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TI/TOL 2D is used to build models of mechanical assemblies which may then be used to make quantitative estimates of the effects of manufacturing variations on assembly performance. Therefore, modeling skills are an essential requirement for tolerance analysis. Without correct modeling, the predicted assembly tolerances will be unreliable, possibly resulting in assembly problems, unexpected increases in the number of rejects, increased costs and reduced quality. This chapter will overview and define the necessary elements for correct modeling.


Assembly tolerance specifications are very different from component tolerance specifications. Component tolerances are applied to each dimension of the individual component parts of an assembly. The variation is monitored and limits are set to control production quality. Individual parts are inspected before assembly and judged good or bad as they meet or exceed the specification limits. For example, the diameter of a shaft may be specified as 0.75 ± 0.010 in. The variation in the diameter depends only on the processes used to produce the shaft.

Assembly tolerances apply to assemblies of parts. The resulting dimension variation is caused by the combined effect of two or more component dimensions. Assembly limits are set to control assembly processes or meet engineering performance requirements. Inspection takes place after assembly. For example, the clearance between a hole and a shaft may be specified as 0.015 ± 0.010 in. The variation in clearance depends on both the shaft variation and the hole variation, that is, C = H - S

Figure 2-1. A simple assembly tolerance specification.

A very important task in assembly tolerance modeling is the specification of design limits on the critical resultant dimensions of the assembly, such as clearances or gaps. Overall assembly dimensions vary as the result of tolerance stack-up of the various component dimensions. These component dimensions are represented by vector chains in the TI/TOL 2D model. Each contributes to the variation of the resultant assembly dimension. The designer must choose suitable upper and lower limits for those assembly dimensions which are critical to the performance of the design.

Tolerance Stack-up

Tolerance accumulation or stack-up may be estimated from one of the expressions in Table 2-1. All four of these approximations are available in the TI/TOL 2D Analyzer. Which one you use depends upon customer requirements, available process data, and desired accuracy.

Table 2-1. Assembly Tolerance Accumulation Formulas

Worst Case
Assures 100% assembly acceptance if all parts are within specification. Costly design model. Requires excessively tight component tolerances
Root Sum Square
Assumes Normal distribution and ±3sigma tolerances. Some fraction of assemblies will not meet specification. May adjust ZASM to obtain desired acceptance fraction. Less costly. Permits looser component tolerances.
Six Sigma Assembly Drift
The same as the Root Sum Square equation with Zp (equal to the number of process standard deviations in each tolerance) replacing Zi.
Six Sigma Component Drift
Most realistic estimates. Accounts for process mean shifts and their long-term affects on assembly distribution.

In the above table, dU is the predicted variation in the resultant assembly dimension, dxi is the variation in a component dimension, partialdiffU/partialdiffxi is the sensitivity that a variation in dxi has on U, TASM is the design limit for variations dU. ZASM and Zi are the number of standard deviations corresponding to the assembly and component tolerance limits. Zp is the number of process standard deviations in the assign component tolerance. Cpki is the process capability index and is a measure of the shift in the process mean.

The TI/TOL 2D Analyzer calculates the sensitivities from the model and predicts the tolerance accumulation of the assembly variables of interest. If limits have been set for an assembly variable, the computed variance of the variable is used to calculate the number of assemblies which will be out of specification. This is shown graphically in Figure 2-2.

Figure 2-2. Determining the number of out-of-specification assemblies.

Types of Assembly Specifications

TI/TOL 2D supports six types of assembly specifications. The four open loop specification types require the user to create two specification endpoints for each specification. The closed loop specifications use kinematic closure lengths or angles from the closed loop(s).



Assembly Clearance or Gap
The variation in the distance between two points on an assembly. Measured when all the parts have been pushed together so the clearance is in one place. Also used for the length of a stack of parts.

The variation in the location of a selected point on an assembly, both horizontal and vertical.
Relative Orientation
The angular variation between two lines, edges, or axes on two different parts of an assembly.
The deviation from parallel between a surface edge or axis of one part and a specified surface edge or axis on another part in an assembly.

Figure 2-3a. Definition of open loop assembly specifications.


Dependent Length
The variation in the location of the point of contact between two mating parts.

Dependent Angle
The variation in the angle between two mating parts in an assembly.

Figure 2-3b. Definition of closed loop assembly specifications.

The corresponding TI/TOL 2D symbols for the design specifications described above are shown in Figure 2-4.

Figure 2-4. TI/TOL 2D Design specification symbols.


The TI/TOL 2D assembly tolerance model consists of vector loops and kinematic joints which are overlaid on a Pro/Engineer assembly drawing. The vectors represent component dimensions whose variation contributes to assembly variations. The kinematic joints describe degrees of freedom in the assembly. Closed vector loops are used by TI/TOL 2D to solve for kinematic variations in the degrees of freedom. Open vector loops are used to solve for assembly variations not associated with joint degrees of freedom, such as gaps or position. The procedure for creating a TI/TOL 2D model is outlined in Figure 2-5.

Figure 2-5. TI/TOL 2D modeling procedure.


Because each part in the assembly model is created separately in Pro/Engineer, the TI/TOL 2D modeler is able to distinguish between the individual parts. A datum reference frame (DRF) is defined on each part. A DRF is a local coordinate system used to locate all other datums, features and joints on the part.

Two types of DRFs are available for 2-D problems:

  1. cylindrical
  2. rectangular

The TI/TOL 2D symbols for both cylindrical and rectangular DRFs are shown in Figure 2-6.

Figure 2-6. Cylindrical and Rectangular DRF Symbols.

A DRF consists of one axis if the DRF is cylindrical or two axes if the DRF is rectangular. The axes define the origin of the local coordinate system of the part, which is used to locate features on that part. Figure 2-7 illustrates each part in a sample assembly with its associated part name and DRF.

Figure 2-7. Assembly with Parts and DRFs defined.

The DRF axes should correspond to the primary, secondary, and tertiary datum planes for a part. During production all part features are ultimately referenced to these specified datum planes used to fixture the part. Selection of the location for the DRF origin determines to a great extent which part dimensions will contribute to assembly variations, and how robust a design is to manufacturing variation.

Feature datums can also be defined for different parts of an assembly. Feature datums are used as secondary DRFs to locate features of a part and contact points between mating parts. Like DRFs feature datums can be either rectangular or cylindrical. The TI/TOL 2D symbols for cylindrical and rectangular feature datums are shown in Figure 2-8.

Figure 2-8. Cylindrical and rectangular feature datum symbols.

TI/TOL 2D requires the user to create an assembly datum point at each DRF and feature datum location prior to creating the tolerance modeling symbols. Each assembly datum point must be created independently of all other assembly datum points.


The points of contact between mating assembly parts are called joints. A joint may involve point contact, line contact, or surface contact between two parts. A joint defines a kinematic pair which constrains the relative motion between two mating parts. For example, a block on a plane is constrained to slide parallel to the plane.

In 2-D a wide variety of assembly conditions may be modeled with just a few joint types. The joint type depends on the kind of contact between parts and the degrees of freedom which it allows. An unconstrained rigid body has six degrees of freedom (DOF): three translational and three rotational. In 2-D each part has only three degrees of freedom: translation in the X-direction, translation in the Y-direction, and rotation about the Z direction. Kinematic joints constrain motion so they reduce the degrees of freedom (DOF). Commonly occurring joint types are shown with their associated degrees of freedom in Figure 2-9. The corresponding TI/TOL 2D symbol is in the upper right-hand corner of each picture.

Figure 2-9. Joint types with associated degrees of freedom.

The information required to define a joint includes the joint type, global location, orientation of the joint axes, names of the two parts in contact with each other, and the joint's location relative to each mating part's DRF.

Joints must be defined at all part interfaces, or contact points and correctly oriented to accurately represent the interaction between parts and their associated degrees of freedom. All joints which allow relative part motion about an axis or along a plane require a feature surface or assembly datum plane to specify the orientation of the joint's sliding plane. Specific requirements for each joint type are summarized in the next section (Loops). Figure 2-10 identifies the sliding plane for a cylindrical slider joint.

Figure 2-10. Sliding contact due to assembly variations.

As an example of a kinematic variation, suppose the height of the right hand step that supports the block in Figure 2-10 was produced at the high end of its tolerance. The block would tip up, causing the cylinder to slide up the wall slightly. This sliding variation along the wall may be estimated statistically in terms of all the dimensional variations of the component dimensions.

Each joint must also be located relative to two DRFs, that is, relative to both parts connected by the joint. The joint is located relative to a DRF by a chain of vectors called a datum path. Each vector in a datum path must be either a controlled dimension, for which the designer may specify a tolerance, or a kinematic assembly vector, which adjusts at assembly time. There can be at most one kinematic assembly vector included in a single joint's set of datum paths. In other words, if one datum path for a joint contains an assembly vector, the other path can't have any assembly vectors in it.

Figure 2-11 illustrates the datum paths for the joint between the Block and the Cylinder. Path 1 points to the Cylinder DRF, while Path 2 leads to the Block DRF. Note that the path to the Cylinder DRF is a single vector, corresponding to the cylinder radius, which is a designer-specified dimension. In contrast, Path 2 consists of two vectors, the first lying on the sliding plane and the second corresponding to the block thickness. The first vector is a kinematic assembly variable which locates the point of contact. It is not a machined dimension, but depends on the size and position of the other parts at assembly time.

Intermediate points along the datum path are feature datums. In this case the feature datum is a reference point on the sliding plane from which the point of contact is located. It is also the point at which the vector path changes direction. Each vector in a datum path must end at a feature datum, except the last vector, which must end at a DRF.

Figure 2-11. DRF paths for a cylindrical slider joint.

TI/TOL 2D requires the user to create an assembly datum point at each joint location prior to creating the tolerance modeling symbols.

Degrees of Freedom

Each modeling element in TI TOL has specific kinematic degrees of freedom associated with it. The total number of degrees of freedom in a model can be no more than three times the number of closed loops.

In a kinematic assembly, joints are used to constrain certain degrees of freedom within the assembly. When performing tolerance analysis, a different paradigm is helpful. A joint constrains certain degrees of freedom, but still allows others to remain. Variations due to small kinematic adjustments are associated with these remaining degrees of freedom (DOF). Therefore modeling elements are discussed in terms of the degrees of freedom they introduce.

Datum Reference Frames and Feature Datums

Rectangular: Rectangular DRFs and feature datums introduce no degrees of freedom into the assembly.

Cylindrical: Cylindrical DRFs and feature datums introduce a rotational degree of freedom into the tolerance model. They are usually used in conjunction with cylindrical slider joints and parallel cylinders joints. Both of these joints allow a kinematic rotation, but the center of rotation is not at the point of contact between the two parts. Instead, the rotation occurs at a cylindrical datum placed at the joint's center of curvature.

Occasionally, however, a cylindrical datum introduces a redundant or even a non-existent rotation into the model. For example, if a revolute joint is located at the same point as a cylindrical DRF and is connected to that DRF, then there is a redundant rotational degree of freedom at that location. One of them must be turned off (see the clutch tutorial, chapter 4). In other cases, there may be vectors which pass in and out of a cylindrical datum with no rotational variation occurring between them. In this case as well, the rotation must be turned off. A good test for this condition is to examine the vectors that join at the cylindrical datum. If a kinematic adjustment within the assembly will cause the angle between the vectors to change, the rotational degree of freedom is valid. If not, consider turning it off.

Kinematic Joints

Rigid: A rigid joint introduces no degrees of freedom into the model. It is used to join parts the user wants to model as having no kinematic adjustment between them.

Planar: A planar joint introduces a single translational degree of freedom into the model. It is used to model kinematic adjustments that involve only translation, such as between two flat plates. However, it can be used to model any situation that requires a translational degree of freedom.

When creating a planar joint, the user must define a "sliding plane." The user must then define a path back to the part DRFs in such a way that a vector is created parallel to the sliding plane. If the DRF is not on the sliding plane, this is accomplished by defining a path which passes through a feature datum that lies on the sliding plane. If TI TOL cannot find a vector in the sliding plane orientation, it will automatically turn off the translational degree of freedom.

Cylindrical Slider: A cylindrical slider joint introduces a translational degree of freedom into the model. It is always linked to a cylindrical DRF or feature datum, so there is a rotational degree of freedom associated with this joint type (though not directly at the joint). It is used to model the contact between a rounded surface and a flat surface. As with a planar joint, the user is prompted for a sliding plane. If there is no vector on the path back to the DRF that lies in the sliding plane, TI TOL automatically turns its translation off.

Edge Slider: A edge slider joint introduces a translational and a rotational degree of freedom into the model. It also prompts the user for a sliding plane. If no vector on the paths back to the DRFs lies in that plane, the translational degree of freedom is automatically turned off. This joint type is used to model the contact between an edge and a flat surface.

Parallel Cylinders: A parallel cylinders joint introduces no degrees of freedom into the model. However, it is always linked to two cylindrical DRFs or feature datums, so there may be up to two rotational degrees of freedom associated with this joint type (though not at the joint). This joint type is used to model the contact between two curved surfaces that have parallel center lines. Occasionally, one (or even both) of the cylindrical datums to which this joint is linked introduces a redundant or even non-existent degree of freedom into the model (refer to Cylindrical datums, above). The user needs to determine whether or not to leave the rotational degrees of freedom active.

Revolute: A revolute joint introduces a rotational degree of freedom into the model. It is used to model the contact between parts joined by a pin, a shaft, or any other device that allows only rotation between the parts. If it is located at the same point as a cylindrical DRF or feature datum and is linked to that same datum, there will probably be a redundant rotational degree of freedom at that location. Either the revolute joint's or the cylindrical datum's degree of freedom must be turned off.


The goal of the tolerance modeling process is to obtain a set of relationships representing the assembly in terms of its geometry and kinematics. These relationships are produced by developing a set of vector loops connecting contact joints. The loops may be either open or closed.

Closed loops start and end at the same location and represent kinematic constraints on the assembly. One kinematic constraint states that all parts in the assembly must maintain contact in order for the tolerance model to be valid.

Open loops are used to determine assembly resultants of interest such as a clearance, orientation or position. For example, a fan blade must have a certain clearance in an assembly to operate.

A valid set of loops, whether open or closed, is not unique since each loop may follow a variety of possible paths. Figure 2-12 shows a simple assembly requiring three closed vector loops to describe the adjustable kinematic variables.

Figure 2-12. Three loops are required to describe the variations in this simple assembly.

As a vector loop threads its way through an assembly, it passes from mating-part to mating-part, always passing through the joints that connect the parts. Each joint has an incoming vector and outgoing vector. Similarly, in crossing a part, it enters through one joint (the "incoming joint") and leaves through another joint (the "outgoing joint"). The paths across the parts follow the associated datum paths created with each joint. The two datum paths are connected in a continuous chain, starting at the incoming joint, following its datum path to the DRF, then passing through the DRF, following the second datum path and terminating at the outgoing joint. Figure 2-13 illustrates a typical path across a part.

Figure 2-13. Datum paths define the path across a part.

Rules for Creating a Valid Set of Closed Vector Loops

  1. The set of loops must pass through every part and every joint in the assembly.
  2. No single loop may pass through a part or joint more than once, but it must start and end at the same point.
  3. There must be enough closed loops to solve for all of the kinematic variables - one loop for every three variables.
  4. A vector loop must pass from one part to the next part through a common joint.
  5. The path across a part must pass through the datum reference frame (DRF), following the datum path from the incoming joint to the DRF and then following the datum path from the outgoing joint to the DRF, only in reverse.
  6. If the path across the part doubles back on itself with an equal and opposite vector, the two vectors cancel each other and must be equivalenced in the modeler.

Some joints place requirements on the vector loops passing through them, as shown in Figure 2-14. These requirements assure that vectors and angles corresponding to the kinematic variables will be present in the assembly model so their variations can be calculated.

Figure 2-14. Required vector paths through joints.

Vector Loop / Joint Requirements

  1. For a cylindrical slider joint, either the incoming or outgoing vector must be normal to the sliding plane and end at the center of the cylinder.
  2. For joints having a sliding plane (planar, cylindrical slider or edge slider), either the incoming or outgoing vector must lie in the sliding plane.
  3. For cylindrical joints (parallel cylinders in contact), the path through the joint must start at the center of one cylinder and end at the center of the mating cylinder, passing through the contact point.

Loops may be defined manually or automatically. Manual definition involves identifying a sequential list of joints for each loop, then specifying tolerances on all independent dimensions in the loop. In contrast to manual definition, automatic definition requires only the specification of tolerances since the loop paths are generated automatically. It is also possible to define one or more loops manually, then generate the rest automatically.

Once a complete loop is formed from joints and datum paths, tolerances are specified on those vector lengths which represent manufactured dimensions. That is, tolerances must be specified on those vectors which correspond to manufactured feature dimensions, such as the length of a side or the radius of a corner. Dimensions or angles which correspond to kinematic variables should not be assigned tolerances. Their variations are determined by kinematic analysis.


Besides size variations, parts may exhibit variations in shape or form, called geometric or feature variations. For example, a machined cylinder will not be perfectly circular. It may be slightly elliptical or lobed, but it must be sufficiently circular to properly perform its function. Feature variations also include variations in location and orientation. Feature controls are the mechanism used to constrain feature variations to fall within acceptable limits.

The ANSI Y14.5 specification was established to standardize control of form and feature variations. Most geometric tolerances are based on an imaginary set of two surfaces, such as parallel planes or concentric cylinders, which define the region of acceptable surface variation. Individual geometric tolerances are shown with their associated symbols in Figure 2-15.

Figure 2-15. TI/TOL 2D Geometric Tolerance Symbols

To model feature variations, the part and joints to which the geometric tolerance applies are identified, and the tolerance zone width or diameter is specified. Since a joint generally involves two mating surfaces, there are usually two form tolerances applied at each joint, one for each surface variation. If more than one joint is in contact with the same feature, such as two joints on a plane, the same form tolerance should be applied to all joints in contact with that feature.

Sometimes there may be more than one geometric tolerance specified for a single surface. That does not necessarily mean that there are two independent sources of variation produced by that surface. The two specifications are primarily given for inspection purposes to meet different functional requirements. For example, a surface may have a 0.010 parallelism tolerance specified along with a 0.005 flatness tolerance. The parallelism may be necessary for alignment, while the flatness may be required for a sliding fit. The produced surface variation will simply be measured by two methods to determine the two resulting effects.

Approximating Form Variations

Form variations only introduce variation into an assembly at the points of contact between mating surfaces. Figure 2-16 shows a cylinder in contact with a plane. In one assembly, the cylinder might rest on a peak of the surface. In the next assembly, the cylinder could be down in a valley. Thus, the surface waviness in the plane results in a translational variation normal to the surface.

Similarly, the cylinder is probably not perfectly round, but exhibits waviness or lobing. If, during assembly, it is placed with a lobe at the contact point, the cylinder center will be higher than average. If a low point on the surface of the cylinder is in contact, the center will be lower. Thus, we see that both surfaces produce independent translational variations which are normal to the surface at the point of contact (see Figure 2-16 below).

Figure 2-16. Translational and rotational variations caused by form variation.

In contrast, a block on a plane produces rotational variation, as shown in Figure 2-16, since one corner of the block may be higher than the other. The magnitude of the variation depends on the length of the block and the amplitude and period of the waviness.

It is interesting to note that the same flatness specification on the planar surface in Figure 2-16 produces distinctly different variations depending on the type of joint or surface contact. Point or line contact, such as occurs in cylindrical slider, parallel cylinder or edge joints, results in translational variation which propagates into an assembly in a direction normal to the surface. Surface or area contact, such as occurs in planar joints, produces rotational variation about an axis normal to the page. As far as assembly tolerance analysis is concerned, it does not matter whether the form tolerance specified on the plane is parallelism, flatness, perpendicularity or angularity. Since each one is represented by a tolerance zone formed by two parallel planes bounding the surface, the effect on the assembly will be the same--a point contact will produce translational variation, while surface contact produces rotational variation .

For point contacts, TI/TOL 2D inserts a vector into the vector loop, normal to the surfaces, having zero nominal length ±dalpha/2, where dalpha is the width of the specified tolerance zone. For surface contacts, TI/TOL 2D inserts a relative rotation ±dØ. The rotational variation ±dØ may be estimated from the relation:

dØ = tan -1 f(dalpha,L)

where, dalpha is the specified ANSI Y14.5 flatness tolerance and L is the length of contact between the block and plane. The value ±dalpha/2 or ±dØ is taken to be the ±3sigma limits of a normal distribution, so the probability of such an extreme value occurring is low.

Rules for Form Tolerances

1. There are generally not more than two feature variations produced at each joint.

2. The type of variation depends on the joint surfaces:

Translational variation,
Cylindrical slider
Parallel cylinders
Edge slider
Rotational variation,
Pin joint

3. For more than one joint on the same surface, be sure to apply the same feature tolerance to each joint that touches the surface.

Approximating Position Variations

Position tolerance is used to specify the limits of location of a feature, such as a hole, in a 2-D plane. Two methods are commonly used in practice:

  1. Rectangular tolerancing
  2. True position tolerancing.

In rectangular tolerancing, the x and y coordinates of a feature are specified relative to a set of datum axes. The tolerances ±dx and ±dy are also added following established dimensioning conventions. The resulting tolerance zone is rectangular, as shown in Figure 2-17 Many object to rectangular tolerancing because the permissible position error varies with direction. It is equal to dx in the x direction, dy in the y direction, and R(dx2 + dy2) in the direction of the diagonal of the rectangular tolerance zone.

True position tolerancing, as defined in the ANSI Y14.5 standard, establishes a non-directional tolerance represented by a circular tolerance zone, also shown in Figure 2-17. This is a more realistic representation for many locating processes. No tolerances are applied to the x and y dimensions in this case. A box is drawn around both dimensions indicating that they are basic dimensions. The position tolerance is placed in a feature control block directly under the hole size specification, as shown, along with the true position symbol. The value of the tolerance specifies the diameter of the tolerance zone.

Fig. 2-17. Comparison of rectangular position and true position tolerancing.

Position variations can accumulate and propagate in an assembly the same as dimensional variations. However, the non-directional nature of position variations is difficult to model in a vector model, since vectors are uni-directional. The effect may be approximated as follows:

where dU is the resultant assembly variation, partials and are the sensitivities to x and y position variations, and dalpha is half the diameter of the position tolerance zone. The sensitivities are calculated automatically by the TI/TOL 2D analysis software.

Approximating Profile Variations

Variations in the profile of a surface introduce variation into an assembly at the point of contact on the surface. Thus, the variation acts normal to the surface with a magnitude equal to ±1/2 the specified profile tolerance, as shown in Figure 2-18. It is represented by adding a vector normal to the surface, having zero nominal length plus or minus 1/2 the tolerance zone. TI/TOL 2D adds this vector at each joint location where a profile tolerance is specified by the designer.

Figure 2-18. Profile variation acts normal to the surface.

Rules for Position and Profile Tolerances

  1. When you have applied position tolerance to a feature, you should not apply size tolerances to the corresponding basic dimensions. The same is true of applied profile and angularity tolerances.
  2. Position variation is non-directional with magnitude equal to half the diameter of the specified position tolerance zone.
  3. Profile variation acts normal to the surface at the point of contact. The magnitude is ±1/2 the width of the tolerance zone.

Figure 2-19 shows an assembly with geometric tolerances added at the mating surfaces.

Figure 2-19. Geometric tolerances added at mating surfaces of an assembly.

All of the possible combinations of joints and geometric tolerances are summarized below. The type(s) of variation introduced by each combination is also indicated.

Table 2-2: Summary of variations caused by geometric tolerances at kinematic joints.

  Rigid Planar Edge
Flatness R R Tn Tn    
Straightness R R Tn Tn Tn  
Circularity Tn   Tn Tn Tn  
Cylindricity Tn   Tn Tn Tn  
Perpendicularity R R Tn Tn    
Angularity R R Tn Tn    
Parallelism R R Tn Tn    
Profile R R Tn Tn Tn  
Runout Tn, R Tn, R Tn Tn Tn  
Concentricity Tx, Ty Tn, R Tn     Tx, Ty
True Position Tx, Ty   Tx, Ty     Tx, Ty

R = Rotational degree of freedom
Tn = Translation normal to the contact surface
Tx = Independent translation in the X-direction
Ty = Independent translation in the Y-direction
blank box = geometric tolerance not allowed at the joint type.


The following set of guidelines will help in setting up most tolerance analysis models. Completing these steps on paper before attempting to create a tolerance model in the computer (at least for the first few assemblies modeled) will greatly reduce the amount of time spent re-working and fixing models.

1. Determine the critical assembly variable(s). This will require the person creating the model to choose an appropriate 2-D reference plane through the assembly. All modeling elements will be located in this plane.

2. Determine what parts in the assembly will be included in the model. Many assemblies include parts that do not influence the critical assembly variation. Eliminating them will help create a simpler, more comprehensible model. For each part remaining in the assembly, locate the datum reference frame (DRF) location, and determine if the DRF is rectangular or cylindrical. In general, parts manufactured by a turning process have cylindrical DRFs.

3. Determine the type and location of all kinematic joints in the assembly. Each joint can be used to link only two parts. Define vector paths from each joint back to the DRFs of the parts it joins. If the joint has a translational degree of freedom, be sure one path contains a dependent (non-dimensioned) vector in the sliding plane. The remaining vectors must be dimensioned lengths (manufactured dimensions). Use feature datums to define the intermediate dimensions if the paths back to the DRF include multiple vectors. Remember to connect cylindrical slider and parallel cylinder joints to cylindrical DRFs or feature datums.

4. If the critical assembly variable(s) is an angle or length defined by the joint paths, it is referred to as a "dependent" length or angle, and a "closed loop specification" is all that needs to be applied to the model. If the critical assembly variable is not defined by the joint paths, an "open loop specification" will probably be required to analyze it. If so, determine the point or plane whose variation is of interest (the "moving" point), and the point or plane you wish to measure the variation relative to (the "fixed" point). Create a specification endpoint at the "fixed point" first and the "moving point" second in order to form an open loop between those two locations. The order in which these specification endpoints is created is very important when creating "Gap" or "Position" specifications.

5. Create a network diagram to determine the number of closed vector loops that must be generated to analyze this assembly. Forming a network diagram provides a good way to check the validity of the loops created by the "Autoloop" option.

To build a network diagram, draw a circle for each part in the assembly and label it. For each contact joint between two parts, draw a line connecting their respective circles. If, for example, part_1 and part_2 join in two locations, there will be two lines connecting their respective circles. Continue this process until each joint in the assembly is represented by a line between two parts. Do not duplicate joints. If joint lines cross, re-arrange the parts so they don't. The number of closed loops is equal to the number of "holes," i.e. regions bounded by the lines (joints) in the network diagram. The number of variables that can be solved for in the assembly is equal to three times the number of loops.

6. Add up the number of degrees of freedom (DOF) present in the model. Include one degree of freedom for each cylindrical DRF, cylindrical feature datum, planar joint, cylindrical slider joint, and revolute joint. Include two degrees of freedom for each edge slider joint. The degrees of freedom present should not exceed the number of variables that can be solved for (found in step 5). If the number of degrees of freedom present in the model exceeds the number of variables that can be solved for, check the assembly for redundant and non-existent DOFs and turn them off. If there are still too many DOFs, declare an appropriate rotational DOF to be the input angle, and turn off its degree of freedom. If there are still too many DOFs, it's likely a mistake has been made at an earlier stage in the modeling process.

7. Determine which manufactured length dimensions are equivalent. Two radii of a circular part are likely to be equivalent (i.e. if one is over-sized, the other is over-sized). Two or more vectors that have common endpoint locations but are defined by different modeling elements are probably equivalent. For example, if one joint path to DRF_1 includes a vector from feature datum 4 to DRF_1, and another joint path to DRF_1 includes a vector from feature datum 5 to DRF_1, and feature datums 4 and 5 are located at the same point, vectors 4_1 and 5_1 are redundant. Redundant vectors must be equivalenced in order to have valid results.

8. Apply tolerances to all manufactured dimensions. If the effects of surface variations of one or more parts need to be included in the model, apply appropriate geometric tolerances at the joints of those parts.

9. If desired, rename the part DRFs, joints, and/or vectors in the model.

At this point, the assembly datum planes and points required to build the model should be well defined. In addition, the general shape of the loops that need to be created, as well as any modifications the loops require (such as equivalencing vectors) will be apparent. Remember that the 2-D reference plane used to create the vector model must be an assembly datum plane. There must be an assembly datum point defined on the reference plane at each modeling element (DRF, feature datum, joint, and specification endpoint) location in order to create those modeling elements. Each assembly datum point must be created independently (i.e. they cannot be created as a "group").

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