchasing the truck, you can see the truck, hear the truck and have confidence in the vehicle that you're looking at before you even get here so when you do get here there's absolutely no surprises and you can make a smart and informed buying decision from wherever you're at and if this video helps you make that decision let your salesman know that you saw the video, it was helpful and that Brett sent you. Dualy tub looks really good, I didn't see any major rock chips or scuffs like that, it does have the 3M tape on there. Back rims are in excellent condition, I didn't see any scuffs or scrapes. Very little corrosion on there. Back tires are just as new as the front tires and the frame and underbody is in very nice clean condition as well. It's like that all the way underneath. That's all the remaining factory exhaust so it hasn't been altered in any way. Boy that frame looks like it's brand new. Cad corners, lower rockers all look really good. Comes with a nice set of Weston Pro Tracks chrome step bars. Get the fender flares on there with the Laramie package coming around to the back. Rear bumper is in nice shape, no major dents or dings on that. Those have the backup parking sensors, full towing package, which includes the receiver hitch, four pin and seven pin wiring. The tailgate is in excellent shape. Has a locking tailgate backup camera. Spray and bed liner has a shock down assist tailgate, also has this ladder attachment to it. This is a pro coat, looks like this truck was in Louisiana, at least that's where the spray and bed liner was done. Has a fifth wheel bed rails in it and seven pin wiring back here, so it's all set if you wanted to put a fifth wheel in here. Put her in and away you go. And going down the driver side just as clean as that passenger side. No major dents or dings in the box, looks really good. And for full disclosure this back rim is in excellent shape as well. Down the rest of this side of the truck, very nice and clean. I didn't see any dents or dings. Has the heated fold up tow mirrors. They fold up like so. They have built in directional signals in them. They also fold in, enter and go system inside the Laramie package. Gives you the tan leather interior, there are no rips, there are no tears on the seats. They're in excellent condition. Both these front seats are heated and cooled memory driver seat lumbar and a very nice set of factory formats in this truck. Auto headlamps, audio controls in the back of the steering wheel, power windows, power locks, power mirrors. These mirrors do power fold in and I always like showing both sides so that you know that both sides are working properly. We'll hop inside. Check out the miles and radio and everything that this truck has to offer on the interior. You can see that this one has 20,718 miles on it. Comes with the digital speedometer, compass, outside temperature display, range to empty and trailer brake and you can change those controls to read whatever you want. Instrument cluster is very nice and clean. Comes with the heated leather wrap steering wheel, cruise controls on the right, Bluetooth and information center controls on the left, and audio controls on the back of the steering wheel. This one has the six speed, 68 RFE automatic transmission with the optional tap shift comes with the 8.4 3C radio, AMFM and Sirius XM radio capabilities on here. Also the factory navigation system, there's your Sirius, then you can do your climate controls here as well. You can do the climate controls up there, notice you do have the dual climate controls and if you don't like doing them up there, no worries. You can do the climate controls down here as well as volume and tune controls, turned out four wheel drive, factory brake controller, stability control, tow haul mode, factory exhaust brake, front to rear parking sensors, you can turn those off and then more heated and cooled seat and heated steering wheel buttons right there. 115 volt, 150 watt plug in, this one does have the Alpine premium sound system, get the little badge on the dash there, keyless entry with the remote start, passenger side, floor mat and seat are in excellent condition, no rips or tears in there, smells extremely clean inside this truck, I don't think it's ever been smoked in and the headliner is in excellent shape as well. You do get map lights up there, power sign, rear window button, home link buttons for your garage doors, security systems and lighting systems and assist and 911 buttons in the rearview mirror. We'll take a quick look at the back seats and then we'll start it up and take a look under the hood. Back seats just as clean as the front seats, these seats are heated on the outboard side, no rips or tears back here, it does have the latch child safety system for any child car seats you may have, power slide rear window, you get these speakers in the ceiling part of that Alpine sound system, these seats fold up for extra storage, get a load floor which is really nice, if you got muddy boots or tools put them on there you're not going to recognize leather seats or if you got pets put both sides down they got a nice flat place delay, you get storage under here, there's a receiver hitch reducer, factory floor mat back here is in nice shape, get in-floor storage with removable bins for easy cleaning, that's really nice, you can put ice and drinks in there, it's like you have too many coolers wherever you go and then you have the heated seat buttons back here and a 12 volt power point, underneath that passenger seat is your factory subwoofer as well, back of the door or back doors have the child safety locks and the bottoms of the doors are in excellent condition, we'll start it up and take a quick look under the hood, starts right up, no check engine lights or anything like that, I would personally like to thank you for checking out the video today and hopefully from this HD video you will have been able to verify the quality and condition of the truck all the way around, sometimes opening the hood helps show under the hood, see all the lights are working nicely, under the hood we have the 6.7 liter Cummins diesel, engine bay is very clean, runs very smooth, once again this truck has been fully safety done and inspected by our service shop as a fresh oil and filter change, all the foods have been checked and topped off and this truck is 100% ready to go, there's your emissions sticker and to see more pictures of this truck or one of our 450 new and used cars, trucks, SUVs, minivans, Wranglers, half tons, three quarter tons, one tons, you name it we got it, go to that website right there www.summitauto.com, full pictures and descriptions of every single vehicle from two locations, all at summitauto.com, if you'd like to check out more HD videos you can go to youtube.com slash summitauto, remember to like, subscribe and share on this video and all the videos that you see there, click the bell notifications, you'll get updates every time we do videos here at Summit Auto, in fact in a second you will see a link to subscribe to our YouTube channel, the upper left link to more Ram 3500 truck videos like this from the upper right, link to this vehicle on our website in the lower left and a link to one of our latest YouTube videos in the lower right, click those check us out and we're super excited to help you with this ultra clean 2017 Ram 3500, crew cab, longbox, dual rear rear rear me and white pearl, thanks again for checking out the video remember to like, subscribe and share


Welcome back everyone, we are going to continue our discussion from the last class in which we saw basically we learn about Duhamel integral and we also saw that how to obtain solution of a single degree of freedom system subject to any arbitrary excitation ok. Now we are going to discuss some of the specific cases of pulse excitation as well as step forces and then see whether how we can obtain the analytical response of a single degree of freedom system. Last class we saw that if we have some arbitrary excitation ok, so if we have the equation a single degree of freedom system which is represented by let us say this differential equation ok and I am not considering damping here for the time being and if P t is some arbitrary excitation ok. So, we derived a integral which is called a convolution integral ok to find out the response to any arbitrary excitation P of t and the idea behind that was that if I have let us say any random ok any arbitrary varying load or excitation P t then the response at any time t ok to get this response we divided this force into very small duration forces ok. And we said that these forces small duration forces would basically represent a impulse ok. So, they would represent each of them an impulse and their response at time t would be some of responses due to each of impulses ok and as we know the response due to an impulse is basically free vibration with the initial velocity ok and we sum up all those responses up to time t to get the response u t at time t. So, basically we derived the expression first for Duhamel convolution integral and we saw that we can substitute the value of unit impulse response function to get the Duhamel integral ok. So, I am just going to write the final expression ok. So, for an undamped system we got this expression ok. So, we can utilize this expression to get the response ok and similarly for damped system as well we can write down the Duhamel integral. But now I would have omega t here in this there would be a e to the power minus eta this term here times sin omega t t minus tau d tau ok. And we discussed that in some cases Duhamel integral is very useful to get the response ok analytical response of function u t something like this ok and in some cases you can use the conventional method of solving the differential equation and it will purely depends on the problem ok. So, we will have to inspect the problem and look at the integrand that you have inside that integration to see which would be easier ok. So, utilizing this basically we obtain the response to a step force which is represented something like this ok. So, force sorry there is no ok. So, this is a step force ok which suddenly applied low of load of magnitude p naught and then it is maintained over time as p naught ok.  And then we found out what is the response due to a linearly increasing force which this say this is p t here and this is t and let us say this force is represented by p naught t by t r where p naught is basically this amplitude which is achieved over time duration of t r ok. So, during the linearly increasing linearly increasing phase we try to obtain the solution u t as u s t naught ok times t by t r times sin omega n ok times t divided by omega n t r. So, as you can see it is vibrating at frequency omega n around its static solution which is represented by this here. So, u s t is represented by this p naught t by k t r which is basically this function here ok. So, this is. So, we saw that t r by t n is a very important parameter here and we are further going to discuss that today ok. So, let us now discuss we stopped at this point and let us now discuss what happens to the response during the constant force phase ok. So, what we are going to do today is basically a step force ok with finite rise time all right and this is a better representation of force that is applied in real life. So, if you think about it is almost impossible to apply a sudden force there is always some duration however small it may be ok over which this p naught or the amplitude of the force that is achieved ok. So, in general I can represent any force that reaches to its maximum value p naught through a step force with a finite rise time and we can also call it a ramp force ok. So, basically the problem statement that we have here is basically I am applying a force which goes to its amplitude p naught over a time duration t r ok and then it is maintained at that value p naught ok. So, if I want to find out the response to this step force let us see how to do that ok. So, our differential equation would be mu dot plus k u that is equal to p t ok and this is basically equal to p naught t by t r for t smaller than t r ok. So, for this zone here ok I am representing this function ok all right and for a constant force phase of course it would be p naught for t greater than t r. So, for this we have already obtained the solution and let me write that again ok I have obtained the solution as u s t naught times t by t r minus sin omega n t r divided by omega n t r ok and this is for t smaller than t r. Now, let us obtain solution for the second part now if you look at it ok the second part we have something like this ok and we know this is nothing but the differential equation for a step force that we had done previous to this step force with the finite rise time. So, basically what I am trying to do here the total response at any time after the rise time t r would be ok I am going to divide this force into 2. So, this and this here ok. So, let us say and this is p naught ok. So, up to this point my single degree of freedom system would have achieved some displacement and velocity ok. So, if you represent this instead of p t if you say let us say this is u t it would have achieved some u t here and some u dot ok. Let us say not u t, let us say this is u t r and then u dot t r ok and for a linear system if I am representing the total solution ok to this step function as a sum of solution to this linearly increasing phase up to time t r and then step force from point t r onwards. It would be basically with this any initial condition it would undergo free vibration right that we already know. However, note that the motion starts here at t r instead of t 0 ok and in this case we already know what is the solution for this one is it it we know that ok for this step force the solution is basically ok. So, this is step force and the solution I can write it as u s t naught 1 minus cos omega n t ok and again remember that this is starting at t equal to t r. So, this t I will have to shift to t r ok. So, if I sum it up basically what I have here the total response can be represented as free vibration response due to linearly increasing phase ok and because it is starting at t equal to t r and not t equal to 0 I am basically shifting it by t r. So, that by time ok variable is represented by now t minus t r ok and this I will write it like that sin omega n minus t minus t r ok plus there is a constant force phase ok for which the response I can again write it as u s t naught ok 1 minus cos omega n t minus t r all right. Now, this this is the free vibration phase response due to free vibration and this is the constant force phase. So, this is the response due to free vibration this is the constant force phase. Now, in this I can substitute the value of u of t r and u dot of t r by substituting t equal to t r in this expression first and then differentiating it and then again substituting t equal to t r ok. So, I will do that what I am going to do here just write the final expression ok.  So, this is the expression that I have and if you look at it this is nothing this is a constant here ok you can call it a and this is another constant is b ok and these are coefficients to sin omega n t minus t r and cos omega n minus t r. So, this is basically something of form a cos theta and then b sin theta ok and we will utilize this property later when we have to find out the maximum value of u t ok.  So, if you look at it here ok the I can write down my u t divided by u s t naught as this function that I have inside this one ok sin omega n and I can do may be one more thing remember omega n t r I can write it as 2 pi t r by t n ok. So, let me substitute that here so that I get this one as ok t n divided by 2 pi t r and then inside I would have the sin term 2 pi t r by t n ok then this cos term here which would be cos 2 pi t by t minus 2 pi t r by t n ok alright. So, if you look at this carefully ok my system is vibrating at frequency as frequency omega n or the time period t n alright. So, that is cleared vibrates at this natural frequency ok and my response actually depends on the value of t r divided by t n note that it does not individually depends on t r or t n ok. If you look at it it is always in the ratio form anywhere in this expression here for u t divided by u s t naught ok. So, it depends on the ratio. So, I might have different ratio of t r by t r for example, 1 divided by 2 or 2 divided by 4 as long as the ratio is still the same I would get similar kind of dynamic response for my system ok. So, that is an important point that the response actually depends on the value of t r which is the rise time divided by the time period of the system ok. So, that need to be kept in mind ok. So, if we want to plot this for different value of t r by t n ok I will have ok the response of the system ok. So, let me just copy that and then show you how does it look like ok. So, if you look at carefully here ok. So, what these plots basically represent the ratio of the dynamic displacement versus peak static displacement ok and these have been plotted for different values of t r by t n ok and the horizontal axis is basically the normalized time ok. So, if you look at it then my t r is very small like this ok in this case. Note that there are two lines here the dotted line is actually it is static displacement which is nothing but u static of time variation of the static displacement as p t divided by k ok. So, this is the static displacement assuming my system does not have any mass so that I can write it as this and which for this case like you know is simply this dotted line that has been shown here ok. The dynamic displacement is basically shown here using the solid line ok. So, the solid line is my dynamic displacement. Now, let us start with the smallest value of t r by t n ok. So, let me draw a representation of this one. When t r by t n is something like this ok it is starting from 0, but the rise time is very small ok. The response that I see it is similar to a step force and that should be obvious considering ok for very small value of t r by t n ok. The force actually resembles a step force. So, this is p naught by k displacement and then my system basically vibrates around the static equilibrium position which is around the point p naught by k ok. So, I can see here that my solution or my system actually vibrates around the static equilibrium position ok and the vibration, the amplitude, the amplification or let us say difference between the dynamic and the static it is quite large ok. Now, as I increase that as I go from t r by t n to this value that I have here ok. So, as I go from t r by t n to t r by t n equal to 0.5, I see that there is still lot of vibration. So, there is still like you know significant difference between static and dynamic. However, in this case that difference has actually been reduced ok compared to t r by t n equal to 0.2. So, as I increase my ok as I increase the rise time ok, the difference between the static displacement and the dynamic peak displacement actually reduces ok and that is further enforced by the fact when you look at t r by t n equal to response for t r by t n equal to 1.5 and then further response for t r by t n equal to 2.5 ok. For the value very large like t r by t n equal to 2.5 you can see the dynamic solution almost trailing the static solution. So, it is almost like you know close to very close to the static solution. So, that means if I have rise time which is ok which is very very large ok which is very very large then my dynamic solution is almost close to the static solution and I do not see much difference or I do not see much amplification of the response ok. So, in this case depending upon the t r t r by t n ok the dynamic characteristic of the response is determined. So, t r by t n is a very it is a very important parameter to characterize the response to a step force with finite rise time ok. There are two additional cases when t r by t n is equal to 1 and when t r by t n is equal to 2 or for a matter of case whenever you will have t r by t n equal to n ok where n is equal to 1 2 3 so on integer values what you will see ok at the end of the linearly increasing phase you do not get any oscillation ok. So, no vibration during the constant force phase ok and why is that happens is actually when you have this condition your velocity actually is equal to 0 at time t equal to t r ok. So, for that case the velocity is equal to 0 basically if you substitute in the expression for ut you will get ut is equal to ust naught ok which would be constant and not vibrating with time. And if you want to imagine this physically think it like the equilibrium position under the action of load p naught is actually at a displacement p naught by k. So, if you apply a static load of p naught the equilibrium position is p naught by k and let us say I have a spring which is ok vibrating around this p naught by k. Now, when it comes to this displacement p naught by k and if your velocity is 0 then what do you think will happen? Remember in previous instances of free vibration ok when you had system undergoing through the equilibrium position it had the maximum velocity and 0 displacement measured from the equilibrium position. So, if I measure the displacement from the equilibrium position when it crosses here it will have 0 displacement and it would be in equilibrium because that is the equilibrium under the load p naught. So, if it is in equilibrium and there is no velocity there then there is nothing that would take the system beyond this and then bring it back here ok. So, and that happens for very specific case when t r by t n is equal to any integer ok. So, for those cases when the system passes through equilibrium basically it has 0 velocity and then there is nothing that would take the system further on to continue the vibration ok. So, that is why for these cases you get this kind of response no vibration during the constant force phase ok alright. If this is clear now let us look at if you remember for a harmonic excitation ok we had defined a function r d which we said is the ratio of maximum peak dynamic displacement divided by the peak static displacement ok. And that was a function of omega by omega n and this was for the harmonic case. Now I want to do something similar for a step force with finite rise time or for the ramp load ok. However, there is no frequency here there is no excitation frequency in this case ok. So, let us see what do we get ok as the value of unit by US t naught ok and then we will try to go more into the expression ok.  Now if you look at it as I previously stated this is a constant and this is also a constant only this and this ok sin and cos functions are basically functions of time. So, you can say this is A and this is B and this is as sin theta and let us say this is cos theta ok. So, this is something like this A sin theta minus cos theta and again this is also a constant. So, maximum value of the dynamic displacement basically depends on the maximum value of this expression here and as we know if we have a function of the type A sin theta minus or plus cos B cos theta the maximum value is always A square plus B square ok ok because I have this term sin theta minus phi whatever I can write it and the maximum value is actually A square plus B square under root ok. So, I am going to do that ok I am substituted here ok and when I simplify this I will find out that my U naught by U s t naught is actually this expression here ok I can just I am just going to write the final expression ok. So, this is phi sorry this is t r here phi t r divided by t n and this divided by phi t r by t n. So, again if you look at carefully and let us say this is my r d now ok this depends on the value of t r by t n. So, like for a harmonic excitation of a single degree of freedom system the parameter on which the response modification factor ok depends on is omega by omega n for a ramp loading that parameter is t r by t n ok and I can go ahead ok I can go ahead and plot this function ok r d ok function as a value as a function of basically t r by t n. So, for different values of t r by t n I am going to basically plot this function ok now I can do that mathematically or before doing that mathematically let us see for extreme cases what happens I have already told you when t r by t n is very very small ok it is very very small ok for those cases this is almost like a step function ok this is a suddenly applied force ok suddenly applied force ok and for suddenly applied force we saw that my ut was approximately 2 times us t naught ok that means r d was actually equal to 2 and that we saw that from this graph ok if you look at it it was almost close to 2 r d here was almost close to 2 here ok. So, this is one extreme case other extreme cases when t r is much much greater than 1 ok. So, this is this would be like a slowly applied force ok. So, for slowly applied force we saw that ut was almost same as the static displacement. So, the peak basically the sorry this should be here u naught not not ut ok and here also they should be u naught. So, u naught is almost equal to us t naught ok because my dynamic displacement was actually ok very very much comparable to the static displacement throughout the time. So, r d is actually equal to 1 ok. So, for these two extreme cases we know that a very small value of t r by t n but for very large value of t r by t n what is the value of r t ok. Now, we also said that if t r by t n ok equal to some integer ok then for those cases there is no vibration due to constant during the constant force phase ok and if there is no vibration it means that static and dynamic are basically same. So, again for those cases as well u naught the peak dynamic displacement is actually equal to peak static displacements r d is again equal to 1 ok. So, we utilizing these three cases situations or scenarios let us see if we are able to plot r d versus t r by t n ok all right. So, we have let us say we have said that we have said that when t r by t n is very small the value is actually 2. So, it is somewhere around here when it is very large ok then it is close to 1 and we have also said that it is equal to 1 at integer value of t r by t n. So, let us say this is 1, this is 2, this is 3, this is 4 and so on. So, basically the actual response looks like something like this ok with this decreasing with time ok all right. So, depending upon the value of t r by t n ok we can calculate r d and the same thing can also be obtained using you know the expression for r d ok, but we have just tried to obtain this using the analytical or discussion that we just presented. Now, a point here to note that how much would be the r d how much different it would be from the static displacement if this is 1 let us say this is 2 this is basically static. It depends on the value of t r by t n ok and remember these t r by t n decides ok whether it is going to be a suddenly applied force or whether it is going to be a slowly applied force ok. So, whether it would have dynamic effect or whether it would be almost same as static and it would behave like a static ok. Now, this is a very important parameter because most of the loads when it is said that the load that is being applied is let us say 10 kilo Newton ok before this ok before this course you did not care about knowing how the load was applied. So, what was the time variation of the load and if somebody says to you that something is applied slowly it does not simply means that it takes long time to actually apply that load you also need to know ok whether the time how long is the time with respect to the time period of the system ok. So, that is a that is an important parameter. So, I will give you an example ok.  So, let us say I have two bridges ok one is ok one bridge is basically a rope bridge ok which is very flexible and the second one I have is actually a concrete bridge ok. Now, let us say for the rope bridge my frequency is around 2 second sorry time period is around 2 second and for concrete bridge let us say it is around 0.1 second ok which is typical you know because it is very flexible the rope bridge it would have ok very large time period compared to concrete bridge which has a very small time period of 0.1 second. Now, whether something is slowly applied or whether it is you know suddenly applied depends on this ratio.  So, that the dynamic effect is very very less ok. So, let us say I can do that for let us say T r by T n if this ratio is very very high ok then I can say that there is no dynamic amplification ok my RAD would be equal to 1 ok. So, it would behave like static. Now, to achieve the static type condition ok in this case the rise time has to be 2 times 10 20 second and in this case the rise time could be just 2 second alright. So, if you see that here ok for this rope because it is very flexible you need to be applying the load at a much much slower pace or over a very large duration of time to achieve similar kind of response ok compared to concrete bridge in which you can apply the load much quickly and still be able to obtain similar kind of response ok and like you know you can observe this realistically as well. If you jump on a rope bridge it will start oscillating. So, you will see much more dynamic effect ok. So, remember you are applying similar kind of load over a similar duration ok.  However, because the time period of the two bridges are different you would see basically different response ok. So, keep that in mind that whether something is static or dynamic ok it depends on the ratio TR by TM and slow whether something is applied slowly or suddenly ok. It is not only a function of time ok times relative it is actually depends on in a structural dynamics whether something is slow or fast it depends on the ratio of the duration with respect to a time period of the system ok. So, this is something that you need to keep in mind TR by TM which is a very important parameter alright. So, with this we saw that how to get the response of a system subject to a ramp loading or basically a step forward with a finite rise time ok. Now, what we are going to discuss a different type of excitation which is basically called pulse excitations ok. So, we are going to get into pulse excitation ok. Pulse excitation is basically any type of excitation that has a finite duration of application ok and then it is 0 ok. So, you have a load that is applied over a finite duration and then it is 0 before and after that. So, some of the examples would be ok a load that is applied like this ok or a load let us say which is applied like this or load that is applied like this ok or you could have a different type of load even this is ok. So, any type of load that is applied only over a finite duration of time is called a pulse load and as you can imagine if you apply a pulse load then the system does not actually achieved like in a steady state response it would always be vibrating and depending upon whether the system has damping or not it might come to rest for a damped system or might might always keep on vibrating ok with the initial conditions provided by this pulse type motion ok. So, you know there are different type of living several type of pulse type excitation ok. What we are going to study first is basically a very common type of pulse excitation which is rectangular pulse excitation. So, for a rectangular pulse excitation basically I have a rectangular pulse force of magnitude or say the amplitude P naught applied over a time duration of TD. So, now there is a new parameter remember there was a rise time in the previous case that we had discussed now that there is a different parameter it is called TD ok. So, the load rectangular pulse is applied suddenly it is kept constant till the time duration TD and then it is released. So, that force is 0 and the system that we have to find out the response of the system subject to this type of loading ok. So, I can again go ahead and write down the equation of motion as mu double dot plus k u equal to P of t where P of t is basically P naught for t smaller than TD and equal to 0 for t greater than TD ok. And in terms of solution we are going to follow the same procedure that what we have been doing till now ok. So, first let us do the forced vibration phase ok. So, for pulse type excitation we would always divide our response in forced vibration phase ok and then the free vibration phase ok and then we will try to obtain the response. So, this is basically my forced vibration phase and this is my free vibration phase ok. So, for this case let us first obtain the response for forced vibration phase and see what do we get ok. So, for forced vibration the equation that I have is this and we already know the response for ok this type of loading ok. We had already derived this, this is nothing but a step force ok. So, the response for this we have already obtained as ut ok divided by u s t naught this we had obtained as 1 minus cos omega n t which can be further written as 1 minus cos 2 pi t by t n ok. And remember this response is valid only for time is smaller than the TD which is the time duration of the pulse ok. Once this is clear let us obtain the response for free vibration phase ok. So, we want to obtain the response for free vibration phase and for free vibration phase it is nothing but a free vibration with initial conditions as u of TD and u dot of TD ok that was imparted to the system due to the forced vibration ok. So, we are going to utilize these initial conditions and find out the response in the free vibration.  And remember now because the initial conditions are at TD ok I am going to write this equation like this ok. So, this is my expression. So, I can go ahead and substitute these values ok from the expression here ok and simplify it to get the final response ok. So, final response that I get here is actually cos omega n t minus TD minus cos omega n t. Remember this is the response for t greater than TD ok. We can further simplify this and write omega n t equal to 2 pi by sorry 2 pi t by divided by t n ok. So, that would give us this expression here ok 2 pi TD divided by t n times sin 2 pi t by t n minus half TD by t n ok. So, this is the response that we get for t greater than TD ok. And if you look at it this is nothing but this sin function ok. So, let us call this sin theta or whatever and it is multiplied with this coefficient here which is constant ok. So, it does not depend on time it is actually a function of TD by t n ok. So, the normalized response if you look at here ok this is actually UAT by UST0 ok. This system actually vibrates again at its natural frequency ok and it depends on the ratio TD by t n ok. So, like for the case when we had a forced vibration of single degree of freedom system subject to harmonic excitation there the important parameter was omega by omega n for a step function with finite rise time or ramp function the parameter was the rise time divided by the time period ok. For pulse type motions we will see the important parameter is actually TD by t n ok. So, basically my system when applied to this oscillates about its static position ok at time period of oscillation of t n ok. And basically we can plot these functions ok we can plot UT by UST0 ok for different value of TD by t n and I am going to just bring that figure just to explain you how does it look like. So, if you look at here ok depending upon the value of TD by t n all right we have obtained basically different response histories here ok. So, I have obtained UT by UST0 ok and what I have done here basically combined the response for UT by UST0 ok for free for forced vibration phase and the free vibration phase ok and these are the curves that we get ok. And if you look at this carefully depending upon the value of TD by t n ok the maximum can occur during the forced phase ok or during the free vibration phase like in these two cases ok. And this happens when TD by t n actually below the value of 1 by 2 ok like in these cases then the peak occurs after the value of TD ok. Let us get into that ok we have these responses let us let us get into that one thing that we just need to focus here that when we applied the load p naught over the time duration TD my system actually starts vibrating about a shifted equilibrium position which is p naught by k and when I remove the load it come back to its original position which is U equal to 0 and then it vibrates about this position ok. Even though the amplitude is different the same type of behavior is opted for different value of TD by t n except again for these cases where TD by t n is actually an integer when that is the case then what happens at the end of the forced vibration phase we basically get U TD equal to 0 and U dot TD equal to 0 ok when TD by t n is equal to any integer ok 1 2 3 like that ok. So, for those cases ok after the completion of the forced vibration phase we do not see any response ok. So, the system just sits there ok and which can again be interpreted in terms of you know vibration about equilibrium that if you are having a system that is vibrating and it comes to equilibrium and at that point if you have 0 velocity and it is in equilibrium then there would not be any further oscillation because you do not have any velocity or you do not have any initial condition to take the system above that because your force is now 0 ok and if your initial conditions are also 0 then there would not be any subsequent motion. So, that is why for these cases we obtain behavior which is something like this ok. So, just see that how the resistance behave subject to different values of TD by t n. Now, let us come back ok to finding the maximum response or finding the maximum value of this ok. So, finding maximum this which is nothing, but peak dynamic displacement divided by peak static displacement and as previously described this is basically the response modification factor ok and this response modification factor ok. Let us obtain that for free vibration phase and forced vibration phase and overall RAD would be maximum ok of both ok. So, let us first obtain RAD ok. Let us see what is the value of RAD for forced vibration ok. The value of RAD was ok we had the expression 1 minus cos 2 pi t by t n ok. So, this was my expression. So, the maximum value of this function is what it is equal to which occurs at cos 2 pi by t by t n when it is equal to minus 1 then I get the maximum response as this one and for that to occur ok. So, it could be minus 1 minus 2 and so on ok. So, for those that condition let us see what is my t by t n ok. If I substitute it here I would basically get 2 pi t by t n as pi 2 pi and so on. So, my t by t n is actually t n by 2 ok then t n 3 t n by 2 and like that ok. So, basically the maximum response occurs at these durations ok and for at least one maximum to occur ok at least one maximum to occur can you imagine that my t by t n should at least be greater than t n by 2 ok or that scenario let us say this is t d here sorry not t ok. So, just correct that ok. Can you imagine ok it would achieve a value of 2 only if t d by t n is greater than t n by 2 ok. So, the maximum of this function the maximum of r d during forced vibration phase would be 2 only if ok t d by t n is greater than this expression here ok. So, here maybe I should not write t d by t n just write t d here ok that gives me from this expression I have become this ok. So, it has to be greater than half that means I would have at least one peak of function that I have here and let us try to understand that graphically as well. So, I have this function 1 and this function minus cos 2 pi t by t n. So, basically what I am saying this is 1 here ok and then I am going to draw minus cos 2 pi t d by t n. Let us see how does it look like ok this is a function basically which starts from minus 1 ok and it goes to a value of ok basically something like this ok. So, for it to achieve a maximum value if I sum this up it would look like something like this shifted by ok. So, this is the first peak that occurs and that occurs at t n by 2. The second peak occurs at 3 n by 3 t n by 2 and so on ok and if I sum this up this would look like this whole thing would be shifted by one upwards ok. So, it would look like something like this ok this is the response to during the forced vibration phase all right. So, let us say this is u here this is the value is equal to 2 ok and I will say that this is u naught by u s t naught. So, this is 2 and this is t by t n ok or if you t it will write it as t here then it would be t n by 2 3 n by 2. So, I am just writing here as half ok 3 by 2 and so on. So, what I see here this would achieve a value of 2 only if my t d by t n is greater than this value half ok. So, I hope you understand this is the expression that I have for RAD and this expression can achieve a value up to 2 provided this condition is satisfied ok. So, cos 2 by t t by t n at least becomes minus 1 minus 2 and like that ok. For that to happen my t d has to be greater than t n by 2. So, I will write RAD is equal to 2 if my t d is greater than t n by 2. So, the question comes well what happens if t d by ok if t d is smaller than t n by 2 well then the maximum response would be during the force vibration phase whatever the response as at t equal to t d is ok which basically we have cos 2 by t d by t n. So, it would be just same value and you put the value of t d here. So, if you look at here if you substitute t d by t n equal to half then it becomes 2 ok, but the duration always might not be large enough so that you can have at least one peak during the forced vibration phase ok. So, I have obtained two expressions, two conditional expression for the maximum during the forced vibration phase. Let us now go into the free vibration phase. So, for free vibration phase basically we know that ok system undergoes free vibration with initial conditions as ut and ut dot t naught. And I know that if a system is undergoing free vibration with these initial condition the amplitude of vibration can always be obtained as using this expression here ok and this is from your free vibration class ok it is. So, in that case it was u 0 square here that initial condition is at t equal to t d ok. So, this is here and you can substitute the value of u of t d and u dot of t d to obtain the expression for u 0 ok. When you substitute this you basically get this as 2 of u s t naught ok times sin of ok this under root terms is actually square. So, when you take the root it comes as sin of pi t d by t n ok. So, that may already becomes u 0 u t naught this becomes 2 sin pi t by t n ok alright. So, basically what did we see here ok we saw that during the forced vibration phase ok depending upon the value of t d by t n ok the maximum could be 2 or maximum could be this ok. If t d is smaller than t n by 2 then what happens actually ok then what happens in that case my response is still increasing ok. And this response is basically ok. So, this is let us say ut ok this response here is basically your u of t d m then there is some velocity here as well, but the maximum response then occurs during the free vibration phase ok. And this is for the case when let us say t d by t n is here ok this is smaller than half here ok. If this is greater than half then this peak would be inside that ok for that case. So, the response is. So, for that case ok when t d by t n is smaller than half the maximum occurs during the free vibration phase and that maximum is actually this value. So, if we combine both ok what do we actually get ok. So, if we combine the forced vibration phase and the free vibration phase I can get one expression for R d which would be equal to R d equal to U s t naught ok. This should be equal to 2 sin pi t d by t n ok this is for t d by t n is smaller than half ok. So, as I said this is going to govern the peak is going to occur during the free vibration phase if t d by t n is smaller than half. And if it is greater than half then the peak occurs during the forced vibration phase and it is equal to 2 ok. And you can go ahead and plot these two functions ok. So, I am just going to get you the final figure that we have here. Let me just copy this. So, this is basically what I am saying ok. Let me write this here. This is forced vibration remember this is for free responses actually ok free responses for the free response I have 2 of sin pi t d by t n ok. Forced response is basically when it is smaller than half t d by t n is smaller than half this value is basically here which is 1 minus cos pi t d 2 pi t d sorry 2 pi t d by t n and if it is greater than half it is 2. So, if you take the envelope of both these functions you get basically the overall maxima. In this one this is governed by the free response ok and this function is basically 2 sin pi t d by t n and this is basically the value equal to 2 alright. So, if you look at it we are able to obtain the r d as a function of t d by t n ok. And this is similar when we have obtained r d for a harmonic excitation as function of omega by omega n or for a ramp loading we had obtained this as function of t r by t n. In this case we are obtaining as a function of t d which is the load duration divided by the time period of the system ok. And this is the plot of peak response versus some function of t either directly as a function of t n or some multiple of t n it is called a response spectra ok. So, plot of peak response of a single degree of freedom system versus its time period is called response spectra ok and it is a very useful tool that is used by designers in industry. For example, not everybody is going to obtain the differential equation then go through all those calculation and get the peak response ok. If this chart is available ok. So, let us say this chart is given to us for a rectangular pulse loading then say you have a structure of time period t n alright.  So, that you can find out the t d by t n ok.  So, basically p naught and t d is known to you ok and t n is also known to you. So, if those parameters are known to you do not have to solve any differential equation you can find out the maximum value of u naught as r d times u s t naught ok and r d can be directly obtained here. So, let us say for any given value of t d by t n you can just go here and then find out what is the value of t d by t n ok. And if you know u naught then you can also find out the equivalent static force in the structure as k u naught assuming that it is represented as a single degree of freedom system. So, I am saying if you can represent this building as a single degree of freedom system ok something like this with k and m then for that case ok. The equivalent static force is ok k times the peak dining displacement which we can further obtain as k times r d and u s t naught which is basically p naught by the stiffness of the structure. So, the equivalent static force is basically r d times the peak static force ok alright. So, with this the discussion on rectangular pulse duration actually concludes we are going to extend the same procedure to find out the response for other type of pulses in the subsequent classes ok alright. Thank you very much.


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Mike, the decision to start dogs, is that more to see and have a comparison contrast in the league, or did you think he gave you a better chance in this game tonight? Well, I'm probably a little bit of both last week, just kind of seeing. Really, it was a great opportunity to evaluate Josh, and then we'll make a decision going forward. I thought he did some good things. I thought he certainly liked to have a couple of throws back, but we'll kind of see where things are here in a couple of days. When did you guys know he was going to be the guy tonight? Early in the week, I told both of them. How'd you like how you play, maybe on defense, keeping the skin close to the penalties? Maybe mistakes end up dooming you at the end? Well, we just beat ourselves too much. I mean, it's too hard to beat the Cowboys or good football teams when you beat yourself, but I thought we competed. I was extremely proud of the way that the 47 guys we had up tonight competed, certainly without mistake. We gave ourselves a chance, but then again, we just did too many things that hurt us and stalled drives or allowed drives to continue. It didn't get the drive stop with a field goal down there in the red zone. Once you made the decision to go to thousands, was there ever any thoughts, OK, if it doesn't work out, to go back to Malik in this game? Does it say anything about Malik's progress, the fact that you bring in a guy from the practice squad? I mean, I think that we're excited about having Malik. We are. We're excited about some of the things that he's done, and we'll continue to just digest this over the weekend and make a decision. Malik has worked hard, but then we just have to see where we're at with the quarterback position here going down to the last week of the season. Panic from some guys in some big moments with some drops or with trade, draping themselves around a guy on 3rd and 19 to give up 51 yards. There's some signs of panic in moments like that. Then just got a penalty, a critical penalty. Panic.  They made a mistake. You had three pass plays of over 30 yards. That's something that hasn't been a part of your offense the last couple of weeks. I thought we protected well.  I mean, there were times when it was obviously passing downs. But I thought there was times for Josh to progress through and move the ball downfield. I think trailing got held. I think his left arm got grabbed. Same guy that called Paul's 50-yard pass interference penalty didn't see it the same way on our sidelines. I don't know what to tell you. But I thought there were opportunities where we could have helped Josh out as well. And we couldn't come down with a catch. But there was also other times where we made plays down the field and hit the screen versus pressure and all those things. I thought Hassan had one of the better runs of the year on 4th and 2, ran over a really good player and fell forward for a first down. How about on the rubbing penalties that don't seem pretty similar, races and plans? You know, those are judgment calls, and I think that was bad judgment. You had a lot of scissors to make on players who played who didn't. Did Kevin, by chance, make the case that he wanted to play? I just think he played tonight with the two picks. Well, I value Kevin's opinion. I value his role on his football team as a captain. And you have to make decisions. I felt like just where we were in that case that he was going to play wanted to play. And that was the plan. And then I thought he played well. I thought he took advantage of his opportunities when the ball came his way. And whether it was deflected or tipped or whatever, I thought he tried to tackle and come down and support. But all those things aside, I appreciate his leadership. And it was a different week, the whole circumstances with next week and trying to prepare a team. And again, that's, I guess, I'm proud of the way that they fought. I'm sure that everybody expected us to just lay down. And I know that we didn't do that. Mike, you talked about needing to find a way to win and find some momentum these weeks before you get to Jacksonville and not focus on the importance of that game. Obviously, you didn't get the win tonight. Or was there anything that you took away as maybe positive momentum moving into this? Yeah. I mean, there was things when we didn't beat ourselves or we didn't extend drives with penalties or third downs a little disappointing. I thought we stopped the run with whoever we had in there, whatever group, whatever front, stopped the run. The one had the one big one. It would have been much less than that if we would have just done our job. Been where we were supposed to be. We wouldn't have given up the long one. I think that was a positives. Certainly, the penalties are negative, being able to avoid some of those. Offensively, I thought we got into a rhythm. I thought we kept competing. We need to run the football better. But man, that's kind of been where we've been the last couple of weeks, is doing some good things, just not consistently enough. And then, unfortunately, beating ourselves in the margin for error when you do that is pretty thin. Have you seen Josh's play come some life into the passing day for you?  I thought that was a good ball. And obviously needed some protection to get there. And we'll see where it went and watch the tape. But I thought that there were some good, you know. I wish Traylon would have been able to come down with that one. It was a little behind them. But I think Traylon expects to make that catch. But I thought the timing on it was good and it was going to the right spot. What's the fact that he had this productive of a night dobs on such a short turnaround? May get that much more impressive that he's only been here? Eight days or nine days, I think. He was excited for this opportunity. Get his first start. You know, I think something that we've always said that, you know, since I've been here is that it doesn't really matter how you end up here or how you get here. The only thing that matters is what you do when you're here. And we've had a lot of guys, you know, take advantage of those opportunities. And, you know, I think Josh wanted to try to do the best that he could. Unfortunately, we lost. And I'm sure we'd like to have the interception back. But we did some good things, just not well enough. What has to happen for you guys to be able to flip a switch and the streak win the division? I mean, I think we just have to get some guys back healthy, clean up some of the little details that show up. You know, you only get one opportunity sometimes with a call on offense or defense versus the same exact look. It's going to happen in the game.  And I think we've had far too many of those. But we'll get them cleaned up. We'll have to eliminate big plays on defense to run the football, take care of it. Same thing we always say, just do a little bit more consistently. And I'm excited. Excited to watch these guys walk back in a building next week, energized, recovered, rested. I'm excited.  You had the same result this week. But do you feel like as if you got that extra more from everybody that you kind of challenged? I mean, I think so. I mean, again, I haven't watched the tape. But I felt like the guys, you know, some of the leadership was really good. Had some good performances, just, you know, not good enough. And, you know, just buying into the meetings, buying into the details, never knowing when you're going to get your opportunity to make a play and be ready to make it. Are you operationally smoother tonight? Well, you know, I think without looking at the tape, maybe.  Everybody is responsible for that, you know, whether the formation gets called right or wrong or play. Like, you have to know the game plan well enough to know where you line up and hear the call. And, you know, there were some mistakes. Like there is every week. I know you don't want to put anybody on Azure Reserve, but was it horrible to line and did you get a chance to talk to him? Yeah, of course I get a chance to talk to Ryan. Yeah, I think it's always difficult, you know, but I think you have to be realistic as well, where we're at, you know, and what was a realistic expectation, you know, for Ryan to come back and how long that would be. And what that may look like when he did come back. So, you know, that was the decision that we made. He was obviously disappointed, talked about his toughness, no different than all the other guys that we've had to put on our hour. Josh, when did you find out that you were going to start? So, after the last game, it was Christmas Eve, and I was actually driving home.  We just want to go through the game plan. We're trying to figure out which way we're going to go, but if you're playing, I just want you to see the stuff as soon as possible.  Had a car, finally. So drove back up, and then we started diving to the game plan, and then the next day, they said, hey, we're probably going to start you. So you're going to take all the routes throughout practice, and that's the way we went. How big a challenge to start a game after just being here eight days, and how in your mind you think it is? Yeah, I think it's a unique situation, and it's a wonderful opportunity that I really embraced and had a lot of fun with. So there's obviously things that I know that I can do better and be better, especially situationally, looking back through the game. And we'll dive into that, because I know we've got a big one coming up. But I thought through the preparation and the guys, that's just how we fought throughout the game. It was great to see. I mean, shoot, we're throwing routes out there on the field that we haven't even thrown in practice with it being a short week. So just being able to make those plays, make critical plays throughout the game will propel us into this upcoming week, where we actually have a full week to prepare for the game. What was it like for you to wait in six years for an opportunity to start a game, and then finally, because you did it? Yeah, man, it was a long time. That's for sure. It definitely was a long time. A lot of working out, and you're like, dang, I don't even know. Especially, you're in Pittsburgh for so long, and Ben's playing at a high level still. So you're like, dang, I don't know when I'm going to get a shot. So every week, though, I give everyone the analogies like this. It's like, you know, you have algebra tests every Friday. You know, you have to prepare for the test. And you prepare. You do all the homework on Monday, Tuesday, Wednesday. You go through the process. You do the extra work. You go to after hours and everything. And then Friday comes, and then you just don't take the test. And then you repeat process. And you never know when you'll have to take the test. So you're not prepared every single week as if I'm going to start, whether I'm dressing, not dressing, starting, not starting. As I said, you know, it's a thing I could definitely do better out there on the field. But I love how I fought personally. I love how the team fought. Took advantage of the opportunity. I'm excited for the next one. And finally, getting the opportunity to start. Did you have those early anxious moments? And if so, what wasn't that you did to allow yourself to settle in, but it seemed like that's what happened? Yeah, I think for sure, when you haven't played a live snap of football since August, and then I was December, that's a long time, especially when you're in a new team, new receivers, new timing, new offense. So yeah, I mean, they're there. I'm human. They're going to be there. And in those moments, I just try to take a deep breath, stay present, stay in the moment, lock in on the game plan, and just the same thing I told the guys was, just take it one play at a time. If you do that, you focus on that play, make it the best play possible, move on to the next one, then you won't get overwhelmed by the moment. So there's a lot of guys out there, not just me, making their first starts or first time back on the field in a long time. And just watching how the guys responded to adversity, responded with success, it was great to see. And it definitely was a building opportunity for us. I know you had a relation with Malik before you got here. What was he like through this process? Helping you or supporting you or however that went? Definitely, he's been great. Just helping, especially a new quarterback, a guy that's been in there, being able to just talk about procedural things, such as cadence, snap count, communication, code words to the guys, especially when we're going no huddle and up tempo. So he's been a great resource, so has Kevin Hogan in the room as well. We have a really good room, the guys really gel well, and it's been good to rely on those guys as well as Pat and Todd, too, to help with my preparation.  Like, is there a feeling in the room that something's turning tide-wise and that in 10 days can change the result? Well, we control our own destiny. And every single person in the locker room understands that. We go out and play a good game. We play how we know how to play football. We don't hurt ourselves. We go play good, clean football in Jacksonville next Sunday, whenever the game is, and we come out with a win. Then we're in the playoffs, and we're rolling from there. So we control our own destiny. Guys understand that. Obviously, every time you step on the field, you expect to win, and you want to win. So it's tough when you don't get the job done. So now we'll go through our process. We'll review the film, review the tape. We have a longer week for preparation for the Jacksonville game, and we control our own destiny. We go out there, we play the football, we know we can. Then we're in the playoffs, and AOC South champs. First start for you in a while, without having the opportunity to go back and look at the tape, what do you think of the thing or maybe a couple of things you're going to be focused on this week? For sure, situationally, as I look back throughout the game, we fumbled the ball with the ball on the ground in the red zone when we have points, and you just can't do that. That's an opportunity where either get the ball out, take a good clean sack, kick the field goal, get points. Then it's an even tighter game, going to the fourth quarter. So little things like that. I took a sack on a keeper in the second quarter, throw the ball away. So there's little things situationally that I'm aware of that I know I can get better. Of course, that comes with reps and being out there and getting hit for the first time in a while. So I'm not too worried about those. I know I'll clean them up. I'll be excited to dive into the tape with the coaches and continue to grow, improve as quickly as possible, and give the team the best chance that we have to win. Sorry, Josh, how much difference might it be having a Derek Henry possibly available in that game in Jacksonville? No doubt. I mean, Ohio is the best player on offense. So getting a chance to get him out there healthy and playing the football he can will be tremendous for us. I thought the young guys, though, today did a great job toting the rock situationally, tempo, getting the pile moving forward. They did a great job. And obviously, having Derek back will be a great asset for the offense. How do you end up with a football after your first career touchdown? And did Robert Woods know that? No chance he knew. There's no chance he knew. Everyone was obviously extremely excited. So I just ran out. I was like, yo, can I keep this? This is my first one, please. And he's like, yeah, you got it. You're good. And then we just celebrated one from there. So definitely have it. I'm going to give it to the fans so they can take it back home and find a nice spot for it. You go from kind of waiting for an opportunity like this for so long to now getting the opportunity and now looking at potentially a biggest opportunity there is, playing for a spot in the playoffs. How good was it to get one under your belt before that big one for all the marbles? 100%. It was great to get out there. Obviously, just get the jitters out. You're about to get actually hit sitting in the pocket. It's completely different than when you're at practice. They can't hit you. They run by, yell, sack. It was a whole different environment, especially then playing situational football from there. As I said, the stuff that I know I have to clean up. So it was good to get out there and get this opportunity because the next one's for all the marbles, man.  You dream about playing playoff football in the NFL. And to have that opportunity is one you never take for granted because just in my career, my six years, I've been to the playoffs, thankfully, three times. But the three years I haven't, the two years I haven't, I guess this is year six. Those are tough years. So you put in a lot of hard work throughout the season. So we go out there. We control our own destiny. It's good to get these reps. Now let's make the most of the next opportunity. Josh, until the line, did you ever feel like you had a chance to show what you can do in this league and go get the start? No, I didn't. Honestly, I didn't. Just given my circumstances and the lack of opportunity I've had, you know, it's definitely been tough because especially when you pour everything into it and you don't get the opportunity, it's always tough. But yeah, you know, I was thinking about it earlier today. It's like your current position is not your destiny. You keep working hard. The opportunities will come. You never know when they will come, but you for sure better be ready when they do come. You expect the start next week? That's up to the coaches, you know. I think they're going to put whoever gives us the best opportunity to win out there. And I respect their decision. So we'll see when we regroup over the weekend and we'll go from there.