Part V: Closed-form solutions to Loop Closure Equations

Transcription

1 Pat V: Closed-fom solutions to Loop Closue Equations This section will eview the closed-fom solutions techniques fo loop closue equations. The following thee cases will be consideed. ) Two unknown angles in the loop closue equation ) One unknown angle and one unknown length 3) Two unknown lengths. Review vecto notation and basic opeations y x ME 360 Couse Notes S. Canfield Pat V -

2 Table: D Vecto Desciptions and Opeations Catesian Complex Pola ME 360 Couse Notes S. Canfield Pat V -

3 Closed-Fom Solutions to the Vecto Loop Closue Equation A geneal loop closue equation may look something like: whee i ae vectos in the chain. The loop closue equation is a vecto equation (and epesents scala equations) and can be solved if only scala unknowns exist in the equation. Thee possible cases exist: case ) two unknown angles case ) one unknown angle, one unknown length case 3) two unknown lengths Solution fo case ) Two unknown angles: A geneal loop closue eq. with two unknown angles may look like: // // /? /? // 0 (c.) Hee the unknowns ae 3 and 4. Fo geneality, call the vectos with unknown angles ua and ua: // // /? /? // ua 0 (c.) ua 5 Step 0: Reaange to collect all the knowns into a single vecto: (c.) with: ME 360 Couse Notes S. Canfield Pat V -3

4 Step : Now, ewite Eq. c. with one of the unknown angles isolated on one side (c.3) Step : Expand Eq. c.3 into eal and x and y components (eal and imaginay pats) (c.4) Step 3: Squae and add Eq. c..4 to eliminate one unknown angle (ua) (c.5) Step 4: Use the tan ½ angle identity to yield an algebaic (quadatic) equation: t t cua, s, t tan ua ua (c.6) t t (c.7) (c.8) ME 360 Couse Notes S. Canfield Pat V -4

5 Step 5: Solve fo t: b b 4ac t (c.9) a Note: the two solutions yield the two banches of the mechanism. Step 6: Solve fo the unknown angle fom the tan-/ identity (ua) (c.0) Step 7: Solve fo the othe unknown angle, ua: Note: The use of an atan function (quadant sensitive) ensues solution of unique angle value. (c.) ME 360 Couse Notes S. Canfield Pat V -5

6 Discuss: Physical Intepetation of solutions: ) solutions: ) Valid solutions: 3 4 O O4 Othe notes: ) Assumes all vecto tavel in the same diection ) If you have a vecto taveling in the opposite diection, use these equations but include a negative on the associated vecto magnitude tem in the equations ME 360 Couse Notes S. Canfield Pat V -6

7 Example : Conside the fou-ba shown below as a schematic, with vecto model, loop closue equation and known 3 4 O O4 The loop closue equation becomes, // o // /? /? (c.) Knowns: ; 6; ua 8; ua 90 ; 40 ; 5; Fist (Step 0 above): Collect all knowns into a single vecto: ME 360 Couse Notes S. Canfield Pat V -7

8 Second: Solve fo a, b, c: (Step 4) Thid: Solve fo t: (Step 5) Fouth: Solve fo angle, 4: (Step 6) Fifth: Solve fo 3 (Step 7) ME 360 Couse Notes S. Canfield Pat V -8

9 Automating the solution pocess fo one common case of mechanisms, the fouba linkage. Since the fouba is such a common mechanism, automating the solution pocess is wothwhile. A few ways to do this might include: Wite a function that etuns the solution based on an input set of link lengths and diving angle Ceate a fouba class that has the functionality of solving the position poblem. An example of the fist of these methods is biefly descibed hee. The syntax is based on pogamming in Matlab. Fist, ceate a function in matlab that will be called to solve the fouba, (fo example, fba.m). This function will accept the link lengths, gound link angle, input angle and banch, and etun the unknown angles in the fouba. % fba.m (theta3, theta4) = fba(,,3,4,theta,theta,banch); theta3= theta4= etun ME 360 Couse Notes S. Canfield Pat V -9

10 Solution fo case ) One unknown angle, one unknown link length: Two possibilities exist in this case, the unknown angle and length exist in the same vecto, o they do not. In the fist case, the solution is tivial (unknown vecto esults fom vecto addition). In the second case, fo example as shown in the loop closue equation shown below, the pocedue is as follows.?/ // /? // (c.) with the fist tailing supescipt indicating known/unknown length, the second indicating known/unknown angle. Fo geneality, call the vecto with unknown length ul and the vecto with unknown angle ua:?/ // /? // ul 0 (c.) ua 4 Step 0: Collect knowns into a single vecto: With Step : Rewite the loop closue equation with the unknown angle isolated on one side of the equation. (c.) Step : Expand Eq. c.3 into eal and x and y components (eal and imaginay pats) (c.3) ME 360 Couse Notes S. Canfield Pat V -0

11 Step 3: Use the squae and add technique to emove the unknown angle. (c.4) Step 4: This equation is now quadatic in the emaining unknown (), solve fo the two oots. Note, the two oots coespond to the two possible banches of this loop. (c.5) Step 5: Finally, solve fo the emaining unknown angle using the atan function as befoe. ME 360 Couse Notes S. Canfield Pat V -

12 Discuss: Physical Intepetation of solutions: ) solutions: ) Valid solutions: ME 360 Couse Notes S. Canfield Pat V -

13 Example of Solution fo case : The figue below shows a linkage schematic with vecto model, loop closue equation and known paametes. Solve fo the unknowns in this model The loop closue equation becomes, // // /??/ // // /??/ o ua ul 0 5; 3; 3 ua 8; Knowns: 50 ; 0 ; (c.) (c.) Fist (Step 0 above): Collect all knowns into a single vecto: Second: Solve fo a, b, c: (Step 4) ME 360 Couse Notes S. Canfield Pat V -3

14 Thid: Solve fo ul: (Step 4, second pat) Fouth: Solve fo angle, 3: (Step 5) ME 360 Couse Notes S. Canfield Pat V -4

15 Solution fo case 3) Two unknown lengths: In case 3, the unknown lengths in the loop closue equation esult in a set of two linea scala equations and can be solved diectly with linea algeba. The following example demonstates this case. The vecto loop equation is,?/?/ // // (c3.) with the unknowns hee in and. Fo geneality, call the vectos with unknown length ul, ul.?/?/ // // ul ul Step 0: Collect the knowns into a single vecto:?/ // /? ul ul k 0 with cos cos kx ky 3 sin 3 k k sin 3 kx 4 a tan ky ky, kx 4 (c3.) Step : Isolate the unknowns on one side of the equation the knowns on the othe;?/?/ // (c3.3) ul ul k Step : Expand this equation into its scala components, c c c ul ul s ul ul ul ul s ul ul k s Step 3: Cast these equations into matix fom, cul culul k ck s ul sul ul k sk o Ax b k k k (c3.4) (c3.5) ME 360 Couse Notes S. Canfield Pat V -5