ME 30 THEOY OFD MACHİNES I SOLVED POBLEM SET 2 POBLEM. For the given mechanisms, find the degree of freedom F. Show the link numbers on the figures. Number the links and label the joints on the figure. Write the number of links, number of joints and types of joints in the mechanisms. Indicate if there is any redundant degree of freedom. a) Planar mechanism. Assume that there is no slip at the cam joint. 4 2 P 3 5 6 P, C s 7 8 9 C p λ = 3 Disregard the spring and the joints related to the spring. l = 9, j = 2 (8, 2P, C p (no slip), C s ) f i = 8 + 2 + + 2 = 3 F = 3(9 2 ) + 3 = 2 + 3 = 2
b) Planar mechanism. Assume that the loader is stationary. 9 8 7 6 4 P 3 P 2 5 λ = 3 l = 9, j = (9, 2P) f i = F = 3(9 ) + = 2 The two freedoms are the raising and turning of the bucket 9, which are controlled by the motions of the two pistons 3 and 6. POBLEM 2. The shaft coupling shown is a spatial mechanism and obeys the general degree of freedom equation. It is known that its degree of freedom is F=. As shown in the figure, the joints between links and 2, and links and 4 are revolute joints, and the joint between links 2 and 3 is prismatic joint. State the type, degree of freedom and the types of relative motion (i.e. rotation and/or translation) of the joint between links 3 and 4.
Solution: POBLEM 3. In the planar mechanism given below BED 90 and A0CD 90 and the link lengths are labelled as A o A = r 2, AB = r 3, AE = a 3, A o C = r and CD = a. a) Find the degree of freedom, the number of independent loops, and the total number of required joint variables (position variables). b) Choose a sufficient number of revolute joints which when disconnected yield an open-loop system. Indicate those joints. Assign the joint variables and show them clearly. c) Write the necessary number of independent loop closure equations using vectors described by directed lines such as SQ, S, etc. d) Using complex numbers, re-write these loop closure equations in terms of the joint variables and the fixed parameters of the mechanism.
Solution: POBLEM 4. For the following double slider mechanism with dimensions OA = b, OC = c, BD = b3 where s4 is the input joint variable. a) Write down the loop closure equations using point to point vectors and using complex numbers. b) Write down the scalar components of the loop closure equations. c) Solve the unknown joint variables analytically in terms of the link lengths and the input joint variable.
2 2 3 D γ3 s s4 4 C O Solution: a) LCE via point to point vectors : OA AD DB OC CB 2 ( 2 3 ) LCE via complex numbers: ib s e b e c is 3 4 b) Scalar components of LCE : eal part : s cos 2 b3 cos( 2 3) c Imaginary part : b s sin 2 b3 sin( 2 3) s4 c) Let us rearrange the above equations as below: s sin s b b sin( ) 2 4 3 2 3 s cos c b cos( ) 2 3 2 3 Divide side by side: sin 2 s4 b b3 sin( 2 3) cos c b cos( ) 2 3 2 3 Cross multiplying gives: c sin b sin cos( ) s b cos b cos sin( ) or 2 3 2 2 3 4 2 3 2 2 3
s b c b cos sin sin( )cos cos( )sin 0 4 2 2 3 2 3 2 2 3 2 sin sin cos cos sin : Since Acos Bsin C where A s b B c 2 2 4 C b sin 3 3 If we let A Dcos B Dsin where 2 2 D A B φ = atan 2 (B, A) Then: θ 2 = φ ± cos C D Once 2 is found from the above expression; s can be found from either s c b cos( ) 3 2 3 (if cos 2 or s s b sin( ) b 0 s ) 0 4 3 2 3 (if sin 2 0 s ) 0 POBLEM 5. In the single d.o.f. mechanism shown, the joint variables are assigned as θ 2, θ 5, s 54, θ 6 and s 63 which correspond to disconnecting the revolute joints at A and B. The link dimensions are given as A o A = r 2 = 7cm, AB = r 3 = 24cm, AE = a 3 = 7cm, A o C = r = 47cm, CD = a = 26cm, BED 90 and A0CD 90. The loop closure equations in complex numbers are LCE r 2 e iθ 2 = ir + a + s 63 e iθ 6 + a 3 ie iθ 6 LCE 2 s 54 e iθ 5 = a + s 63 e iθ 6 + (r 3 + a 3 )ie iθ 6
By solving the Loop Closure Equations analytically, determine the joint variables when θ 2 = 20 o. Find all solutions corresponding to different assembly configurations of the mechanism. (Assume that there are no physical limitations of the prismatic joints). Solution: