short course using abaqus 2010.pdf - Faculty of Mechanical

86 Pages·2010·17.88 MB·English

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CONTENTS
1. INTRODUCTION
- Overview of Brake NVH
- Overview of ABAQUS

2. STABILITY ANALYSIS
- Complex eigenvalue analysis
- Dynamic transient analysis

3. ABAQUS/CAE: Getting Started
a: Complex Eigenvalue Analysis
b: Dynamic Transient Analysis
- Parts: Sketch or Importing CAD model
- Property: Assigning material properties
- Assembly: Integration of the parts
- Step: Selecting types of analysis
- Interaction: Identify and assigning
contact properties
- Load: Assigning loads and boundary
Conditions
- Mesh: Selecting and assigning seeds
and element types
- Job: Creating, checking and submitting
job
- Visualization: Extracting results

4. MODAL TESTING AND ANALYSIS
- Experimental approach
- Validation

5. SQUEAL TESTS
- Experimental approach
- Validation

BIBLIOGRAPHY

Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
SECTION 1: INTRODUCTION
1.1 OVERVIEW OF BRAKE NVH
Since vehicle comfort has become such an important factor to indicate the
quality of a passenger car, eliminating or reducing the noise and vibration of a
vehicle structure and system seems to provide a leading edge in the market to
vehicle manufacturers. With progress made towards other aspects of vehicle
design refinement against vehicle vibration and noise through improvement,
refinement in brake vibration and noise is inevitable. This can be seen from the
literature where the awareness on the brake vibration and noise issues started
as early as 1930’s.

Early studies of brake noise and vibration attempted to identify techniques for
eliminating and/or reducing the noise and vibration and later gradually focused
on its generation mechanisms. Since then, problem of the noise and vibration
in brake has been studied with experimental, analytical and computational
methods, but there is as yet no method to completely suppress brake noise and
vibration in general and squeal in particular. Moreover, a complete
understanding on the problem has still not been achieved. Papinniemi et al
(2002) suggested that these are due to complexity of the mechanisms itself and
competitive nature of automotive industry which limits the amount of
cooperative research, i.e., published in the open literature.

There are several categories of brake noise and vibration and they can be
defined according to the mechanism of generation. Lang and Smales (1983)
categorised various types of vibration problems such as judder, groan, hum,
squeal, squeak and wire-brush. In a recent review, Ouyang et al (2005)
suggested three major categories of brakes noise and vibration as follows:
creep-groan, judder and squeal. Of these categories, squeal is the most irritant
and annoying both to the customers and environment, and is expensive to the
vehicle manufacturers due to warranty payouts. Squeal has been the primary
subject of past studies on disc brake noise and vibration. It is well accepted
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
that brakes squeal is due to friction - induced vibration or self-excited vibration
via a rotating disc. Brake squeal frequently occurs at frequency above 1 kHz
(Lang and Smales, 1983) and is described as sound pressure level above 78 dB
(Eriksson, 2000). Table 1 shows category of brake NVH and mechanisms.
Detailed review on the brake squeal can be found in (Kinkaid et al, 2003).

Table 1: Classification of brake noise (Nik Husin, 2009)
Type Frequency Mechanism Description
Creep-groan 200-500 Hz Stick-slip motion between Occur at very
the friction material and low brake line
rotor surface pressure and
vehicle speed
Hot judder or < 1 kHz Periodic features on the The noise
rumble rotor surface that result in produced is
cyclic brake torques multiple of the
wheel speed
Squeal > 1 kHz Five major categories . Mechanism of
I. Stick slip generation
II. Negative damping depend on
III. Geometric/Kinematics characteristics
(GK) constraint and operating
IV. Modal coupling conditions of
V. Hammering excitation brake system

A number of theories have been proposed in the literature to explain the brake
squeal phenomenon. An early experimental investigation found that variation in
friction coefficient in the contact interface was the cause for brake to squeal. In
Mills (1938) hypothesised that squeal was associated with the negative gradient
characteristics of dynamic friction coefficient against the sliding velocity.
Sinclair (1955) through his mathematical model showed that the presence of
such mechanism led to unstable oscillations and gave rise to self-excited
vibration in the system. Later, Fosberry and Holubecki (1955, 1961) suggested
that the disc brake tended to squeal when either a static coefficient of friction
was higher than the dynamic coefficient or a dynamic coefficient decreases with
increase of speed present in the contact interface. The above mechanisms are
also referred to as “stick-slip” and “negative damping” respectively in current
terminology. Spurr (1961-1962) seemed to be a first researcher who turns away
from the above theories. He proposed a new theory of brake squeal called
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
sprag-slip, by which the unstable oscillation in the system could also occur
even with constant friction coefficient. North proposed a very promising theory
of squeal where he developed a binary flutter model which more closely
resembled disc brake assembly. The eight-degree of freedom model (1972) and
later the two-degree of freedom model (1976) considered not only geometrical
characteristics of the brake components but also took into account friction
coupling and stiffness of the interactive components. North’s 2-DOFs model
consisted of the disc layered between two flexible pads that had both
translational and rotational stiffness. The distinctive features of this theory
were the presence of the disc and the friction forces produced by pressure of
brake pads, and the presence of two independent disc modes.

Brake squeal problem has been studied over the years either by experimental,
analytical and numerical approaches alone or a combination of them.
Analytical approaches have proved to be insufficient to provide a complete
understanding of squeal behaviour as well as in providing a design tool that
could predict and suppress squeal. Furthermore, analytical approaches are
frequently limited to the certain parameter of interest. Analytical approaches,
despite its limitation, are nevertheless quite useful in briefly explaining
instability of the system. Those drawbacks could be overcome with the
application of numerical approaches via the finite element method that allows
generation of models with a large number of degrees of freedom. Numerical
approaches also admit deformability in the model that was quite frequently
treated as rigid in analytical approaches. Experimental approaches are
essential for not only to quantify the nature of squeal and the various operating
conditions affecting it but also to provide results confirmation of the numerical
approaches as well as noise quality confirmation of the brakes before releasing
to the market.

With the advances in the computer technology recent years, more complicated
and complete FE models can be easily built as well as quick turn-around in
simulation time. Advancement in the contact formulation and algorithm much
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
help engineers and researchers to obtain more reliable and accurate
representation of contact pressure distribution. This parameter is significantly
important either for the complex eigenvalue analysis or the dynamic transient
analysis. In recent years, the complex eigenvalue becomes the most preferred
method in the brake research community to study brake squeal than the
transient analysis. The positive real parts of the complex eigenvalue indicate
the degree of instability of the linear model of a disc brake and are thought to
show the likelihood of squeal occurrence or the noise intensity (Liles, 1989). On
the other hand, instability in the disc brake can be associated with an initially
divergent vibration response using transient analysis. Liles (1989) was the early
researcher who incorporated complex eigenvalue analysis with the finite
element method whilst Nagy et al. (1994) pioneered dynamic transient analysis
with the finite element method. Complex eigenvalue analysis allows all unstable
frequencies to be found in one run for one set of operating conditions and hence
is very efficient. However, not all unstable frequencies thus obtained can be
observed in experiments. Transient analysis is able to predict true unstable
frequencies (those found in experiments) in principle if the system model is
correct. However it is very time-consuming. Moreover it does not provide any
information on unstable modes.

1.2 OVERVIEW OF ABAQUS
ABAQUS is a commercial software package for finite element analysis developed
by HKS Inc of Rhode Island, USA and now marketed under the SIMULIA brand
of Dassault Systemes S.A .The ABAQUS product suite consists of three core
products: ABAQUS/Standard, ABAQUS/Explicit and ABAQUS/CAE as shown
in Figure 1. ABAQUS/Standard is a general-purpose solver using a traditional
implicit integration scheme to solve finite element analyses. ABAQUS/Explicit
uses an explicit integration scheme to solve highly nonlinear transient dynamic
and quasi-static analyses. ABAQUS/CAE provides an integrated modeling
(preprocessing) and visualization (post processing) environment for the analysis
products. The ABAQUS products use the open-source scripting language
Python for scripting and customization. ABAQUS/CAE uses the fox-toolkit for
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
GUI development. ABAQUS is used in the automotive, aerospace, and industrial
product industries. The product is popular with academic and research
institutions due to the wide material modeling capability, and the program's
ability to be customized. ABAQUS also provides a good collection of
multiphysics capabilities, such as coupled acoustic-structural, piezoelectric,
and structural-pore capabilities, making it attractive for production-level
simulations where multiple fields need to be coupled (Source:
http://en.wikipedia.org/wiki/Abaqus).
ABAQUS/CAE is a complete ABAQUS environment that provides a simple,
consistent interface for creating, submitting, monitoring, and evaluating results
from ABAQUS/Standard and ABAQUS/Explicit simulations. ABAQUS/CAE is
divided into modules, where each module defines a logical aspect of the
modeling process; for example, defining the geometry, defining material
properties, and generating a mesh. As you move from module to module, you
build the model from which ABAQUS/CAE generates an input file that you
submit to the ABAQUS/Standard or ABAQUS/Explicit analysis product. The
analysis product performs the analysis, sends information to ABAQUS/CAE to
allow you to monitor the progress of the job, and generates an output database.
Finally, you use the Visualization module of ABAQUS/CAE (also licensed
separately as ABAQUS/Viewer) to read the output database and view the
results of your analysis. ABAQUS/Viewer provides graphical display of ABAQUS
finite element models and results. ABAQUS/Viewer is incorporated into
ABAQUS/CAE as the Visualization module. (Source: ABAQUS Manual)
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
Figure 1: ABAQUS Products (Source: ABAQUS Manual)

SECTION 2: STABILITY ANALYSIS
The complex eigenvalue analysis that available in ABAQUS is commonly utilized
to determine disc brake assembly stability. The positive real parts of the
complex eigenvalue indicate the degree of instability of the disc brake assembly
and are thought to indicate the likelihood of squeal occurrence. The essence of
this method lies in the asymmetric stiffness matrix that is derived from the
contact stiffness and the friction coefficient at the disc/pads interface.

The transient analysis is not as mature as the complex eigenvalue analysis in
the finite element method. Instability in the disc brake system through this
analysis can be found with an initially divergent vibration time response. This
time domain information then can be converted to frequency domain
information using the Fast Fourier Transform (FFT) technique.

2.1 COMPLEX EIGENVALUE ANALYSIS
Typically, the complex eigenvalue analysis is simulated in the implicit version
while the explicit version is used for the transient analysis. In order to perform
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
the complex eigenvalue analysis using ABAQUS, four main steps are required
(Kung et al, 2003). They are given as follows:
• Nonlinear static analysis for applying brake-line pressure
• Nonlinear static analysis to impose rotational speed on the disc
• Normal mode analysis to extract natural frequency of undamped
system
• Complex eigenvalue analysis that incorporates the effect of friction
coupling
In this analysis, the complex eigenvalue are solved using the subspace
projection method. The eigenvalue problem can be given in the following form:

(λ2M+λC+K)y=0 (1)

where K is the unsymmetrical (due to friction) stiffness matrix. This
unsymmetrical stiffness matrix leads to complex eigenvalue and eigenvectors.
In the third step stated above, the symmetric eigenvalue problem is first solved,
by dropping damping matrix C and the unsymmetrical contributions to the
stiffness matrix K, to find the projection subspace. Therefore the eigenvalue λ
becomes a pure imaginary whereλ= iω, and the eigenvalue problem now is
similar to the equation (1.0).

This symmetric eigenvalue problem then is solved using subspace eigensolver
(ABAQUS, 2003). The next step is that the original matrices are projected in the
subspace of real eigenvectors and given as follows:

M∗ = [z ,z ,...,z ]TM[z ,z ,...,z ], (2a)
1 2 n 1 2 n

C∗ = [z ,z ,...,z ]TC[z ,z ,...,z ], (2b)
1 2 n 1 2 n

K∗ = [z ,z ,...,z ]TK[z ,z ,...,z ], (2c)
1 2 n 1 2 n
Copyright © 2010 by: Abd Rahim Abu Bakar, UTM
Now the eigenvalue problem is expressed in the following form:

(λ2M∗+λC∗+K∗)y∗ =0 (3)

The reduced complex eigenvalue problem is then solved using the QZ method
for a generalized nonsymmetrical eigenvalue problem. The eigenvectors of the
original system are recovered by the following:

yk =[z ,z ,...,z ]y*k (4)
1 2 n

where yk is the approximation of the k-th eigenvector of the original system.

The linearised shear stress that takes into account friction effect and the
inclusion of µ−ν characteristics is expressed in the following form (Bajer et al,
2003):

 ∂µ  ∂µ . µp .
dτi = µ + ∂p pnidp + ∂γ& pninjdγj+ γ& (δij − ninj)dγj (5)
where
τ - tangential stress,
I
µ = µ(γ&, p) - friction coefficient,
p - contact pressure,
&
γ
n = ,i = 1,2 - normalised slip direction (1 is the radial
i γ&
direction and 2 is the circumferential
direction)
γ& = γ&2 + γ& 2 - equivalent slip rate/ sliding velocity,
1 2
γ& - relative slip/sliding velocity.

Copyright © 2010 by: Abd Rahim Abu Bakar, UTM