Spring angles at static equilibrium on application of force

[ANKIT SAJWAN]

Homework Statement

Two linear springs with stiffness Kh, Kv are attached to a single mass m at right angle to each other, i.e; one of the spring(Kh) is in the horizontal direction(x-axis) and the other(Kv) is in the vertical direction(y-axis).

A constant force F is being applied to the mass, inclined at an angle theta with the x-axis.

I need to find the angle which the springs Kh, Kv will subtend with the x-axis and y-axis respectively at the postion of static equilibrium.

Homework Equations

F = K * x

The Attempt at a Solution

I attempted the problem by assuming that the horizontal component of F will just compress the Kh spring by some distance, say a , while the Kv spring will not have any effect as the force is perpendicular to the linear spring. Did the similar thing for vertical component of F.

My query is, is it correct to assume that the horizontal component of F will not have any effect on Kv, as from geometry we can see that if the spring Kh moves by distance a, then the spring Kv will elongate.

 

[CWatters]

Try making several drawings showing possible outcomes.

Is there more info in the question? A drawing perhaps?

 

[ANKIT SAJWAN]

Initial condition at the instant of application of force

 

[CWatters]

Now draw it displaced to a likely position.

Have to go out for a few hours. Will try and get back to this later.

 

[Dr.D]

Are you concerned with large displacements, or is your problem limited to small displacements?

 

[ANKIT SAJWAN]

@Dr.D : Actually force is acting almost vertically i.e; angle theta is almost 90 degree, so we can assume very small displacement in the horizontal direction, but can't say about the vertical displacement whether it can be considered small or not

 

[ANKIT SAJWAN]

@CWatters : This is how it would look at the equilibrium positions, with the springs forces developed, i wish to know those two angles, aplha1 and aplha2

 

[CWatters]

So can you see how both springs have h and v components.

 

[ANKIT SAJWAN]

@CWatters : Yes i can see that, and i have used lami's theorem to obtain some sets of equations. But these equations which I've obtained are very complicated and obtaining the angles aplha1,aplha2 in terms of F & theta is not possible.

 

[CWatters]

Two possibilities..

Post your equations and we'll see if someone from the maths section can help solve them.
or
If one or more of the spring constants are large compared to the applied force then one or more of the angles α₁ and α₂ might be small enough that you can make a small angle approximation eg Sin(α1) ≈ 0.

What information is known/provided in the problem?

 

[Dr.D]

This looks like a good problem to attach via the Principle of Virtual Work. I have not solved the problem, but I think I would probably choose x and y displacements as my generalized coordinates, and then express everything (including the angles) in terms of those two variables.

 

[ANKIT SAJWAN]

@Dr.D @CWatters

I tried to do it with principle of virtual work, and obtained a final equation in x(horizontal displacement of mass)

I have attached images

 

[Dr.D]

Sorry to say, but I can't see your work. I'm an old man with weak eyes, and it would take me a full day to decipher that (time I need for other things).

 

[ANKIT SAJWAN]

Haha I'm so sorry, I thought it uploaded.
I have attached the images links, please have a look!

https://ibb.co/k1MhLb
https://ibb.co/gr7KYw

 

[Dr.D]

The image was there, but the contrast is very poor in your images, and the hand writing is difficult to read. If you want help, please put your work in a more readable form.