7.4 - Kinematic Analysis - Bottle Opener

Kinematic Analysis

Overview

Our botle opener is actuated through the sue of a single DOF four bar crank rocker. Actuation is handled by a DFRobot FIT0186 12V DC gear motor. Some of the specs include: 251 RPM no-load, an internal gear ratio of 43.8:1, and 18kg cm. Stall torque. This drives our crank continuously through 360 degrees of rotation. This rotary input is then converted into oscillating rocking motion of our ourpur link. The bottle opener hook is mounted at the end of our rocker where it locks into and levers off the crown camp. The four links are defined with measurements as follows:

Mobility Analysis

Using the Gruebler formula for a planar mechanism:

As the four-bar linkage has 4 links and 4 revolute joints as well as no half joints our final calculation comes out to 1 DOF. This confirms that our mechanism requires a single actuator in order to fully define the morion of all of our links. Additionally, it promises that no extra inputs are required.

Using the Grashof Condition:

S = L2 = 69.693 mm

L = L3 = 138.40 mm

P = L1 = 125.00 mm

Q = L4 = 100.00 mm

Since our mechanism satisfies the Grashof condition we know this satisfied Grashof Class 1. Additionally, as the shortest link is adjacent to the grounded link we know that this is a crank-rocker configuration where the crank rotates fully and our rocker oscillates through a high torque arc.

Position Analysis

Our position analysis was performed using our standard four-bar vector loop equations.

Open

• theta 3 (coupler angle) oscillates between around 30-55 degrees over our full crank rotation which reflects our couplers limited angular movement in relation to ground

• theta 4 (rocker angle) oscillates between approximately 70-100 degrees giving us a total stroke of around 30 degrees which reflects the overal angular range over which the opener hook travels per each cycle

• The Path of point B traces a total arc of around 177 mm which represents the total spatial trajectory of our hook tip

• Transmission angle seems to peak around 55 degrees and reaches a minimum of around 30 degrees over the operating range:

  LaTeX | Mosaic

Velocity Analysis

Velcoity anlaysis was calculated by using differentiation with the vector loop equation to produce our standard four-bar equations

Open

Our results were normalized to w2 at approximately 1 rad/s, after applying some of the formulas below:

We found that w3 ranges approximately between -0.23 and 0.10 rad/s and our w4 ranges approximately between -0.45 and 0.25 rad/s per our unit input. It is imporant to note that our magnitude peaks at approximately 0.45 meaning our angular velocity is around 45% of the cranks at that configuration. Additionally found that our maximum angular rocking velocity under no-load was approximately 11.8 rad/s.

Using our linear velocity equations at point B or our hook tip, we find that it can reach up to ~1.18 m/s.

Acceleration Analysis

Acceleration analysis was performed by differentiating our veolcity equations which yielded our standard four bar acceleration for alpha 3 and alpha 4 as functions of theta 2 in which we set our alpha 2 to zero to account for constant speed motor input.

Open

From our plots we could tell that our alpha 3 varies between around -0.15 and 0.28 rad/s^2 and that our alpha 4 varies between approximately -0.25 and 0.45 rad/s^2. At the motors operating speed of approximately w2 ~ 26.3 rad/s the acceleration is scaled by a factor of w2^2. It is imporant to note that the large centripetal acceleration can spike the loads on our joints during operation.

Force Analysis

The motor delivers a stall torque of approximately:

This torque is applied at our crank pivot point (O2) which acts through our crank arm of L2. Using power conservation and assuming no frictional loss:

We find that when mv ~ 0.45 our torque multiplication is around 2.2 which gives our rocker a torque of around 3.9 Nm. As the mechanism approaches a toggle configuration our mv would approach 0 and our mt would approach infinity.

The corresponding force output would be approximately 39 N at mid cycle which is significantly higher as the mechanism approaches the toggle point during specific points during the prying. It is worth noting that operating our motors at the full 7A stall current allows our mechanism to operate somewhat as expected through lower transmission angle portions of our rocker stroke that may other stall our system.

Analysis

A full animation of our crank-rocker mechanism sweeping through its 360 degree range of motion is shown below:

Open

This animation depicts that our crank-rocker behaves as intended in which the continous rotation of L2 is constained by the oscillation of L4 through its ~30 degree arc. This results in the path of point B as our hook tip traces an approximately 177 mm. working arc across our bottle cap.