Mechanism Design

By

Jakob Anderson – Design Spec. & constraints, Type synthesis, Optimization Yashovardhan Sand – Transmission angle optimization, Analysis Dylan Rainer – Dimensional synthesis, Modeling, Analysis

ME 5243 – Advanced Mechanism DesignDr. Ahmet Sezen

EXECUTIVE SUMMARY

A wakeboard tower is a device that elevates the pulling position of the rope that is used to tow a wakeboarder. Since most towers are large fixed structures, storage is a challenge. A mechanism for a compact collapsible wakeboard tower was designed per constraints and specifications. After type synthesis, a Watt-1 BKC was chosen. Dimensional synthesis was performed to solve for all possible link lengths and geometries. Optimization was performed to narrow the solutions a final solution was chosen. Velocity, acceleration, and transmission angle analysis bolstered our final chosen design.

BACKGROUND

The elevated towing position given by a wakeboard tower reduces the downward pull of the rope on the airborne wakeboarder and allows the rider to achieve more launch and longer air hang time. A wakeboard tower must be very solid since there is a lot of force generated from pulling the wakeboarder out of the water. It is also common for wakeboard racks (and sometimes speakers) to be attached to the tower. This allows for equipment to be stored out of the boat for more room inside the boat.

PROBLEM

Most towers are structures and do not fold down/collapse.

With the tower upright, it is hard to get the boat under a boat lift or in a garage for safe storage

Since most towers are structures, the only way to put the boat under the lift or into a normal sized garage is to disassemble the tower. This takes time each time you want to put the boat on the lift or into your garage.

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PROJECT DESCRIPTION

In this project a collapsible wakeboard tower mechanism for a Glastron speed boat is designed so that storage can be attained simply and efficiently. The Glastron speed boat has existing fixed pivot locations that are 91” apart from a pre-existing tower that does not collapse. This newly designed collapsible tower will utilize these fixed pivot locations so the old ones do not need to be removed and new ones do not need to be mounted/drilled. Since the wakeboard tower will consist of two identical mechanisms, connected by a coupler rod, only one mechanism will be designed (one side).

Figure 2

CONSTRAINTS AND SPECIFICATIONS

Human Powered

Four fixed pivot locations (two on each side) See figure 2 for fixed pivot locations.

Each set of fixed pivots are 91 inches apart

Three positions for motion generation

One driving link

In the folded position the mechanism should neither go lower than 20” below the fixed pivots to avoid contact with water, nor be higher than 20 inches above the fixed pivots

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TYPE SYNTHESIS

After writing the constraints and specifications, we began type synthesis by considering how many links were required. According to the motion generation specifications, it would be possible to create the desired motion of the coupler link with a simple four-bar mechanism since the motion is not all that complex.

However, in this application, the folded down position of the coupler link is specified so close to one of the ground pivots that are prescribed, the mechanism must extend extremely high in the first position, and the mechanism needs to be extremely compact in the third position. According these constraints, a four-bar mechanism simply will not work due to the fact that very large links would be needed and it would not have the ability to be compact enough. In order to obtain the compact position and reach the extended initial condition, a six-bar mechanism was desired.

When analyzing the possible six-bar types, a Watt-I inversion of the Watt BKC was chosen with its coupler link located 2 away from ground on both ends due to its ability to extend a very large distance and retract into an extremely compact position for storage. A Watt-II, Stephenson-I, Stephenson-II, or Stephenson III BKC would not be able to reach this height while offering the same extremely compact design that is desired.

Conventional Watt-I BKC

DIMENSIONAL SYNTHESIS

The known variables are: P1, P2, P3, αj, Ao, Bo, δ 2, δ 3

The first solved loop was Z1, Z2, Z3. Our closed loop equations are:

Z1 (e iψ 2−1 )+Z 2 (ei φ 2−1 )+Z3 (e i α 2−1 )=δ2

Z1 (e iψ 3−1 )+Z 2 (ei φ3−1 )+Z3 (ei α 3−1 )=δ3

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There are four equations and ten unknowns so we have six free choices. We choose Z3=7+5i, ψ2=42.25° ,

ψ3=68.6 °, and ground pivot specification. This way we can put the equation into standard form, and we have a path generation with prescribed timing and ground pivot specification. In standard form this is:

Z1 (e iψ 2−1 )+Z 2 (ei φ 2−1 )=δ2−Z 3 (ei α 2−1 )=δ 2'

Z1 (e iψ 3−1 )+Z 2 (ei φ3−1 )=δ3−Z 3 (e i α3−1 )=δ 3'

For ground pivot specification we define vectors from the specified position to each “precision position”, where these positions are located at δ 2

' and δ 3' .

R1=P1−Z3−Bo

R2=R1+δ 2'

R3=R1+δ 3'

The augmented matrix of this is:

| 1 1 R1ei ψ 2 ei φ2 R2e i ψ3 ei φ3 R3|=0

After opening up the determinate we get the following equation:

(−R3 e iψ2+R2ei ψ 3 )+(R3−R1eiψ 3 )e i φ2+(−R2+R1ei ψ 2 )ei φ3=0

Which can be rewritten as:

D 1+D2e i φ2+D 3e i φ3=0

To solve for φ2 and φ3 we can use:

φ2=cos−1(D 32−D 12−D 22

2D1D2 )+∆1−∆2=−91.3 °

φ3=angle(−D 1e∆1−D2 e∆2eψ 2

D 3e i ∆3 )=−128.8°

Using the standard form equation, Z1 and Z2 can be found using matrix inversion now that we know φ2 and φ3.

[Z 1Z 2]=[ei ψ 2−1 ei φ2−1

ei ψ 3−1 ei φ3−1] [δ2'

δ3' ]=[−34.65+40.82i

−57.14+24.17 i ]

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The known variables are: P1, P2, P3, αj, Ao, Bo, δ 2, δ 3, Z1, Z2, Z3, ψ2, ψ3, φ2,φ3

The second solved loop was Z8, Z9, Z4. Our closed loop equations are:

Z9 (ei γ 2−1 )+Z8 (ei ψ 2−1 )+Z 4 (e i α2−1 )=δ2

Z9 (ei γ 3−1 )+Z8 (ei ψ 3−1 )+Z 4 (e i α3−1 )=δ3

There are four equations and eight unknowns so we have four free choices. We choose Z4=5i because we want this side of the coupler to be vertical at the first precision position, and to put the equation in standard form. We also must choose ground pivot specification, because Z8 is at the same ground pivot as Z1. This is also path generation with prescribed timing. In Standard form this looks like:

Z9 (ei γ 2−1 )+Z8 (ei ψ 2−1 )=δ2−Z 4 (ei α 2−1 )=δ 2' '

Z9 (ei γ 3−1 )+Z8 (ei ψ 3−1 )=δ3−Z 4 (ei α 3−1 )=δ 3' '

Again, for ground pivot specification we define vectors from the specified position to each “precision positions”.

R1=P1−Z 4−Bo

R2=R1+δ 2' '

R 3=R1+δ 3' '

The augmented matrix of this is:

| 1 1 R1ei ψ 2 e i γ2 R2e i ψ3 e i γ3 R3|=0

After opening up the determinate we get the following equation:

(−R3e i ψ2+R2ei ψ 3 )+(R3−R1e iψ 3 )e i γ2+(−R2+R1ei ψ 2 )ei γ 3=0

Which can be rewritten as:

D 1+D2ei γ2+D 3e i γ3=0

To solve for γ2 and γ3 we can use:

γ2=cos−1( D32−D 12−D 22

2D 1D 2 )+∆ 1−∆2=−103.36 °

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γ3=angle(−D 1e∆1−D 2e∆ 2eψ2

D3e i ∆3 )=−146.89 °

Using the standard form equation, Z8 and Z9 can be found using matrix inversion now that we know γ2 and γ3.

[Z 8Z 9]=[e i ψ2−1 e i γ2−1

e i ψ3−1 e i γ3−1][δ 2' '

δ 3' ' ]=[−28.63+46.09i

−56.16+18.90 i ]The known variables are: P1, P2, P3, αj, Ao, Bo, δ 2, δ 3, Z1, Z2, Z3, Z4, Z8, Z9, ψ2, ψ3, φ2,φ3, γ2, γ3

The third solved loop was Z6, Z5, Z4. Our closed loop equations are:

Z5 (e i γ 2−1 )+Z6 (e i β 2−1 )=δ2−Z 4 (ei α 2−1 )=δ 2' '

Z5 (e i γ3−1 )+Z6 (e i β 3−1 )=δ3−Z 4 (ei α 3−1 )=δ 3' '

There are four equations and six unknowns so we have two free choices. We want to choose ground pivot specification. Also, this is a motion generation task.

Again, for ground pivot specification we define vectors from the specified position to each “precision positions”.

R1=P1−Z 4−Ao

R2=R1+δ 2' '

R 3=R1+δ 3' '

The augmented matrix of this is:

| 1 1 R1ei β 2 e i γ 2 R2ei β 3 e i γ 3 R3|=0

After opening up the determinate we get the following equation:

(R3e i γ2−R2e i γ3 )+ (−R3+R1ei γ 3 )ei β2+(R2−R1ei γ2 )e i β3=0

Which can be rewritten as:

D 1+D2e i β2+D 3e i β3=0

To solve for β2 and β3 we can use:

β2=cos−1(D32−D12−D22

2D1D 2 )−∆ 1+∆2=−2.25 °

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β3=angle(−D 1e∆1−D 2e∆2 eψ2

D 3e i ∆3 )=−39.49 °

Using the standard form equation, Z5 and Z6 can be found using matrix inversion now that we know β2 and β3.

[Z 6Z 5]=[e i β 2−1 e i γ 2−1

e i β 3−1 e i γ 3−1][δ 2' '

δ 3' ' ]=[ 38.23+40.82i

−31.86+23.81i ]The remaining vectors can be found by using closed loop triangles as follows:

Z7=Z 9−Z 5=−24.29−4.91 i

Z10=Z 1−Z8=−6.02−5.27 i

Z11=Z 3−Z 4=7

OPTIMIZATION STRATEGY

To ensure the best mechanism was selected for this application, we employed an optimization strategy based on many constraints. Since the ultimate design is a mechanism that is very compact, we wanted all links to have y-coordinates (based on our chosen axis) to be between -15” - 20” in the third position. We also wanted the coupler link to be 70” above the axis in the second position and no link higher than 80” above the axis in any of the positions. In order for this mechanism to be compact and not extend past the end of the boat, we chose our psi2 and psi3 to be between 0 and 90 degrees (so the driving ternary link would fold inward/toward the front of the boat). Finally, alpha 3 was chosen to be -90 degrees so the coupler link would lay “flat” on the padded surface of the engine compartment in the back of the boat. This also ensured that the rope “hitch” at the 1st position would not be protruding vertically for someone to trip on.

In order to obtain the solutions that fit our constraints, a MATLAB code was written to find all possible solutions and store only valid solutions into a matrix to be utilized afterward. Since we only cared about the height of the coupler link in the second position and not the rotation specifically, alpha 2 was varied from -120 degrees to 120 degrees. For each alpha 2 value, MATLAB varied psi2 from 0 to 90 degrees and for each psi2, psi 3 was varied from 0 to 90 degrees.

This set of for loops computed the possible 3-position motion generation Watt-I six bars that met the design specifications. If a solution was found that met all the specified criteria, the code would store that solution into a matrix. The code then output the matrix of all valid solutions, based on the constraints specified, to an XLS file and conditional formatting was used to narrow constraints based on the number of acceptable solutions.

ANALYSIS

The solutions that fit the narrowed constraints were then plotted in MATLAB to visualize the mechanisms. After visualizing all the narrowed solutions, two mechanisms were chosen, and rendered in Creo and the transmission angles

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were calculated. The final solution that was chosen was the smoothest operating mechanism with the transmission angles that made the most sense.

For a six-bar mechanism there are two transmission angles used to analyze the transmission of load and torque to the input and output links. The acceptable design range for the transmission angle is between 45o and 135o [1]. Any values outside this range will result in low mechanical advantages and mechanism lock ups. The two transmission angles are shown in the diagram below.

ψ1 and ψ2 are the transmission angles for the two 4 bar circuits.

Based on the conventionally designed Watt-I 6 bar linkage BKC the transmission angles obtained did not allow the mechanism to operate smoothly and caused it to lock up at the precision position 2. The transmission angles for that mechanism is shown below in table 1.

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Transmission angles for a Watt-I mechanism

PRECISION POINT 1

(Degrees)

PRECISION POINT 2

(Degrees)

PRECISION POINT 3

(Degrees)

Ψ1 67.6 30.5 24

Ψ2 20.28 47.8 97.26

Table 1. Transmission angle for failed mechanism

As mentioned in the dimensional synthesis, the alpha 3 value was chosen as -90o. For this mechanism α 2 was chosen as

80o, ψ2 was chosen at 86o and ψ3 was chosen at 70o. This value set was selected form the solution space obtained through the optimization strategy mentioned above. All related mechanisms in the solution space gave similar transmission angle values resulting in lock-ups at multiple positions. This implied that a circuit defect was present in the mechanism. A line diagram of this mechanism drawn in MATLAB is presented in Appendix B.

This led to a modification of the positioning of links in the mechanism. The ternary grounded link flipped 180o about the z axis. A line diagram of the mechanism is shown in Appendix A.

This mechanism was animated in Creo and it showed smooth motion. The transmission angles were also calculated for each precision position, and are presented in table 2, below.

PRECISION POINT 1

(Degrees)

PRECISION POINT 2

(Degrees)

PRECISION POINT 3

(Degrees)

Ψ1 110.4 103.97 34.1

Ψ2 22.94 127.9 61.75

Table 2. Transmission angles for the chosen mechanism

Based on values obtained, one of the transmission angles for the first precision point and third precision point is below the 45o limit. This is chosen as an acceptable situation since it is desired that the mechanism lock in its complete extended form and its collapsed state.

Velocity, acceleration, and force analysis was performed on the chosen mechanism, too. In Creo the mechanism assembly was analyzed for coupler velocity and acceleration, and several joints were analyzed for radial force. The analysis was conducted assuming a 2 deg/s input on the driving ternary Z1-Z8-Z10.

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Modified Watt-I BKC

Velocity of coupler Acceleration of coupler

Force on coupler joint Force on joint between Z6 and ground

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Force on joint between ternary Force on joint between Z6 and ternary

All graphs produced above are with respect to time. The mechanism was animated in Creo and these values were obtained. The analyzed time period correlates to the movement of the mechanism from the first precision point to the third precision point. Since the cable for the wakeboarder will be attached to the coupler the analysis was done on the coupler. The velocity analysis shows low velocities of the coupler when the mechanism is collapsed and driven from the ternary link pinned to the boat. A small acceleration value and small change in the acceleration of the coupler shows enhanced usability and ease of storage. The force on the coupler joint and the ternary bound to the coupler is low until the mechanism is in the second precision point but increases as it gets closer to the completely collapsed position. Low force values on the joints will ease bearing choices for the actual mechanism, and we only see a maximum radial force of about 620N. A spike in the force values just before the mechanism is collapsed is seen due to smaller transmission angles around precision point three. This is acceptable since mechanism locking is favored in the stored position.

CONCLUSION

After analyzing the final mechanism, we can see that this Watt-1 six-bar mechanism fits all constraints, hits all prescribed precision points on the same circuit, and will work when it is built and applied in a commercial application. The stresses at each pivot point are within current bearings load ratings and they will be able to handle these stresses. The success of this mechanism is due to the good transmission angles of the final chosen design and its smooth operation. It is obvious from our final solution that a four-bar mechanism in this application would not have been as compact as the current design. It is also clear that no other type of six-bar inversion or BKC would be able to achieve this compact storage position while reaching the specified extended position as well as being able to clear the passengers heads while in the boat when the tower is being lowered or extended. This is due to us choosing an appropriate second position and the correct type of mechanism. It is important to note that this design is only one side of the total wakeboard tower and that the total tower will consist of one mechanism on each side of the boat connected at the coupler link.

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APPENDIX A – FINAL MECHANISM

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APPENDIX B – FAILED MECHANISM

Precision point 1

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Precision point 2

Precision point 3

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