Optimization Methods Unit5

7/29/2019 Optimization Methods Unit5

1/16

MEE552 Optimization Methods

Unit 5 Optimization Design of Machine Elements


2/16

General Characteristics of Mechanical Elements1. All mechanical elements must satisfy certain functional requirements

of the structure or machine wherein they are used.

A functional requirement is a condition which must be satisfied bythe mechanical element if the structure or the machine is to function

properly.

In the strict sense, functional requirements can either be positive andgenerally specified, or negative and generally implied.

Ex: Design of a mechanical extension spring

In the design of a mechanical extension spring the function requirements ofthe machine from the spring might be a certain maximum force Fmax and acertain force gradient k. The values Fmax and k would be positive functionrequirements, since they must be directly obtained for proper functioning of

the machine and they are conditions which we want. Maximum shearing stress in the spring wire should be equal to or less than an

allowable shearing strength of the material, using an acceptable criterion forfailure due to fatigue.

However, this specification is really for a negative functional

requirement, since it states a condition which we do not want.


3/16

Negative functional requirements are really undesirable effects,where as positive functional requirements are really desirable

effects. One example of functional requirements is shown in figure. In

figure, in order for the mechanical structure to function properly, thebeam must transmit a maximum force Fmax and it must have aspecified length L.

Functional requirements are influenced by factors which are externalto the element being designed, and this is generally true formechanical elements.

The functional requirements of an element are not always

independent of each other.

Simple Beam

General Characteristics of Mechanical Elements


4/16

Hence functional requirements, if they can be changed at all fromtheir specified values, are often dependent on each other as the

results of conditions external to the element. Functional requirement is not always limited to a specific single

value to obtain a satisfactory mechanical design. There might be asatisfactory feasible range over which a functional requirementcould have a satisfactory value.

Associated with functional requirements we often have limitequations, which define a satisfactory range for values of thefunctional requirement.

The limit values for any range might very well be rigidly (loosely

defined) established by explicitly defined unchangeable factorsexternal to the element under design.



5/16

2. At least to some extent, all mechanical elements have certain undesirable effects.

Associated with all mechanical elements is the very important undesirable effectof cost. Other common undesirable effects of importance might be deflections,

stresses, weight, space occupancy, vibration, force and the like, and the degree ofsignificance for nay such undesirable effect depends on the particular problem athand.

In order to have a satisfactory design of a mechanical element, feasible values forall undesirable effects must be limited to tolerable ranges.

Maximum shearing stress max at any point in the mechanical element must beequal to or less than an allowable shearing stress related to a material strength,using the maximum shearing stress theory of failure.

Limit equations exist for all undesirable effects, and they must be adhered to inthe design procedure for obtainment of a satisfactory design.

Limitations on deflections, weight, force, vibrations and other undesirable effectsmight be either rigid or loose, depending on the particular mechanical element

being designed and its particular intended application.

The tolerable limit for an undesirable effect can be determined only from a studyof the particular element and its particular application, including the significanceattached to the occurrence of the undesirable effect on the entire mechanical

structure.



6/16

General Characteristics of Mechanical Elements3. All mechanical elements are uniquely defined by a complete

specification of the material and the geometry of the element.

Many of the important properties of a material are given by thematerial parameters found in the literature, such as the modulus ofelasticity, Poissons ratio, weight density and unit weight cost.

Various machining properties, the extent of availability and therestriction to standard sizes are important considerations.

Generally, in the conduct of a design study the selection of amaterial for a mechanical element will be based primarily on thematerial parameters which are explicitly defined.

The geometry of a mechanical element can always be uniquelydefined by specifying certain independent geometrical parameters.Also certain manufacturing error limits are specified.

The significant geometry is explicitly defined only by specifyingtheoretical values for geometrical parameters and limits onmanufacturing errors, and these tolerances must be adhered to in theactual manufacturing process for the element.


7/16

Functional Requirements

A functional requirement is a condition which must be satisfied by the mechanicalelement if the structure or the machine is to function properly.

It can define as the selection of materials and geometry which satisfies specifiedand implied functional requirements while remaining within the confines ofinherently unavoidable limitations.

Adequate design: any possible design solution represents a synthesis whichmerely satisfies the functional requirements within the confines of the existinglimitations.

In order to obtain a more explicit method of design, the overall objective must bemore clearly defined.

Associated with any mechanical element are certain inherently unavoidableundesirable effects.

For instance, in any specific mechanical element such effects as stresses, deflections,

vibrations, space occupancy, weight, and cost might be undesirable and the degree ofsignificance for each effect depends on the particular application.

Hence a tolerable limit will exist for the permissible degrees of each undesirable effect.

Associated with any mechanical element are certain desirable effects, such aspower transmission capability, energy absorption capability, momentary overloadcapacity, varying force transmission capability, speed capability, usable length of

life and factor of safety.


8/16

Functional Requirements

An explicit overall objective in mechanical design is obtained based on any one ofdesirable and undesirable effects.

Conduct method of design as to minimize the most significant undesirable effector to maximize the most significant desirable effect.

Introduce an additional design objective in an explicit design procedure forselecting materials and geometry.

Optimum design: a mechanical design which is the best possible one from thestandpoint of the most significant effect.

The method of optimum design is an explicit method of design. Also, it should beemphasized that the method does not ignore any of the functional requirements orany of the existing limitations.

Satisfactory design is determined in an explicit manner, it is the best possibledesign from the standpoint of the most significant effect and all other undesirable

effects are within their permissible ranges of magnitude.


9/16

Aspects of Design Solution

Adequate design solution

It is characterized by traditional influences and cut-try techniques

It is entirely satisfactory from the standpoint of the overall objectives in the statementof the problem.

Optimum Design Solution

It will illustrate the possibility of reaping previously hidden benefits by such anapproach.

It will explicitly determine values for geometry and material and illusrate the effects of

existing limitations on the optimum design. Desirable design solution

It is illustrate to an improvement in basic geometrical shape or to an improvementin manufacturing technique.

Illustrating the value of the optimum design approach in providing an explicit

method of synthesis and simultaneously result.


10/16

Typical Design Equations

Design equations generally express either functional requirements or undesirable effects interms of parameters which can be classified according to three basic groups. Functional requirement parameters (or undesirable effect parameters) other than the one

explicitly expressed by the design equation. Material parameters

Geometrical parameters

Functional requirement parameters generally are primarily influenced by factors external tothe element, and depending on the particular mechanical structure, these parameters canpossibly be dependent upon each other.

In a typical design equation, the specified functional requirement parameter group is

independent of both the material parameter group and the geometrical parameter group. The material parameters in a design equation are generally not independent of each other.

The characteristic parameters of any specific material can be thought of as independent ofboth the functional requirement parameters and the geometrical parameters in a typicalmechanical element design equation.

It is always possible and desirable to select independent geometrical parameters which

uniquely define the geometry of the mechanical element. Redundancy in defining the geometry of the element generally should be avoided in thedesign procedure. Hence, choosing independent geometrical parameters will allow theindependent variation of each in the design procedure.

Although geometrical parameters which are independent of each other can always beselected, the geometrical parameter group in a typical design equation is often dependenton the specified functional requirement parameter group and the material parameter group.


11/16

Typical Design Equations

Summery of typical skeleton design equation, which figuratively

expresses the quantity B.


12/16

Optimum Design

The optimum design of a mechanical element is the selection of thematerial and the values for the independent geometrical parameters

with the explicit objective of either minimizing an undesirable effector of maximizing a functional requirement, making certain in thedesign procedure that the element satisfied other functionalrequirements and that other undesirable effects are kept withintolerable limits.

It should be conducted on the basis of the most significant quantityto be minimized or to be maximized, and the particular basis ofoptimum design depends on the particular application in its

particular structural or machine setting.

Primary Design Equation The most important design equation is the one which expresses the quantity

upon which the particular optimum design is based, since essentially itcontrols the procedure of design. For any particular case this equation will beknown as the primary design equation.

It will be determined by the most significant functional requirement or by themost significant undesirable effect


13/16

Subsidiary Design Equation In the optimum design of a mechanical element, the design

equations other than the primary design equation will be calledthe subsidiary design equations.

Hence the subsidiary design equations generally express eitherfunctional requirements or significant undesirable effects,whether they are directly specified or indirectly implied.

Limit Equations Limitations on stress in a mechanical element are imposed by

certain material properties called strengths, in conjunction withan appropriate theory of failure.

Limitations on geometry are imposed by certain functionalrequirements of the mechanical structure or machine as by spacerestrictions, by practical manufacturing limitations in conjunctionwith material characteristics and by availability of standard sizes.

Limitations on functional requirements are imposed by factorspeculiar to the entire mechanical structure or machine.

Optimum Design


14/16

Method of Optimum Design: Procedural StepsStep1: Draw a free-hand sketch of the mechanical element showing the significant

basic geometry. Select the independent geometrical parameters which will be usedfor uniquely defining the geometry of the element.

Setp2: Decide on the significant explicit objective for conduct of design, write theprimary design equation which expresses the optimum design quantity. It is interms of functional requirement parameters, material parameters and goemetrical

parameters.

Step3: Write all subsidiary design equations, which express functional requirementsand significant undesirable effects, other than the primary design equation.

Step4: Write all limit equations, for functional requirement parameters, forsignificant undesirable effect parameters, for material parameters and forgeometrical parameters. Indicate any limits which are definitely of the loose type.

Setp5: Combine all subsidiary design equations with the primary design equation byeliminating an unlimited and unspecified common parameter from the primary

design equation for each subsidiary design equation.It is suggested that geometrical parameters be eliminated, since they can be variedindependently of eah other and generally then can be varied continuously; theseare important desired characteristics for eliminated parameters because of step 9,in view of the fact that steps 7 and 8 necessarily will be made first.


15/16

Method of Optimum Design: Procedural Steps

Setp6: Using the developed primary design equation from step 5, roughly determinethe variation of the optimum design quantity with respect to each independent

parameter or independent parameter group in the primary design equation.

Step7: Considering the variations from step 6 together with the previously writtenlimit equations, determine the optimum value for each independent parameter orindependent parameter group in the developed primary design equation; becauseof the explicit objective initially chosen for conduct of design, the optimum valueswill be uniquely determined.

It is important to remember that limit values can be substituted in the procedure of

design only for parameters or parameter groups which are independent of theother terms in the developed primary design equation.

The primary design equation will be expressed in terms of functional requirementparameters, material parameters, limits, and occasionally, specific optimum valuesother than limits; this will be called the final primary design equation.

Setp8: From the final primary design equation, select the optimum material on the

basis of the material parameter group in that equation. This group will be calledthe material selection factor, and its value will determine the optimum availablematerial.

Occasionally it will be impossible to separate the material parameters from thegeometrical parameters in the developed primary design equation.


16/16

Method of Optimum Design: Procedural Steps

Step9: From the final primary design equation at the end of step 7,many of the optimum values fo the geometrical parameters of the

mechanical element will already have been determined.However, in order to determine the optimum values for thegeometrical parameters which were eliminated form the primarydesign equation in step 5, merely solve for the parameter valuesusing the original subsidiary design equations at this time.

Optimization Methods Unit5

Documents

Transcript of Optimization Methods Unit5