ENGR 990 Engineeing Mathematics pplication of Tigonometic Functions in Mechanical Engineeing: Pat II Poblem: Find the coodinates of the end-point of a two-link plana obot am Given: The lengths of the links and B and the angles Find: The coodinates of the end-point B The coodinates of B ma be found b adding the coodinates of elative to and the coodinates of B elative to cos( ) cos( ) and sin( ) sin( ) Eample : Given: The lengths and angles of a two link plana obot ae 30 (deg), 60 (deg) 3 (ft), (ft), Find: The Catesian coodinates and of B using: a) a calculato, and b) the values listed above fo commonl used angles a) Using a calculato to evaluate the sine and cosine functions: cos( ) cos( ) 3 cos(30) cos(60) 598 3598 (ft) sin( ) sin( ) 3 sin(30) sin(60) 5 73 33 (ft) b) Using the values fo commonl used angles: 3 cos( ) cos( ) 3 598 3598 (ft) 3 sin( ) sin( ) 3 5 73 33 (ft) Kamman ENGR 990 Engineeing Mathematics page: /5
Eample : Given: The lengths and angles of a two link plana obot ae 30 (deg), 0 (deg) 3 (ft), (ft), Find: The Catesian coodinates and of B using: a) a calculato, and b) the values listed above fo commonl used angles a) Using a calculato to evaluate the sine and cosine functions: cos( ) cos( ) 3 cos(30) cos(0) 598 598 (ft) sin( ) sin( ) 3 sin(30) sin(0) 5 73 33 (ft) b) Using the values fo commonl used angles: Note fist that 0 80 60 (deg), so cos(0) cos(60) and sin(0) sin(60) 3 3 3 cos( ) cos( ) 3 ( ) 598 598 (ft) sin( ) sin( ) 3 5 73 33 (ft) Kamman ENGR 990 Engineeing Mathematics page: /5
Invese Poblem: Find the angles of the links of the obot am given the endpoint position Given: The coodinates of the end point B and the lengths of the links and B Find: The link angles Fist, we can calculate the length using the Pthagoean Theoem Then, we can appl the law of cosines to tiangle B to find the angle o cos( ) cos We can appl the law of cosines again to find the angle cos( ) cos Finall, the link angles ma now be found b noting a) tan( ) / tan ( / ) b) Kamman ENGR 990 Engineeing Mathematics page: 3/5
Eample 3: Given: The coodinates of the end point B and the lengths of the links and B ae 5 (ft), 35 (ft), 3 (ft), and (ft) Find: The link angles Following the appoach outlined above, a) b) 5 35 38079 (ft) 3 38079 338079 cos( ) 3 38079 34 (deg) cos 338079 0548 (ad) 3 38079 978 (deg) c) 3 3 cos( ) cos 3 696 (ad) d) 3540 (deg) 0678 (ad) tan( 548) 35 / 5 tan (35 / 5) 548 8 (deg) 696 0678 0633 (ad) Check: We can now use the calculated link angles to check the position of the endpoint Does it match ou equied position? cos( ) cos( ) 3 cos(0678) cos(0633) 4455 09457 4998 5 (ft) sin( ) sin( ) 3sin(0678) sin(0633) 7377 763 35 (ft) Kamman ENGR 990 Engineeing Mathematics page: 4/5
Elbow-down and Elbow-up Positions Note that the above answes could be intepeted in two was, the elbow-down position and the elbow-up position as illustated in the following diagams Elbow-down position Elbow-up position Note on calculato usage: When calculating tan ( ) sin ( ), cos ( ) and, ou calculato will place the esults in specific quadants as outlined in the table So, ou calculato does not alwas place the angle into the coect quadant Note that in the above eample, we used the law of cosines (and hence cos ( ) ) to calculate the angles of the tiangle B, and ou calculato gave angles in the ange 0 What if we had used the law of sines to calculate the angle? sin( ) sin( ) sin sin( ) / sin 38079 sin(0548) / 879 (deg) 4449 (ad) Note that this is not the coect esult s we know fom ou wok above, the coect esult is in the second quadant So, 4449 6967 (ad) This is ve close to the esult we found above Function Range Quadants sin ( ) cos ( ) tan ( ) I, IV 0 I, II I, IV Kamman ENGR 990 Engineeing Mathematics page: 5/5