VPI,
Before taking your comments in turn, it might be worthwhile taking a moment to examine what happens in a steady climb.
The most important point to note is that lift is less than weight in a steady climb. This can be illustrated by considering an aircraft with a thrust:weight ratio of considerably more than 1. In level flight the wings carry all of the weight, so wing lift equals weight (if we ignore forces produced by other parts). In a vertical climb (nose pointing straight up) the thrust carries all of the weight. So the wings must produce zero lift. This means that in a steady climb the lift must be equal to the (weight x the cosine of the angle of climb). And the thrust required is equal to the drag plus (weight x (1 - the cosine of the angle of climb)).
Excess thrust is that part of the thrust that is not being used to oppose the drag. So in a steady climb the excess thrust is equal to that part of the weight that is being carried by the engines. The greater the excess thrust, the greater will be the proportion of the weight that can be carried by the engines, and hence the greater the angle of climb. So the best angle of climb is achieved by flying at the speed at which excess thrust is greatest.
To understand why rate of climb is related to excess power we need to remember that power is the rate at which work is done. When an aircraft climbs work is done in two ways; To push it forward against the drag and to increase its altitude. The total work done is equal to the (drag x distance moved through the air) plus the (weight x the increase in altitude). So the rate at which work is done or power is expended, is (drag xTAS) plus (weight x ROC).
The fist part of this (the drag x TAS) is the power required to maintain TAS. The second part (weight x ROC) utilises the remaining (or excess) part of the power available. Excess power is the power available minus the power required to maintain TAS. Only this excess power is available to increase altitude. Now if power expended in increasing altitude is equal to (weight x ROC), then ROC is equal to the excess power divided by the weight. So for any given weight the best ROC is achieved at the speed at which excess power is maximum.
Now taking your comments in turn.
Quote - Is it safe to say that once climbing at Vmd, a reduced climb rate would surely result with a decrease of airspeed and therefore increase in drag…
It depends on the type of aircraft (or rather propulsion system).
TURBOJET AIRCRAFT
For a turbojet with its almost constant thrust, best angle of climb occurs at Vmd. Any increase or decrease in speed will increase drag, reduce excess thrust and reduce angle of climb. This is because (as you imply) if drag increases and thrust is constant then excess thrust and angle of climb will decrease.
But the best rate of climb requires best excess power and this occurs at a higher speed (about 1.32 Vmd). So increasing speed above Vmd initially increases ROC up to its optimum value, then decreases it.
PROP AIRCRAFT
For a prop aircraft thrust decreases rapidly as speed increases. So although Vmd gives the lowest drag it does not give the highest thrust or excess thrust. Best angle of climb occurs at a speed (Vx) slightly lower than the minimum power speed Vmp. When decelerating from Vmd to Vx, thrust increases faster than drag. So angle of climb increases up to its optimum (at Vx) then decreases again. ROC depends upon excess power. Although Vmd gives minimum drag, the minimum power required occurs at the considerably lower speed of Vmp. So when decelerating from Vmd, ROC increases up to its optimum at the best ROC speed (Vy), which is just above Vmp, then decreases again at lower speeds.
So for your P3. whether you are after best angle of climb or best ROC, you should be flying slower than Vmd. Both increasing altitude and increasing ambient temperature increase the power required and decrease the power available. The overall effect is a reduced ROC and a reduced best ROC speed.
Quote - …..gaining more lift at increased speed - lift increases with the square of speed ….
It is true that if you increase speed whilst maintaining a constant angle of attack, lift will increase, causing the aircraft to climb. But this is not going to produce either the best angle of climb nor the best rate of climb. The problem is that at speeds above Vmp, the greater lift comes at the expense of greater power required. At speeds above Vmd the greater lift comes at the expense of higher drag, which means higher thrust required.
TURBOJET AIRCRAFT
For turbojets the thrust is almost constant speed and the power available increases almost linearly with increasing speed. So as speed increases towards Vmd, the reducing drag increases excess thrust and angle of climb. As speed increases above Vx (which is Vmd), the increasing drag reduces both excess thrust and angle of climb. As speed increases up Vy (which is slightly higher than Vmd), rate of climb increases, then decreases at higher speeds.
PROP AIRCRAFT
For propeller aircraft thrust decreases from the moment you let the brakes off and power available increases up to some speed between Vmp and Vmd then decreases at higher speeds. So as speed increases from Vx (just below Vmp) the angle of climb decreases and above Vy (just above Vmp), the rate of climb also decreases.
Quote ….. This (3 engine) 190 Kts seems bizarre…..
I cannot comment on speeds that are specific to the P3 but your query regarding the 3 engine best climb speed appears to illustrate the way asymmetric power alters Vmd, Vmp, Vx and Vy. With one engine out the drag is increased due to the dead prop (even when feathered) and the increased trim drag caused by the need to overcome asymmetric thrust. This increases both drag and power required, but decreases Vmp, Vmd, Vx and Vy. The overall effect is a reduced best angle of climb and ROC and reduced best climb speeds.
The magnitude of these effects depends upon which engine has failed. Assuming each is able to produce the same thrust and power, then the one which causes the greatest asymmetric thrust will cause the greatest reduction in climb performance and the lowest best climb speed. Assuming the P3 has right handed propellers (not counter or contra-rotating), then the left outer will have the biggest effect and the right inner (I think) will have the least effect. The figure of 190 Kts you quote might simply be the average or worst case figure. You could test this in the simulator by noting the ROC when trying different combinations of failed engine and speed. If I am right you will find that climb rate and best climb speed are highest with the right inner dead and lowest with the left outer dead.