Parametric Analysis of Break Squeal using Finite Element Method

Modal analysis is used to determine vibration characteristics such as natural frequencies and mode shapes of a structure or a machine component. The frequencies obtained from the modal solution have real and imaginary parts due to the presence of an unsymmetric stiffness matrix. The imaginary frequency reflects the damped frequency, and the real frequency indicates whether the mode is stable or not. There are mainly two types of Modal Analysis. They are,

• Complex Eigen Value Analysis (CEA)

• Transient Dynamic Analysis (TDA)

Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag (a liquid's viscosity can hinder an oscillatory system, causing it to slow down) in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in

other oscillating systems such as those that occur in biological systems and bikes. Not to be confused with friction, which is a dissipative force acting on a system. Friction can cause or be a factor of damping. The damping ratio is a dimensionless measure describing how a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g., frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next.

where the system's equation of motion is

and the corresponding critical damping coefficient is

Complex Eigenvalue Analysis (CEA):

The classical method for instability prediction in squeal analysis can be done by calculation of unstable eigen frequencies of linearized equation of motion around the equilibrium point. Thus, the first step is to find the

equilibrium point (0) which can be done using Eq. (1). .

0 = + (1) Where is the external force such as the pressure on back plates and is the nonlinear

force at the contact interface of disc and pad. Having the

equilibrium point, the perturbation around the equilibrium point can be defined. Where M, C and K are the mass, damping and stiffness matrix, respectively. Regarding that

nl = . , the equation of motion can be written as

+ ( ) = (4) In Eq. (4), nl

represents the nonlinear stiffness generated by friction force

between pad and disc. Considering ( ) = . and

= 0, the closed form of Eq. (4) can be written as ( 2 +

+ nl) = 0 (5) Where is the complex

eigenvalue and A is the corresponding eigenvector. In fact,

complex eigenvalue analysis is solving Eq. (5). The

eigenvalues, can be expressed as = ± , where =

1, in which is the real part of complex eigenvalue, nominate as Re (), represents the stability of the system, and is the imaginary part of complex eigenvalue, nominated as Im(), represents the mode frequency. The

Eq. (6) expresses the generalized displacement of the

system, . ( ) = () . (A1 ± A2 ) (6)

Where A1 and A2 are constants calculated by initial

conditions (e.g., (0) = 0 and (0) = 0). According to Eq. (6), if is negative, decreases exponentially with time and the system will be stable. Otherwise, when is positive, increases with time and the system is then

unstable. It is essential to have all eigenvalues on the left-

hand side, to have a stable system. Although, all the eigenvalues with positive real part represent the instability but the squeal may not happen in all of them.

Transient Dynamic Analysis (TDA):

Transient means, something that fades with time. The dynamic analysis in time domain is called Transient Dynamic Analysis. You give the time history of loading and then you will get the time history of response. That means, load vs. time will be the input,Response(displacement, stresses, velocities, acceleration) vs. time will be the result. In dynamic analysis, there is frequency response analysis and transient response analysis. One is where you give loading as a function of frequency and the other as a function of time. In this project, the transient analysis is performed using Dynamic, Explicit in Abaqus/Explicit which uses central difference method as time integration. The differential

equation for structural dynamics can be considered as (

) + ( ) + () = () Such second order differential

equation needs two initial conditions in order to be solved

in a non-generic state. Let denote (0) = 0 and

(0) = 0 as initial conditions. Considering () = , the

foundation of central difference algorithm can be written as

= (+1 1)/ 2 In which is the time step. The above equation can be interpreted as the derivative at time is

approximated as the slope of the line passing through the

function at 1 and +1. Using the Tylor series for the

+1 and 1, we have +1 = + . + 2 2. ,

1 =

. + 2 2. By adding above two equations,

triangular matrix. Calculate the starting displacement from

Taylor series 1 = 0 . 0 + 2 2 0. Step 1: Loop for each time step, n=1n. Step 2 Calculate the right-

hand side of the iteration. = ( 2 2

) (1 2 1 2 ) 1. Step 3 Solve for displacement at the next time step. (1 2 + 1 2 ).

+1 = . Step 4 Evaluate the set of velocity and acceleration. Step 5 Set + 1 and continue to the next

step.

ADVANTAGES & DISADVANTAGES:

The advantage of CEA is that it gives mode shape of unstable eigenvalues. In other words, it gives unstable modes that can be used for vibration refinement in terms of geometrical revisions in brake system. CEA uses the linearized equation of motion around the equilibrium point, thus it can be considered as a limitation of this method because it is valid only at the vicinity of the equilibrium point. Furthermore, it is a steady state method that is not able to model the transient behavior and nonlinearities of squeal phenomena. In addition, it uses some assumptions such as a constant contact area between disc and pads and linear friction law. In fact, these assumptions are not compatible with reality. The transient analysis gives only the frequencies that may contribute to brake squeal together with the dominating displacement field. It does not give any information about the unstable mode shapes at a given frequency. Therefore, having knowledge about the brake squeal mode shape, is essential to do structural modification and prevent brake squeal as well. On the other hand, due to the time dependency of moving load on disc, it is better to use TDA for modelling the transient variables of model such as displacement, velocity and acceleration. The main drawback of the transient analysis is mainly long computational time and having convergence for each iteration. Time-domain simulations require huge data storage in case of using small time step. In this case, it is hard to handle it by commercial software such as

can be derived as

= (+1 2 + 1)/ 2 By

ABAQUS. The computational time and data storage will

substituting the first and second derivative, the discrete

governing equation can be written as (1/ 2. + 1 /2. )

. +1 = ( 2 / 2.

) . (1 / 2. 1 /2. ) 1 Central difference

method is conditionally stable provided that the step size h

is smaller than the critical step size that can be calculated as

below: L min= minimum element length /, =

/. As a rule, in explicit transient analyses, the small

value for leads to the verytime-consuming calculation that

limit generating detailed FE models in terms of time, file volume and software capability for handling very large result files. Therefore, there is a trade-off between the time step () and total simulation duration that should be managed in a correct way. Step 0: Input mass (M), damping (C), stiffness (K) Calculate LU factorization of M such that M=LU Note that LU factorization of matrix M means to define it as M=LU in which L: lower triangular matrix U: upper triangular matrix Input initial conditions

0, 0 and step size, Calculate the initial acceleration from

0 = 1. [ (0) 0 0]. Calculate LU factorization of 1 2 + 1 2 such that 1 2 + 1 2,

= in which L: lower triangular matrix U: upper

be a serious problem, particularly, when the FRF up to high

frequencies is desirable so that the time steps should be very small in the central difference integration to give the Discrete Fourier Transform (DFT) in high frequencies with reasonable accuracy.

V. COMPONENTS DESCRIPTION:

Brake system-functions:

The fundamental functions of a brake system can be summarized as follows. Reducing the speed of vehicle and if

necessary to a requested stationary position (normal braking) Prevent unwanted acceleration while travelling downhill. Maintain the vehicle at stationary condition by parking brake. Conduct the vehicle to full stop with high deceleration braking (emergency braking). Ensure vehicle stability (under and over-steering and maintain tire friction)The principal type of brake used to generate braking force at the wheels is called friction brake. Friction braking converts the potential and kinetic energy into heat. Friction brakes can be categorized into two types: disc brakes and drum brakes. Figure depicts a disc brake assembly with suspension and part of the subframe. The brake assembly consists mainly of a brake disc, a calliper and a pair of pads with shims and under layer designed to generate a brake torque. During braking, the calliper pushes the pads onto disc by hydraulic piston. The friction force generated at the frictional surfaces between the brake pads and rotating brake disc generates the required braking torque transferred to the wheel/tire to stop the vehicle. Disc brakes are used in all front axle of passenger cars and in some cases can also be found in the rear axle. Despite their higher cost as compared to drum brakes, disc brakes are more robust and havebetter cooling performance.

Drum brake:

Drum brakes are radial brakes combining a brake shoe mounted on the stub axle and a rotating brake drum mounted on the axle. Drum brakes are composed of two brake shoes (seldom one only) that are pressed outward against the friction surface of the drum by the action of hydraulic piston during braking. When the braking operation is done, a spring pulls back the brake shoes to ensure a clearance between the surface of drum and brake pad. Drum brakes are less sensitive to external 3 dust and rain as it is a closed component and is cheaper than disc brakes. However, drum brakes suffer from poor cooling characteristics and early cracking.

Automotive friction brakes are grouped according to their basic designs into two classes: Drum brakes use brake shoes that are pushed out in a radial direction against a brake drum. Disc brakes use pads that are pressed axially against a rotor or disco Advantages of disc brakes over drum brakes have led to their universal use on passenger- car and light- truck front axles, many rear axles, and medium-weight trucks on both axles.

Types of calipers:

There are mainly two types of calipers based on difference in their design they are Fixed calipers and Floating Calipers.

Fixed-caliper design, bolted solidly to the flange, has either two or four pistons which push the pads out. Fixed-caliper disc brakes have more balanced inner and outer pad wear with less pad taper than floating caliper designs. They require no anchor or integral knuckle for shoe support.

A typical floating caliper disc brake, one or two pistons are used on the inboard side only. The hydraulic pressure forcing the piston and pad toward the rotor also forces the piston housing (wheel cylinder) in the opposite direction to apply the outboard pad against the rotor. Floating caliper brakes offer a number of advantages over fixed caliper designs. They are easier 10 package in the wheel since they do not have a piston on the outboard or wheel side. They have a lower brake fluid operating temperature than the fixed caliper and, hence, lower brake fluid vaporization potential. They also have fewer leak points and are easier to bleed in service.

A Simplified Disc Brake System:

The simplified FE model of a brake system, see Figure 3, is composed of brake disc, a pair of brake pads. Here, the brake pads include the back plate but no shim neither friction material under layer is modelled. The back plates are made of steel, see Figure 4, to push the pad onto the brake disc and generate the requested friction torque. The normal force and frictional torque, I e tangential friction force, between the brake pads and brake disc may excite the brake disc to vibrate. The friction force makes the problem as nonlinear vibration so that the linear vibration theory is not able to model the squeal problem properly.

Brake Noises and Vibrations:

There are various kinds of brake noises and vibrations with numerous terminologies to nominate the phenomena in the literature that are probably inconsistent. One of the terminologies. Noises and vibrations can be classified that are based on frequency range. Some types of brake noise and vibration can be summarized.

VI Current Analysis Approach:

The complex eigenvalue analysis is the preferred method in the brake research community because it is more mature than the dynamic transient analysis. Furthermore, the complex eigenvalue analysis can provide much faster solutions than the dynamic transient analysis. To perform the complex eigenvalue analysis using ABAQUS, four main steps are required as follows. Nonlinear static analysis for applying brake-line pressure. Nonlinear static analysis to impose the 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. Since the equivalent stiffness matrix is asymmetrical because of friction, a complex eigenvalue analysis (CEA) is required. Equation can be written as a general eigenvalue problem. A.X = .X Where is the eigenvalue and X the eigenvector. Both are complex valuated. Especially, the eigenvalue may be written. = a + ib.

Complex Eigenvalue Extraction Procedure:

To extract complex eigen values in ABAQUS following procedure need to b followed: Step 1: Importing 3D CAD parts of brake system (Rotor, Pad, Caliper, Back plate, Fasteners and Knuckle) into the part module. Step 2: By using Property Module give the necessary properties like Elastic Modulus(E), Density(), and Poisson ratio() etc. to the respective parts. The property values for the parts are listed below. List of materials and their properties for case 1:

List of materials and their properties for case 1

Step 3: Select Assembly Module and assemble all components. Step4: create a general static step from Step Module. In the Model Tree, double-click Steps to create a new step. In the Create Step dialog box that appears, enter PRESSURE as the name and select Initial as the step after which the new step will be inserted. Choose Static, General from the list of available general procedures and click Continue. Accept all default setting for the step definition. Step5: create a general static step using Step Module. In the Model Tree, double-click Steps to create a new step. In the Create Step dialog box that appears, enter ROTOR MOTION as the name and select PRESSURE as the step after which the new step will be inserted. Choose Static, General from the list of available general procedures and click Continue. Step 6: Create Frequency step using Step Module. In the Model Tree, double-click Steps. In the Create Step dialog box that appears, enter FREQ as the name. Select frequency from the list of linear perturbation procedures and click Continue. Step 7: Now, create a complex eigenvalue extraction step to extract the eigenmodes after the natural frequency extraction step FREQ. In the Model Tree, double-click Steps. In the Create Step dialog box that appears, enter COMPLEX FREQ as the name. Select Complex frequency from the list of linear perturbation procedures and click Continue. In the step editor, request eigenvalues. In the Other tabbed page of the editor, note that the unsymmetric matrix solver is selected. Click OK. Step 8: Define an interaction property with a coefficient of friction of 0.4 and apply it to the predefined contact pairs in the ROTOR MOTION step. In the Model Tree, double-click Interaction Properties. In the Create Interaction Property dialog box that appears, enter PAD_ROTOR-FRICTION as the name and select Contact as the type. Click Continue. In the Create Interaction Property dialog box that appears, enter PAD_ROTOR- FRICTION as the name and select Contact as the type. Click Continue. In the Edit Contact Property dialog box that appears, select Mechanical Tangential Behavior. Choose Penalty as the friction formulation and enter 0.4 as the coefficient of friction. Click OK. In the Model Tree, click mouse button 3 on Interactions and select Manager from the menu that appears. In the Interaction Manager, select the cell corresponding to the first interaction under the step named ROTOR MOTION and click Edit. In Edit

Interaction dialog box that appears, change the contact

interaction property to PAD_ROTOR-FRICTION and click OK. Repeat steps e and f for the other interactions. Step 9: Predefined motion fields are not currently supported in Abaqus/CAE; so, we will use the keyword editor to specify an angular velocity of 5.0 rad/sec about the Z axis. The predefined set ROTOR will be used for this purpose. From the main menu bar, select Model Edit Keywords brake. In the Edit keywords dialog box appears, scroll down to find the *STATIC option block associated with the ROTOR MOTION step and select it. Click Add After at the bottom of the dialog box and enter the following text. MOTION, ROTATION ROTOR, 5., 0,0,0, 0,0,1 Note: the first entry is the angular velocity and the next two are coordinates that define the axis of rotation. Click OK. Step 10: Applying Loads and boundary conditions to the given brake system using Load Module.

Pressure applied on both sides & Boundary condition

Step 11: Build Finite element model by meshing the components by using Mesh Module. Apply Tetrahedral mesh to all the components. Step 12: Now create job by using Job Module and run the simulation. Output: The real (EIGREAL) and imaginary (EIGIMAG) parts of the eigenvalues, (a and b) frequencies in cycles/time (EIGFREQ); and damping ratios (DAMPRATIO = – 2a/|b|) are written automatically to the data (.dat) file and to the output database (.odb) file as history data. In addition, you can request that the generalized displacements (GU), which are the modes of the projected system, be written to the output database file. Output variables such as stress, strain, and displacement (which represent mode shapes) are also available for each eigenvalue; these quantities are perturbation values and represent mode shapes, not absolute values. Finally, two unstable modes are formed at 54th and 89th modes with squeal frequencies of 6096.5Hz and 8821.9Hz respectively as shown in figure.

VII. Results and Conclusion:

The focus of this study is on the reduction of low frequency squeals which ranges from 1.5 kHz to 10 kHz. To cover such a wide frequency range, 200 modes are requested in the complex eigenvalue calculation. In this case study, the basic parameters chosen for the parametric studies are friction coefficient and brake disc rotation speed. The friction coefficient = 0.4, and brake disc rotation speed is 5 rad/s. The results of complex eigenvalue analysis are in the form of = a +ib where a is the real part and b is the imaginary part of the eigenvalue which presents frequency of the unstable mode. All calculated eigenvalues are then plotted on a complex plane as shown in Figure. The squeal

frequencies of 3 and 8 kHz from the test data are all predicted in the complex eigenvalue results as unstable modes. CASE-1: Finding component contributions through component contribution factor plug in at unstable modes.

Damping ratio Vs Mode number Plot (Case -1)

Results around 1st Unstable mode (Case-1)

Results around 2nd unstable mode (Case-1)

CCF plots:

Fig.17 CCF plot at 54th mode

CCF plot at 89th mode

Results from CCF plots found that Caliper and Rotor are contributing more for forming ofinstabilities in the brake system.

CASE-2:

Reduction of unstable modes by varying Material and Geometric parameters of the Caliper. From the literature review it is seen that instability of the most unstable modes are reduced by varying material properties and geometric properties of the Caliper. New material list for optimization of squeal.

List of Materials and their properties for case 2

When Caliper is changed from Thermoplastics to Aluminium alloys and changes in geometry of the caliper, following results are obtained.

Fig.19 Damping ratio Vs Mode Number plot (case 2)

Fig.21 Mode shape at 44th Mode(case-2.1)

Fig.22 CCF for unstable mode: 44(5810.6 Hz)

Results around 2nd unstable mode (Case-2)

Fig.23 Mode shape at 54th Mode(case-2.2)

Table 5. Results around 1st unstable mode (Case-2)

Results and Conclusion:

Table 7. Results around 3rd unstable mode (Case-2)

Fig.24 CCF for unstable mode: 54(6462.4 Hz

Fig.25 Mode shape at 62nd Mode(case-2.3)

Fig.26 CCF for unstable mode: 62(6922.1 Hz)

Verifying Brake Squeal Simulations:

Verifying via correlation with test results or results from other programs. Checking the mass of the assembly of the brake system. Mass of the brake system from simulation results =12Kg. Approximate mass of the original brake system =12 to 15Kg. Assembly level correlation. Checked how the natural frequencies of the whole system correlate with test data or other available results.

This paper explores a proper way of predicting squeal frequencies using complex eigenvalue analysis. The complex eigenvalue analysis is performed with an implicit version of ABAQUS. The Process of Eigen Value analysis Provides the cause of Noise generation in Disc brake assembly. Unstable modes are predicted by using complex eigen value analysis. Instabilities present in the system are reduced by changing material of caliper from Thermoplastics to Aluminium alloys and outer geometry of the Caliper. With changes in brake disc structure especially the vane pattern and length of the flange it is possible to further reduce and eliminated the unstable modes at frequencies. Thus, instabilities present in the system are eliminated for low squeal frequency range.

Recommendations for Future Work:

The following suggestions are made for future work. Squeal analysis can be performed by varying parameters such as brake pressure, brake temperature, wear etc. The materials of the assembly can be optimized by composite materials. Dynamometer Experimental analysis may produce optimum results. Combining the data of modal analysis and thermal analysis would produce optimum results.

References:

1. Prof. V Kumar, Automotive Brake Disc Design to Suppress High Frequency Brake Squeal Noise. IJERT ISSN:2278-0181,Volume 03, Issue 01 -(January 2014)

2. Abd rahim abubakar and huajiang Ouyang Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal. Research Gate, International Journal of Vehicle Noise and Vibration (IJVNV), Vol. 2, No. 2, 2006.

3. En-cheng. liu, shih-wei kung et al., Effect of chamfered brake pad patterns on the vibration squeal response of disc brake system December 2008Conference: 3rd International Symposium on Advanced Fluid/Solid Science and Technology in Experimental Mechanics At: Tainan, Taiwan

4. Zaidi bin mohd ripin Analysis of disc brake squeal using the finite element method Department of Mechanical Engineering, University of Leeds, United Kingdom September 1995

5. Mir arash Keshavarz, Brake squeal analysis in time domain using abaqus finite element simulation of brake squeal for passenger cars masters thesis in automotive engineering programme.

6. Crolla, D A. and Lang, A.M. Brake Noise and Vibration

   – The State of the Art, Vehicle Tribology, Leeds-Lyon 17, Tribology Series 18, (Dowson, D. ; Taylor, C

   M. and Godet, M. , ed.) Elsevier Science Pub., Sept. 1991, pp. 165 -174.

7. Case for Support Internal Report of Mechanical Engineering Department, Leed University, Sept. 1990.

8. WHICH? CAR-Guide to New and Used Cars 1995, Consumer Association, London 1995.

9. Smales, H Friction Materials – Black Art or Science, Proc. I.Mech.E. Vol. 209, No. D3, Part D: Journal of Automotive Engineering, 1995, pp. 151 -157.

10. Murakami, H.; Tsunada, N.;Kitamura, M. A Study Concerned with Mechanism of Disc brake Squeal, SAE Paper841233.

11. Fieldhouse, J.D. and Newcomb, T.P. An Investigation into Disc Brake Squeal Using Holographic Interferometry, 3rd Intl EAEC Conference on Vehicle Dynamics and Powertrain Engineering – EAEC Paper No. 91084, Strasbourg, June 1991

12. Lamarque, P.V. Brake Squeak: The Experience of Manufacturers and Operators: Report No. 8500B Inst. Auto. Engrs. Research and Standardization Committee 1935.

13. Fosberry, R A C. and Holubecki, Z. An Investigation of the Causes and Nature of Brake Squeal, MIRA Report 1955/2.

14. Fosberry, RAC and Holubecki, Z. Some Experiments on the Prevention of Brake Squeal, MIRA Report 1957/1.

15. Sinclair, D. Frictional Vibrations, Journal of Applied Mechanics, 1955, pp 207 – 214.

16. SAE Paper 2002-01-0922 Modal Coupling and Its Effect on Brake Squeal Research and Vehicle Technology, Ford Motor Co.

17. Investigation of the Effects on Braking Performance of Different Brake Rotor Designs -Paper by Mirza Grebovic

18. Liles, G. D., Analysis of Disc Brake Squeal Using Finite Element Methods, SAE Paper 891150.

19. Nack, W. V., and A. M. Joshi, Friction Induced Vibration: Brake Moan, SAE Paper 951095.

20. Hamzeh, O. N., W. Tworzydlo, H. J. Chang, and S. Fryska, Analysis of Friction- Induced Instabilities in a Simplified Aircraft Brake, SAE Brake Colloquium, 1999.

21. Kung, S.-W., K. B. Dunlap, and R. S. Ballinger, Complex Eigenvalue Analysis for Reducing Low Frequency Squeal, SAE Paper 2000-01-0444.

22. Moirot, F. C., A. Nehme, and Q. C. Nguyen, Numerical Simulation to Detect Low-Frequency Squeal Propensity, SAE Paper 2000-01-2767.

23. Shi, T. S., O. Dessouki, T. Warzecha, W. K. Chang, and A. Jayasundera, Advances in Complex Eigenvalue Analysis for Brake Noise, SAE Paper 2001- 01-1603.

24. Lee, L., K. Xu, B. Malott, M. Matsuzaki, and G. Lou, A Systematic Approach to Brake Squeal Simulation Using MacNeal Method, SAE Paper 2002-01-2610.

25. Ouyang, H.-J., W. V. Nack, Y. Yuan, and F. Chen, On Automotive Disc Brake Squeal-Part II: Simulation and Analysis, SAE Paper 2003-01-0684.

26. Bajer, A., V. Belsky, and L.-J. Zeng, Combining a Nonlinear Static Analysis and Complex Eigenvalue Extraction in Brake Squeal Simulation, SAE Paper 2003-01- 3349.

27. Kung, S.-W., G. Stelzer, V. Belsky, and A. Bajer, Brake Squeal Analysis Incorporating Contact Conditions and Other Nonlinear Effects, SAE Paper 2003- 01-3343.

28. Bajer, A., V. Belsky, and S.-W. Kung, The Influence of Friction-Induced Damping and Nonlinear Effects on Brake Squeal Analysis, SAE Paper 2004-01- 2794.

29. Abendroth, H., Advances in Brake NVH Test Equipment, Automotive Engineering International, pp. 60, February 1999.