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Pre-CATS is used to build a model of a mechanical assembly that may then be used by CATS.BYU to make quantitative estimates of the effects of manufacturing variations on assembly performance. Modeling skills then, 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 section 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., as shown in Fig. 2-1. 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 which are represented by vector chains in the Pre-CATS model. Each contributes its share to the overall variation of the resultant assembly dimension. The designer must assign upper and lower limits to 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 CATS.BYU. Which one you use depends upon customer requirements, process data available 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 +/-3[[sigma]] tolerances. Some fraction of assemblies will not meet specification. Less costly. Permits looser component tolerances.
General Root
Sum Square
More versatile. May adjust ZASM to obtain desired quality.
Six Sigma
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, [[partialdiff]]U/[[partialdiff]]xi 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. Cpki is the process capability index and is a measure of the shift in the process mean.

CATS.BYU calculates the sensitivities from the tolerance model and predicts the tolerance accumulation of the assembly variables of interest. If specified limits have been set for an assembly variable, the computed distribution 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

There are more types of assembly specifications than just clearances or gaps. Those currently supported by Pre-CATS and CATS.BYU are assembly clearance or gap, and position. These specifications are shown graphically in Figure 2-3.


Assembly Clearance or Gap
The variation in distance between two points on an assembly, as measured when all the parts have been pushed together so all the clearance is in one place. Also applicable to the overall length of a stack of parts.

The variation in the location of a selected point on an assembly, both horizontal and vertical.

Global 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.


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

Variable Angle
The variation in angle between two mating parts in an assembly.
Figure 2-3. Definition of assembly specifications.


The Pre-CATS assembly tolerance model consists of vector loops and kinematic joints which are overlaid on a CAD assembly drawing. The vectors represent component dimensions whose variation contributes to assembly variations. The kinematic joints describe degrees of freedom or adjustable dimensions in the assembly, whose variation is determined by accumulated manufacturing variations in the assembly. Closed vector loops are used to solve for the kinematic variations. Open vector loops are used to solve for the variation in assembly specifications, such as gaps or position. The procedure for creating an assembly tolerance model begins with identifying the component parts, defining kinematic joints, and then creating vector loops connecting contacting joints. Feature control tolerances are then added, followed by design specifications. These modeling steps are discussed in the sections below.

Note that in the following subsections, 2-D figures are used. The discussion reflects 3-D concepts, but refers to the 2-D figures for clarity.


The Pre-CATS modeler must be able to distinguish between individual parts in the assembly model. Thus, each part is assigned a unique part name. In addition, a datum reference frame (DRF) is defined on each part. A DRF is a local coordinate system used to specify the location of all other datums, features and joints on the part.

Three types of DRF's are available for 3-D problems:

  1. cylindrical
  2. rectangular
  3. spherical

A DRF consists of a location point and at least two axes. The axes define the origin of the local coordinate system attached to the part, which are used to locate features on that part. Figure 2-4 illustrates each part in a sample assembly with its associated part name and datum reference frame (DRF).

Figure 2-4. Assembly with Parts and DRF's 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. Moving the DRF to a different location may result in a design that is more robust to manufacturing variation. Thus, the selection of the DRF axes may determine how the part needs to be fixtured during production to assure an assembly that is robust to manufacturing variation.


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 relationship 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 3-D, a wide variety of assembly conditions may be modeled with just a few joint types. The joint type determines 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. Kinematic joints constrain motion so they reduce the degrees of freedom (DOF). Commonly occurring joint types are shown in Figure 2-5.

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. A graphical dictionary of joints is included in Appendix B, showing the required joint axes with their associated degrees of freedom.

Figure 2-5. 3-D joint types.

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. For example, defining the location of a revolute joint or pin joint between two parts is not sufficient; the rotation axis must also be specified. Similarly, all joints which allow relative part motion about an axis or plane require information specifying the direction of the axis or plane.

Figure 2-6 identifies the sliding plane for a cylindrical slider joint, that is, the direction in which the cylinder and the ground may adjust kinematically relative to each other due to assembly variations. The joint origin and a second point identified by an X are used to define a vector describing this direction.

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

Figure 2-6. Sliding contact due to assembly variations

Each joint must also be located relative to two DRF's, 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 dimension, which adjusts at assembly time.

Figure 2-7 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 called feature datums. They are indicated by a small square symbol, such as the one located at the corner of the block in Fig. 2-7. 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 2-7. DRF paths for a cylindrical slider joint


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. For example, a kinematic constraint may state 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. Fig. 2-8 shows a simple assembly requiring three vector loops to describe the adjustable kinematic variables.

Figure 2-8. Three loops are required to describe the variations in this simple assembly.

Rules for Creating a Valid Set of 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 may start and end at the same point.
  3. There must be enough loops to solve for all of the kinematic variables - one loop for every six variables in 3-D, or every three variables in 2-D.

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. Fig. 2-9 illustrates a typical path across a part.

Figure 2-9. Datum paths define the path across a part.

Rules for Vector Paths

  1. A vector loop must pass from one part to the next mating part through a common joint.
  2. 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.
  3. If the path across the part doubles back on itself with an equal and opposite vector, the two vectors cancel each other and will be omitted from the loop.

Some joints place requirements on the vector loops passing through them. 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. The required vector paths through the joints are illustrated in the joint dictionary in Appendix B.

Vector Loop / Joint Requirements

  1. For a cylindrical slider or spherical slider joint, either the incoming or outgoing vector must be normal to the sliding plane.
  2. For joints having a sliding plane (planar, prismatic, cylindrical slider, edge slider, or spherical slider), either the incoming or outgoing vector must lie in the sliding plane.
  3. For cylindrical contact joints (parallel cylinders or crossed cylinders), 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.

Once a complete loop is formed from joints and datum paths, tolerances are specified on those vector lengths which are independent of assembly adjustments. 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. Tolerances may also be optionally applied to any angles between the vectors in an assembly which are independent (the angle between two adjacent vectors on the same part). Dimensions or angles which correspond to kinematic variables adjust during assembly. Their variations are determined by kinematic analysis.

Example 1 - Locking Computer Tape Hub Assembly

An assembly model is shown in Figure 2-11. The figure shows a radial cross section of a locking computer tape hub assembly. The vertical center line indicates the axis of rotation. In this example, the plunger slides vertically downward while the wedge on its outer diameter pushes the arm outward until it locks against the inside diameter of a computer tape reel. We wish to analyze the assembly in the extreme position shown, with the plunger resting on the base. In this position we would like to assure an interference fit between the rubber pad and the reel (shown here as a clearance for clarity).

There are three datum reference frames (DRF) located on the axis for the three axially symmetric parts: the Plunger, the Reel, and the Base. Joint 1, a rigid joint, joins the Plunger to the Base. It has been placed along the line of motion of the Plunger, on top of its DRF. In the position shown, resting on the Base, it no longer has a degree of freedom. Joint 2, a planar joint, permits horizontal sliding contact between the Arm and the Base. The joint lies on top of the DRF for the Arm. Joint 3 is a cylindrical slider placed at the point of contact between the Arm and the Plunger.

Figure 2-11. Locking computer tape hub assembly.

Joints and feature datums.

Two loops describe the variation in this assembly, one closed and one open, as shown in Fig. 2-12. The open loop is set up to compute the gap between the Arm and Reel. The Gap is the resultant of a one-dimensional loop. It is the difference between RT, the radius of the tape reel, and RL, the resultant radius of the tape hub. The nominal value of RL, along with dependent variables u, [[Phi]] and the Gap, may be determined by an accurate CAD layout. However, the variations in the Gap, RL, u and [[Phi]] are wanted as well as the nominal values. The resultant variation may be computed by CATS in terms of the independent variables and their sensitivities using a WC, RSS or Six Sigma analysis.

Figure 2-12. Locking computer tape hub assembly.

Completed loops.

Example 2 - Remote Positioner

Figure 2-13 shows a remote positioning mechanism consisting of five parts joined by pin joints, forming two parallelograms. Its function is to control the position and orientation of the point P, suspended inside a test chamber. As Part 1 is rotated about its pivot point at the origin of the X-Y axes, Part 5 performs a parallel rotation about point P. However, if there are variations in the lengths of any of the parts, errors in the position and orientation of point P will result. The magnitude of these errors may be estimated by statistical analysis of a vector kinematic model.

This example illustrates two additional assembly specifications. Design limits may be placed on the variations in position and orientation of P. Comparing the computed values with the specified limits permits us to predict the accuracy of the mechanism and determine which component tolerances would be most effective to tighten.

Figure 2-13. A remote positioning device, showing the possible errors in position and orientation.

Figure 2-14. Closed loops for determining the kinematic variables.

Figure 2-15. Open loop for determining the variations at point P.

One joint on each part is selected to be the datum reference location (DRF) for that part. Other joints and features are located relative to the reference joint on each part.

The vector model of the assembly consists of two closed loops, shown in Fig. 2-14, and a single open loop, shown in Fig. 2-15. The closed loops may be analyzed to determine the variation in the kinematic variables. The open loop may be analyzed to determine the variations in the position and orientation at point P.

There are six rotational kinematic variables corresponding to the rotational degrees of freedom at pin joints 2 through 7. Joint 1 is a kinematic input and is used as the reference angle, therefore it is assumed to be known. The two closed loops produce six scalar equations which relate the six angles to the various link lengths. When tolerances are specified on the link lengths, the equations may be solved for the resulting six angular variations.

The preceding examples demonstrate the application of four joint types: planar, rigid, cylindrical slider and revolute. Subsequent examples will employ the remaining joint types: edge slider and parallel cylinder.

The examples also demonstrated the application of three assembly specifications: gap, position and orientation. All three are open loop specifications. In the chapters which follow, closed loop specifications for variable length and angle will be applied.


Besides size variations, parts may exhibit variations in shape or form, called feature variations. For example, a machined cylinder will not be perfectly circular. It may be slightly elliptical or lobed, but it must be circular enough 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 feature controls 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 feature control types are shown with their associated symbols in Figure 2-16.

Figure 2-16. ANSI Y14.5 Feature Control Symbols

To model feature variations, the type, part and joints to which the feature control applies are identified, and the tolerance zone width 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 ANSI feature 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 2-17 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 lobbing. 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.

Figure 2-17 Translational variations caused by form variations.

Figure 2-18 Rotational variations caused by form variations.

In contrast, a block on a plane produces rotational variation, as shown in Fig. 2-18, 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 Figures 2-17 and 2-18 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, CATS inserts a vector into the vector loop, normal to the surfaces, having zero nominal length +/-d[[alpha]]/2, where d[[alpha]] is the width of the specified tolerance zone. For surface contacts, CATS inserts a relative rotation +/-dØ. The rotational variation +/-dØ may be estimated from the relation

dØ = tan -1 f(d[[alpha]],L)

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

3-D Form Variations

In 3-D, the effect of form variations is more complex. As illustrated in Figure 2-19, variation in flatness simultaneously transmits both a translational and rotational variation through a cylindrical slider joint. The translational variation is seen by looking down the cylinder axis. The rotational variation occurs about an axis in the plane which is normal to the cylinder. Similarly, each 3-D joint has a unique response to variations in the contacting surfaces.

Figure 2-19 Affect of form variation on a 3-D joint.


Once the tolerance modeling has been completed, Pre-CATS generates a tolerance model file in CATS/PC Neutral File format, containing a representation of the essential geometry and kinematic relations for the entire assembly. The model file can then be used by the CATS/PC analysis software to generate a set of algebraic equations to determine the values of the dependent variables. The dependent variables include the variation in the kinematic variables, or the small adjustments that occur at assembly time, and the variation in other accumulative assembly dimensions, such as gaps, position or orientation.

When the analysis is complete, the results are recorded in the model file, which is passed back to the modeler. The dependent tolerances, which were unknown when the assembly model was created, are displayed by the modeler for the engineer to review and evaluate. The model file may be viewed by entering into a text editor and calling the file.


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