Question

We wish to determine the moment at the shoulder that is required to perform the arm motion (shoulder abduction) depicted in Figure 3 below. This motion may be modelled with a simple, single- element linkage system as shown in the figure. The shoulder joint is represented by a simple pin support centred at O. The arm and the carried weight are represented as a rigid body consisting of a rod and a cuboid. In the rigid body model, the length of the rod, L is 50 [cm], its mass is 4.2 [kg], and its centre of mass is located 25 [cm] from point O. The relevant dimensions of the cuboid are b = 15 [cm] and c= 10 [cm], and it has a mass of 2 [kg] which acts 57.5 [cm] from the point o. b= 15 [cm] c = 10 [cm] L = 50 [cm] Mo 1350 0° Figure 3: Shoulder abduction. Part(a) [2 marks] The angular displacement of the rigid body ranges from 0 = 0° (the vertical as shown in the figure) to e = 135º and can be modelled using simple harmonic motion. Assuming a rate of 20 [reps/min), write down an expression for angular displacement, [rad] as a function of time, t [s]. You may assume that the motion starts with an angular displacement of 135º. Hint: The angular displacement, e can be expressed as = offset + @ocos(ot), where @offset is an offset angle. When offset is added to the simple harmonic motion expression, it forces e to range from 0° to 135º.

Answer 1

Giveni In mbor 1.2kg the rigid body model Length, Som and moss con of bar 25cm from o cuboid masse mir axg (g) 57.5cm b= 15cm (El Din porto mass moment inertia the 00 mo for uboid system L 50cm ме“, 745 mpery 135* Obout poin) 0. To = mych + Io Tood t I cuboid mc (L+ b )2 4.2x65)?+ 9x 1.575)? 3 - 1.0 1125 Kg.my 3 radius of gyration of bur cuboid system. (K) from inertia balonie My K² = To K = 0.347 m 4-2+2)k? = 1.01125 angulor velocity w, and acceleration by We know from newton and low Text = Toa (9x9.81*.575C0145 + 4.2x9.81*:25 (0)95) = 1.0112582 2 = 15 rado When o= 135' law a Saw Qat dF now dw 1ste in of time angular displacement o dt do fda= sista 0= 7.5++