Stress Analysis Design Project
|Phil Aufdencamp||Gloriana Redondo||Si Hoon|
Overview of the Mechanism
Our mechanism was constructed to be able to lift a 1 lb sliding weight by 2 inches at a distance of 30 inches from the mechanism’s anchoring position. The mechanism uses a base to support the arm that is needed to extend to the location of the sliding weight and to balance the moment created by the weight of the arm as it lifts the sliding weight. At the end of the arm a servomotor is mounted to provide the torque needed to lift the weight off the post. The servomotor uses a lever arm that is placed under the sliding weight so that the lever arm’s rotation causes a
vertical displacement of the sliding weight on the cylindrical post. The lever arm has a length of 13 inches and uses an ounce of weight accumulated 6.5 inches from where the torque is applied by the servomotor, to serve as counterweight to the moment created by the sliding weight on the lever arm’s end.
Our mechanism consists of four parts: the base, the arm, the servomotor, and the lever arm. The base, located 30 inches from the sliding weight, is a 13.00” by 3.25” by 5.50” mix of vertical columns with a variety of cross braces whose purpose is to maintain the structure’s shape. Since the weight of the arm causes a moment on the base, these braces are used to minimize the amount of bending on the columns of the base and on the connections between the base and the base plate. The base also has two angle pieces that are located underneath the horizontal piece where the arm is screwed to the base. These are pre-stressed to eliminate the flex between the arm and the base. Then on the base plate screws were strategically placed on the point where the moment caused by the arm lifted the base plate the most to increase the amount of contact area between the mechanism and the platform (the table), thus increasing the rigidity of the structure when clamped.
The arm is a 2.25” by 3.00” triangular shaped structure that extends 23.00” from the base.
Since the arm needed to be so long, in order to distribute the tensile and compression stresses caused by the moment applied from lifting the sliding weight, triangle trusses were used. These triangular trusses are superior to a rectangular shaped arm with trusses since in this case the mechanism experienced bending stresses more than it did torsion. Also a triangular arm was preferable since it made the mechanism lighter. At the end of the arm the servomotor was mounted with a lever arm that extended the 7” to the sliding weight. The lever arm (13” in length) made contact with the 1lb. weight at one end and then 6” from the pivot point had a counterweight of an ounce. This counterweight created an opposite moment to that of the sliding weight and thus decreased the moment applied to the arm, and as a result the bending.
Theoretical Predictions of Performance
Percent of Maximum Servo Torque
The Percent of Maximum Servo Torque used in the device is defined by this expression: K = U/M, where U is the Torque that we required based on our lever arm, the weight we have to move, and M is the theoretical maximum value of 76 oz-in. We can calculate U by doing a sum of moments about the mounting point between the servo and the lever arm.
The moment in the center is U, the value of actual applied torque which we are solving for, the force on the left is mg, the force of gravity due to the counterweight. The moment it applies is simply Mc = (l) (mg), where l is the distance from the servo to the center of mass of the counterweight. The moment due to the weight we are trying to lift, the force on the right, is simply Mw = (L) (Mg) where L is the distance from the servo to the point of contact with the weight, and M is the mass of the weight we are lifting. Using these values for our variables
m = 1 oz
M = 16 oz
l = 7 in
L = 4.5 in
and the expression
U = MgL – mgl
We find that U = 65 oz-in, and K = 65/72 = 90.3% While this percentage of maximum servo torque seems relatively high, we needed to keep the overall mass of the lever arm (and therefore the counterweight and its distance from the fulcrum) low in order to reduce the force on the arm and minimize undue deflection.
Theoretical Lifting Height
-This analysis will be performed assuming a perfectly rigid arm, and a lever arm that doesn’t bind or catch anything as it swings through its travel. The consequences of the second assumption are fairly straightforward; the consequences of the first assumption is that any motion of the weight
relative to the servo is entirely translated to upwards motion of the weight, and not deflecting the arm and moving the servo down. We are also going to assume that the servo begins at 45 degrees below the horizontal, and ends at 45 degrees above it.
Applying the constraints of our assumptions, some basic geometry, and physical dimensions of the lever arm, we know the following
Angle CBA = 45 degrees
Angle CBD = 45 degrees
Angle ABD = 45 degrees
AC = CD
AC = BAsin(CBA)
BA = L
L = 4.5 in
Observing the total theoretical lift height Htheo to be AC + CD, or 2AC, and plugging in some numbers
Htheo = 2Lsin(45 deg)
Htheo = 7.65 in
Triangular Arm – The cross section of our main arm is a triangle that is roughly twice as tall as it is wide. This cross section combines the height necessary to make a structure strong in bending, with the larger perimeter necessary to make a structure strong in torsion to deal with the induced moment caused by the asymmetrical loading of the arm.
Base – The overall superstructure of the base is unique, in that it’s not a standard box, but nor is it a triangle. It’s a borderline organic structure that is actually quite efficient given the rather unconventional manner which we loaded it in.
Prestressed Members – We made extensive use of prestressed components in our structure, by intentionally fabricating them to be ~ 5% longer or shorter depending on the application. This allowed us to guarantee that it was loaded in the way we designed it, and also removed a lot of the “settling” that was occurring in our previous iterations.
Servo Arm – The servo arm was fabricated with a high degree of precision, which allowed us to load the servo according to our math with a low factor of safety, reducing overall weight. The notch minimized any binding that would have occurred at the interface between the weight and the arm.