Analysis of Suspended Mass/Pulley System: Model #6, Study Guides, Projects, Research of Engineering

Download Analysis of Suspended Mass/Pulley System: Model #6 and more Study Guides, Projects, Research Engineering in PDF only on Docsity! Analysis Project: Model #6 Suspended Mass / Pulley System Mike Hedman EGR 345 December 7, 1998 Analysis Project: Model #6 Suspended Mass / Pulley System Mike Hedman EGR 345 December 7, 1998 Executive Summary Model number six is a system made up of a 10-kg mass, which will be called a “block”, suspended from a damping device and a spring. There is a second spring, which matches the spring above the block, connected below the block. This spring is also attached to two pulleys. A rope, which is wrapped around the pulleys, is the input for the system. For the output, the motion of the block was analyzed. To find the motion of the block, the damper above the block was given a value of 100 N.s/m. The system was found to oscillate, which was not desired. Therefore, a damper had to be found which would correct the system to get the fastest output response without oscillations. The damper chosen was the precision dial adjustable hydraulic shock absorber as found on page 754 of the McMaster-Carr Company catalog (volume #104). The shock absorber carries the part number, 6518k81 and a price of $163.17. Introduction The purpose of this paper is to analysis the model assigned to find an output, from the system, for a given input (see figure 1). The output, in this case, was the vertical displacement x1 of the 10-kg mass m1. The input was a tension on the rope, which caused a force on the mass m1. Because the design objective for the project was to get the fastest output response without overshoot or oscillation, the damper was considered to be variable. This gave an adjustable parameter to achieve the design objective. Modeling and Analysis For purposes of simplifying the model two assumptions were made. First, the rope will not stretch. The second assumption is the two pulleys’ masses can be neglected because they are very small compared to the mass m1. So the mass of each pulley could be neglected, each is considered hollow with a long heavy bar attached. This allows the moment of inertia to be high with little mass. Equation (1) is derived for the vertical displacement of the mass m1. The free-body diagram for the mass m1 is shown in figure 3. The forces are summed to be positive if they are in the vertically down direction. Note the positive direction for the equation is the same as the value x1 in figure 1. Therefore, the inertial force is negative in the equation, since it opposes motion. When the rope is pulled, the mass m1 feels the downward force from the lower spring, and gravity also causes a downward force. The spring and damper on top of the mass m1 resist the downward motion. Figure 3 – FBD for mass m1 (1) Figure 4 shows the FBD for the upper pulley. The sum of the moments around the upper pulley caused by the roped wrapped over the top of it yields equation (2). The spring ks2 reacts by resisting the downward motion, caused by pulling the rope. It is also an equal but opposite force as applied to the mass m1. The rope causes a downward tension T1. The force T2 is also downward and is anchored. This causes the pulley’s vertical motion. As stated before, this spring also causes a short lag in the movement of the mass m1. Figure 4 – Upper pulley FBD (2) m 2t x1 d d 2  k d t x1 k s1 k s2 x1 d d  k s2 x2  m g T 1 r 1  T 2 r 1  j 1 2t  1 d d 2   Since the upper pulley can move vertically, equation (3) is the motion of the pulley vertically. The upper pulley’s mass is assumed very small compared to the mass m1, so in the equation the mass is zero. (3) Equation (4) is the sum of moments about the lower pulley (See figure 5) also caused by the rope wrapped underneath it. Because the lower pulley is anchored to the ground it can not translate, so this is the only equation needed for motion. The moment (F)(r2) is resisted by the tension in the rope and the inertial force. In the free-body diagram for the lower pulley, the reaction forces of the anchoring pin are neglected because they have no effect on the desired output information. Figure 5 – Lower pulley FBD T 1 T 2 k s2 x2 x1 0 