Parametric Study of Engine Rigid Body Modes

Transcription

1 Parametric Study of Engine Rigid Body Modes Basem Alzahabi and Samir Nashef C. S. Mott Engineering and Science Center Dept. Mechanical Engineering Kettering University 17 West Third Avenue Flint, Michigan, 4854, USA ABSTRACT Powertrain mounting systems serve a number of functions for the overall vehicle. First and foremost, the mounting system must maintain the position of the powertrain in the system, control the overall motion of the powertrain, and also prevent the engine, transmission, and accessories from contacting other components of the vehicle, thereby avoiding damage to any vehicle systems from the potential impact. The powertrain-mount system design is a compromise between isolating the powertrain vibration from the vehicle, achieving the rigid body mode targets for vehicle ride and handling, and restraining the powertrain motions within vehicle packaging constraints. Therefore, investigating the powertrain vibration characteristics for a particular mounting system is very critical in determining initial mount locations and spring rates for a new powertrain or vehicle architecture. In this paper, a parametric study of the powertrain s vibration characteristics is presented. In this investigation, the natural frequencies of the rigid body modes, kinetic energy, and the strain energy of the mounts in different directions are calculated. The kinetic energy of each mode is used to identify its corresponding rigid body mode. The impact of spring rate changes and variation of engine mount location is studied, and the critical parameters affecting the modal characteristic were also identified. The validity and accuracy of the analytical solution is verified with the numerical results obtained from finite element analysis. NOMENCLATURE [M] [K] {u&} & {u} x c y c z c α β γ m c I k a Mass Matrix Stiffness Matrix Powertrain's Center of Gravity's Acceleration vector Powertrain's Center of Gravity's Displacement vector Center of Gravity's For/Aft translational displacement. Center of Gravity's Lateral translational displacement. Center of Gravity's Vertical translational displacement. Roll angular displacement. Pitch angular displacement. Yaw angular displacement. Powertrain's mass. Powertrain's mass moment of inertia of the corresponding subscript direction. Stiffness value in the corresponding subscript direction and mount number. Distance between the corresponding subscript mount number and the CG in its subscript direction.

2 ω n Ke Mode shape's natural frequency. Kinetic Energy. SUBSCRIPTS. i x y z xy, xz, yz Mount's number. Property (Stiffness, Location, or Inertia) in the Longitudinal X direction. Property (Stiffness, Location, or Inertia) in the Transverse Y direction. Property (Stiffness, Location, or Inertia) in the Vertical Z direction. Off-diagonal Mass moment of Inertia elements relative to the vehicle's coordinate system. INTRODUCTION The analysis of the powertrain system's mode shapes vibration behavior can lead to a better design model in terms of noise and vibration transmissibility. The powertrain represents a considerable mass of the total mass of the vehicle, and can be simplified as a single rigid body. Powertrain mounts are required to be stiff enough and positioned well, not only to support the powertrain, but also to restrain its movement, therefore minimizing noise transmission to driver compartment. Previous studies [1] of powertrain systems determined the contribution of powertrain mounting system to the overall noise transmission under various driving conditions. Natural frequencies of the powertrain assembly along with the corresponding rigid body modes [2] of the system are indicators of to which degree the powertrain meets the vehicle driving requirements [1,]. Realizing that some properties of the powertrain can modified, mount properties such as stiffnesses and positions are viewed as variables that can be evaluated to provide satisfactory vibration mode results. In this study, the vibration characteristics of the powertrain system are formulated as an eigenvalue problem. The six natural frequencies and their corresponding rigid body modes are obtained by solving the eigenvalue problem. The six mode shapes are rarely pure in a single direction, i.e. vertical, lateral, for/aft, roll, pitch, or yaw. The coordinate coupling rate is a measure of how a mode shape can be described as a combination of motion in different directions [,6]. The sensitivity of the vibration characteristics, i.e. natural frequencies and normal modes, due to changes in mounts' rates and position has been evaluated. First, the eigenvalue problem is developed by writing the dynamic equation of motion of the powertrain [4]. Solution of the eigenvalue problem is obtained by utilizing a Matlab script. The script allows for variation in the input parameters i.e. stiffness rates and mounts' locations. The results were finally compared with FEA results obtained using MSC.NASTRAN and found to be in a very good agreement with the analytical results. ANALYTICAL DERIVATION To study the powertrain's six rigid body modes, the powertrain is simplified as a concentrated mass and inertia properties relative to the center of gravity location [4,5]. Each mount can also be simplified as a combination of three uniaxial scalar linear springs acting along the three axial directions in the mount location [4,5] (see Figure (1)). Therefore, the powertrain-mount system is reduced to a mass-spring system in which a concentrated mass is connected to the uniaxial springs by rigid elements. Figure 2 Kinematic representation of a bushing Applying Newton s Second Law [4] yields a six second-order differential equations for the displacement vector {u}. These dynamic equations of motion can be written in the matrix form as:

3 [ M ]{ u& } + [ K]{ u} = {} (1) Where, M is the mass matrix, K is the equivalent stiffness, and {u} is the displacement vector of the powertrain's center of gravity. mc [ M] = m c m c I x I I xy xz I I y I xy yz I xz I yz I z (2) K11 K 22 Symmetric K [ K ] = () K 42 K 4 K 44 K 51 K5 K54 K55 K 61 K 62 K64 K65 K 66 The components of the Stiffness matrix are presented in appendix A at the end of the paper [4]. The displacement components of the center of gravity along the six degrees of freedom can be represented as: [ x y z α β γ ] T { u} = (4) c In order to determine the decoupling rates of the various mode shapes, the percentage kinetic energy for each particular degree of freedom is calculated [5,6]. By changing the mount position and stiffness values, decoupling rates can be varied over a desired frequency range of a particular mode shape. The next section will address an illustrative application to the previous derivations, followed by the observations and remarks on the results. ILLUSTRATIVE EXAMPLE A parametric study of a powertrain of known inertia properties Table (1), mount stiffnesses Table (2), and positions (Table ), is performed. The normal mode analysis will be calculated first to find the frequency and decoupling rates. Then the sensitivity of the system to changes in the mount parameters will be studied. The mount parameter sensitivity is studied by setting a range of values for the position and stiffness parameters of the three mounts. The derived results are presented in the form of charts and tables of the parameter changes Vs natural frequencies and decoupling rates. Analyzing these results will help in determining suitable change required to improve the base-line design for the rigid body mode requirement. The analysis was performed using a Matlab script developed for this purpose. To confirm accuracy of the analysis, the results were compared to the ones obtained from the FE analysis. The system s six mode shapes were described as For/Aft for X translational, Lateral for Y translational, Vertical for Z translational, Roll for X rotational, Pitch for Y rotational, and Yaw for Z rotational modes. Some ideal values that had been suggested to match handling safety and comfortable driving conditions requirements address that the most important modes of Vertical, Pitch, and Roll need high decoupling rates +8%, and natural frequencies staying between 7-15 [Hz] [1,]. Tables (1, 2, and ) include the base-line design properties and parameters. c c Powertrain Properties Mass (m c ) [Kg] 6. Mass Moment of Inertia (I x ) [Kg.mm 2 ] 1.74 x 1 7 Mass Moment of Inertia (I y ) [Kg.mm 2 ] 2.48 x 1 7 Mass Moment of Inertia (I z ) [Kg.mm 2 ].1 x 1 7 Table 1 Powertrain properties Relative to the C.G.

4 X direction (k x ) [N.mm -1 ] Mounts' spring rates Y direction (k y ) [N.mm -1 ] Z direction (k z ) [N.mm -1 ] Mount No Mount No Mount No Table 2 Powertrain mounts' spring rates Center of Gravity Positions of the powertrain C.Gnd Mounts Longitudinal X direction Transverse Y direction Vertical Z direction Mount No Mount No Mount No Table Baseline powertrain C.Gnd mounts locations relative to the vehicle coordinate system. The analyses were performed using MATLAB and MSC.NASTRAN. They both showed identical results for natural frequencies and their attributed decoupling rates. Table (4) presents the resulting values derived from both approaches. Natural Frequency [Hz] Kinetic Energy Distribution [%] Fore/Aft Lateral Vertical Roll Pitch Yaw % <1% <1% <1% 5.8% 1.5% 7.7 <1% 7.6% 5.% 17.8% 1.4% 1.7% % 6.% 6.% 1.9% 6.4% 22.9% % 4.8% 24.8% <1% <1% 68.4% 11.4.% 4.% 7.2%.% 76.% 6.% 14.6 <1% 11.5% 2.5% 76.8% 9.5% <1% Table 4 - Natural frequencies and coupling rates of the powertrain system (MATLAB and MSC.NASTRAN). As seen in this table, for each natural frequency value, the highest corresponding decoupling rate reflects the attributed mode to this natural frequency value, i.e. for the first value 5.21 Hz the mode is obviously is the For/Aft mode shape with 92.% of the kinetic energy acts in this degree of freedom direction, and so on. For some design variables, it could be the case when a specific mode is dominating two or more natural frequencies,

5 this means that an equal number of mode shapes will not be represented; this case is called a mode-crossing case and will be addressed in the graphs that follow. The decoupling rates for this design require optimizing as can be observed from Table (4). For each mount, there are three position variables and three stiffness variables, which lead to eighteen controllable variables. The changes of positions were constrained to a range of '±1 mm' from the nominal position value with a step of '8. mm', stiffness constraint ranges were set to '±8.E4 N.m -1 ' with a step of '6.4E N.m -1 '. I. Sensitivity to Location Change The location change has been represented as 'a', which is the difference between the original and new positions of the mount relative to the vehicle's coordinate system. Mount No.1 Variation in Longitudinal X Direction (a x ) 1 : - Natural Frequencies: Figure (2) shows that over the range of (-416,-196) mm, there is barely any change in the natural frequencies of the six mode shapes. The Vertical mode ceases to be represented beyond (a x ) 1 =-25 due to the fact that its decoupling rate becomes very low to be considered a mode. A wider range for (a x ) 1 can show how the Vertical mode will engage in a mode-crossing case with another mode when its decoupling rate starts to increase again. - Decoupling Rates: Examination of Figure (-a) shows the translational For/Aft and Lateral modes experiencing slight decline over the range, while the vertical mode declines steeply to 5% at (a x ) 1 =-25. Figure (-b) illustrates the rotational modes, while the Roll keeps increasing in a slight rate, the Pitch mode reaches a maximum value at (a x ) 1 =-1, and the Yaw mode keeps declining to a minimum point at (a x ) 1 =-27. The change of parameter (a x ) 1 is obvious to be much critical regarding the decoupling rates than the natural frequency values for all the six mode shapes. Decoupling Rate [%] Figure 2. Natural Frequency for the six mode shaper relative to the change in (a x ) 1 Figure. Decoupling Rates for the six mode shaper relative to the change in (a x ) 1

6 Variation in Transverse Y Direction (a y ) 1 : - Natural Frequencies: Slight change in natural frequency values has been identified for the mode shapes. - Decoupling Rates: While slight increase for the For/Aft mode was identified, Roll, Pitch and Vertical modes have experienced high rates of increase in decoupling rates over the range. The Yaw mode declines over the range to reach a minimum point at (a y ) 1 = 9. The analysis shows that the higher end of the range offers an optimal solution for all the six mode shapes when compared to the nominal design value. Variation in Vertical Z Direction (a z ) 1 : - Natural Frequencies: Slight decline for Roll mode by (1 [Hz]) over the range can be noticed, while the Lateral mode increases by (1 [Hz]). For the other modes, there is only an insignificant change in value. - Decoupling Rates: Yaw and Vertical modes start at high decoupling rates at the lower end of the range, i.e. (a z ) 1 =-225 but decline steeply over the range. On the other hand, there is only slight change in decoupling rate for the rest modes. The lower end of the range offers an optimal decoupling rate for all the mode shapes compared to the nominal value of the parameter. Mount No.2 Variation in Longitudinal X Direction (a x ) 2 : - Natural Frequencies: A small increase by (1 [Hz]) could be identified in Roll mode over the range, while both the Pitch and Yaw modes increase by (2 [Hz]). The analysis identifies a mode-crossing situation occurring between the Yaw and Vertical modes at (a x ) 2 = 26., the natural frequency values for both modes are too close to have the mode-crossing causing any significant switching of values. - Decoupling Rates: This parameter shows a positive effect on all the six mode shapes at the lower end of its range. While the For/Aft and Lateral modes show no significant change in decoupling rates over the range, Roll mode decline steeply over the range. While all Vertical, Pitch and Yaw modes reach a maximum peak at (a x ) 2 = 22, both Vertical and Pitch modes reach a minimum point and engage in mode-crossing at (a x ) 2 = 26. as has been stated before. Variation in Transverse Y Direction (a y ) 2 : - Natural Frequencies: The analysis shows that the Roll mode could be increased by (2 [Hz]) over the range, where the rest of the modes experience slight changes only. Concerning the natural frequency this parameter is an example of the case when only one property needs to be modified while the rest of properties are desired to stay constant. - Decoupling Rates: Vertical and Lateral modes keep increasing over the range in parallel curves over the range. Steep decline could be noticed for the Pitch mode over the range, while slight changes take place for the rest of the modes. Variation in Vertical Z Direction (a z ) 2 : - Natural Frequencies: Slight changes for Yaw and Vertical modes but with a mode-crossing taking place between them at (a z ) 2 =-27, Roll mode declines by (1 [Hz]) over the range, and only slight changes for the rest of the modes. - Decoupling Rates: Another mode-crossing case could be identified between Vertical and Yaw modes when they both reach a minimum point at (a z ) 2 =-27. For the rest of the modes, only slight changes take place. The highest end of the range shows to have a positive effect on all mode shapes regarding decoupling rates compared to the nominal design value for this parameter. Mount No.

7 Variation in Longitudinal X Direction (a x ) : - Natural Frequencies: The analysis shows that this parameter has no significant effect on the natural frequencies. - Decoupling Rates: The highest end of the range is found to enhancing the decoupling rates for all mode shapes. The Pitch mode experiences a minimum point at (a x ) = 15 while the rest of the modes experience steady increase over the range. Variation in Transverse Y Direction (a y ) : - Natural Frequencies: As Figure (4) shows, the Roll mode declines over the range by (2.5 [Hz]). A modecrossing case takes place between the Yaw and Vertical modes at (a y ) =-44 corresponds to the minimum value of decoupling rates for both modes as will be mentioned for Figure (5). Decoupling Rate [%] Figure 4. Natural Frequency for the six mode shaper relative to the change in (a y ) Figure 5. Decoupling Rates for the six mode shaper relative to the change in (a y ) - Decoupling Rates: Observing Figure (5-a) we can see that there is almost no change of decoupling rates for both For/Aft and Lateral modes over the range. The Vertical mode keeps declining to reach a minimum point at (a y ) =-44 where the mode-crossing with the Yaw mode takes place. The Yaw mode at Figure (5-b) behave in the same fashion as the Vertical mode by declining to reach a minimum point at (a y ) =-44 and then increase to reach almost the same value of the beginning of the range at the end of it, where the Roll and Pitch modes keep declining over the range having almost the same values over the whole range. The lower end of the range shows to have a positive effect on the decoupling rates for all mode shapes. On the other hand, the parameter shows to be affecting the Roll mode only regarding the natural frequency value.

8 Variation in Vertical Z Direction (a z ) : - Natural Frequencies: Slight changes for all modes over the range. The analysis shows Yaw and the Vertical modes engaging in a mode-crossing situation at (az)=-1 and the Yaw mode ceases to exist due the extremely low value for its decoupling rate at that point, and the Vertical mode crosses to take its value making safe to predict that the Yaw mode will switch modes with the Vertical mode once its decoupling rate becomes high enough somewhere beyond the range upper limit. - Decoupling Rates: There is almost no change for the For/Aft, Roll or Lateral modes. All Pitch, Yaw, and Vertical modes decline over the range where the Vertical mode reaches its minimum value at (az)=-1 'the same point where the mode-crossing takes place', at the same point the Yaw reaches a very low decoupling rate where it can no longer be represented as an acting mode below this point. II. Sensitivity to Stiffness Change Mount No.1 Variation in Longitudinal X Direction (k x ) 1 : - Natural Frequencies: The analysis shows this parameter having slight effect on the For/Aft mode, with no change for the rest of the mode shapes. - Decoupling Rates: For decoupling rates as well, this parameter shows no significant effect on any of the mode shapes. Variation in Transverse Y Direction (k y ) 1 : - Natural Frequencies: A mode-crossing case takes place between Yaw and Vertical modes at (k y ) 1 =2.9 N.m -1 which coincides with minimum points in decoupling rate curves for both modes. There has been no significant change in natural frequency for the rest of the mode shapes. - Decoupling Rates: Both Yaw and Vertical modes keep increasing over the range. The highest limit of the range is shown to have an enhancing effect on the decoupling rates for Yaw and Vertical modes only while keeping all decoupling rates for the rest of the modes constant. Variation in Vertical Z Direction (k z ) 1 : - Natural Frequencies: A mode-crossing case takes place between Yaw and Vertical modes at (k z ) 1 =5.2 N.m -1 with slight change in value for all of the mode shapes. - Decoupling Rates: Yaw and Vertical modes decline slightly to a minimum point at (k z ) 1 =5.2 N.m -1 where the mode-crossing takes place. Mount No.2 Variation in Longitudinal X Direction (k x ) 2 : - Natural Frequencies: A small effect could be recorded for this parameter on the For/Aft and Pitch modes that increase by (1[Hz]) over the range. All the rest of mode shapes do not show any change in natural frequency values over the range. - Decoupling Rates: For decoupling rates also, the analysis shows that this parameter has a little effect on the all mode shapes. Variation in Transverse Y Direction (k y ) 2 : - Natural Frequencies: No significant change for any of the six modes has been monitored.

9 - Decoupling Rates: Changing the value of this parameter over the range showed a very limited effect on all the six mode shapes regarding decoupling rates. Variation in Vertical Z Direction (k z ) 2 : - Natural Frequencies: From Figure (6) we can see slight increase in natural frequency value for the Roll and Pitch modes with no significant change for the rest of mode shapes. - Decoupling Rates: We can see from Figure (7-a) that the Vertical mode keeps declining over the range losing about 4% of the value at the beginning and slight increase for the Lateral mode. Figure (7-b) shows slight decline of value for the Pitch and Yaw modes over the range with almost no change in value for the Roll mode. Decoupling Rate [%] Figure 6. Natural Frequency for the six mode shaper relative to the change in (k z ) 2 Figure 7. Decoupling Rates for the six mode shaper relative to the change in (k z ) 2 Mount No. Variation in Longitudinal X Direction (k x ) : - Natural Frequencies: From Figure (8) we can notice no change in the natural frequency values over the whole range except for a mode crossing between the Yaw and Vertical modes that takes place at (k x ) =1.4e5 N.m -1 which -as we will see from Figure (9)- coincides with a minimum point for decoupling rate curves for both modes.

10 - Decoupling Rates: From Figure (9-a) we can see that the For/Aft and Lateral modes experience no change over the whole range, where the Vertical mode reaches a minimum point at (k x ) =1.4e5 N.m -1 corresponding to the mode crossing case with the Yaw mode. Both Yaw and Vertical modes keep increasing over the range with no change for any of the other modes. Decoupling Rate [%] Figure 8. Natural Frequency for the six mode shaper relative to the change in (k x ) Figure 9. Decoupling Rates for the six mode shaper relative to the change in (k x ) Variation in Transverse Y Direction (k y ) : - Natural Frequencies: Slight changes of natural frequency values take place only for all the six mode shapes. - Decoupling Rates: For the decoupling rates, changing the parameter has shown no effect on any of the six mode shapes. Variation in Vertical Z Direction (k z ) : - Natural Frequencies: The analysis showed the Roll mode increasing over the range by (1.5 [Hz]) with almost no change for the rest of the mode shapes. - Decoupling Rates: While the Vertical mode declines over the range by 4%, there is slight effect on the rest of the mode shapes concerning the decoupling rates. CONCLUDING REMARKS Changes in natural frequencies and decoupling rates with respect to the change in the stiffness and location of the mounts has been presented in the previous section. By observing those graphs and the comments on the

11 results it can be seen that changes in positions are much more crucial in terms of affecting both natural frequency and decoupling rates, although, it is obvious that changes in natural frequencies are not as drastic as changes in decoupling rates. In the examples presented, changes in stiffnesses appear to affect specific modes only out of the existing six modes. Although the change in the resulting values is slow through the range, it is useful when the intension is to optimize specific modes and to keep the values attributed to the other modes as Figure (9) shows for example. On the other hand, position parameters usually affect all the six modes in terms of natural frequency and decoupling rates. The effect is larger in terms of the decoupling rates. Changes of parameter (a x ) 1 (the longitudinal position of mount one with respect to the C.G.) appear to be quite effective, but at the same time very sensitive. Another phenomenon observed in the graphs is the "Mode-Crossing" case, which is a single step in parameter value that causes the natural frequency of a specific mode to switch dramatically to another mode's natural frequency value and take each other s values. The case is accompanied by the point that both decoupling rates curvatures of the two degrees of freedom reach a global minimum point. It can also be noticed that one of the two curves ceases being represented in the graph during this transient area, this is due to the fact that the natural frequency that was related to it does not have any obvious direction attributed to it anymore because its decoupling rate becomes very low, as Figure (2),Figure () show for Vertical curve. Although its value in this transient area is predictable in the case where only two modes are engaged in mode crossing case, as for the case of Vertical mode in Figure (2), This is not always the case. Those observations apply to all of the eighteen parameters investigated in terms of the general behavior with respect to the change in mount position or stiffness. Attempting to optimize the system with those eighteen variables and two design objectives natural frequency and decoupling rates is a challenging task. This paper developed an approach aiming to clarify the most effective parameters to target in order to simplify any optimization analysis. The approach showed coherent results with those obtained using MSC.NASTRAN code. APPENDIX A This appendix includes the stiffness matrix that has been represented by equation () for the powertrain-mount system that had been studied in the paper [4]. Component Content Component Content Component Content K 11 k x i 44 i= 1 2 k kz i yi 61 2 K y z + i i K k xi yi K 22 k y i 51 i= 1 K k x i z K i 62 k yi xi K k z i 5 i= 1 K k z i x K i 64 k xi yi zi K 42 k y i z K i 54 k z x i i y K i 65 k xi yi zi K 4 k z i yi 55 2 k kz i xi 66 2 K x z + i i 2 K k x y + i i yi 2 xi k Table 5 Non zero components of the stiffness matrix K.

12 APPENDIX B This appendix contains the complete analysis results for the rest of the parameters that were investigated in the study. Figure-1 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a x) 2. Figure-11 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a x). Figure-12 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a y) 1. Figure-1 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a y) 2.

13 Figure-14 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a z) 1. Figure-15 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a z) 2. Figure-16 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (a z). Figure-17 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k x) 1.

14 Figure-18 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k x) 2. Figure-19 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k y) 1. Figure-8 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k y) 2. Figure-8 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k y).

15 Figure-8 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k z) 1. Figure-8 Natural Frequency and Decoupling rates for six mode shapes relative to the change in (k z). REFERENCES [1] L. Piancastelli, P. Barnard, and N. Powell, ADAMS Custom Interface for Powertrain Mounting System Design, ADAMS User Conference Proceedings, 2. [2] W. Thomson, M. D. Dahleh, Mechanical Vibration Theory of Vibration with Application, Prentice Hall 199/1998, ISBN X. [] B. Alzahabi, A. Mazzei, L. Natarajan, Investigation of Powertrain Rigid Body Modes, IMAC XXI 2, Society of Experimental Mechanics, International Conference on Experimental Mechanics. [4] C. M. Harris, Shock and Vibration Handbook, fourth edition, McGraw-Hill, ISBN [5] S. M. Pandit, Y-X. Yao, Z-Q. Hu, Dynamic Properties of the Rigid Body and Supports from Vibration Measurements, Journal of Vibration and Acoustics, Transaction of the ASME, Vol. 116, JULY [6] M. R. Hatch, Vibration Simulation Using MATLAB and ANSYS, Chapman & Hall/CRC 21, ISBN