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A ½-hp electric motor, running at 1,800 rpm, will be used to drive
a grinding wheel operating at 600 rpm. A flat belt-and-pulley drive
system configuration has been selected

 r = radius of
 d = diameter of
 φ= angle of
 n = pulley
 Drive motor will have a 2-inch-diameter pulley mounted to its
  1/2-inch output shaft
 Candidate designs should be able to utilize the full horsepower
 Customer desires a compact system design
 Drive pulley will slip first, before the driven pulley
 Purchasing department has located a vendor that can provide a flat
  belt that can withstand a maximum 30-pound tensile load
 Coefficient of friction between the belt and pulley is 0.3
 Other design engineers in your group will design the mountings,
  bearings, and protective equipment
 Parametric design efforts should focus on distance between
  centers and driven-pulley diameter
    BELT-AND-PULLEY Assignment
   Determine:
     – Solution Evaluation Parameters (SEP)
     – Design Variables
     – Problem Definition Parameters

   Develop a plan for solving the design problem
     – In developing the plan discuss and give the formulas that
       would be needed
     Solution Evaluation Parameters
   Principal function of the pulley:
     – transform the power of the motor from a high speed to low
          smaller motor torque is converted to a larger torque,

           (conservation of energy law)
   Principal function failure - if the belt slipped or if the belt broke
    owing to excessive tension
   Customer would be more satisfied with a compact design.
   Since we know that the tension forces in the belt are limited by the
    amount of friction between the belt on the driver pulley, up to the
    point of impending slip, we could determine the torque that the
    belt can deliver to the pulley, Tb and compare it with the
    maximum torque, Tm that the motor can supply.
   Calculate the maximum belt tension, Fb to make sure that it does
    not exceed the 35 (lbs.) limit
Solution Evaluation Parameters
                    Design Variables
   Value of the center distance, c,
     – affects the compactness of the design
     – is to be determined by the designer
     – ???Center distance is increased, more of the belt wraps around
       the pulley increasing the ability of the belt to grip the pulley
       and thereby satisfy the torque requirements of the motor
      Problem Definition Parameters
   "givens" that define design problem conditions
     – friction coefficient, belt strength, motor power, and motor
       pulley diameter
         Plan for Solving the Design
 Using analytical relations from physics and mathematics we can
  use a hand calculator or build a spreadsheet to analyze a variety of
  engineering characteristics, including:
  1. grinding wheel pulley speed, n2
  2. angle of wrap as a function of the center distance, c
   3. belt torque, Tb'
  4. maximum belt tension, F1
  5. slack-side belt tension, F2
  6. initial tension (before torque is applied), Fi
 Then we will check that the constraints are not violated.
  Specifically, we will make sure that the belt will deliver the full
  motor torque to the grinding-wheel pulley and that the belt tension
  does not exceed the belt strength limit.
        Generating and Analyzing
Model the behavior of the system using relations from physics
and mathematics and develop a system of equations to

   Motor will deliver power W(hp), to a pulley rotating at n rpm
    when producing a torque T m lb.. ft. according to equation

   Belt is not permitted to slip on the pulleys, the pulley speeds
    are related to the ratio of the pulley diameters.

   Determine the angle of wrap φ1 using basic geometric
         Generating and Analyzing
   Belt
    – Coefficient of friction f
    – In contact with the pulley for an angle of wrap φ
    – maximum belt tension F1 on the taut side of the pulley
    – F1 related to the belt tension F2 on the slack side of the

   Static equilibrium, free-body diagram of motor pulley,
    – sum the moments about the bearing B
        obtain the torque Tb, delivered by the
          belt to the driver pulley of radius r1
           Generating and Analyzing
   Tm - Maximum torque delivered by the motor

          Tb ≥ 17.52 (lb. ∙ in.)

   n2 - driven-pulley speed > as a function of the design
    variable, diameter d2

    – grinding-wheel speed has been specified as 600 rpm
Generating an Initial Value of the
      Design Variable c
   Customer more satisfied with a compact design → distance
    between the pulley centers c to be small.
    – The closest that the two pulleys can be is when their radii are
       almost touching, theoretically speaking

   φ1 - Angle of wrap on the driving pulley

   F1 - tensile force that satisfies the motor torque constraint
Generating an Initial Value of the
      Design Variable c
   F1 - to satisfy the torque constraint, a 37.5-lb. tension will be
    necessary → exceeds the belt strength constraint of 35 lbs.
    – center distance of 4 in. is an infeasible value
Finding Feasible Values

                  Safety factor, n

                  Best values
                  Trade-offs
   Choose best design candidate from feasible designs
    – Select one criteria
    – Select multiple criteria
       Compactness vs. safety

   Weighted-rating method
        Step 1: Establish a set of evaluation criteria.
        Step 2: Rate the feasible designs for each criterion.
        Step 3: Weight the ratings according to importance.
        Step 4: Sum the weighted ratings to calculate an overall
                     weighted rating.
  Step 1. Evaluation Criteria,
Importance Weights, Satisfaction
   Evaluation criteria often developed from solution evaluation
    parameters or from engineering characteristics ("voice of the
    customer" )

   Identify the customer's:
    1. Functional requirements for the product
    2. Key engineering characteristics (that measure how well the
         functions are performed)
    3. Importance of each functional requirement
  Step 1. Evaluation Criteria,
Importance Weights, Satisfaction
   Belt-and-pulley example
    – As long as motor horsepower fully utilized
       → two most important engineering characteristics belt tension
          (no slip) and center distance (compactness)
           – solution evaluation parameters

   Customer considers belt tension as "very important" and center
    distance as "important"
    – Assume that we interpret "very important" with a weight of 0.6
        and "important" with a weight of 0.4
Step 2. Rate Feasible Designs for
         Each Criterion

   Rating scale 1-5
   Customer Satisfaction curves
    – Link each criterion or SEP to a value of satisfaction
    – Graphs – curve fit
    – Rating scale 0-1
                Satisfaction Curves
                      Belt-Pulley Designs
   Two-point formula

   Substitute satisfaction
    S for y, and solution
    evaluation parameter
    for x
     – Smax = 1
     – Smin = 0

              Decreasing Curve

              Increasing Curve
          Step 3: Weight Rating by
   Multiply satisfaction rating by the importance weight
Step 4: Calculate Overall Weighted
   Sum of the importance weighted satisfaction levels to obtain Q,
    overall satisfaction

   For a center distance of 6 in.
         Overall Weighted Ratings
   Sum of the importance weighted satisfaction levels to obtain Q,
    overall satisfaction
            Planning and Scheduling
   Planning
     – Identify & order key activities
     – Top 10 – 20 critical activities
         Break down into sub-activities

            – Break down into sub-sub-activities

   Scheduling
     – Create a time frame for the plan
               Timelines - Schedules
   Bar chart - Gantt Chart
     – No connectivity between tasks

   Network logic diagram - Critical Pathway
     – Dependent project activities (tasks) are connected with arrows
       and time durations
Gantt Chart
Network Logic Diagram
                Critical-Path Method
   1. An activity - time-consuming effort that is required to perform
      part of a project. An activity is shown on an arrow diagram by
      a line with an arrowhead pointing in the direction of progress
      in completion of the project.

   2. An event - the end of one activity and the beginning of another.
      An event is a point of accomplishment and/or decision. A
      circle is used to designate an event.
                Critical-Path Method
Logic restrictions
   1. An activity cannot be started until its tail event is reached.

   2. An event cannot be reached until all activities leading to it are

   3. An event dependent on another event preceding it, even though
      the two events are not linked by an activity, has a dummy
      activity denoted by
    Longest Time Through Network
               Methodology - Parameters
1. Earliest start time (ES): The earliest time an activity can begin when
                             all preceding activities are completed as
                             rapidly as possible.
2. Latest start time (LS): The latest time an activity can be initiated
                            without delaying the minimum completion
                            time for the project.
3. Earliest finish time (EF): EF = ES +D, where D is the duration of
                            each activity.
4. (Latest finish time LF): LF = LS +D
5. Total float (TF): The slack between the earliest and latest start times.
                            TF = LS-ES. An activity on the critical path
                            total float.
    Time durations - same time units & most likely estimate of time
Calculation of ES
Calculation of LS
        Calculation of EF, LF, TF

Critical path is defined by the activities with zero total float
Modified Bar Chart