# Parametric

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```					 BELT-AND-PULLEY EXAMPLE
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

pulley
d = diameter of
pulley
φ= angle of
wrap
n = pulley
speed
BELT-AND-PULLEY EXAMPLE
 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
available
 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
speed.
 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
Problem
 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
analyze

   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
relations
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
pulley:

   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
Redesigning
Finding Feasible Values

   Safety factor, n

   Best values
Evaluating
   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
Importance
   Multiply satisfaction rating by the importance weight
Step 4: Calculate Overall Weighted
Rating
   Sum of the importance weighted satisfaction levels to obtain Q,
overall satisfaction

   For a center distance of 6 in.
Overall Weighted Ratings
Belt-and-Pulley
   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

   Network logic diagram - Critical Pathway
– Dependent project activities (tasks) are connected with arrows
and time durations
Gantt Chart
Network Logic Diagram
Critical-Path Method
Elements
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
complete.

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

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