Force required to lift a platform (over the range of angles)

[AlexKud]

Homework Statement

Hi, this is not really an academic homework but it is something I'm trying to solve for my future project. I have a background in electrical engineering hence I'm having a hard time solving the forces problem.

https://dl.dropboxusercontent.com/u/43124243/Force_problem.png I'm trying to calculate the force in N over the range of operational angles (15-40°) required to lift the platform and steps. I want to print a chart in Excel, so I'm trying to find the formula for calculating the required force.

The mass of steps (F_N1) - 700 kg
The length of steps (A→B) - 2000 mm
The mass of platform (F_N2) - 300 kg
The length of platform (B→C) - 500 mm
The fixed position of linear actuator (D) - 3900 mm from steps' pivot point
Operational angles (α) - 15-40°

Homework Equations

I assume that this equation is relevant

F = m × g × cos (θ)

The Attempt at a Solution

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Here is my poor attempt at the solution:

e.g. @ angle 40°

F = (½ × m_steps + m_platform) × g × cos(α)
F = (½ × 700 + 300) × 9.8 × cos(40°)
F = 4.88 kN

I suspect that this part is correct although I do not know how to add the linear actuator's tilt angle (β) into the equation. I can find the angle β itself via SOHCAHTOA rule but I'm not sure what to do with it next. Any help will be much appreciated.

 

[andrevdh]

your pictures did not upload correctly Alex - you have the option to edit your post at the bottom of your text

 

[AlexKud]

your pictures did not upload correctly Alex - you have the option to edit your post at the bottom of your text

Thanks, andrevdh. I've just updated it.

 

[haruspex]

F = (½ × msteps + mplatform) × g × cos(α)

There is no actual force that corresponds to that calculation. What force were you trying to calculate with that?
If you choose a rigid body, steps plus platform, say, there are three basic equations you can write to express equilibrium:
- the balance of forces in the horizontal direction (sum of horizontal forces on it =0)
- the balance of forces in the vertical direction
- balance of moments about some axis; for this problem the pivot point looks like a good axis to use.
In general, you might or might not need all three equations. In the present case, just the moments equation looks to be enough.

I'm puzzled about two of the lengths quoted. They imply AD is nearly double AB, whereas the diagram clearly shows AB is the longer.

The length of steps (A→B) - 2000 mm

The fixed position of linear actuator (D) - 3900 mm from steps' pivot point

 

[AlexKud]

balance of moments about some axis; for this problem the pivot point looks like a good axis to use.

Can you help me with the exact equation please? As I've mentioned I'm not that good in physics.

I'm puzzled about two of the lengths quoted.

haruspex, the values are random. I'm interested in method not so much in the answers.

 

[haruspex]

Can you help me with the exact equation please? As I've mentioned I'm not that good in physics.
haruspex, the values are random. I'm interested in method not so much in the answers.

Ok.
The are several ways to get the moment of a force about a point. One is to multiply the force by the distance from the line of action of the force to the axis.
Taking the axis as A, the weight of the steps, mg, acts through a vertical line half way along AE. Can you write that calculation out?
Similarly, the weight of the platform acts through a vertical line that comes down somewhere to the left of B (in the diagram). What is the distance from that line to A?
These two torques add up, being both anticlockwise.
They are balanced by the clockwise torque from the compression in BD. This one is a bit trickier. Can you figure out angle ABD, and from that find the distance from A to the line BD (projected)?

 

[andrevdh]

Except for the mass we get that the forces are aligned along the "rods"
The "rods" are either in compression or tension, so there are forces at the connecting points pointing either inwards or outwards (for some the direction are clear, for others not so it does not matter which direction you set it put after solving they would come out being either + or -)
I makes sense to add F1 and F2 to counteract the weights, but there are no such forces present
The platform is a bit of a dark horse
What I suspect is going on is that its centre of mass is located at B and that it is manually pivoted at the various angles so that it is orientated horizontally or there is another triangle at point B?
Change you diagram to conform to these suggestions Alex
I also suspect that your lengths are in cm not mm

 

[haruspex]

also suspect that your lengths are in cm not mm

That would make it huge. An overall length of about 4m seems reasonable.

 

[andrevdh]

Platform 5 meters and steps 20 meters - huge?

 

[Nidum]

The mechanism as drawn may not function at smaller values of angle alpha .

Before doing any more theoretical analysis do some diagrams to ensure that the mechanism is kinetically possible and that the lifting force is always acting in an effective direction .

 

[haruspex]

Platform 5 meters and steps 20 meters - huge?

Yes. The platform is likely intended for one person to stand on. Half a meter would be cosy, but adequate. I cannot think of an application that would require a 5m platform.

 

[BvU]

I suspect there are some typos in the dimensions. Platform length, 3900 mm AD versus 2000 mm AB etc.
Could you check, @AlexKud ?

 

[andrevdh]

I thought that are the typical dimensions of an airplane boarding platform?

 

[haruspex]

I thought that are the typical dimensions of an airplane boarding platform?

In my experience, the platform at the top of boarding steps is less than 2m in length.
The airstairs at http://www.industrialmaintenanceplatforms.com/aircraft-passenger-stairs-15f2830.html are quoted as 10m in overall length.

I suspect there are some typos in the dimensions

I queried that earlier. See reply at post #5.