- #1

petterg

- 162

- 7

The scenario is:

I'm having an under powered car, and the need to get a boat trailer up from a local boat ramp frequently. The problem is not the car power, but the clutch ability to get the power to the wheels. The car is able to get the trailer up, but I know the clutch will burn out pretty soon if I don't find a way to reduce the load when starting.

My plan is to create a kind of car ramp with the same angle as the slope of the boat ramp, and put them behind the trailers wheels (under water line) when backing down the ramp. That way the trailer wheels will start on leveled surface when the car does the hill start.

Now I'm trying to calculate how long theses car ramps need to be. To do so, I'll need to calculate the speed the car/trailer will have when the trailer wheels comes off the ramp.

The information I know is:

The hill angle is H=10 deg

Weight on wheels starting in the hill is M1=1600kg

Weight on wheels starting horizontally on the leveled car ramps is M2=1000kg

I assume the power transferred by the clutch when not fully engaged, the power that is accelerating the car/trailer is P=60kW.

The length of the ramps is S, and the goal is to find the speed (V) as a function of the driven distance S. At the starting point speed is 0.

Other variables used in formulas:

t: time

a: acceleration

The formulas I'm trying to use:

(1) Kinetic energy (0.5*m*v^2) at S: (energy of car + energy of trailer)

Ek = 0.5 * M1 * V^2 + 0.5 * M2 * V^2

(2) Potential energy (m*g*h) at S: (energy of car + energy of trailer)

Ep = M1 * g * sin(H) * S + M2 * g * 0 = M1 * g * sin(H) * S

(3) Added energy (E=P*t)

E = Ek + Ep = P * t

(4) Acceleration vs distance

S = v0*t + 0.5 * a * t^2 = 0.5 * a * t^2

(5) Force vs acceleration (m*a)

F = (M1+M2) * a + M1 * sin(H) * g

(6) Power vs Force

P = F * S/t

-------------------

Trying to make use of the formulas:

(7) Rewriting (6)

F = P * t /S

(8) Rewriting (4)

a = 2 * S/t^2

(9) Insert (8) into (5)

F = (M1+M2) * 2 * S / t^2 + M1 * sin(H) * g

(10) Solving (7) and (9) for t

P * t /S = (M1+M2) * 2 * S / t^2 + M1 * sin(H) * g

P/S * t ^3 - M1 * sin(H) * g *t^2 - (M1+M2) * 2 * S = 0

Substituting x = M1 * sin(H) * g * S/(3*P) and y = x^3 + (M1+M2) * S^2 / P

t = (y+(y^2+(-x^2)^3)^(1/2))^(1/3) + (y-(y^2+(-x^2)^3)^(1/2))^(1/3) + x

(11) Inserting (1) and (2) into (3) and solving for V:

0.5 * (M1+M2) * V^2 + M1 * g * sin(H) * S = P*t

V^2 = 2*P*t / (M1+M2) - M1 * g * sin(H) * S / (M1+M2)

V = [ 2*P*t / (M1+M2) - M1 * g * sin(H) * S / (M1+M2) ]^0.5

Now, when inserting numbers into (10), say S=1m, I get that the longer it takes to drive that 1m, the faster the speed. That doesn't make sense! The same thing happens when inserting numbers in (11).