stiffness part 1

7/29/2019 Stiffness Part 1

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Introduction and Definitions

1.1 BASIC NOTIONS

1.1.1 Stiffness

Stiffness is the capacity of a mechanical system to sustain loads without excessive

changes of its geometry (deformations). It is one of the most important design

criteria for mechanical components and systems. Although strength is considered

the most important design criterion, there are many cases in which stresses in

components and their connections are significantly below the allowable levels,

and dimensions as well as performance characteristics of mechanical systems

and their components are determined by stiffness requirements. Typical examplesof such mechanical systems are aircraft wings, and frames/beds of production

machinery (machine tools, presses, etc.), in which stresses frequently do not ex-

ceed 37 MPa (5001,000 psi). Another stiffness-critical group of mechanical

components is power transmission components, especially shafts, whose defor-

mations may lead to failures of gears and belts while stresses in the shafts caused

by the payload are relatively low.

Recently, great advances in improving strength of mechanical systems and

components were achieved. The main reasons for such advances are development

of high strength structural metals and other materials, better understanding of

fracture/failure phenomena, and development of better techniques for stress anal-

ysis and computation, which resulted in the reduction of safety factors. These

advances often result in reduction of cross sections of the structural components.

Since the loads in the structures (unless they are weight-induced) do not change,structural deformations in the systems using high strength materials and/or de-

signed with reduced safety factors are becoming more pronounced. It is important

Copyright 1999 by Marcel Dekker, Inc. All Rights Reserved.


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to note that while the strength of structural metals can be greatly improved by

selection of alloying materials and of heat treatment procedures (as much as 5

7 times for steel and aluminum), modulus of elasticity (Youngs modulus) is not

very sensitive to alloying and to heat treatment. For example, the Youngs modu-

lus of stainless steels is even 515% lower than that of carbon steels (see Table

1.1). As a result, stiffness can be modified (enhanced) only by proper selection

of the component geometry (shape and size) and its interaction with other compo-

nents.

Stiffness effects on performance of mechanical systems are due to influence

of deformations on static and fatigue strength, wear resistance, efficiency (friction

losses), accuracy, dynamic/vibration stability, and manufacturability. The impor-

tance of the stiffness criterion is increasing due to:

1. Increasing accuracy requirements (especially due to increasing speeds

and efficiency of machines and other mechanical systems)2. Increasing use of high strength materials resulting in the reduced cross

sections and, accordingly, in increasing structural deformations

3. Better analytical techniques resulting in smaller safety factors, which

also result in the reduced cross sections and increasing deformations

4. Increasing importance of dynamic characteristics of machines since their

increased speed and power, combined with lighter structures, may result

in intense resonances and in the development of self-excited vibrations

(chatter, stick-slip, etc.)

Factors 24 are especially pronounced for surface and flying vehicles (cars,

airplanes, rockets, etc.) in which the strength resources of the materials are uti-

lized to the maximum in order to reduce weight.

Stiffness is a complex parameter of a system. At each point, there are gener-ally different values of stiffness kxx, kyy, kzz in three orthogonal directions of a

selected coordinate frame, three values of interaxial stiffness kxy, kxz, kyz related

to deformations along one axis (first subscript) caused by forces acting along an

orthogonal axis (second subscript), and also three values of angular stiffness

about the x, y, and z axes. If the interaxial stiffnesses vanish, kxy kxz kyz

0, then x, y, z are the principal stiffness axes. These definitions are important

since in some cases several components of the stiffness tensor are important; in

special cases, ratios of the stiffness values in the orthogonal directions determine

dynamic stability of the system. Such is the case of chatter instability of some

machining operations [1]. Chatter stability in these operations increases if the

cutting and/or the friction force vector is oriented in a certain way relative to

principal stiffness axes x and y. Another case is vibration isolation. Improper

stiffness ratios in vibration isolators and machinery mounts may cause undesir-able intermodal coupling in vibration isolation systems (see Article 1).

Main effects of an inadequate stiffness are absolute deformations of some



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Table 1.1 Youngs Modulus and Density of Structural Materials

E E/Material (105 MPa) (103 kg/m3) (107 m2/s 2)

(a) Homogeneous Materials

Graphite 7.5 2.25 33.4Diamond 18.0 5.6 32Boron carbide, BC 4.50 2.4 19Silicon carbide, SiC 5.6 3.2 17.5Carbon, C 3.6 2.25 16.0Beryllium, Be 2.9 1.9 15.3Boron, B 3.8 2.5 15.2Sapphire 4.75 4.5 10.1Alumina, Al2 O3 3.9 4.0 9.8Lockalloy (62% Be 38% Al) 1.90 2.1 9.1

Kevlar 49 1.3 1.44 9.0Titanium carbide, TiC 4.04.5 5.76.0 7.09.1Silicone, Si 1.1 2.3 4.8Tungsten carbide, WC 5.50 16.0 3.4Aluminum/Lithium (97% Al 3% Li) 0.82 2.75 3.0

Molybdenum, Mo 3.20 10.2 3.0Glass 0.7 2.5 2.8Steel, Fe 2.10 7.8 2.7Titanium, Ti 1.16 4.4 2.6Aluminum, Al (wrought) 0.72 2.8 2.6Aluminum, Al (cast) 0.65 2.6 2.5Steel, stainless (.08 0.2% C, 17% Cr, 7% Ni) 1.83 7.7 2.4Magnesium, Mg 0.45 1.9 2.4Wood (along fiber) 0.110.15 0.410.82 2.61.8Marble 0.55 2.8 2.0

Tungsten (W 2 to 4% Ni, Cu) 3.50 18.0 1.9Granite 0.48 2.7 1.8Beryllium copper 1.3 8.2 1.6Polypropylene 0.08 0.9 0.9Nylon 0.04 1.1 0.36Paper 0.010.02 0.5 0.20.4

(b) Composite Materials

HTS graphite/5208 epoxy 1.72 1.55 11.1Boron/5505 epoxy 2.07 1.99 10.4Boron/6601 Al 2.14 2.6 8.2Lanxide NX 6201 (Al SiC) 2.0 2.95 6.8T50 graphite/2011 Al 1.6 2.58 6.2Kevlar 49/resin 0.76 1.38 5.580% Al 20% Al2O3 powder 0.97 2.93 3.3

Melram (80% Mg, 6.5% Zn, 12% SiC) 0.64 2.02 3.2E glass/1002 epoxy 0.39 1.8 2.2



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components of the system and/or relative displacements between two or several

components. Such deformations/displacements can cause:

Geometric distortions (inaccuracies)

Change of actual loads and friction conditions, which may lead to reduced

efficiency, accelerated wear, and/or fretting corrosion

Dynamic instability (self-excited vibrations)

Increased amplitudes of forced vibrations

Inadequate stiffness of transmission shafts may cause some specific effects.

The resulting linear and angular deformations determine behavior of bearings

(angular deformations cause stress concentrations and increased vibrations in

antifriction bearings and may distort lubrication and friction conditions in sliding

bearings); gears and worm transmissions (angular and linear deformations lead

to distortions of the meshing process resulting in stress concentrations and varia-tions in the instantaneous transmission ratios causing increasing dynamic loads);

and traction drives (angular deformations cause stress concentrations and chang-

ing friction conditions).

It is worthwhile to introduce some more definitions related to stiffness:

Structural stiffness due to deformations of a part or a component considered

as beam, plate, shell, etc.

Contact stiffness due to deformations in a connection between two compo-

nents (contact deformations may exceed structural deformations in pre-

cision systems)

Compliance e 1/k, defined as a reciprocal parameter to stiffness k (ratio

of deformation to force causing this deformation)

Linear stiffness vs. nonlinear stiffness (see Ch. 3)Hardening vs. softening nonlinear stiffness (see Ch. 3)

Static stiffness kst (stiffness measured during a very slow loading process,

such as a periodic loading with a frequency less than 0.5 Hz) vs. dynamic

stiffness kdyn, which is measured under faster changing loads. Dynamic

stiffness is characterized by a dynamic stiffness coefficient Kdyn kdyn/kst.

Usually Kdyn 1 and depends on frequency and/or amplitude of load

and/or amplitude of vibration displacement (see Ch. 3). In many cases,

especially for fibrous and elastomeric materials Kdyn is inversely corre-

lated with damping, e.g., see Fig. 3.2 and Table 1 in Article 1.

1.1.2 Damping

Damping is the capacity of a mechanical system to reduce intensity of a vibratoryprocess. The damping capacity can be due to interactions with outside systems,



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or due to internal performance-related interactions. The damping effect for a vi-

bratory process is achieved by transforming (dissipating) mechanical energy of

the vibratory motion into other types of energy, most frequently heat, which can

be evacuated from the system. If the vibratory process represents self-excited

vibrations (e.g., chatter), the advent of the vibratory process can be prevented by

an adequate damping capacity of the system.

In the equations of motion to vibratory systems (e.g., see Appendix 1), both

intensity and character of energy dissipation are characterized by coefficients at

the first derivative (by time) of vibratory displacements. These coefficients can

be constant (linear or viscous damping) or dependent on amplitude and/or fre-

quency of the vibratory motion (nonlinear damping). There are various mecha-

nisms of vibratory energy dissipation which can be present in mechanical sys-

tems, some of which are briefly explored in Appendix 1.

Since the constant coefficient at the time-derivative of the vibratory displace-ment term results in a linear differential equation, which is easy to solve and to

analyze, such systems are very popular in textbooks on vibration. However, the

constant damping coefficient describes a so-called viscous mechanism of energy

dissipation that can be realized, for example, by a piston moving with a relatively

slow velocity inside a conforming cylinder with a relatively large clearance be-

tween the piston and the cylinder walls, so that the resistance force due to viscous

friction has a direction opposite to the velocity vector and is proportional to the

relative velocity between the cylinder and the piston. In real-life applications

such schematic and conditions are not often materialized. The most frequently

observed energy dissipation mechanisms are hysteretic behavior or structural ma-

terials; friction conditions similar to coulomb (dry) friction whereas the friction

(resistance) force is directionally opposed to the velocity vector but does not

depend (or depends weakly) on the vibratory velocity magnitude; damping injoints where the vibratory force is directed perpendicularly to the joint surface

and causes squeezing of the lubricating oil through the very thin clearance be-

tween the contacting surfaces (thus, with a very high velocity) during one-half

of the vibratory cycle and sucking it back during the other half of the cycle; and

damping due to impact interactions between the contacting surfaces. Some of

these mechanisms are analytically described in Appendix 1.

Effects of damping on performance of mechanical systems are due to reduc-

tion of intensity of undesirable resonances; acceleration of decay (settling) of

transient vibration excited by abrupt changes in motion parameters of mechanical

components (start/stop conditions of moving tables in machine tools and of robot

links, engagement/disengagement between a cutting tool and the machined part,

etc.); prevention or alleviation of self-excited vibrations; prevention of impacts

between vibrating parts when their amplitudes are reduced by damping; potentialfor reduction of heat generation, and thus for increase in efficiency due to reduced



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peak vibratory velocities of components having frictional or microimpacting in-

teractions; reduction of noise generation and of harmful vibrations transmitted

to human operators; and more.

It is important to note that while damping is associated with transforming

mechanical energy of the vibratory component into heat, increase of damping

capacity of mechanical system does not necessarily result in a greater heat genera-

tion. Damping enhancement is, first of all, changing the dynamic status of the

system and, unless the displacement amplitude is specified (for example, like

inside a compensating coupling connecting misaligned shafts; see Section 8.5.2),

most probably would cause a reduction in the heat generation. This somewhat

paradoxical statement is definitely true in application to mechanical systems

prone to development of self-excited vibrations, since enhancement of damping

in the system would prevent starting of the vibratory process, and thus the heat

generation, which is usually caused by vibratory displacements. This statementis also true for a system subjected to transient vibration. Since the initial displace-

ment of mass m in Fig. A.1.1 and the natural frequency of the system do not

significantly depend on damping in the system, a higher damping would result

in smaller second, third, etc. amplitudes of the decaying vibrations, and thus in

a lower energy dissipation.

Less obvious is the case of forced vibration when force F F0sin t isapplied to mass m in Fig. A.1.1. Lets consider the system in which mass m is

attached to the frame by a rubber flexible element combining both stiffness and

damping properties (hysteresis damping, r 1; see Appendix 1). If amplitude

of mass m is A, then the maximum potential energy of deformation of the flexible

element is

V k A2

2(1.1)

The amount of energy dissipated (transformed into heat) in the damper c or in

the rubber flexible element is

V V kA2

2(1.2)

At the resonance, amplitude Ares of mass m is, from formula (A.1.19b) at 0 and from (A1.18)

Ares F0

k k2

F0

kk A0

A0 2

(1.3a)



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where A0 F0/k static ( 0) deflection of the flexible element, logarith-mic decrement, and for not very high damping

2 (1.3b)

Thus, the energy dissipation at resonance (or the maximum energy dissipation

in the system) is decreasing with increasing damping capacity (increasing Y).

This result, although at the first sight paradoxical, does not depend on the

character (mechanism) of damping in the system and can be easily explained.

The resonance amplitude is inversely proportional to the damping parameter (,, etc.) because the increasing damping shifts an equilibrium inside the dynamicsystem between the excitation (given, constant amplitude), elastic (displacement-

proportional), inertia (acceleration-proportional), and damping (velocity-propor-

tional) forces. The amount of energy dissipation is a secondary effect of thisequilibrium; the energy dissipation is directly proportional to the square of the

vibration amplitude. Although this effect of decreasing energy dissipation with

increasing damping is especially important at the resonance where vibratory am-

plitudes are the greatest and energy dissipation is most pronounced, it is not as

significant in the areas outside of the resonance where the amplitudes are not

strongly dependent on the damping magnitude (see Fig. A.1.3).

Effects of damping on performance of mechanical system are somewhat sim-

ilar to the effects of stiffness, as presented in Section 1.1.1. Damping influences,

directly or indirectly, the following parameters of mechanical systems, among

others:

1. Fatigue strength. Increasing damping leads to reduction of strain and

stress amplitudes if the loading regime is close to a resonance. It is even more

important for high-frequency components of strain/stress processes, which arefrequently intensified due to resonances of inevitable high frequency components

of the excitation force(s) and/or nonlinear responses of the system with higher

natural frequencies of the system.

2. Wear resistance. High (resonance) vibratory velocities, especially asso-

ciated with high-frequency parasitic microvibrations, may significantly accelerate

the wear process. High damping in the system alleviates these effects.

3. Efficiency (friction losses). Depending on vibration parameters (ampli-

tudes, frequencies, and, especially, directivity), vibrations can increase or reduce

friction. In the former case, increasing damping can improve efficiency.

4. Accuracy and surface finish of parts machined on machined tools. Al-

though surface finish of the machined surface is directly affected by vibrations,

accuracy (both dimensions and macrogeometry) may be directly influenced by

low-frequency vibrations, e.g., transmitted from the environment (see Article 1)or may be indirectly affected by changing geometry of the cutting tool whose

sharp edge(s) are fast wearing out under chatter- or microvibrations. The latter



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are especially dangerous for brittle cutting materials such as ceramic and diamond

tools.

5. Dynamic/vibration stability of mechanical systems can be radically en-

hanced by introducing damping into the system.

6. Manufacturability, especially of low-stiffness parts, can be limited by

their dynamic instability, chatter, and resonance vibrations during processing.

Damping enhancement of the part and/or of the fixtures used in its processing

can significantly improve manufacturability.

Importance of the damping criterion is increasing with the increasing impor-

tance of the stiffness criterion as discussed in Section 1.1.1 due to:

a. Increasing accuracy requirements

b, c. Increasing use of high strength materials and decreasing safety fac-

tors, which result in lower stiffness and thus higher probability of

vibration excitationd. Increasing importance of dynamic characteristics

e. Increasing awareness of noise and vibration pollution

Main sources of damping in mechanical systems are:

a. Energy dissipation in structural materials

b. Energy dissipation in joints/contacts between components (both in mov-

ing joints, such as guideways, and in stationary joints)

c. Energy dissipation in special damping devices (couplings, vibration iso-

lators, dampers, dynamic vibration absorbers, etc.). These devices may

employ viscous (or electromagnetic) dampers in which relative vibratory

motion between component generates a viscous (velocity dependent)

resistance force; special high-damping materials, such as elastomers or

shape memory metals (see Table 1.2); specially designed (vibroim-

pact) mechanisms in which coimpacting between two surfaces resultsin dissipation of vibratory energy (see Appendix 1); etc.

1.2 INFLUENCE OF STIFFNESS ON STRENGTH ANDLENGTH OF SERVICE

This influence can materialize in several ways:

Inadequate or excessive stiffness of parts may lead to overloading of associ-

ated parts or to a nonuniform stress distribution

Inadequate stiffness may significantly influence strength if loss of stability

(buckling) of some component occurs

Impact/vibratory loads are significantly dependent on stiffnessExcessive stiffness of some elements in statically indeterminate systems may

lead to overloading of the associated elements


stiffness part 1

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