OPTIMAL DESIGN OF A DAMPED ARBOR FOR HEAVY DUTY MILLING ( M.Tech Machine Design Project )-2

5. DAMPING 5.1Definition of Damping In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscil...

5. DAMPING

5.1Definition of Damping

In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator. In mechanics, friction is one such damping effect. In engineering terms, damping may be mathematically modeled as a force synchronous with the velocity of the object but opposite in direction to it. If such force is also proportional to the velocity, as for a simple mechanical viscous damper (dashpot), the force F may be related to the velocity v by F= -cv , where c is the viscous damping coefficient, given in units of newton-seconds per meter.

Figure: 5.1.1 Mass spring damper system

An ideal mass-spring-damper system with mass m (kg), spring constant k (N/m) and viscous damper of damping coefficient c (in N-s/ m or kg/s) is subject to an oscillatory force and a damping force,

                         F_s   = -kx              F_d = -cv = -c(dx/dt) = -cẋ    

  Treating the mass as a free body and applying Newton's second law, the total force F_tot on the body

  F_tot = ma = m(d2x/dt2) = mẍ.

Since F_tot = Fs + Fd, then => mẍ = -kx + -cẋ

This differential equation may be rearranged into

ẍ +ẋ + x = 0.          ω_o =  √k/m               ζ = c / 2√mk

ω₀, is the (undamped) natural frequency of the system and ζ, is called the damping ratio.

5.2 Types of Damping

Three main types of damping are present in any mechanical system:

1) Internal damping (of material)

2) Structural damping (at joints and interfaces)

3) Fluid damping (through fluid-structure interactions)

5.2.1 Material (Internal) damping

Internal damping of materials originates from the energy dissipation associated with microstructure defects, such as grain boundaries and impurities; thermo elastic effects caused by local temperature gradients resulting from non uniform stresses, as in vibrating beams eddy current effects in ferromagnetic materials; dislocation motion in metals; and chain motion in polymers. Several models have been employed to represent energy dissipation caused by internal damping. This variety of models is primarily a result of the vast range of engineering materials no single model can satisfactorily represent the internal damping characteristics of all materials.

5.2.2 Structural damping

Rubbing friction or contact among different elements in a mechanical system causes structural damping. Since the dissipation of energy depends on the particular characteristics of the mechanical system, it is very difficult to define a model that represents perfectly structural damping.

 The Coulomb-friction model is as a rule used to describe energy dissipation caused by rubbing friction. Regarding structural damping (caused by contact or impacts at joins), energy dissipation is determined by means of the coefficient of restitution of the two components that are in contact. Assuming an ideal Coulomb friction, the damping force at a join can be expressed through the following expression:

f =c.sgn( q& )

where:

f = damping force,

q& = relative displacement at the joint,

c= friction parameter

and the signum function is defined by:

sgn (x) = 1 for x ≥ 0

sgn (x) = -1 for x < 0

5.2.3 Fluid damping

When a material is immersed in a fluid and there is relative motion between the fluid and the material, as a result the latter is subjected to a drag force. This force causes an energy dissipation that is known as fluid damping.

The damping phenomenon can be applied to the machine tool systems in two ways :

1. Passive damping

2. Active damping

Passive damping refers to energy dissipation within the structure by add on damping devices such as isolator, by structural joints and supports, or by structural member's internal damping. Active damping refers to energy dissipation from the system by external means, such as controlled actuator.

5.3 Damping in machine tools

Damping in machine tools basically is derived from two sources--material damping and interfacial slip damping. Material damping is the damping inherent in the materials of which the machine is constructed. The magnitude of material damping is small comparing to the total damping in machine tools. A typical damping ratio value for material damping in machine tools is 0.003. It accounts for approximately 10% of the total damping. The interfacial damping results from the contacting surfaces at bolted joints and sliding joints. This type of damping accounts for approximately 90% of the total damping. Among the two types of joints, sliding joints contribute most of the damping. Welded joints usually provide very small damping which may be neglected when considering damping in joints.

System/Materials	Loss Factor
Welded Metal structure	0.0001 to 0.001
Bolted Metal structure	0.001 to 0.01
Aluminium	0.0001
Brass, Bronze	0.001
Beryllium	0.002
Lead	0.5 to 0.002
Glass	0.002
Steel	0.0001
Iron	0.0006
Tin	0.002
Copper	0.002
Plexiglas TM	0.03
Wood, Fiberboard	0.02

Table: 5.3.1 Typical damping values of different materials

5.4 Effects on Work Material Properties

Mechanical properties of the workpiece may be affected with a built-up edge or dull tool. Arbor Milling can create an untempered martensitic layer on the surface of heat-treated alloy steels, about 0.001 in. thick. Other materials are affected very little by arbor in milling.

5.5 Springs

A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. There are a large number of spring designs; in everyday usage the term often refers to coil springs.

Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication. Some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current (because of its low electrical resistance). The kinds of device in which plastic materials have been used successfully are, first, those which store energy; secondly, those which absorb energy by permanently deforming or fracturing; and thirdly, those cushioning devices which dissipate energy through frictional heat.

Springs can be classified depending on how the load force is applied to them

·         Tension/extension spring – the spring is designed to operate with a tension load, so the spring stretches as the load is applied to it.

·         Compression spring – is designed to operate with a compression load, so the spring gets shorter as the load is applied to it.

·         Torsion spring – unlike the above types in which the load is an axial force, the load applied to a torsion spring is a torque or twisting force, and the end of the spring rotates through an angle as the load is applied.

·         Constant spring - supported load will remain the same throughout deflection cycle

·         Variable spring - resistance of the coil to load varies during compression

·         Ideal Spring – the notional spring used in physics: it has no weight, mass, or damping losses.

In this project SS springs are used for the damping in the Arbor which has stiffness value of    2.02X10⁶ N/m. For applications requiring more specialized springs, Lee Spring has the advanced technology to provide custom LeeP Compression Springs to meet the most exacting specifications. Springs are rust-proof, recyclable and lightweight. Standard polycarbonate compression springs are available in include heavy gauge steel metal ends Applications include suspension system, vibrations, punching, stamping, engine mounts &bushings.

 5.6 Mass Material

Cemented tungsten is used for mass material, it is placed between the two springs in side Arbor. It has a high density of 16X10³ kg/m³. Arbor is connected to the tool and it is used for milling operation that removes the work piece material. Where tool is subjected to impact loadings during the operation. Here comes the vibration effect to the tool and the arbor connected to it besides tool holder. Mass material is the one which controls the vibrations occur in milling operation by providing the damping effect, where spring acts as the elastic material that absorbs the load and minimizes the max amplitude.

Cemented Tungsten

Material			Cemented Tungsten
Property	Minimum Value (S.I.)	Maximum Value (S.I.)	Units (S.I.)	Minimum Value (Imp.)	Maximum Value (Imp.)	Units (Imp.)
Density	15.03	15.88	Mg/m3	952.027	991.357	lb/ft3
Poisson's Ratio	0.2	0.22		0.2	0.22	NULL
Tensile Strength	370	530	MPa	53.664	76.87	Ksi
Young's Modulus	600	686	GPa	87.0226	99.4958	106 psi

Table: 5.6.1 Cemented Tungsten properties

  

6. MODELING OF VIBRATION SYSTEM OF DAMPED ARBOR

6.1 Model of damped arbor

As shown in Fig.6.1.1, chatter stability is affected by the dynamic stiffness of cutting tools, cutting conditions, material property of work piece, and so on. A block diagram of the system representing self excited chatter vibration of a cutting process is shown in Fig.6.2.1 According to the diagram, the initial uncut chip thickness which is determined by cutting conditions is input into the system and converted to a cutting force. The cutting force causes tool displacement which affects both an instantaneous chip load and chip load at the next cutting with a time delay.

Figure: 6.1.1 Damped Arbor

   

A model of a typical damped arbor, composed of a mass damper inside a hollow space close to the tool side end, is shown in Fig.6.2.1 The mass damper can increase the dynamic stiffness or decrease the compliance of the structure, resulting in the decrease in the displacement.

Figure: 6.1.2 Block diagram of a system of self excited chatter vibration.

6.2 Optimization of a mass damper for a damped arbor with generalized profile

A straight arbor body with a tapered hollow space, a generalized shape of the arbor to be optimized in this study, is shown in Fig.6.2.1

Fig: 6.2.1 Model for dynamic stiffness of the hollow tool arbor.

The left end of the arbor with length of l_s represents the contribution of the spindle and tool holder to the vibration system and static displacement. Its length, Young’s modulus Es ,and density ρ_s are determined. For numerical analysis of vibration of the arbor, the body and the additional body are divided into n disk elements. As shown in the figure, the i th element at a position z_i from the fixed left end has an outer diameter D_i ,inner diameter d_i  and thickness Δz. Its weight w_i  and geometrical moment of inertia I_i are expressed as follows. density as taken from the material properties which is 8 x 10⁶ kg/m³

w_{i =}  density x volume

= 3.616kgs

I_i = (D_i⁴- d_i⁴)

Where ρ_a is the density of the arbor body. Moment M_i, bending angle increment Δθ_i, and

deflection u_i  at position z_i for a vertical static force P applied to the right end of the arbor

expressed as

I_i =  (D_i⁴- ( x m x h x 26 x D₀⁴-49 x d_i + 21 x D₀² x d_i² - 9 x d_i³ + (d_i⁴)))

= (47⁴- ( x 1.8 x 40 x 26 x 32⁴ - 49 x 40 + 21 x 32² x 40² - 9 x 40³  + (40⁴)))

= 3.8x10⁷  kg/m³

M_i = P(L - Z_i)

where L is the length of the arbor including the additional body and a cutter. Young’s modulus

E_i = E_s for z_i < l_s and E_i = Ea for  z_i ≥ l . Thus the deflection of the n th element at the end of the tool is

Then, the equations of the motion are expressed as follows;

k₁ x₁ + c₁ x₁ + m₁ x₁ − k₂ (x₂ −α ⋅ x₁ )− c₂ (x₂ −α ⋅ x₁ )= f₁

k₂ ( x ₂ −α x₁ )+ c₂ (x₂ −α x₁ )+ m₂ x₂ = 0.

Namely,

This equation is then expressed in a formulation

[K ]{X }+ [C ]{X }+ [M ]{X }= {F }.

Then,                                              

{X }= [φ ]{F}.

Where the compliance of the vibration system is given by

[φ ] = ([K ] + jω[C] − ω²[M ])⁻¹.

Here damping coefficients c₁ and c₂ is given by

       Fig: 6.2.2  Consideration of the connecting position between tool holder and the mass.

          Before the optimization, some sizes associated with the arbor were determined in advance as shown in Fig.6.2.3 The length and diameter of the damped arbor are 400 mm and 47 mm, respectively. The former includes the length of the additional body 50 mm and that of the milling cutter 50 mm. In the hollow space, both ends of the mass are supported by 10 mm-thick elastic material. The radial gap between the tool arbor and the mass is 1mm. Let l₁, d_a and d_b denote length of the hollow space, hollow diameter at the tool side end and hollow diameter at the tool holder side end, respectively. The material of the mass is cemented tungsten which has a high density of 16×10³ kg/m³.

Fig: 6.2.3 Geometry of damped arbor.

First, Young’s modulus and density of the additional body were determined so that the maximum amplitude and natural frequency calculated for a solid arbor were consistent with measured results. Then, stiffness k₂ that minimizes the maximum negative real part value of the compliance R_n_max was obtained for a particular length of straight hollow space, l₁ = 150mm and d_a = d_b = 36 mm. Next, the length and diameters of the straight and tapered hollow spaces were optimized by using the optimized stiffness of supporting rubbers k₂__opt .As a method for optimal design of a two degrees of freedom system, the fixed point theory is usually used to minimize the maximum absolute value of compliance ϕ_max.

However, the conditions for maximizing the chatter free depth of cut are different from the conditions for minimizing ϕ_max . As factors of self excited chatter vibration, regenerative chatter and mode coupling are known. In this research, chatter suppression performance is assumed to be evaluated by maximum negative real part value of the compliance by average tooth angle approach in order to establish the optimal design method of damped tool arbor.

  

6.3 Analytical Result of Selecting Optimum Damped Arbor

6.3.1 Cylindrical hollow space

In order that the maximum amplitude and natural frequency agree with the measured results, material properties of the spindle and tool holder are given artificially. In this research, the material property of the 50 mm area of the spindle side of the arbor such as Young's modules E_s, density ρ_s, and damping ratio was tuned. First, the static compliance of the solid arbor, which has the same geometry as the objective arbor (diameter 47 mm, length 400mm including cutting tool) was identified.

Static compliance was calculated from the displacement at the end of the arbor when static force was loaded. The static compliance which is measured by at the end of the arbor was 4.39×10^-6 N/m. The compliance at low frequency corresponds to the static compliance which is expressed as 1/K_s where K_s is static stiffness. In order that the static compliance and natural frequency agree within 5% error, it is assumed that the element in an area with length of 50 mm at the end of machine tool side has Young’s modulus 1.9×102 MPa and density 2×10¹¹ kg/m³.Damping ratio is assumed to be 0.04 in order that the absolute value of the amplitude agrees with the calculation results within 5% error.

The cylindrical hollow properties are taken from reference 2.

      Figure: 6.3.1 Static compliance of solid arbor

The modal parameters of the mass are damping ratio ζ₂ = 0.463, m₂ =1.74 kg.

The relationship between spring constant k₂ and R_n__max(k₂) is shown in Fig. 6.3.2 This figure indicates that the minimum value of R_{n_max}(k₂) is given at a spring constant of 2.02×106N/m. It is confirmed that the value of k₂ that minimizes R_{n_max} is quite different from that which minimizes φ_min. According to Fig. 6.3.2, the optimum spring constant k₂__opt is 2.02×106 N/m, which is used in the calculations described below.

Fig: 6.3.2 Relation between spring constant and maximum negative real part of compliance

Misalignment of the center of arbor and mass causes the unbalance of the mass during arbor revolution, and it may affect the natural vibration and forced vibration of the arbor.

When the spindle revolution speed is S=636 min-¹, oscillation frequency due to the unbalance is around 11 Hz. Since it is very small compared with the natural frequency of the arbor. Effects of the unbalance on the natural vibration of the arbor is very small. Unbalance mass mu is expressed as;

m_u = m2 e/ra

                         

where r_a is the radius of the mass and e is the distance of the center of arbor and mass. If the inner diameter of hollow space is da=36 mm, outer diameter of the mass is r_a=17 mm since the gap between hollow space and mass is 1mm. If the unbalance is e=1mm, the m_u value finds from above equation, m_u= 0.102 kg. Centrifugal force F_u due to the unbalance of the mass is expressed as

F_u = m_ue(2π(s/60))

The spindle revolution speed S=636 min^-1 derives F_u=0.45 N which is considered to be very small compared with cutting forces. Thus, the effects of the misalignment of the center of arbor and mass is assumed to be negligible in this configuration.

6.3.2 Tapered hollow space

The absolute and real part values of the compliance calculated for three spring constants, k₂ = 1.01×10⁶ N/m, 2.02×10⁶ N/m and 4.04×10⁶ N/m and remaining values are same as per the hollow arbor. The relationships between R_n__max−K_s in the case of the tapered and straight hollow of l₁ is compared. A vertical line for the threshold of the static stiffness is also shown in each graph of Fig. 6.3.3.  The smaller diameter d_b of the tapered hollow space changes from 36 mm to 28 mm, while the larger diameter d_a is always 40 mm. These results indicate that the tapered hollow space always decreases |R_{n_max}| if the static stiffness is constant. The improved value increases with increasing length l₁ and decreasing smaller diameter d_b. The best point in the R_{n_max}−K_s plane for each length l₁ is indicated by an open circle. Accordingly, the optimum values of l₁ and d_b for a tapered hollow space were determined to be  l₁ = 150 mm, d_b = 32 mm, respectively. The  optimal point for the straight hollow space indicated by a triangle is very close to that for the tapered hollow space. Therefore, the damped arbor with the optimum shape of the straight hollow space with diameter of 36 mm and length of 150 mm is called the optimum damped arbor hereafter and used in the experiments and analysis described below. It should be noted that if the threshold for the static stiffness is higher than the value set in this study, a tapered hollow space must be adopted because the gap of compliance between the two types of hollow space is large.

Fig: 6.3.3  Static and dynamic stiffness

7. CATIA INTRODUCTION

CATIA (Computer Aided Three-dimensional Interactive Application) is a multiplatform CAD/CAM/CAE commercial software suite developed by the French company Dassault Systems directed by Bernard Charles. Written in the C++ programming language, CATIA is the cornerstone of the Dassault Systems software suite.

CATIA (Computer Aided Three-Dimensional Interactive Application) started as an in-house development in 1977 by French aircraft manufacturer Avions Marcel Dassault, at that time customer of the CAD/CAM CAD software to develop Dassault's Mirage fighter jet. It was later adopted in the aerospace, automotive, shipbuilding, and other industries.

7.1 Scope Of Application

multiple stages of product development (CAx), including conceptualization, design (CAD), engineering (CAE) and manufacturing (CAM). CATIA facilitates collaborative Commonly referred to as a 3D Product Lifecycle Management software suite, CATIA supports engineering across disciplines around its 3DEXPERIENCE platform, including surfacing & shape design, electrical fluid & electronics systems design, mechanical engineering and systems engineering.

CATIA facilitates the design of electronic, electrical, and distributed systems such as fluid and HVAC systems, all the way to the production of documentation for manufacturing

7.2 Mechanical engineering

CATIA enables the creation of 3D parts, from 3D sketches, sheet metal, composites, molded, forged or tooling parts up to the definition of mechanical assemblies. The software provides advanced technologies for mechanical surfacing & BIW. It provides tools to complete product definition, including functional tolerances as well as kinematics definition. CATIA provides a wide range of applications for tooling design, for both generic tooling and mold & die.

7.2.1 Design

CATIA offers a solution to shape design, styling, surfacing workflow and visualization to create, modify, and validate complex innovative shapes from industrial design to Class-A surfacing with the ICEM surfacing technologies. CATIA supports multiple stages of product design whether started from scratch or from 2D sketches. CATIA is able to read and produce STEP format files for reverse engineering and surface reuse.

7.2.2 Fluid systems

CATIA offers a solution to facilitate the design and manufacturing of routed systems including tubing, piping, Heating, Ventilating & Air Conditioning (HVAC). Capabilities include requirements capture, 2D diagrams for defining hydraulic, pneumatic and HVAC systems, as well as Piping and Instrumentation Diagram (P&ID). Powerful capabilities are provided that enables these 2D diagrams to be used to drive the interactive 3D routing and placing of system components, in the context of the digital mockup of the complete product or process plant, through to the delivery of manufacturing information including reports and piping isometric drawings.

7.3 Solid Modeling:

Solid modeling (or modeling) is a consistent set of principles for mathematical and computer modeling of three-dimensional solids. Solid modeling is distinguished from related areas of geometric modeling and computer graphics by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of computer-aided design and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.

The use of solid modeling techniques allows for the automation of several difficult engineering calculations that are carried out as a part of the design process. Simulation, planning, and verification of processes such as machining and assembly were one of the main catalysts for the development of solid modeling. More recently, the range of supported manufacturing applications has been greatly expanded to include sheet metal manufacturing, injection molding, welding, pipe routing etc.

Beyond traditional manufacturing, solid modeling techniques serve as the foundation for rapid prototyping, digital data archival and reverse engineering by reconstructing solids from sampled points on physical objects, mechanical analysis using finite elements, motion planning and NC path verification, kinematic and dynamic analysis of mechanisms, and so on.

7.4 Benefits of CATIA:

1. It is much faster and more accurate than any CAD system.

2. Once design is complete, 2-D and 3-D views are readily obtainable.

3. The ability to change in late design process is possible.

4. It provides a very accurate representation of model specifying all the other        dimensions hidden geometry etc.

5. It provides a greater flexibility for change, for example, if we like to change the dimensions in design assembly, manufacturing etc. will automatically change.

6. It provides clear 3-D Model which are easy to visualize or model created and & it Also decrease the time required for the assembly to a large extent.

7.5 Design of Components

Three components are taken in this project to determine the optimum design that reduces the vibration effect in the arbor.

1) Solid arbor

2) Hollow rectangle arbor with damping material

3) Hollow tapered arbor with damping material

The following screenshots contains the step by step procedure of all the three product design in CATIA V5.

Solid Arbor

Figure: 7.5.1 Circle creation

1) Circle is generated in the sketcher module.

Figure: 7.5.2 Extruding 

1) Solid arbor is created part module by adding the material axially to the circle drawn.

2) Length 400mm is taken, which provides you the complete solid arbor including tool holder

Cylindrical  Hollow with Damping material in an Arbor

Figure 7.5.3 Cylindrical Hollow Modeling 

1)      Solid arbor is created and rectangle section is taken according to the hollow dimension

Figure 7.5.4 Inserting Damper Mass

1) Groove option is taken and material is removed inside the solid arbor

Figure 7.5.5 Wireframe Model

1) Mass has been developed in the hollow space of the arbor by using shaft option

Figure 7.5.6 Spring

1) Spring has been generated by providing sectional wire diameter and length of the spring.

2) By using Rib option, spring is generated by selection the profile and center curve.

Figure 7.5.7 center curve

1) Complete hollow cylindrical arbor with damping material and springs in both the sides are represented in wireframe mode of view, in which inside parts are visible.

Tapered Hollow with Damping Material in an Arbor

Figure 7.5.8 Damping Material

1) Groove option is taken and material is removed according to the tapered section inside the solid arbor

Figure 7.5.9 Tapering Inside

1) Mass has been developed in the tapered hollow space of the arbor by using shaft option.

Figure 7.5.10 Wire Frame Model

1) Complete hollow tapered arbor with damping material with springs in both the sides are represented in wireframe mode of view, in which inside parts are visible.

2) Springs are assembled on the either side of the mass material in assembly module, where complete arbor body is shown in the above picture.

8. INTRODUCTION TO ANSYS

ANSYS is general-purpose finite element analysis (FEA) software package.  Finite Element Analysis is a numerical method of deconstructing a complex system into very small pieces (of user-designated size) called elements. The software implements equations that govern the behaviour of these elements and solves them all; creating a comprehensive explanation of how the system acts as a whole. These results then can be presented in tabulated, or graphical forms.  This type of analysis is typically used for the design and optimization of a system far too complex to analyze by hand.  Systems that may fit into this category are too complex due to their geometry, scale, or governing equations. 

8.1 Introduction To Ansys

ANSYS is the standard FEA teaching tool within the Mechanical Engineering Department at many colleges. ANSYS is also used in Civil and Electrical Engineering, as well as the Physics and Chemistry departments. 

ANSYS provides a cost-effective way to explore the performance of products or processes in a virtual environment. This type of product development is termed virtual prototyping.  

With virtual prototyping techniques, users can iterate various scenarios to optimize the product long before the manufacturing is started. This enables a reduction in the level of risk, and in the cost of ineffective designs. The multifaceted nature of ANSYS also provides a means to ensure that users are able to see the effect of a design on the whole behavior of the product, be it electromagnetic, thermal, mechanical etc.

8.2 Generic Steps to Solving any Problem in ANSYS

Like solving any problem analytically, you need to define (1) your solution domain, (2) the physical model, (3) boundary conditions and (4) the physical properties. You then solve the problem and present the results. In numerical methods, the main difference is an extra step called mesh generation. This is the step that divides the complex model into small elements that become solvable in an otherwise too complex situation. Below describes the processes in terminology slightly more attune to the software.

Build Geometry Construct a two or three dimensional representation of the object to be modeled and tested using the work plane coordinate system within ANSYS. 

Define Material Properties

Now that the part exists, define a library of the necessary materials that compose the object (or project) being modeled.  This includes thermal and mechanical properties. 

Generate Mesh

At this point ANSYS understands the makeup of the part.  Now define how the modeled system should be broken down into finite pieces.  

Apply Loads

Once the system is fully designed, the last task is to burden the system with constraints, such as physical loadings or boundary conditions.    

Obtain Solution

This is actually a step, because ANSYS needs to understand within what state (steady state, transient… etc.) the problem must be solved. 

Present the Results

After the solution has been obtained, there are many ways to present ANSYS’ results, choose from many options such as tables, graphs, and contour plots. 

8.3 Specific Capabilities of ANSYS

Structural - Structural analysis is probably the most common application of the finite element method as it implies bridges and buildings, naval, aeronautical, and mechanical structures such as ship hulls, aircraft bodies, and machine housings, as well as mechanical components such as pistons, machine parts, and tools.

Static Analysis - Used to determine displacements, stresses, etc. under static loading conditions. ANSYS can compute both linear and nonlinear static analyses. Nonlinearities can include plasticity, stress stiffening, large deflection, large strain, hyper elasticity, contact surfaces, and creep. 

Transient Dynamic Analysis - Used to determine the response of a structure to arbitrarily time-varying loads. All nonlinearities mentioned under Static Analysis above are allowed. 

Buckling Analysis - Used to calculate the buckling loads and determine the buckling mode shape. Both linear (eigenvalue) buckling and nonlinear buckling analyses are possible.  

In addition to the above analysis types, several special-purpose features are available such as Fracture mechanics, Composite material analysis, Fatigue, and both p-Method and Beam analyses.

Thermal 

ANSYS is capable of both steady state and transient analysis of any solid with thermal boundary conditions.

Steady-state thermal analyses calculate the effects of steady thermal loads on a system or component.

Users often perform a steady-state analysis before doing a transient thermal analysis, to help establish initial conditions. A steady-state analysis also can be the last step of a transient thermal analysis; performed after all transient effects have diminished. ANSYS can be used to determine temperatures, thermal gradients, heat flow rates, and heat fluxes.

 Modal Analysis - A modal analysis is typically used to determine the vibration characteristics (natural frequencies and mode shapes) of a structure or a machine component while it is being designed. It can also serve as a starting point for another, more detailed, dynamic analysis, such as a harmonic response or full transient dynamic analysis.

Modal analyses, while being one of the most basic dynamic analysis types available in ANSYS, can also be more computationally time consuming than a typical static analysis.  A reduced solver, utilizing automatically or manually selected master degrees of freedom is used to drastically reduce the problem size and solution time.

Harmonic Analysis: Used extensively by companies who produce rotating machinery, ANSYS Harmonic analysis is used to predict the sustained dynamic behaviour of structures to consistent cyclic loading. A harmonic analysis can be used to verify whether or not a machine design will successfully overcome resonance, fatigue, and other harmful effects of forced vibrations.

Vibrational Analysis

Modal analysis is used to determine a structure’s vibration characteristics natural frequencies and mode shapes. It is the most fundamental of all dynamic analysis types and is generally the starting point for other, more detailed dynamic analyses.

Geometry & Meshing

Same considerations as a static analysis:

Include as many details as necessary to sufficiently represent model geometry.

A fine mesh will be needed to resolve complex mode shapes.

Both Young’s modulus and density are required.

Linear elements and material properties only.  Nonlinearities are ignored.

8.4 Model Analysis

Analysis of Solid Arbor

In this project natural frequency of a tool arbor is analyzed by using finite element analysis software ANSYS . As shown in Fig.5, the length of the arbor body and cutting tool is 400mm and the diameter is 47mm. The left end of the arbor is BT holder and it is connected to a spindle of a machine tool.

 In order to consider the influence of flexibility of the holder and machine tool, material properties of the left end of the model with the length of 50mm were given artificially. This part represents the contribution of the holder and machine tool to the natural frequency, and the material properties as Young's modulus and density are defined in accordance with the measured dynamic stiffness at the right end of the arbor. The area of 350mm length of the arbor body and tool which is made of chromium molybdenum steel (JIS-SCM435) has a 47 mm diameter. Density of the arbor body and tool is 7.8×10³ kg/m3, Young's modulus is 2.1×10² GPa respectively.

                                      

Figure8.4.1 Solid Arbor

In this case, natural frequency of the arbor  of mode 1 and mode 2 are 82Hz, 98Hz respectively.

These results are taking from Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.7, No.3, (Onozuka,etal.,2013).

8.5 Theoretical Calculations of Natural Frequency of Circular Hollow Model

Natural Frequency: The frequency at which a system oscillates when not subjected to a continuous or repeated external force. Newton's second law is the first basis for examining the motion of the system. As shown in Fig. 8.5.1 the deformation

Figure: 8.5.1 Spring-Mass System and Free-Body Diagram

k∆= w = mg

By measuring the displacement x from the static equilibrium position, the forces acting on m are  k(∆+x)  and w. With x chosen to be positive in the downward direction, all quantities  force, velocity, and acceleration are also positive in the downward direction.

The natural period of the oscillation is established from ω_n  = 2 π, or

T=2𐍀√m/k

and the natural frequency is 

f_n =  =1/T = 1/2𐍀√k/m

Where:
f_n = natural frequency in hertz (cycles/second)
k = stiffness of the spring (Newtons/meter or N/m)
m = mass(kg)

We have to take values are

k₂ = 2.2x10⁶ N/m

m₂ = 1.75Kg        

density = 8x10³ Kg/m³

Natural frequency of the 1st order mode fd₁ is expressed as the following equation.      

On the other hand, the 2nd order mode of the damper is assumed to be twisting. The weight is supported at both ends by springs that have a spring constant of k₂/2. Twisting stiffness K₂, the moment of inertia of the cylindrical weight around y-axis J₂, is expressed as follows

where d₂ is the diameter and l₂ is the length of the weight.

Frequency of natural vibration of twisting mode fd₂ is expressed as follows.

 f_d2 =1/2𐍀 √ k2/j2

= 298 Hz

 f₁ = 173 Hz

f₂ = 298 Hz

8.6 Analysis Of Arbor With Circular Hollow Model

In this research natural frequency of a tool arbor is analyzed by using finite element analysis software ANSYS . The length of the arbor body and cutting tool is 400mm and the diameter is 47mm. The left end of the arbor is BT holder and it is connected to a spindle of a machine tool.  In order to consider the influence of flexibility of the holder and machine tool, material properties of the left end of the model with the length of 50mm were given artificially.

This part represents the contribution of the holder and machine tool to the natural frequency, and the material properties as Young's modulus and density are defined in accordance with the measured dynamic stiffness at the right end of the arbor. The area of 400mm length of the arbor body and tool which is made of chromium molybdenum steel (JIS-SCM435) has a 47 mm diameter. Density of the arbor body and tool is 7.8×10³ kg/m³, Young's modulus is  2.1×10² GPa respectively. Hollow circular space is designed with a diameter of 28 mm and length of 150mm made of cemented tungsten in the arbor body. This is Circular mass material used for damping in the hollow space provided in side the Arbor

Figure: 8.6.1  Model of Circular Hollow Arbor

Dynamic stiffness of the damped arbor in which damper as shown in Fig.8.8.2 is installed in its hollow space is analyzed. The spring constant of the damper is k =2.2 ×10⁶ N/m and the damping ratio is ζ =0.3. The figure of the vibration mode shows the displacement and deformation of the arbor body, spring, and weight when the displacement of the arbor body is maximized in the x direction. In this figure, relative displacement of the nodal points of the analytical model is drawn by being expanded, and this figure exaggerates the relative displacement and deformation of each part of the damped arbor. The frequencies of the natural vibration for the 1st order mode and 2nd order mode are increased by the increase in the stiffness of damper.

Figure: 8.6.2  Mode-1

Figure: 8.6.3  Mode-2

The frequencies of two natural vibration modes are also indicated. These modes are different from the natural modes of the arbor body or damper. This figure shows that the frequency of the 1st order natural vibration mode of the system is 168 Hz. The frequency of  2nd  mode  is 292 Hz.

Mode	Frequency [Hz]
1.	168
2.	292

Table: 8.6.1 Natural frequencies of Circular Hollow Arbor

By using these equations, natural frequencies fd₁ and fd₂ are calculated where the mass of the weight m₂ = 1.75 kg. Frequencies of the 1st and 2nd order modes of natural frequencies are measured. From these results so that the analyzed result agree with the measured frequencies within 5% error.

8.7 Theoretical Calculations of Natural Frequency of Tapered Hollow Model

Natural Frequency: The frequency at which a system oscillates when not subjected to a continuous or repeated external force.

Newton's second law is the first basis for examining the motion of the system.

k∆= w = mg

By measuring the displacement x from the static equilibrium position, the forces acting on m are  k(∆+x)  and w. With x chosen to be positive in the downward direction, all quantities  force, velocity, and acceleration are also positive in the downward direction.

The natural period of the oscillation is established from ω_n  = 2 π, or

T= 2𐍀√m/k

and the natural frequency is 

f_n =  =1/T = 1/2𐍀√k/m

Where:
f_n = natural frequency in hertz (cycles/second)
k = stiffness of the spring (Newtons/meter or N/m)
m = mass(kg)

We have to take values are

k₂ = 2.2x10⁶ N/m

m₂ = 1.86Kg

density = 16x10³ Kg/m³

Natural frequency of the 1st order mode fd1 is expressed as the following equation.      

= 2.188 kgs

Where d₂ is the diameter and l₂ is the length of the weight.

Frequency of natural vibration of twisting mode fd₂ is expressed as follows.

 f_d2 =1/2𐍀 √ k2/j2

= 322 Hz

f₁ = 184 Hz &  f₂ = 322 Hz

8.8 Analysis of Arbor with Tapered Model

In this research, dynamic stiffness of a tool arbor is analyzed by using finite element analysis software ANSYS . As shown in Fig.8.6.1, the length of the arbor body and cutting tool is 400mm and the diameter is 47mm. The left end of the arbor is BT holder and it is connected to a spindle of a machine tool.  In order to consider the influence of flexibility of the holder and machine tool, material properties of the left end of the model with the length of 50mm were given artificially. This part represents the contribution of the holder and machine tool to the dynamic stiffness, and the material properties as Young's modulus and density are defined in accordance with the measured dynamic stiffness at the right end of the arbor.

The area of 350mm length of the arbor body and tool which is made of chromium molybdenum steel (JIS-SCM435) has a 47 mm diameter. Density of the arbor body and tool is 7.8×10³ kg/m3, Young's modulus is 2.1×10² GPa respectively. Hollow tapered space is designed with a diameter of one end is 40mm and other end is 32mm and length of 150mm in the arbor body. This is Tapered mass material used for damping in the hollow space provided in side the Arbor.

Analysis is performed in the solution task of ANSYS. These analyses is  applied to the finite element tapered mass arbor model. First of all, model analysis is done and the deflection of the beam is obtained under single force acting on to the node of the beam. The static deflection of the beam is calculated to normalize the deflection of the beam under the moving load obtained from the model analysis. The nodal deflection results of the beam under a single force.

Dynamic stiffness of the damped arbor in which damper is installed in its tapered hollow space is analyzed. Fig. 8.6.1 shows the real part of dynamic stiffness under conditions in which the spring constant of the damper is k =2.2 ×10⁶ N/m and the damping ratio is ζ =0.3. The figure of the vibration mode shows the displacement and deformation of the arbor body, spring, and weight when the displacement of the arbor body is maximized in the x direction. In this figure, relative displacement of the nodal points of the analytical model is drawn by being expanded, and this figure exaggerates the relative displacement and deformation of each part of the damped arbor.

 The frequencies of six natural vibration modes are also indicated. These modes are different from the natural modes of the arbor body or damper. This figures shows that the frequencies of the 1^th and 2^nd order natural vibration modes of the system are 175 Hz and 312 Hz.

Figure: 8.8.1  Mode-1

Figure: 8.8.2  Mode-2

The step is to perform a modal analysis. 2 vibration modes and corresponding mode shapes are calculated for the dynamic response of the arbor under a moving load. 2 natural frequencies are given in the following table (Table8.8.1).

Mode	Frequency [Hz]
1.	175
2.	312

Table: 8.8.1 Natural frequencies of Tapered Hollow Arbor

By using these equations, natural frequencies f_d1 and f_d2 are calculated where the mass of the weight m₂ = 1.5 kg. Frequencies of the 1st and 2nd order modes of natural frequencies are measured. From these results so that the analyzed result agree with the measured frequencies within 5% error.

9. FABRICATION OF TAPERED HOLLOW DAMPERED ARBOR

In this project  Arbor is made of Chromium Molybdenum steel (AISI 4140) material has taken as a 50mm diameter and 500mm length. It's density is 8.0x10³kg/m³, Young's modulus is 2.1x10²GPa. The outer diameter of the body is 47mm is machined and tapered hollow space of 40 to 32 diameter and 150mm length is done by on the lathe machine. The usual starting point for drilling with a centre lathe is to use a countersink bit. This is used to drill slightly into the material and creates a starting point for other drills that are going to be used. Attempting to drill with a traditional drill bit without countersinking first will lead to the drill bit slipping straight away. It is not possible to drill a hole successfully or safely without using a centre drill first.

Once a hole has been produced by a centre drill, machine twist drills can be used to enlarge the hole and if necessary to drill all the way through. If a large diameter hole is needed then a small hole is drilled first (eg. 4mm dia). Then the hole is enlarged approximately 2mm at a time. Trying to drill a large diameter hole in one go will inevitably lead to the drill bit over heating and then jamming in the material. This is potentially dangerous.

Figure: 9.1  Fabrication process

One end side of arbor is machined inclined for gripping by a milling chuck and its length is 50mm. Inside of the hole surface is finished by surface finish. Then the damped arbor is ready for machining the material with given feed rate and spindle rotation.

Figure: 9.2 Tapered hollow arbor

  

10.DYNAMIC ANALYSIS IN LABVIEW

The past years have witnessed several severe bridge collapses around the world. These disastrous events have led to hundreds of lives and billions of dollars lost. Unfortunate accidents like these can be avoided in the future through better design, validation, and monitoring of critical structures such as bridges. For this reason, modal analysis has become increasingly popular in R&D and more importantly in online real-time monitoring systems. National Instruments continues to provide modal analysis solutions from laboratory test to online monitoring. This paper discusses in detail the hardware architecture as well as the specific modal parameter extraction algorithms used in industry to uncover the modal parameters of a structure.

10.1 Introduction

Modal analysis is the study of the dynamic properties of a structure under vibration excitation. Through modal analysis, structural engineers can extract a structure’s modal parameters (dynamic properties). The modal parameters, including natural frequency, damping ratio, and mode shape, are the fundamental elements that describe the movement and response of a structure to ambient excitation as well as forced excitation. Knowing these modal parameters helps structural engineers understand a structure’s response to ambient conditions as well as perform design validation.

10.2 Industry Trends

There are two types of modal analysis performed in industry today.

1. Operational modal analysis.

2. Experimental modal analysis.

Experimental modal analysis is the most commonly used form of modal analysis today. It is the traditional method in which an individual uses a device, such as a hammer, to excite a structure and then measures the response. Then, the transfer function is calculated and certain modal parameter extraction algorithms are used to extract the dynamic properties of the structure.

Experimental modal analysis has been very useful in design validation and finite element analysis (FEA) verification; however, it has not proved useful in monitoring the long-term health of a large structure such as a bridge. For this reason, operational modal analysis has become the focus of innovation. Operational modal analysis is strictly concerned with the operation of a structure. This type of analysis focuses on monitoring the modal parameters of a structure and looking for trends in the data as warnings for failure. With operational modal analysis, structural engineers can proactively monitor the health of a bridge to avoid these catastrophic failures in the future.

10.3 Experimental Modal Analysis

Experimental modal analysis is the field of measuring and analyzing the dynamic response of a structure when excited by a stimulus. Experimental modal analysis is useful in verifying FEA results as well as determining the modal parameters of a structure. Performing experimental modal analysis is a four step process:

Figure 10.3 Experimental Modal Analysis Process

10.3.1 Vibration Sensors (Accelerometers)

Vibration sensors, known as accelerometers, must be properly placed on a structure to record the vibration response of a structure due to a known excitation by either a shaker system or an impulse hammer. These excitation systems are necessary to properly excite the modes of the system which reveal the modes of the structure. The accelerometers must have the frequency range, dynamic range, signal-to-noise ratio, and sensitivity needed for the specific test scenario. Vendors such as PCB Piezotronics work to ensure that the proper sensor is chosen for the application.

Figure: 10.3.1 PCB Accelerometer for Measuring Vibration

10.3.2 Data Acquisition

Specialized data acquisition (DAQ) hardware is needed to properly acquire these vibration signals. The recommended data acquisition hardware is the NI Dynamic Signal Analyzer (DSA), which simultaneously acquires each channel with 24-bit high-resolution delta sigma ADCs. These DSA products have anti-aliasing filters to prevent aliasing and noise from affecting the measurement quality. Finally, they have the proper signal conditioning to power piezoelectric (ICP or IEPE) accelerometers. National Instruments has a variety of platforms available including USB, wireless 802.11g, PXI, and real-time embedded targets.

Figure:10.3.2 National Instruments DAQ Hardware

10.3.3 FRF Analysis

The frequency response function (FRF) compares the stimulus and response to calculate the transfer function of the structure. The result of the FRF is the structure’s magnitude and phase response over a defined frequency range. It shows critical frequencies of the structure, which are more sensitive to excitation. Those critical frequencies are the modes of the structure under test. An example of the magnitude result from an FRF is shown in Figure 10.3.3

Figure: 10.3.3 FRF Results for a Test Scenario

10.3.4 Modal Parameter Extraction

Modal parameter extraction algorithms are used to identify the modal parameters from the FRF data. These algorithms include peak picking, least square complex exponential (LSCE) fit, frequency domain polynomial (FDPI) fit, and FRF synthesis. Each of these algorithms perform the same function of identifying the modal parameters, however, each are optimized for a specific test scenario.

1.      Peak picking is a method used to extract a mode from a precomputed signal’s FRF. It is a frequency domain single-degree-of-freedom (SDOF) modal analysis method and suitable to estimate uncoupled and lightly damped modes. The computation is fast, but the result is sensitive to the frequency shift.

2.      LSCE fit is used to simultaneously extract multiple modes from precomputed signal’s FRF. It is a time domain multiple-degree-of-freedom (MDOF) modal analysis method and suitable for estimating modes in a wide frequency band. It is ideal for lightly damped modes.

3.      FDPI fit is used to simultaneously extract multiple modes from precomputed signal’s FRF. It is a frequency domain MDOF modal analysis method suitable to estimate heavily damped modes, particularly for heavily damped modes in a narrow frequency band.

4.      FRF synthesis is used to create synthetic FRF for testing and evaluation. With computed modal parameters, engineers can compare synthesized FRF and original FRF to verify the resulting estimation.

The end result of each algorithm is identified mode(s). As explained above, each algorithm is used in a certain scenario. Figure 10.3.4  there is a peak around 280 Hz. If the user is interested in identifying that mode, FDPI fit is the correct choice because it is a narrow frequency band. The result of the FDPI algorithm is shown in Figure 10.5.4.

Figure: 10.3.4 FDPI Fit Test Results

 DISCUSSION

·         The rayleigh’s method is coupled in the proposed analysis for the dynamic stiffness assuming that mass give a counter force at the centre of gravity of it. Amplitude ratio is introduced for the calculation of the compliance where the compliance is calculated without the amplitude ratio and compared. It states that mass is connected at the end of the tool when amplitude is not included. So assuming amplitude ratio gives best results.

·         The bending moment will be minimum near the tool and maximum whilst at the tool holder. The kinetic energy is more at the tool end. The tapered hollow damped arbor reduces the kinetic energy more than the cylindrical hollow arbor considering same volume for both the arbors.

·         The stiffness constant affect compliance with the length of the arbor drastically. when the stiffness is increased compliance will be changed for the larger arbor. So optimum stiffness value of larger arbor is considered for the small arbor with negligible deterioration of dynamic stiffness.

 

  

11. CONCLUSION

Main objective is to suppress chatter vibration and improve cutting performance during the machining. The dynamic stiffness is calculated by using Reyleigh's method implementing displacement ratio.

The optimal value of the cylindrical hollow arbor is taken from the reference [2].

The optimal spring constant is obtained by using of the maximum negative value.

            The maximum negative compliance value is calculated for using different spring constants and optimal spring constant is taken which by value of maximum negative compliance is mentioned in above figure.

            By making the optimal stiffness value as a threshold value the different dia

meters are considered for both cylindrical hollow arbor and tapered hollow arbor and compliance values are determined.

           According to figure No. 6.3.3 the dimensions of tapered hollow arbor is measured at the threshold stiffness value and also the volume of both the damped arbors are kept same. It is to be conclude that the diameter of d_b = 40 and d_a = 32 is optimal value for designing of tapered hollow arbor by keeping the inner hollow length 150mm as a constant.

           The natural frequencies of the damped arbor is calculated theoretically in two different modes. The designs of arbors are done damped by using catia v5 and it is dumped into ansys software. Where the model analysis is performed on the body and frequencies are taken for two different modes and compared with the theoretical values of the arbors with which the results are deviated by 5%.

           The dimension of the optimal design is noted and fabricated using lathe machine.  Chromium Molybdenum is used as arbor body material, Tungsten carbide is used as a mass material. The springs are also selected according to the spring constant calculated.

The experimental test is carried on the fabricated tapered hollow arbor and it is concluded that the depth of cut is more when compared to solid and cylindrical hollow arbors. The dynamic stiffness of the arbor also obtained.

           The tapered hollow space can reduce kinematic energy more than the straight one, but the reduction of the static stiffness is smaller for the hollow space than for the straight one when the volumes of both types of spaces are the same. These results can be seen in Fig. 6.3.3 Note that a tapered hollow space with da = 40 mm and db = 32 mm has the same volume as a straight hollow space with da = 36.2 mm.

 

  

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