Work piece dynamics influence on stability in machining

MASTER’S THESIS

2007:162 CIV

Universitetstryckeriet, Luleå

Per Wiklund

Work piece dynamics influence

on stability in machining

MASTER OF SCIENCE PROGRAMME Mechanical Engineering

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Solid Mechanics

Abstract

This project was carried out at Sandvik Coromant in Sandviken. The objective was to investigate how the machining dynamics was influences by the workpiece and

different positions of the fixtures. The project consisted of two parts: investigation of one simple beam model to gain knowledge and understanding of the problem and secondly: study of alternative clamping positions for a jet engine turbine exhaust case. For both cases natural frequencies, mode shapes, frequency response functions and machining stability was investigated. For the beam model the natural frequency changes resulting from machining were found to have specific patterns which were dependent on the boundary conditions. The changing natural frequencies were

reflected on to the stability lobes. In order to maximize the production rate the spindle speed should be altered to natural frequencies changes.

The natural frequencies can change considerably when the beam cross section is machined down. The mode shapes of the beam is however fairly unchanged. This makes it possible to choose a good position of an additional support based on knowledge of the original beam. A position of an additional support at a node point for a particular mode maximizes the natural frequency for the mode one number lower. A position of the support near the middle of the beam minimizes the highest flexibility that occurs and also maximizes the first natural frequency for a prismatic beam.

For the jet engine casing six lower and six upper clamping sections with a 30 degree phase shift was found to have both the highest first natural frequencies and the highest machining stability limits of the investigates cases. Locking the casing in three

Det här examensarbetet är utfört vid Sandvik Coromant i Sandviken . Uppdraget bestod i att undersöka hur arbetsstyckets förändrade dynamik under bearbetning och olika fixturer påverkar bearbetningsdynamiken.

Uppdraget omfattades av två delar, Del ett: undersöka en enkel balkmodell för att få kunskap och förståelse om problemet. Del två: studier av alternativa positioner för fixturer för en jet motor casing. I båda fallen så undersöktes egenfrekvenser,

frekvensresponsfunktioner och bearbetningsstabilitet. För balkmodellen så fanns att förändringen av egenfrekvenserna under bearbetning antog specifika mönster. Dessa förändringar visade sig också vara beroende av randvillkoren för balken. De

förändrade egenfrekvenserna ändrar också stabilitets loberna. Detta gav att för att maximera avverkningen så ska spindelhastigheten ändras med de förändrade egenfrekvenserna.

Egenfrekvenserna kan förändras betydligt för en balk när dess tvärsnitt bearbetas ned. Modformerna för balken är dock relativt oförändrade. Detta medför att det är möjligt att välja en bra placering av ett extra stöd baserat på den ursprungliga balken. Om det extra stödet placeras i en nodpunkt för en specifik mod så maximeras egenfrekvensen för moden med ett nummer lägre. En stödplacering nära mitten för balken minimerar den högsta flexibiliteten som kan uppstå under bearbetning. Denna placering

maximerar också den första egenfrekvensen för en prismatisk balk.

Abstract ...1 1 Introduction...5 1.1 Background ...5 1.2 Objective ...5 1.3 Limitations ...5 1.4 Method ...5 1.5 Company presentation ...6 1.5.1 History...6

1.5.2 The Sandvik group ...6

2 Method ...8

2.1 Vibration fundamentals...8

2.1.1 Free vibrations for a single degree of freedom systems ...8

2.1.2 Response to harmonic excitation for sdof-systems...10

2.1.3 Superposition ...11

2.1.4 Multiple degrees of freedom...11

2.1.4.1 Free vibrations ...11

2.1.5 Distributed parameter systems...15

2.1.6 Modal analysis ...20

2.1.7 Displacement method for Euler-Bernoulli beam - structures ...21

2.1.8 Transfer functions and FRF ...23

2.1.9 Complex stiffness...23

2.1.10 FRF for displacement method...24

2.2 Machining ...25 2.2.1 Mechanics of cutting...25 2.2.2 Cutting forces...26 2.2.3 Stability in machining ...28 2.2.3.1 Influence of workpiece ...34 2.2.4 Cutting data...35

2.2.5 Additional supports for workpiece...36

2.3 Dynamics of a beam during machining ...37

2.4 Method for beam calculations...37

2.4.1 Calculation of machining stability limits...41

2.5 Jet engine casing ...41

3 Results for beam model, no support ...43

3.1 Natural frequencies ...43

3.1.1 Boundary conditions pinned-pinned ...43

3.1.2 Boundary conditions clamped-clamped...44

3.1.3 FEM model comparison...46

3.2 Mode shapes for 50 percent step height...47

3.2.1 Boundary conditions pinned-pinned ...48

3.2.2 Clamped-clamped boundary condition ...49

3.3 FRF results for boundary conditions pinned - pinned ...50

3.3.1.1 No step ...50

3.3.1.2 Step height 30 % ...51

3.3.1.3 Step height 50 % ...53

3.4 FRF and stability limits for boundary conditions clamped – clamped, step height 50 percent...55

3.4.1 FRF ...55

3.4.3 50 % depth of cut down-milling with flexible machine-tool structure 57

4 Conclusions for beam model without additional support ...60

4.1 Natural frequencies ...60

4.2 Mode shapes...62

4.3 Frequency response function ...62

4.4 Machining stability limits ...63

4.5 Optimization of the cutting data ...63

5 Results for beam with additional support ...66

5.1 Natural frequencies ...66

5.1.1 Prismatic beam...66

5.1.2 For beam with step height 50 percent ...67

5.2 Mode shapes...69

5.2.1 For beam with no step...69

5.3 Frequency response function ...71

5.3.1 Prismatic beam...71

5.3.2 For beam with step height 50 percent ...72

5.4 Stability limits in machining for beam with an additional support...75

5.5 FRF comparison...77

6 Conclusions for beam model with additional support ...80

6.1 Natural frequencies ...80

6.1.1 For no step...80

6.1.2 For 50 percent step height...80

6.2 Mode shapes...80

6.3 FRF ...80

6.3.1 No step ...80

6.3.2 Step height of 50 %...81

6.4 Stability lobes...81

7 Example with a jet engine exhaust case...82

7.1 Analytical part...82

7.2 Jet engine casing with three lower clamping sections ...88

7.3 Jet engine casing with six lower clamping sections...92

7.3.1 Calculation of FRF for jet engine casing with six lower clamping sections 95 7.3.2 Machining Stability limits for a jet engine casing with six lower clamping sections...98

7.4 Jet engine with six lower and six upper clamping sections in phase ...100

7.5 Jet engine casing with six lower and six upper clamping sections with a 30° phase shift ...103

7.5.1 Calculation of FRF for jet engine casing with six lower and six upper clamping sections with a 30° phase shift ...106

7.5.2 Stability limits...108

7.6 Jet engine casing with twelve lower clamping sections...110

8 Conclusions for the exhaust case calculations ...113

9 Summary of calculations and work method...117

1 Introduction

1.1 Background

An important and unwanted phenomenon that may occur during machining is chatter vibrations. This is a self-excitation mechanism causing the tool-work piece system to vibrate, resulting in high cutting forces and poor surface finish. When machining thin walled structures it is important to consider the dynamics of both the tool and the workpiece to avoid the appearance of chatter vibrations. The fact that the dynamics of the workpiece changes as material is removed adds more complexity to the problem.

1.2 Objective

The objective is to investigate how the changed dynamic of the workpiece during machining, and different positions if the fixtures influence the machining dynamics. This project consists of two parts, first part: investigation of one simple beam model to gain knowledge and understanding of the problem and the second part: study of alternative clamping positions for a jet engine turbine exhaust case.

1.3 Limitations

The beam model compromises at two different types of boundary conditions for beam end points and for the additional support point in between. For beam model without additional support, the end points were pinned - pinned or clamped - clamped. The boundary conditions for the additional support was of the type pinned. For finite element method calculations the beam endpoints was locked in such way that no motion was allowed in the beam length axis for either of them. For the beam model displacement was limited to one plane only, defined according to Figure 17. Only the first two or in some cases three natural frequencies was considered for the beam model. For the jet engine casing the FRF was calculated only for the first three natural frequencies and only in the areas around anti nodes. All calculations of natural frequencies, mode shapes and FRF were carried out either in the finite element software PRO/Mechanica and ANSYS Workbench or in mathematics software Maple 9.5 or in Microsoft Excel. Stability limits for machining was calculated in CutPro 8.1. The cutting coefficients used in stability limits calculations may not be an exact match for the tool and cutting conditions and therefore the stability limits are only an approximation.

1.4 Method

The early stages of the project consisted of literature studies to gain knowledge of the complex phenomena called chatter vibrations. Other areas studied were cutting processes, dynamics and mathematics.

and stability limits were performed on a model of a jet engine turbine exhaust case. Comparisons of natural frequencies and mode shapes between five different support cases were made, of which two was chosen to be investigated further. For these two cases FRF and stability limits was calculated.

1.5 Company presentation

This chapter contains the history of AB Sandvik Coromant, marketing and

organization. Besides this the position of Sandvik Coromant in the Sandvik group will be mapped out.

1.5.1 History

Göran Fredrik Göransson founded Högbo Stål och Jernwerks AB in 1862 were he developed high quality steel with the Bessemer-method. The company was however struggling with the finances and only four years after the start the company went bankrupt. This setback did not intimidate Göransson and 1868 he founded a new company named Sandvik Jernverk AB. This name was then kept until 1972 when it was chanced to the present name Sandvik AB.

1.5.2 The Sandvik group

Sandvik AB is an engineering industry with focus on high technology and the companies turn over for 2004 were 7 billion USD. The business is located in 130 different countries and has an estimated number of 38 000 employees all over the world. Sandvik AB has three main field areas.

• Sandvik Tooling

• Sandvik Mining and Construction

• Materials Technology

Sandvik Tooling sells and develops tools and tool systems for machining of metal.

The largest clients can be found in the auto and aerospace industry were components with high demands on durability, corrosion and temperature resistant play an important part in production. These material properties complicate machining and put high demands on the tools used to machine these components. The cutting products that Sandvik Tooling produces are drills, milling tools and turning tools and the main materials used in these tools are Cemented carbide, synthetic diamond, ceramics and CBN. These materials have an extreme durability against wear and suits machining in aluminum, steel, titanium and different alloys perfectly.

Sandvik Mining and Construction aims its business toward the mining and

Sandvik Materials Technology is the last large segment in the Sandvik group. This

2 Method

This chapter consists of a summary of vibration fundamentals and basics of metal cutting.

2.1 Vibration fundamentals

The simple structure in Figure 1 has one degree of freedom and the motion of the system can be described with the differential equation 2.1, Inman (2001). The parameter k is the spring stiffness, m the mass and c the viscous damping of the system. This equation is often referred to as the equation of motion. The solutions are well known and are presented below. If the system has no damping it is called an undamped system F kx x c x

m&&+ &+ = 2.1

Figure 1 Single degree of freedom system (sdof)

2.1.1 Free vibrations for a single degree of freedom

systems

For the case of no external force acting on the system the equation of motion becomes. 0

= + +cx kx x

m&& & 2.2

For the undamped case of a sdof system the angular frequency of oscillation is described by equation 2.3.

m k

n =

ω

2.3

0 0 1 tan v x n

ω

φ

₌ − , 2.5 2 2 0 2 0 n v x A

ω

+ = . 2.6

Where x0 and v0 are the displacement and velocity at t =0respectively.

For the under critical damped case, which means that relative damping ζ <1, the angular frequency of oscillationω_d is described by equation 2.7 and is called the damped natural frequency. The relative damping ζ is defined according equation 2.8.

2 1 ζ ω ωd = n − , 2.7 km c 2 = ζ . 2.8

The complete solution is

2.1.2 Response to harmonic excitation for sdof-systems

If the force is harmonic it can be described by equation 2.12, Inman (2001).

t F

F = 0cosω 2.12

The equation of motion becomes

t F kx x c x

m&&+ &+ = ₀cosω 2.13

t f

x x

x&+2ζωn&+ωn = 0cosω

& 2.14

The complete solution consists of a homogenous solution and a particular solution. p h x x x= + 2.15 Undamped case

Under critical damped case

For the under critical damped case the solution becomes

) sin(ω φ ζω ₊ = Ae t x_h nt _{d ,} 2.21 ) cos(ω −θ = X t x_p , 2.22 φ θ sin cos 0 X x A= − , 2.23

θ

ω

ζω

θ

θ

ω

φ

sin ) cos ( ) cos ( arctan 0 0 0 X X x v X x n d − − + − = , 2.24 2 2 2 arctan

ω

ω

ω

ζω

θ

− = n n , 2.25 2 2 2 2 0 ) 2 ( ) (ωn ω ζωnω f X + − = . 2.26

2.1.3 Superposition

Superposition refers to the fact that if x1 and x2 is a solution to a linear differential equation then x1 + x2 is also a solution, Inman (2001). This also yields that if a linear system, for example the equation 1 2

2

f f x x&+ωn = +

& has two different forcing

functions acting on it and if x1 is the solution to 1 2

f x x&+ω_n =

& and x2 tox&&+ω_n2x= f₂. Then the total solution to the system will bex_t =x₁+x₂.

2.1.4 Multiple degrees of freedom

If there are more then one degree of freedom in the system the equation of motion takes the form of 2.27. Herex is the displacement vector, x& and x&& are the vectors of

first and second derivatives of the displacements respectively.

F Kx x C x

M&&+ &+ = 2.27

M is the mass matrix, C is the damping matrix and K the stiffness matrix.

2.1.4.1

Free vibrations

If there is no external force acting on system the equation of motion becomes 0

= +

+Cx Kx

x

t j

ue

x= ω 2.29

Which after differentiating and solving yields a solution for a two degree of freedom system consistent with equation 2.30, Inman (2001).

2 2 2 2 1 1 1 1(sin( ) (sin( ) ) (t A t u A t u x =

ω

+

φ

+

ω

+

φ

. 2.30

Where u1 and u2 are vectors that determines the mode shapes of the structure,

ω

1 and 2

ω

is the angular frequencies for the mode shapes respectively.

Eigenvalue problem

The problem of solving equation 2.28 can be extended and formalized to take

advantage of the mathematical eigenvalue problem. This can be done in several ways, the simplest is perhaps to assume solution on form x=uejω tand arrange the equation according to equation 2.31.

(

M−1K

)

u =

λ

u 2.31

u is the eigenvector and λ is corresponding eigenvalue which relates to natural frequency by λ=ω2. The solution then becomes

∑

= + = n i i i i i u C t x 1 ) sin( ) ( ω φ 2.32

Ci and φiare constants to be determined by initial conditions.

This is not a computational efficient way of solving the response for large systems. Another way to formalize the eigenvalue problem is to transform the equation of motion in to a new set of coordinates as in equation 2.33 and multiply with −1/2

M ,

Inman (2001). As seen in equation 2.36 this becomes the eigenvalue problem. q M x= −1/2 2.33 0 ~ ₌ +Kq q& & , 2.34 2 / 1 2 / 1 ~ _=M− _KM− K 2.35 v v K~ =

λ

2.36

State vector approach

The methods of using eigenvalues for solving multiple degree of freedom systems presented above have the advantage of making it possible to use computer software for fast and easy calculation of response. But none of the methods are especially good for solving problems with damping. However with the state vector approach presented below many vibration problems can be solved such as systems with viscous damping and gyroscopic effects, Frazer et al. (1946).

The equation of motion for a general free vibrating system can be written as 0

= +

+Cx Kx

x

M&& & 2.37

The second order differential equation can be rewritten as system of first order differential equations by using the state vector in equation 2.38.

      = x x y & 2.38

This can be done in more then one way however equation 2.39 shows one way which gives symmetrical matrices.

0 0 0 0 =       +      − y C K K y M K & . 2.39

If the first matrix is called –S and the second R the equation becomes

0 = +

−S y& R y . 2.40

By assuming a solution on the form of equation 2.41 the eigenvalue problem can be formalized to equation 2.42. t e Y C y= λ 2.41

[

R−

λ

S

]

Y =0 2.42

The eigenvalues λ and eigenvector Y can then be solved for. The constants C are determined by the initial conditions. The total solution for the homogenous equation can be written as in equation 2.43.

If E is the matrix of eigenvectors then the constants C can be determined by equation 2.45.

[

Y Y Yn

]

E = ₁, ₂,.., 2.44 ) 0 ( 1 h y E C = − 2.45

The resulting eigenvalues and eigenvectors from the calculations above are on complex form.

State vector approach for calculation for response to harmonic force

If the force acting on the system is harmonically the state vector approach can easily

be used to calculate the response. The equation of motion for such system can be written as equation 2.46 and rewritten with the state vector formulation equation 2.47. t f t f Kx x C x

M&&+ &+ = _ssin

ω

+ _ccos

ω

. 2.46

t f t f y R y S + = ssin

ω

+ ccos

ω

− & 2.47

Assuming a solution on the same form as the forcing function yields equation 2.48. Collecting sin and cos terms gives the solution to a and b in equation 2.49 and equation 2.50. t b t a y_p = sin

ω

+ cos

ω

2.48

[

f S a

]

R b = c +

ω

−1 _2.49

[

SR S R

] [

fs SR fc

]

a =

ω

2 −1 + −1 −

ω

−1 2.50

The total solution is the homogenous solution plus the particular solution.

) ( ) ( 1 t y e Y C t y p n i t i i + =

∑

= λ 2.51

When determine the constants C the conditions of the forcing functions must also be solved for, according to equation 2.52.

2.1.5 Distributed parameter systems

A distributed parameter system has an infinite number of natural frequencies. The mass and stiffness are modeled as distributed throughout the structure in contrast to the lumped-parameter systems above, with a finite number of natural frequencies. Only a few distributed systems have closed form solutions, such as a string with both ends fixed and a uniform, slender beam.

Vibration of string

The equation of motion for a string with both ends fixed and not exposed to any external forces can be seen in equation 2.53, Inman (2001).

2 2 2 2 2 ( , ) ( , ) t t x w x t x w c ∂ ∂ = ∂ ∂ , c=

τ

/

ρ

A. 2.53

Figure 2 Vibrating string

τ is the tension force in the string. By assuming a solution on the form of equation

2.54 equation 2.53 can be rewritten as equation 2.55. ) ( ) ( ) , (x T X x T t W = 2.54 2 ) ( ) ( ) ( ) ( _{= =−}

_σ

′′ t cT t T x X x X && 2.55 2

σ is a constant. This allows equation 2.55 to be written as two equations 2.56 and 2.57, each one depending only on one variable.

0 2 =

+

′′ X

Equation 2.56 has a solution on the form of equation 2.58. x a x a x X( )= 1sin

σ

+ 2cos

σ

2.58

Together with boundary conditionsX(0)=0, X(L)=0 this leads to equation 2.59 sometimes referred to as the eigenfunction and yields the mode shapes of the string.

) sin( ) ( x L n a x X_n = _n π , n=1,2,... 2.59

The solution to equation 2.57 with

L nπ

σ = n=1,2,...can be seen in equation 2.60

ct B ct A t T_n( )= _nsinσ_n + _ncosσ_n 2.60

In equation 2.60 it can be seen that the natural frequencies of the system is equal toσnc.

Linearity give rise to that the total solution can be written as sum according to equation 2.61. ) cos sin sin sin ( ) , ( 1 ct x d ct x c t x w n n n n n n n σ σ σ σ

∑

∞ = + = 2.61

The constants c and _n d are determined by the initial conditions with the use of _n

Fourier series. Forced response of the string can be solved with the use of modal analysis as presented later in the report.

Transverse vibration of a beam

Many structures can be analyzed by modeling them as sections of beams. Beams that fulfill the special properties below are referred to as Euler-Bernoulli beams. The special thing with these beams is that the vibration response can be solved quite easily by separation of variables as for the string. As it can be read in list below the

rotational inertia and shear deformation are ignored. This is a good approximation as long as the height and width of the beam are small compared with its length. There are other models that don’t ignore these effects, such as the Timoshenko beam but these aren’t as easy to solve.

Properties of Euler-Bernoulli beams

• Uniform along its span, or length, and slender

• Plane sections remain plain

• The plane of symmetry of the beam is also the plane of vibration so that rotation and translation are decoupled

• Rotary inertia and shear deformation can be neglected

The equation of motion for free transverse vibrations of an Euler-Bernoulli beam, Figure 3, is somewhat similar to the equation for the string. The equation however contains four spatial derivatives and can be seen in equation 2.62.

0 ) , ( ) , ( 4 4 2 2 2 = ∂ ∂ + ∂ ∂ x t x w c t t x w , A EI c

ρ

= 2.62

Figure 3 Vibrating beam

Assuming a solution on the form of equation 2.63 yields the separable equation 2.64. ) ( ) ( ) , (x t X x T t W = 2.63 2 2 ) ( ) ( ) ( ) ( _{=− =}

_ω

′′′ ′ t T t T x X x X c && 2.64

This allows equation to be rewritten as two separate differential equations each depending only on one variable. The time dependent equation takes the form of equation 2.65 and has a solution according to equation 2.66.

0 2 2 = + c T T&&

ω

2.65 t B t A t

0 ) ( ) ( 2 =       − ′′′ ′ X x c x X

ω

. 2.67

The solution to equation 2.67 becomes

x a x a x a x a x

X( )= ₁sin

β

+ ₂cos

β

+ ₃sinh

β

+ ₄cosh

β

. 2.68

EI A c 2 2 2 4

ω

ρ

ω

β

= = . 2.69

The relations between constants are determined by the boundary conditions which gives the mode shapes and the scaling is determined by initial conditions. The total solution written as a sum can be seen in equation 2.70.

∑

∞ = + = 1 ) ( ) cos sin ( ) , ( n n n n n n t B t X x A t x w

ω

ω

2.70

The figures below show the first three mode shapes for a two combinations of boundary conditions for an Euler-Bernoulli beam.

-1,2 -0,8 -0,4 0 0,4 0,8 1,2 0 0,25 0,5 0,75 1

First mode shape Second mode shape Third mode shape

-1,2 -0,8 -0,4 0 0,4 0,8 1,2 0 0,25 0,5 0,75 1

First mode shape Second mode shape Third mode shape

2.1.6 Modal analysis

If the natural frequencies and mode shapes are known the transient and stationary vibrations can be determined by modal analysis. This holds for structures which are linear.

Discrete modal analysis

This is a method that uses coordinate transformation to decouple the equations of vibration into separate equations which can be solved independently. The equations can then be transformed back to the original coordinate system. For more information see Inman (2001).

Analytical modal analysis

This version of the modal analysis is the equivalent method for analytical functions as the discrete is for matrices. From linearity it is known that any deflection of a

structure can be described by a weighted sum of its mode shapes, Åkesson et al. (1977).

∑

∞ = 1 ) ( ) ( ) , (x t q t X x w _{n n} . 2.71

The time dependent coefficientsq_n(t), referred to as modal displacement, can be seen as the displacement for an imaginary single degree of freedom system. One for each mode shape, with the mass, stiffness and damping according to Figure 6.

Figure 6 Imaginary single degree of freedom system

The calculation of the modal mass mn and modal stiffness kn can be performed according to equation 2.72 and equation 2.73.

The damping can be introduced as equivalent mode dampingc or the relative mode _n damping

ξ

_n. nk n n c c =

ξ

, c_nk =2 m_nk_n =2m_n

ω

_n. 2.74

The modal force Qn can be calculated according to equation 2.75.

∫

= L n n t W x t X x dx Q 0 ) ( ) , ( ) ( 2.75

This results in equation 2.76 which has known solutions. ) (t Q q k q c q

mn&&n + n&n + n n = n 2.76

Or n n n n n n n n n q q Q k q& +2

ξ

ω

& +

ω

2 =

ω

2 / & , n n n m k =

ω

. 2.77

2.1.7 Displacement method for Euler-Bernoulli beam -

structures

This method is sometimes referred to as the exact finite element method and uses simpler elementary cases for Euler-Bernoulli beams, which let more complex models to be broken down in to a combination of those simpler cases, Åkesson et al. (1977). Quite similar to the static method of elementary cases used in solid mechanics. The method states that

P p E(

ω

) = 2.78

[

]

t pj p p p= 1, 2,.., 2.79

[

]

t Pj P P P= ₁, ₂,.., 2.80

from the dynamic elementary cases and Kolouseks functions Åkesson (1977). The Kolouseks functions used later in this report can be seen in equations 2.81 to 2.88.

β

π

= u 2.81 2 / 1 4 2 0 / ( / ) /

ω

ω

π

EI mL

ω

β

= = 2.82

m is the mass per length unit.

) cos cosh 1 /( ) cos sinh sin (cosh 2 u u u u u u u k = − − 2.83 ) cos cosh 1 /( ) sin (sinh 4 2 u u u u u k = − 2.84 ) cos cosh 1 /( ) cos sinh sin (cosh 6 u3 u u u u u u k = + − 2.85 ) cos sinh sin /(cosh ) sin sinh 2 ( 7 u u u u u u u k = − 2.86 u u u u u u u u u

k9= 2(cosh sin +sinh cos )/(cosh sin −sinh cos 2.87 ) cos sinh sin /(cosh ) cos cosh 2 ( 11 u3 u u u u u u k = − 2.88

When working with this method, beam elements with constant cross section should be chosen. The force vector can be acting as distributed and/or in connection points of the beam element. Both the force and the displacement is assumed to be varying harmonically, synchronously, this means that p is the amplitude of the displacement and P the amplitude of the force.

Equation 2.78 can be used to calculate the natural frequencies of the structure for the special case that the vector P is equal to the zero vector equation 2.89. This means that there aren’t any external forces applied to the structure in the connection points or distributed over the length of the beam elements.

0

=

P 2.89

This reduces equation 2.78 to equation 2.90.

0 ) ( p=

E

ω

2.90

The necessary condition for solutions is

[

( )

]

0

For the case when E(

ω

) is a 2x2 matrix equation 2.91 becomes simply 0 ) ( ) ( ) ( ) ( _{22 21 12} 12

ω

E

ω

−E

ω

E

ω

= E . 2.92

2.1.8 Transfer functions and FRF

Frequency response function, FRF, is a method of describing a systems steady state response to harmonic excitation. The FRF can be calculated or measured with use of for example accelerometers or lasers. Here an example for a single degree of freedom system will be shown. Laplace transformation of equation 2.93 with initial conditions equals to zero yields equation 2.94 which represents the transformed solution.

) cos( 0 t F kx x c x

m&&+ &+ =

ω

2.93

(

2 2 2

)

0 )( ) (

ω

+ + + = s k cs ms s F s X 2.94

Of special interest is the transfer function H(s) defined in equation 2.95.

) ( 1 ) ( ) ( 2 H s k cs ms s F s X ₌ + + = 2.95

If s is set equal to jω the transfer function becomes

ω

ω

ω

jc m k j H + − = 1 ) ( 2.96

Equation 2.96 is called the complex frequency response function and describes the systems response to harmonic input. The physical vibration amplitude X0 can be calculated according to equation 2.97 and the phase

φ

between the force and displacement can be found as in equation 2.98.

) ( 0 0 F H j

ω

X = 2.97 ) (

ω

φ

=∠H j 2.98

2.1.9 Complex stiffness

) 1 ( * j k k = −

η

2.99

It can be shown that the energy dissipation for a sdof-system, with complex stiffness, driven by a sinus force corresponds to equation 2.100.

ω

η

k c

= 2.100

The damping depends on the driving frequency of the system. This can be extended to the young’s modulus of a material.

) 1 ( * j E E = +

η

2.101

2.1.10 FRF for displacement method

The frequency response for the displacement method can easy be found from the fundamental equationE(

ω

)p=P. Solving for p yields

P E

p= −1(

ω

) . 2.102

2.2 Machining

Metal cutting is widely used for shaping products and there are several different operations available such as turning, milling, drilling etc. In this report only milling operations are considered.

2.2.1 Mechanics of cutting

There are several different types of milling in Figure 7 shows two commonly used operations, face milling (left picture) and peripheral milling (right picture).

Figure 7 Face milling and peripheral milling.

There are a number of definitions of the tool used throughout this report and they are defined in Figure 8 and Figure 9.

Figure 8 Definitions of tool parameters used in the report

ÄNDRA D till DC

Cutting speed is the speed the cutting edge has relative to the workpiece and can be calculated as in equation 2.103, the diameter of the tool Dc is in mm and the spindle speed n in rpm. 1000

π

c c nD v = 2.103

2.2.2 Cutting forces

The number of variables that influence the cutting process is large, however some of the can be brought together by the concept of mean chip thickness, Lindström (1985). This concept is based on writing the chip area for any geometry of an insert, as the area of rectangle. By doing this the same equations can be used for calculation of for example cutting forces for a number of different insert geometries.

The chip area during cutting can be described with a rectangle as equation 2.104. The quantity he is referred to as the mean chip thickness and is a theoretical quantity that can be calculated according to equation 2.106. be is the active length of the cutting edge and can be approximated as in equation 2.105.

e eh b A= 2.104

(

)

2 180 sin cos 1 r f r a b r r r p e + + − − = ε ε

_κ

κ

κ

π

2.105

κr is the setting angle in degrees and rε is the corner radius.

2 180 sin ) cos 1 ( _{r f} r a f a h r r r p p e + + − − =

π

κ

κ

κ

ε 2.106

Figure 10 shows the forces acting on the tool during cutting. One model for calculation of the cutting forces is Kienzles model which can be seen in equation 2.107 to equation 2.112.

A K

F_c = _c 2.107

Kc is referred to as the specific cutting force coefficient and can be considered as a function of the equivalent chip thickness and material properties according to equation 2.108. Kc1 change due to material strength and mc due to material deformation hardening. c m e c c K h K = 1 − 2.108

The forces acting in radial and axial directions can be expressed according to equation 2.109 and equation 2.111. A K Fr = r 2.109 r m e r r K h K = 1 − 2.110 A K F_a = _a 2.111 a m e a a K h K = ₁ − 2.112

Figure 10 Cutting forces

The tangential force Fc and the radial force Fr are the primary forces acting on the tool.

F

c

F

a

2.2.3 Stability in machining

Chatter vibrations are an important problem that can occur during machining. It is an unstable process, caused by a self-excitation mechanism, resulting in large vibrations, high forces and leaving a rough surface behind. Excitation of vibration modes of the tool-workpiece system during machining is leaving a wavy surface behind. During the following revolution in turning, or next tooth in milling, the vibrating tool meets this newly made wavy surface resulting in a variation of chip thickness and forces. Depending on the phase shift between the two successive waves the amplitude of the vibrations may escalade.

Orthogonal cutting

As a simple model for analytical prediction of chatter an orthogonal cutting model is used. This model can be applied to for example turning operations. The tool cuts a smooth surface and vibrates with one of its natural frequencies leaving wavy surface behind, Figure 11. The next revolution the chip thickness varies as a function of time equation 2.113. T is the period of one revolution, Altintas (2000).

Figure 11 Regenerative waves, from Altintas (2000)

[

( ) ( )

]

)

(t h₀ y t y t T

h = − − − 2.113

The feed cutting force becomes )

( )

(t K a h t

F_f = _f . 2.114

If the workpiece is approximated as a single degree of freedom system the equation of motion can be expressed as 2.115.

[

( ) ( )

]

) (t k a h0 y t T y t F y y c y

m&&+ &+ = f = f + − − 2.115

2 2 2 2 ( ) ( n n n s s k s

ω

ξω

ω

+ + = Φ . 2.116

Laplace transformation of the cutting force and 2.116 yields the transfer function between the dynamic and reference chip thickness, equation 2.117.

) ( ) 1 ( 1 1 ) ( ) ( 0 s e K a s h s h f sT Φ − + = ₋ 2.117

Substituting s for jω makes the system critically stable. Arranging the characteristic equation into real and imaginary parts and equating to zero yields the equations 2.118, 2.119 and 2.120. ) ( ) ( tan c c G H

ω

ω

ψ

= 2.118

ψ is the phase shift of the structure’s transfer function .

ψ

π

ε

=3 +2 2.119

ε is the phase shift between the inner and outer waves. The critical axial depth of cut

can be calculated according to equation 2.120. The equation is valid only for the negative real part of the transfer function.

) ( 2 1 lim c fG K a

ω

− = 2.120

And the corresponding spindle speed can be calculated according to equation 2.121, k is the integer number of waves produced on the surface.

c f k T

π

ε

π

2 2 + = 2.121

Machining stability 0 10 20 30 40 50 60 70 3500 4500 5500 6500 7500 8500 9500 10500 11500 12500 13500 14500 Spindle speed [rpm] A x ia l d e p th o f c u t [m m ]

Figure 12 Example of stability plot

As it can be seen in equation 2.120 the critical axial depth of cut is inversely

proportional to the flexibility of the work piece, or tool. This means that a reduction in height for the peak in a structure’s FRF will allow for deeper cuts with out the

Milling

Milling adds more complexity to derivation of equations for stability limits, due to the fact that the chip varies sinusoidal in thickness as the tool rotates. A model of the milling system can be constructed as in Figure 13, Altintas (2000). With the coordinate transformation v_j =−xsin

φ

_j −ycos

φ

_j, where

φ

is the instantaneous angle of immersion for the tooth. The resulting dynamic chip thickness from inner and outer modulation can be seen in equation 2.122. j represents the tooth number, vj,0 and

vj are the dynamic displacements at previous and present tooth passing respectively, st is the feed rate per tooth and g(

φ

j)is a unit step function that determines if the tooth is active or not.

Figure 13 Chatter vibrations during milling operation

[

sin (

]

( )

)

( _j s_{t j} v_j,0 v_j g _j

h

φ

=

φ

+ −

φ

2.122

Transformation back to x,y coordinates yields

[

sin cos

]

( ) ) ( _j x _j y _j g _j h

φ

= ∆

φ

+∆

φ

φ

, 0 0 y y y x x x − = ∆ − = ∆ 2.123

[ ]

( ) ( ). 2 1 ) (t aK A t t F = _t ∆ 2.125 Where

[

]

∑

− = − + − = 1 0 ) 2 cos 1 ( 2 sin N j j r j j xx g K a

φ

φ

2.126

[

]

∑

− = + + − = 1 0 ) 2 sin ) 2 cos 1 ( N j j r j j xy g K a

φ

φ

2.127

[

]

∑

− = − − = 1 0 ) 2 sin ) 2 cos 1 ( N j j r j j yx g K a

φ

φ

2.128

[

]

∑

− = + − = 1 0 ) 2 cos 1 ( 2 sin N j j r j j yy g K a

φ

φ

2.129

and N is the number of teeth.

As it can be seen from the equations the cutting forces are periodic at the tooth passing frequenciesω = NΩ. This makes it possible to describe by Fourier series. Altintas (2000) showed that the dynamic milling forces have strength only at chatter frequency and at tooth passing frequencies away from the chatter frequency. And their experiments showed that of those frequencies only the chatter frequency excites the structure. This is because the nature of cutting tends to low pass filtering the overtones, see Figure 14. Exceptions can be if the structure has closely spaced modes with natural frequencies distributed at tooth passing frequency intervals. From this knowledge the stability can be calculated. Only the results are presented here, for more details see Altintas (2000). For two orthogonal degrees of freedom the resulting critical depth of cut and the corresponding spindle speed are:

(

ε

π

)

ω

k T c 2 1 ₊ = 2.134

ψ

π

ε

= −2 2.135

( )

κ

ψ

=arctan 2.136 ) ( _c xx i

ω

Φ and Φ_yy(i

ω

_c)are the transfer functions. Λr is the real part of Λ and ΛI is the imaginary part of Λ.

Figure 14 Block diagram of chatter in milling, from Altintas ( 2000)

Additional effects

2.2.3.1

Influence of workpiece

If both the workpiece and the tool are considered to be flexible, the relative transfer function must consist of the combined transfer functions for the tool and workpiece in

x and y direction respectively, Bravo et al. (2005)

2.2.4 Cutting data

When choosing the cutting data there are several aspects to consider such as

• Available machine power

• If the tool is strong enough to withstand the cutting forces

• Demands for tolerances of the machined part

• Minimal machining cost

• Maximal production rate

• Stability in cutting

Endurance

The following equations for endurance arise from Lindström (1985). Taylor’s equation for endurance states that

T

cT C

v α = 2.139

Where T is the endurance time in minutes, α is Taylor exponent and CT is a constant. Equation 2.139 is only valid for a specific set of tool geometry, feed and depth of cut. However the concept of equivalent chip thickness can be used for endurance as well. The equation for endurance time becomes

0 ) ( C h T v mhe e c = α _2.140

The exponent m is a function of he and C0 is a material constant. To simplify, the equation can be reduced to equation 2.141 where m often takes the value of 0.5. 0 C h T vc em = α 2.141

Economic endurance time for minimized machining cost

The economic endurance time for minimized machining cost is presented below. It is based on equation 2.139 and is only valid for a specific feed as stated above.

m v ce K K T 1 160      − =

α

[min] 2.142

α is the inclination of the Taylor line, Kv is the cost for one insert and Km is the machine cost per hour. An expression that takes both cutting speed and chip thickness in consideration can be derived from the concept of equivalent chip thickness and can be seen in equation 2.143. v e c T T T h v p + = 2.143

vb m v v t K K T =60 + 2.144

tvb is the time for changing one insert [minutes].

Life time for maximal production rate

b cm t T       ₋ = 1 1

α

2.145

tb is the time for changing one insert..

α

−

= cm

cm CT

v 2.146

2.2.5 Additional supports for workpiece

From chapter 2 is known that the lowest stability limit, for a particular set of cutting conditions, is set by the maximum value of the negative real part of the transfer function for the tool-workpiece-system. For each separate system this maximum amplitude occurs close to natural frequency. One way to improve stability would be to minimize this maximum value. If the dynamic stiffness is much higher for the tool than for the workpiece one approximate way to try to maximize this absolute stability limit is to minimize the FRF (displacement) peaks for the workpiece. To achieve this additional supports for the workpiece can be added. The frequency response function or flexibility for the structure in Figure 16, varies as the forcing function moves in along the beam. The height for a peak in the FRF corresponding to a particular mode increases as the forcing functions moves towards the antinodes of its mode shape and decreases when moved towards the nodes. If an additional support were to be

2.3

Dynamics of a beam during machining

To get an insight of the dynamic behavior of a workpiece-tool-system during

mashining it’s important to understand what is happening to the natural frequency’s, mode shapes and FRF of the workpiece, through different phases of machining. The model in Figure 15 represents a beam being machined by peripheral milling operation with its axial depth of cut being equal to width of the beam.

Figure 15 Peripheral milling model

As an approximate simple model to illustrate this, a beam with at step in height positioned at a distance from αL from end point was used.

Figure 16 Beam model with modeled step at αL

From chapter 2 it’s known that the main forces during peripheral milling are acting in the plane of cut. In this model the only the y-direction is considered to be flexible because of the much higher stiffness in the x-direction.

It is desired to know how the natural frequencies changes as the step moves along the length-axis of the beam. One would also be interested in if there are any differences in how the natural frequencies changes, depending on the boundary conditions of the beam ends.

2.4 Method for beam calculations

The natural frequencies and mode shapes can be calculated in more then one way, for

choice was made to use both an analytical method, based on so called displacement model for Euler–Bernoulli beam structures and FEM models. This decision was made mainly in believe it would give the author a chance for deeper knowledge and

understanding in the area, but also because the displacement method gives an exact solution to the model with in the approximations made. For both FEM and

displacement calculations the model in Figure 16 used. In FEM the choice of using beam models was made. This was partly to save calculation time due to the many calculations performed and partly to be able to compare the results with the analytical model made. Pro/Mechanica, which has been used for FEM calculations for the beam models, uses an adaptive mesh up to 9 degree order, which degree used is determined by the convergence criteria set. To create a denser mesh points can be added to the model in calculation. The author had problem to get the design studies, which was used a lot in this project, to work when this was done. This resulted in not using the option of additional points for many calculations, resulting in a low number of elements and high degrees of polynomials to get good convergence.

Control calculations for some sets of beams, steps and boundary conditions showed that both methods seemed give the same results. For plotting frequency response functions and mode shapes etc from PRO/Mechanica the results where exported to text files. These files were imported to Excel where they were investigated further and plotted.

For the displacement method the beam with a step can be divided in to two separate pieces both with an unequally but constant height. Two degrees of freedom are introduced in the beam elements connection point, the rotation θ and the “translation”

δ Figure 17.

Figure 17 Degrees of freedom for the beam model connection point

The problem is to find the dynamic stiffness matrix E(

ω

)one method is to first set the displacements are to δ =1and θ =0 It is easily seen that this corresponds to the left elementary case in Figure 18 and the stiffness elements are

9 2 2 9 2 1 21 ) ( ) ) 1 (( ) ( K L EI K L EI E

α

α

ω

− − = 2.148

Next the displacements are set to δ =0andθ =1.This corresponds to the right elementary case in Figure 18. The stiffness elements are

9 2 2 9 2 1 21 ) ( ) ) 1 (( ) ( K L EI K L EI E

α

α

ω

− − = 2.149 7 2 7 1 22 ) 1 ( ) ( K L EI K L EI E

α

α

ω

+ − = 2.150

Figure 18 Dynamic elementary cases

Assembling of the stiffness elements gives the matrixE(

ω

).

            + − − − − − + − = 7 2 7 1 9 2 2 9 2 1 9 2 2 9 2 1 11 3 2 11 3 1 ) 1 ( ) ( ) ) 1 (( ) ( ) ) 1 (( ) ( ) ) 1 (( ) ( K L EI K L EI K L EI K L EI K L EI K L EI K L EI K L EI E

α

α

α

α

α

α

α

α

ω

2.151

As expected the stiffness matrix is symmetric.

From chapter 2 it is known that equation 2.92 can be used to calculate the natural frequencies of a structure with two degrees of freedom.

Because of the complexity of the Kolouseks functions

ω

must be determined

numerically with the help of computer software. With the natural frequencies known the eigenvector can be determined .This is done by putting the known frequencies into equation 2.153 and solving for the displacement vector p.

To make the necessary calculations the mathematics software Maple 9.5 was used.

Method for calculation of mode shapes for displacement method

To calculate the mode shapes for the combined beam model equation 2.155 for mode function can be used. The first step is to calculate the eigenvector for each natural frequency. This is done by solving for the displacement vector p in equation. This is easy, particular in the case when the stiffness matrix E is a two by two matrix, due to the fact that the elements of the matrix E are constant when evaluated at natural calculated frequency.

0 ) ( p=

E

ω

_n 2.153

The solution becomes

12 11 12 11 p E E p =− 2.154

This gives a relation between the angular and vertical displacement in the connection point. The mode function from transverse vibration of an Euler-Bernoulli beam is valid for each beam section.

x a x a x a x a x

X( )= 1sin

β

+ 2cos

β

+ 3sinh

β

+ 4cosh

β

2.155

With the boundary conditions known for each beam section end point and with the solution of eigenvector giving rise to the connection point boundary conditions, the constants in mode function can be solved. One constant can of course not be

determined without initial conditions. The constant remaining undetermined scales the beam section mode shapes and makes it possible to connect the two sections by making the right choice. In the computations made the undetermined constants was solved by the putting the one of the constants equal to a number and solving the other in such way that the deflection in the connection point became equal. The mode shapes were than scaled in such way that the maximum values were equal to 1.

Calculation of FRF

complex modulus is good way to introduce damping compared to for example the difficulties the use of proportional damping would bring.

The FRF for the beam model with complex modulus can be calculated as it was proposed earlier, with the addition that absolute value of the equation must be taken to get the physical vibration amplitude. The calculations of FRF for different support locations have been performed in PRO/Mechanica with the use of beam models. Calculations of the FRF in FEM were carried out by applying a harmonic force of one Newton in the transverse direction at the connection point of the beam sections, and measure the resulting transverse direction displacement in the same point

2.4.1 Calculation of machining stability limits

Calculations of machining stability limits were performed in both in Maple 9.5, as an orthogonal cutting approximation with the displacement model as base, and in the software CutPro 8.1 for the milling process. Only the results from CutPro are presented in this report. The workpiece was considered to be rigid in its length-direction due to the much higher stiffness. The FRF was calculated in

PRO/Mechanica and the resulting displacement/phase values where extracted to text files. The imaginary and real part of the FRF was calculated from the displacement and phase in Excel and was later imported to CutPro to reassemble the FRF which was used as a base for the stability limits calculations. Measured FRF for real tools was used in stability limits calculations for the combined Machine/tool-workpiece structure.

2.5 Jet engine casing

Figure 19 Jet-engine casing

Milling operations on the casing can be performed in several different ways. Face milling in a turn-milling operation is common. However the regenerative effect is probably most likely to appear during operations similar to Figure 20 with a vertical positioned tool for finishing operations.

Clamping sections was set at inner edge of the lower and upper flange (arrows). Each section of the edge being clamped was 6 degrees. Simulations of the casing were made in FEM software PRO/Mechanica and Ansys Workbench and cutting simulation program CutPro 8.1. As a first preparatory phase natural frequencies and mode shapes was calculated for five different cases of clamping sections. From the results two cases was chosen to be investigated further. The next phase consisted of calculation of FRF at different points of the structure for the chosen cases. And the final part was to use the FRF results to calculate the machining stability limits.

3 Results for beam model, no support

This chapter presents the results from calculation of natural frequencies, mode shapes, frequency response functions and stability limits for the beam model without any additional support. Both the displacement method and the FEM software

PRO/Mechanica were used. The step location presented in this chapter is equal to the

α parameter in percentages and the step height is equal to the parameter s in

percentages. The relative values of natural frequencies are compared to a prismatic beam with original step height.

3.1 Natural frequencies

The natural frequencies below are calculated with the use of the displacement method model. The dimensions and material parameters for natural frequency calculation were according to Table 1.

Table 1 Beam dimensions and material parameters

Length [mm] 300

Width [mm] 30

Height [mm] 10

Density [Kg/m3] 7850 Young’s modulus [GPa] 200

3.1.1 Boundary conditions pinned-pinned

The calculated first and second natural frequency for different step height and step locations can be seen in Figure 21 to Figure 24.

100 120 140 160 180 200 220 240 260 F re q u e n c y [ H z ] 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Step location [%] Step heig ht [%_]

50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 0% ₁₀% ₂₀% ₃₀% ₄₀% ₅₀% ₆₀% ₇₀% ₈₀% ₉₀% 100% Step position F re q u e n c y 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

Figure 22 First relative natural frequency for a pinned – pinned beam as a function of position and height of step 400 500 600 700 800 900 1000 1100 F re q u e n c y [ H z ] 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Step location [%] Step heig ht [%_]

Figure 23 Second natural frequency for a pinned – pinned beam as a function of position and height of step 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 0% ₁₀% ₂₀% ₃₀% ₄₀% ₅₀% ₆₀% ₇₀% ₈₀% ₉₀% 100% Step position F re q u e n c y 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

Figure 24 Second relative natural frequency for a pinned – pinned beam as a function of position and height of step

3.1.2 Boundary conditions clamped-clamped

250 300 350 400 450 500 550 600 F re q u e n c y [ H z ] 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Step location [%] Step heig ht [%_]

Figure 25 First natural frequency for a clamped – clamped beam as a function of position and height of step 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 0% ₁₀% ₂₀% ₃₀% ₄₀% ₅₀% ₆₀% ₇₀% ₈₀% ₉₀% 100% Step position F re q u e n c y 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

Figure 26 First relative natural frequency for a clamped – clamped beam as a function of position and height of step 700 800 900 1000 1100 1200 1300 1400 1500 1600 F re q u e n c y [ H z ] 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Step location [%] Step heig ht [%_]

50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 0% ₁₀% ₂₀% ₃₀% ₄₀% ₅₀% ₆₀% ₇₀% ₈₀% ₉₀% 100% Step position F re q u e n c y 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

Figure 28 Second relative natural frequency for a clamped – clamped beam as a function of position and height of step

3.1.3 FEM model comparison

As a check, the results of the natural frequency calculation from the analytical method was compared with results from calculations with FEM software PRO\Mechanica. A beam model with step height 50 percent, length 300 mm, width 30 mm and original height of 10 mm was used. The young’s modulus was set to 200 GPa and the density to 7850 kg/m3 In FEM model the displacement for the beam model was locked in such way that it motion was only allowed in the plane defined in Figure 17. For boundary condition pinned - pinned no motion was allowed in the beam length direction for any of the beam end points. The result can be seen in Figure 29 and Figure 30 plotted together with the results from the displacement method calculations. Figure 31 and Figure 32 shows a comparison between the FEM model and the

displacement method model with a lower length to thickness ratio for the beam for boundary conditions clamped - clamped. As it can be seen the differences between the models becomes larger for a thicker beam. This may be due to the beam model

including shear stiffness.

First natural frequency FEM and displacement method comparsion 0 50 100 150 200 250 300 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Step location F re q u e n c y [ H z ]

FEM Displacement method

Second natural frequency FEM and displacement method comparsion 0 200 400 600 800 1000 1200 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Step location F re q u e n c y [ Hz ]

FEM Displacement method

Figure 30 Second natural frequency comparison, pinned - pinned

First natural frequency comparison, clamped - clamped, 500x70x30 beam 0 200 400 600 800 1000 1200 1400 0 50 100 150 200 250 300 350 400 450 500 Step location [mm] F re q u e n c y [ H z ]

FEM beam model Displacement method

Figure 31 First natural frequency comparison non-slender beam, clamped – clamped

Second natural frequency comparison, clamped - clamped, 500x70x30 beam 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 150 200 250 300 350 400 450 500 Step location [mm] F re q u e n c y [ H z ]

FEM beam model Displacement method

Figure 32 Second natural frequency comparison non-slender beam, clamped – clamped

3.2 Mode shapes for 50 percent step height

α-presented is in transverse direction. For the plots from displacement method model the step moves from 300 to 0 while as for the FEM model it moves from 0 to 300. That’s why the figures from displacement method looks mirrored compared to the figures from FEM calculations.

3.2.1 Boundary conditions pinned-pinned

First mode shape boundary condition pinned-pinned

0 0.2 0.4 0.6 0.8 1 1.2 0 50 100 150 200 250 300 alpha 0.6 alpha 0.8 alpha 0.1

Figure 33 First mode shape for different step positions where α denotes relative position for the step. Observe that the amplitudes are normalized

Second mode shape boundary conditions pinned-pinned

-1.5 -1 -0.5 0 0.5 1 1.5 0 50 100 150 200 250 300

alpha 0.6 alpha 0.8 alpha 0.1