1D Wave Equation: Vertical Rod and Displacement vs. Textbook Solution

I am trying to setup Mathematica to analyze a vertical round rod under its own weight, fixed on one end free on the other. I have the 1D wave equation and a distributed load to represent the self weight of the round rod.

Vertical Rod Layout

The problem is when I compare the Mathematica solution to the textbook solution the two do not agree.

Sample problem is given below.

Y = 199*^9; (*Young's modulus in Pa *) \[Rho] = 7860; (* Steel density in kg/m^3*) dia = 1/39.37; (* 1" dia converted to meters*) c = Sqrt[Y/\[Rho]]; len = 1000; (*length in meters*) tmax = 5; (* Max time for analysis*) area = \[Pi]*dia^2/4; (*Round rod cross sectional area*) wtfactor = \[Rho]*9.81*area/len;  frwt[x_] := \[Rho]*    area*9.81*(1 -       x/len); (*Rod Self weight imposed as a distributed load*) nsol6 = NDSolve[{\!\( \*SubscriptBox[\(\[PartialD]\), \({t, 2}\)]\(z[x, t]\)\) == c^2*\!\( \*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(z[x, t]\)\) + frwt[x] +       NeumannValue[0, x == len],    z[0, t] == 0}, z[x, t], {x, 0, len}, {t, 0, tmax},   Method -> {"FiniteElement",      "MeshOptions" -> {"MaxCellMeasure" -> 10}}   ] fnnsol6[x_, t_] = nsol6[[1, 1, 2]] Plot3D[fnnsol6[x, t], {x, 0, len}, {t, 0, tmax},   PlotLabels -> Automatic, AxesLabel -> Automatic]  deltaL = ((\[Rho]*9.81*len^2)/(  Y*2)) (*Textbook elongation for a vertical rod under self weight*) calcdeltaL =   fnnsol6[len,    5] (*Calculated delta Length from PDE solution.  Should match \ textbook*)  deltaLfunc[x_, l_] := \[Rho]*9.81*   x*(2*len - x)/(2*Y) (*Verified Correct*) xydata = Thread[{Range[0, 1000, 100],      deltaLfunc[x, 1] /. {x -> Range[0, 1000, 100]}}]; xydata2 =   Thread[{Range[0, 1000, 100],     Reverse[a]}]; (*Same answer different calc format for debugging*) Show[Plot[fnnsol6[x, 0], {x, 0, len}, PlotLabels -> {"PDE Val"},    PlotRange -> All   ],  ListLinePlot[xydata2, PlotStyle -> Green, PlotLabels -> {"Correct"}]] 

If you’ve read this far, thank you.

In summary my question is: Is this a Mathematica issue or a PDE setup problem? The PDE is right out of a textbook so I don’t think that’s the problem but Mathematica gives no errors and I am out of troubleshooting ideas so looking for some help.

Thank You

Convert Cloak of Displacement to a static bonus instead of disadvantage

One of my players has a Cloack of Displacement, which grants, for every round, monsters have disadvantage attacks against the player until the first attack hits.

What follows from that power, is that I have to ask the player for every attack, if the monster has disadvantage or not. That is an extra effort I don’t want to do; the information about whether or not he has a benefit is on his side of the table. (Note: as a DM I always ask “does AC X hit you?”)

So, we want to change the Cloak; instead of granting the monster disadvantage on their attack, the Cloak offers him, for every round, a bonus to AC, until he is hit.

Here is my question; what bonus should the Cloak give to the player?

Assume the player is a level 12 PC with an AC of 16. If we need to take the attack bonus of the monsters into account; assume a master needs to roll a 10 to hit.

Does Cloak of Displacement hide your character’s location or do others see two bodies?

Description for Cloak of Displacement (emphasis mine):

While you wear this cloak, it projects an Illusion that makes you appear to be standing in a place near your actual location, causing any creature to have disadvantage on Attack rolls against you. If you take damage, the property ceases to function until the start of your next turn. This property is suppressed while you are Incapacitated, Restrained, or otherwise unable to move.

Does that mean the cloak makes your actual body unable to be seen and projects an illusion of your body nearby, or are both bodies visible?

Calculate initial velocity based on displacement, time and constant acceleration.

“A car has a constant speed along a road. It goes down a hill at a constant acceleration. 50s after it goes down the hill the speed is doubled and 50s later it reaches the end of the 200m hill and is back at a constant speed. Find out the initial velocity and acceleration.”

At first I made relevant graphs to see if I could find some useful information from that but no luck. Then I tried to use the “suvat” equations but we haven’t learned them in class so I’m not allowed to use them, which is why I’m stuck as to how to solve this basic problem.