Cable Tension from UDL | Experiment Setup and Calculation

[Els27]

Homework Statement

Ok guys my first ever post so be nice!

Im doing an experiment using a wind tunnel looking at the effects of vortices on cables. Basically the experiment willconsist of 2 vertical cantilever beams supporting a cable between them (So if you like it looks like a portal frame only the top beam is a cable). The cable's longitudinal axis is perpendicular to the flow. The flow creates a lifting force in the form of a UDL with respect to time. The cable is treated as pinned at each end. I am measuring vibration of the cable by relating the strain in the 2 supports to the amplitude of vibration.

Thats the setup, now the actual problem is determining the model dimensions so that i get a givenstrain value. I am aiming for 0.0001 strain (thats what I've been told to go for given the strain gauge accuracy)

Homework Equations

I have 3 equations for the setup one for the UDL force of the flow on the cable, one for the tension force that UDL puts on the cable (which is transmitted to the supports) and finally one that governs the strain in the pole given a ceratin point load at the top(from tension in cable).

Force of flow per meter

0.5*p*d*u^2*Cl*sin(ws*t)
where p=air density =1.2
d=cable diameter =say 1mm
u=free stream velocity= up to 28m/s
Cl=coefficient of lift= call 1.0
ws = shedding frequency= not needed for now
and t= time in s

Strain in pole(cylindrical support) from stress = moment *y/ moment of Inertia

e=32*P*(L-z)/E*pi*D^3
where e=strain=0.0001
P=point load at top of beam
L= length of pole
L-z= lever arm point load has to where strain gauge is
E=youngs modulus
pi=3.14...
D=diameter of support

finally the equation i currently have (but have beef with) for tension at supports from cable with UDL

w*a^2/2h
where w=UDL force=equation above from flow
a= distance max amplitude of cable = half cable length =0.15m
h = amplitude of displacment

The Attempt at a Solution

Ok this is a lengthy thread i know sorry but need help big time!
So given my equation for strain, i know what strain i need and I've assumed a support diameter of 5mm and steel modulus 200e^6. This gives a point load of 9.82e^-4 N. Given other similar shapes of supports gives point load to the order of e^-4.

So i know what force at the top of the cantilever i need (~9.82e^-4) to give a strain of 0.0001. Now i need the tensile force in the cable that will give me the point load at my supports. The flow force equation is dynamic and all i am interested in is the amplitude (i.e. not the sin(ws*t) bit right?) This gives a max UDL on the cable at some point in time doesn't matter what time that's the force I am interested in =0.5*p*d*u^2*Cl. Now as Cl and p are constant (1.0 and 1.2 respectively) and given the max flow speed 28 m/s the only variable is 'd' and the equation is now 470.4*d. Let's say for now a 1mm cable is used (its going to be around 0.5mm to 1.5mm i think) that gives a UDL of 0.47 N/m.

So now i have a simply supported cable with UDL across entire length. So now what is the tension in the cable and hence the point load at supports? This is where my third equation and major problems come in. My thoughts at this point are given the applied load (the UDL) the cable will deflect i think parabolically is the word but it will remain almost stationary at supports and deflect most in centre. This will be resisted by the stiffness of the cable. the force at one of the pinned end is

P= w*a^2/2*h (this is taken from course notes so i assume they are right)

But one would assume as you increase the deflection 'h' the material would be stretching more and increasing in tension. However the equation says otherwise. (Panic time! and no i don't have a panic button on my keyboard...) I know what P i want to achieve through my strain equation (9.82e^-4)All i need is h the deflection at the position of max amplitude (the centre). I have tried simplifying the string into a 'single mass on a spring' system which gives a max deflection of 0.0018m or 1.8mm .but when plugging the numbers in i get 5.875N and i don't believe that's the case at all! i can't believe that wind flow is causing that much tension in the wire

Does anyone know if my tension form UDL equation is correct?
Can someone make sense of this and tell me where I've gone wrong / give me some pointers please?

All help much appreciated
cheers

Elliot

 

[Jhero]

Dear Elliot,

Thank you for sharing your experiment and questions with the forum. It sounds like you have put a lot of thought and effort into designing your experiment. I will do my best to help you with your questions and concerns.

First of all, let's address your equation for the strain in the pole. It seems like you have the correct equation, but it may be helpful to clarify some of the variables. The moment in this equation is the bending moment, which is equal to the force P multiplied by the distance L-z. The distance L-z is the lever arm, or the distance from the point load to the point where you are measuring the strain. The moment of inertia in this equation is the moment of inertia of the pole, which is a measure of its resistance to bending. This can be calculated using the formula for the moment of inertia of a solid cylinder, which is 1/4*pi*D^4. So your equation should be:

e = 32*P*(L-z)/(E*1/4*pi*D^4)

Also, it may be helpful to plot this equation to get a better understanding of how the strain changes with different values for P and L-z.

Moving on to your question about the tension in the cable. Your equation for the UDL force seems correct, but it may be helpful to plot this equation as well to see how the UDL changes with different values for d and u. Now, for the tension in the cable, you are correct that it will increase as the cable deflects more. The equation you have for this tension (w*a^2/2h) is actually the equation for the maximum tension in the cable. This occurs at the point where the cable is deflected the most, which is at the center point. So if you want to find the tension at the supports, you will need to calculate the tension at the center point and then divide by 2 (since the tension will be evenly distributed along the cable).

The equation for the maximum tension in a simply supported beam with a UDL is actually:

Tmax = w*L^2/8

Where T is the maximum tension, w is the UDL, and L is the length of the beam. So in your case, the maximum tension at the center point of your cable would be:

Tmax = 0.47*0.15^2/8 = 0.0013125 N

And the tension at