28.2 Mechanical Design to Minimize Vibration

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In the development of high-speed and high-precision motion systems, the notion of determinism is

a key consideration (Evan, 1989), which implies that a physical system complies with the law of

cause and effect, and this behavior allows the physical system to be modeled mathematically.

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The governing equations describing the model can then be used to predict the behavior of the

system and thus allow for the compensation for possible errors to meet the demand of a tight error

budget. A mechatronic approach, in which the structural design and the control design are to be

seamlessly integrated (Rankers, 1997), is one of the possible approaches for machine design. This

approach has been adopted efficiently by many scientists and engineers, and the benefits are clearly

evident in the end products, such as the wafer scanner and stepper.

In this section, the key issues to address in a sound mechanical design to keep mechanical vibration to

a minimum will be highlighted. The issue of mechanical design represents a very large area in precision

motion systems. In this section, only key points will be highlighted in general, to enable designers to

design “rigid” structures during the initial phase, even before the physical modeling stage. Design, being

an iterative process, always requires the designer to revisit the drawing board frequently, until an

optimum design is achieved. The section will give qualitative ideas with abundant figures to illustrate key

ideas, rather than using a purely quantitative approach. The reason for this is that, during the initial

phase of a design, intensive quantification is normally not necessary for decision making. Iteration and

optimization, which are normally mathematically intensive, should be addressed during the next stage of

the design process.

28.2.1 Stability and Static Determinacy of Machine Structures

A structure can be a supporting framework that houses all the subassemblies that make up a machine, or

it can be a collection of many smaller structures, or even a single component. The reaction forces of the

high-speed moving parts will excite the structural dynamics resulting in mechanical vibrations. These

vibrations can be attenuated by reducing either the excitation or the response of the structure to that

excitation (Beards, 1983). The first factor can be overcome by relocating the source within the structure

or by isolating it from the structure so that the generated vibration is not transmitted to the structure via

the supports. As for the second factor, changing the mass, the stiffness, or the damping can alter the

structural response. In order to understand the dynamic responses of the structure, the real structure can

be transformed into a physical model, which is usually a simplified model representative of the real

structure. For example, a real machine can be modeled as a number of coupled spring – mass systems. A

derived physical model of the real system can be translated to a mathematical model which can be solved

via software or by hand, thereby allowing engineers from different disciplines to communicate and refine

their portion of the design.

Every designer working on the structure for a machine needs to answer a very important question. Is

the designed structure rigid and stable? A structure is rigid if its shape cannot be changed without

deforming the members in the structure (Fleming, 1997), and a structure is stable if rigid-body

translation or rotation cannot occur. A good way to tell whether a structure is stable or not is the degree

of indeterminacy. A structure is considered statically determinate if all the support reactions and internal

forces in the members can be determined solely by the equations of static equilibrium. Otherwise, the

structure is considered statically indeterminate. Statically indeterminate structures arise owing to the

presence of extra supports, members, reaction forces, or reaction moments. For a structure to be statically

determinate, it must first be constructed correctly and then supported correctly.

28.2.2 Two-Dimensional Structures

Most machine structures, in practice, are three-dimensional. However, it is useful to look at a twodimensional

problem first, before extending the problem to a three-dimensional one. Generally,

machine structures are stationary. Therefore, the sum of the forces and moments acting on it must

be zero, which is in accordance with Newton’s Second Law. In mathematical form,

X

Fx ¼ 0 ð28:1Þ

Vibration Suppression and Monitoring in Precision Motion Systems 28-3

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X

Fy ¼ 0 ð28:2Þ

X

Mz ¼ 0 ð28:3Þ

where Fx and Fy are the forces in the x-axis and

y-axis, respectively, while Mz is the moment

about the z-axis, where the z-axis is pointing

out of the page. For a plane structure, we will

make use of the sign conventions as depicted in

Figure 28.1.

Since the static determinacy of a structure is a

two fold issue, it is possible to proceed without

considering the support first. Each structural

configuration can be tested to verify if the plane

structure satisfies the equation:

2j ¼ m þ 3 ð28:4Þ

where j denotes the number of joints and m denotes the number of members. There are three possible

cases, as follows:

1. If 2j ¼ m þ 3; then the structure is statically determinate.

2. If 2j . m þ 3; then the structure is unstable.

3. If 2j , m þ 3; then the structure is statically indeterminate.

The three conditions are depicted in Figure 28.2.

When a structure is unstable due to member deficiency, it appears that the structure becomes a

four-bar mechanism with one degree of freedom (DoF), as shown in Figure 28.2c. This DoF is

undesirable, since it is the structure that is being designed and not the mechanism. If, however,

there are too many members present, the structure becomes statically indeterminate. Under such a

condition, it will be difficult to assemble the fifth bar of the structure shown in Figure 28.2d, if the

x

y

θ

z

FIGURE 28.1 Sign convention for plane structure.

(a)

j = 3

m = 3

2 j = m + 3

Statically

Determinate

(c)

j = 4

m = 4

2 j > m + 3

Unstable

(b)

j = 4

m = 5

2 j = m + 3

Statically

Determinate

(d)

j = 4

m = 6

2 j < m + 3

Statically

Indeterminate

FIGURE 28.2 Plane structure: (a) and (b) statically determinate structure; (c) unstable structure; (d) statically

indeterminate structure.

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dimension of the fifth bar is not exact. Assembly is probably possible using brute force, but internal

stresses will be built into the structure even without any external loading. When a structure is

statically determinate, it will be stress-free when it is not loaded externally other than by its own

weight. In the event of thermal expansion of its member, owing to an increase in temperature,

statically determinate structures allow expansion of their members, without inducing any stress

resulting from an overconstrained condition due to the redundant members.

The triangle is the basic shape for a plane structure as shown in Figure 28.2a. Statically determinate

plane structure can be expanded from this basic structure, simply by linking two new members to two

different existing joints for every new joint added, as shown in Figure 28.2b. However, the axis of the two

new members must not form a line; in other words, the three joints must not be on the same line, as

shown in Figure 28.3a. It is also noteworthy that the ground constitutes one member as well, and all

joints are pin-joints, as shown in Figure 28.3b.

The second part of structure design lies in its supports. In this aspect, the whole structure can

be treated as a rigid body. A plane structure has three DoF; that is, the plane structure is capable

of motion in the x and y directions, and rotation about the z-axis. Therefore, three members

are needed providing three reactive forces to exactly constrain the plane structure in the plane.

(a) (b)

new members

new joint

extra

member

ground

FIGURE 28.3 (a) Unstable structure; (b) extra member.

Stable

(a) (b)

(c) (d)

Stable

Stable Unstable

FIGURE 28.4 (a), (b), and (c) Stable and exactly constraint supports; (d) unstable support.

Vibration Suppression and Monitoring in Precision Motion Systems 28-5

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Figure 28.4a –c show some possible support for

plane structure, while Figure 28.4d shows an

unstable support scenario.

It should be highlighted that the condition of

having two support members at the same location

can be replaced by a single pin-joint, as shown in

Figure 28.5. It is apparent that, in both cases, they

constitute two reaction forces and do not constrain

the angular motion about the z-axis present in the

plane structure.

The correct number of members in a structure,

as well as the correct number of supports, must be in place for a stable and statically determinate

structure. At this juncture, the issue of where the loads are to be applied on the structure must be

addressed. To this end, it is necessary to examine the members that make up the structure. The

stiffness of a bar member is affected by the way the load is applied with respect to its axial axis, its

cross-sectional geometry (e.g., the diameter, for a round bar) and the modulus of elasticity, E, of

its material. In most cases, the bar is either loaded in tension, compression, or bending, as shown

in Table 28.1.

It is apparent from examining Table 28.1 that the stiffness of a bar is much better in axial loading than

in bending loading. For a value of d ¼ 0:05 m and L ¼ 1:2 m, the ratio of kt=kb is 192. That is, a bar is

192 times stiffer when loaded axially than in bending. Therefore, when designing a rigid and stiff

structure, the members must be loaded in tension or compression, never in bending. At times,

redesigning the way an external load is applied on a structure can greatly improve the stiffness of the

structure. Various configurations are shown in Table 28.2. As a general rule to observe, the loading

point should be located at the joints.

28.2.3 Three-Dimensional Structures

Next, space structures or three-dimensional structures will be considered. These are structures that

are of interest in most applications. In a very general sense, space structures can be perceived as a

combination of many plane structures, arranged in a manner that all the planes are not coplanar.

Therefore, for a space structure to be rigid, every plane structure that makes up the space structure

≈

FIGURE 28.5 Equivalent of a two-member support.

TABLE 28.1 Comparison of Stiffness for Axial Loading versus Bending Loading

Configurations Loading Condition Stiffness (N/m) Normalized Stiffness

(a)

F F

L Tension kt ¼ 0:25pEd2 =L 1

(b)

F F

L Compression kc ¼ 0:25pEd2 =L 1

(c)

F

0.5L

Bending kb ¼ 0:75pEd4 =L3 3ðd=LÞ2

Units: E (N/m2), d (m), L (m), and L .. d:

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must be rigid in its own right. This is one reason to have a good understanding of plane structural

rigidity.

Since machine structures are stationary, the sum of the forces and moments acting on the

machine must be zero, which is in accordance with Newton’s Second Law. Mathematically,

this implies

X

F ¼ 0 ð28:5Þ

X

M ¼ 0 ð28:6Þ

where F and M are three-dimensional force and moment vectors, respectively. The sign conventions

as depicted in Figure 28.1 will be used.

As before, each structural configuration can be tested to verify if the plane structure satisfies the

equation:

3j ¼ m þ 6 ð28:7Þ

TABLE 28.2 Comparison of Stiffness for Various Loading Configurations

Configurations Stiffness (N/m) Normalized Stiffness Compare

(a) F F

L

0.5L F

kt ¼ 0:25pEd2 =L 1 1

(c)

F

L

b b

kb ¼ 0:75pEd4 =L3 3ðd=LÞ2 1/192

(d)

F

L

k^ ¼ 0:5pEd2 sin2b=L sin2b 1/2

(e)

F

L

L

b

b

kcl ¼ 0:047pEd4 =L3 0:1875ðd=LÞ2 1/3072

(f) k. ¼ 0:25pEd2 =L 1 1

b ¼ 458; d ¼ 0:05 m, L ¼ 1:2 m.

Vibration Suppression and Monitoring in Precision Motion Systems 28-7

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where j denotes the number of joints and m

denotes the number of members. There are three

possible cases, as follows:

1. If 3j ¼ m þ 6; then the structure is statically

determinate.

2. If 3j . m þ 6; then the structure is

unstable.

3. If 3j , m þ 6; then the structure is statically

indeterminate.

In the plane structure, the triangle is the

basic shape, and is rigid and statically determinate.

In a space structure, the basic form

for being rigid and statically determinant is

the tetrahedron, which is depicted in

Figure 28.6. Adding a new noncoplanar

joint to the three existing joints of a triangular

plane structure derives the tetrahedron structure.

This new joint is connected to the

existing joints with three new members. By

following this procedure, a rigid and

statically determinate space structure can be

derived.

Other space structures are shown in

Figure 28.7 and Figure 28.8. It is also noteworthy

that the members are connected with

ball joints.

Thus far, the approach to obtain the

tetrahedron space structure from the triangle

plane structure, the pyramid from the tetrahedron,

and the box from the tetrahedron

has been illustrated. The next aspect of the

design is to combine some of these structures.

The structures can be treated as being coupled

together as rigid bodies, and a rigid body

in space has six DoF, i.e., the structure is

capable of translations in the x, y, and z

directions, and rotation about the x, y, and z

axes. Therefore, six members are needed,

providing six reactive forces to exactly constrain

the structure in space. Figure 28.9 shows a

typical gantry configuration, which is used

extensively in many coordinate-measuring

machines (CMM). However, one of the members

is bearing a bending load, which has been

shown earlier to be very detrimental to the

stiffness of the structure. There are alternative

structure configurations as shown in Figure 28.10

and Figure 28.11, although some redesign may

be needed if such a configuration is to be

utilized.

FIGURE 28.6 Basic space structure — the tetrahedron

structure.

FIGURE 28.7 Pyramid structure derived from a

tetrahedron structure.

FIGURE 28.8 Box structure derived from a tetrahedron

structure.

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If the ground is perceived as another rigid

body to which the space structure is to be

coupled, then the design of the supports for a

space structure is similar to those of

coupling two space structures together; that is,

six reactive forces are needed to exactly

constrain the space structure. Some ways to

arrange the six supporting members constraining

a space structure are suggested in

Figure 28.12.

Examples of physical supports offering one,

two, or three reactive forces are shown in

Figure 28.13.

This method of design, known as kinematical

design, requires the use of point contact at the

interfaces. Unfortunately, this method has some

disadvantages:

* Load carrying limitation.

* Stiffness may be too low for application.

* Low damping.

There are, however, ways to overcome the

disadvantages, which are via the semikinematical

approach. This approach is a modification of

the kinematical approach, and it aims to

overcome the limitations of a pure kinematical

design. The direct way is to replace all point

contact with a small area, as shown in

Figure 28.14; doing so decreases the contact

stress, but increases the stiffness and load

carrying capacity. However, the area contact

should be kept reasonably small.

This section has only illustrated some

fundamental concepts in designing rigid and

statically determinate machine structures.

Interested readers may refer to Blanding

(1999) for more details on designing

a machine using the exact constraints

principles.

FIGURE 28.9 Gantry space structure.

FIGURE 28.10 Coupling of a tetrahedron structure to

a box structure with six members.

SUMMARY

The approach taken to reduce the mechanical vibration in precision motion systems is to focus on

a proper mechanical design, based on the determinacy of machine structures in this chapter. The

aim of this approach is to design systems with stable and statistically determinate structures. Twodimensional

structures and three-dimensional structures are considered.

Vibration Suppression and Monitoring in Precision Motion Systems 28-9

© 2005 by Taylor & Francis Group, LLC

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