Position analysis of slider crank like mechanism using loop equations

[miketingscott]

Hi Everyone,

This is my first post so bare with me. Firstly, a bit of background about the problem.
I am designing a near rear drive train for a formula student car which incorporates a chain tensioning device linking the differential to the rear of the chassis.

Essentially I am wanting to quantity the amount of movement in the x direction brought about through the lengthening or shortening of the chain tensioner (a threaded section of metal) which acts at a varying angle dependent on length.

The mechanism follows that of an inverse slider crank mechanism whereby there are two fixed points, one arm of fixed length, L1, which can rotate in a circular motion and an arm of varying length, r1, attached to the rotating arm. A sketch of the system is attached.


I am able to develop the x and y loops in terms of the angles shown in the picture. The part which I am getting REALLY confused by is showing the equations. From looking at other examples this should just be a case of equation the two equations however I have not been able to successfully do this/I end up with too many unknowns.

Ideally what I was hoping to produce was an equation which when inputed with the length of the chain tensioners, r1, would produce a value for x(after some more basic trig/co-ordinate geometry maybe). From looking at what I have so far I would have thought that the equations would be used to produce theta and theta2 which could then be used to calculate the co-ordinate for where r1 would sit, which could then be used to calculate the change in x.

Phi, L1, L2 =constants and are known
R1 would be the input
Theta and theta2 = outputs

If someone could advise the method that I should be using that would be great. I have been reading into this/looking at examples for quite some time and have really struggled to get my head round it.

Thanks,

 

[timthereaper]

I know of some kinematic equations you can use for this, but honestly, I'm a bit confused at the drawing you provided. I'm trying to determine which locations are fixed points and how each fixed-length member must move. Also, maybe there's a simpler input than r1 to consider, so I'm just having trouble seeing how it moves. However, I'll try to describe the method and you can apply it.

For a kinematic position analysis, you need to satisfy equilibrium equations, such as $\sum_i r_x = 0, \sum_i r_y = 0$. If you have things in terms of distances and angles, then equations such as $x_1 = r_1 cos(\theta_1), y_1 = r_1 sin(\theta_1)$ become useful because we can use Euler's identity to kind of translate those constraints into real and imaginary components using $e^{i\theta} = cos(\theta) + i sin(\theta)$. Thus, you can define vectors in a loop that have the form $r_1 e^{i \theta_1}+r_2 e^{i \theta_2}+r_3 e^{i \theta_3}+r_4 e^{i \theta_4}=0$ (this is for a 4-bar linkage). By expanding, the sum of the real parts correspond the x-direction equilibrium constraint and the sum of the imaginary components corresponds to the y-direction equilibrium constraint.

You can define your distances and angles are variable or fixed, depending on your conditions. A fixed length member will have a constant r, and a slider along a vector will have a constant angle. Other equations might be needed to determine how angles might change with position. This approach should yield all the equations necessary to solve for all unknowns.

FYI, you can get velocity and acceleration information from this as well by differentiating the position vectors. Hope this helps!

 

[miketingscott]

Firstly thanks for your reply.

Regarding the picture:

The top point and left point are fixed points. The right hand point is constrained to the top point and due to the length L1 being fixed can be at some point on the consequent arc.

It is the length r that can not only vary in length but also in angle; due to being connected to the bar L1.

I have calculated the X and Y loops which I found to be:

X: L2cos∅+L1cos(1-θ2)-r1cosθ=0

Y: r1sinθ+L1sin(1-θ2)-L2sin∅=0

I have at this stage then not been able to equate the equations(due there being two unknown,θ and θ2.

I would like it to end up at a point where I can say that for an increase of x amount to the length of r1, there would be an increase of x in the x axis, if you understand what I mean. I am pretty sure this is possible from discussions I've had with people at the process I'm following should be correct, but I think it is the trig identities etc I'm struggling with which would maybe allow parts to be canceled out to yield the equations I need?

Thanks again

 

[timthereaper]

So if I've understood you right, then the attached picture would be the vector diagram I would draw (I called it θ₁ instead of θ). Thus, the position analysis would be:
$\vec{r_1}-\vec{r_2}-\vec{r_3}=0$
Using the method I described earlier:
$r_1e^{j\theta_1}-r_2e^{j\theta_2}-r_3e^{j\phi}=0$
Separating the real and imaginary parts and substituting the known values:
$r_1\cos{\theta_1}-L_1\cos{\theta_2}=L_2\cos{\phi}$
$r_1\sin{\theta_1}-L_1\sin{\theta_2}=L_2\sin{\phi}$
The only things you don't know in these equations are θ₁ and θ₂, since you're providing a value for r₁. With 2 equations and 2 unknowns, it's just math from here on out.

 

[miketingscott]

Yes the sketch you have drawn is spot on, I appreciate the time you have put in here.

The equations are the same (minus one sign which was wrong) as mine so I can atleast be sure I understand that part of the problem.

I think it is actually the solving of the two equations, with the trig, that I am getting stuck on.

Taking the equations and calling them 1 and 2:

1: r1cosθ1−L1cosθ2=L2cosϕ
2: r1sinθ1−L1sinθ2=L2sinϕ

I have rearranged 2 to get:

θ1=(L2sinϕ+L1sinθ2)/r1)(sin^-1)

And then I have tried to solve 1 using this to get an equation for θ2 in terms of knowns. Is this a correct method of rearranging?

I can now see that if I can first find a value of θ2 for a given r1 and can then put this substitiute the value into 2 allowing me to find θ1. Once I know θ1 and θ2 it should be some geometric manipulation to find the end point of r1 in terms of x which is what I hope to achieve.

Thanks for your help so far, I can now see how it should pan out. I was getting really confused as have been trying so many different things with it for some time. If you could just let me know if that method of rearranging the equation/trig identities is correct that would be great.

Thanks again!

 

[miketingscott]

I just wanted to upload an additional photo of my working so far.

As you will be able to see, the equation I end up just cancels itself out, so I know I have made a mistake somewhere which is most probably due to the misuse of a trig identity or similar

 

[timthereaper]

I'm sure there are a couple of ways to solve this problem analytically, but in kinematics I always just solved it numerically with Matlab. With more complex linkages, such as four- and five-bar linkages, it's very time consuming if not impossible to find an analytical solution. Besides, as an engineer, I'm after answers and not necessarily about proper math technique :tongue:. If you have access to math software, I suggest using it. If not, post this in the Math forum and see what you get. Good luck on your project!