College of Engineering, Michigan State University,
East Lansing, Ml 48824
Optimal Design and Simulation of Vibrational Isolation Systems Vibration isolation of a rigid body on compliant mounts has many engineering applications. An analysis for solving these problems using a rigid body simulation and a penalty function optimization is discussed. The simulation is used to calculate natural frequencies and mode shapes, which are a function of the mount design parameters. Laboratory testing results are presented which verify the accuracy of the simulation. The optimization procedure penalizes natural frequencies in an undesirable frequency range and also large design changes. This penalty function is minimized by changing the mount design paramters consisting of the location, stiffness, and/or orientation. The result is a set of design parameters defining a vibration isolation system with natural frequencies moved away from the center of the undesirable frequency range. An interactive computer program was written which allows the engineer to use this technique as a design tool.
Introduction Vibration isolation of rigid bodies on compliant mounts is a
common engineering design problem. The problem discussed here is the minimization of forces transmitted through the compliant mounts when external forces are applied to the rigid body. The sources of these external forces may include rotational imbalance and reciprocating masses. Examples include the isolation of machinery such as compressors, electric motors, and automotive engines.
For small damping and frequencies below a mode's natural frequency, the amplitude of the forces transmitted from the rigid body through the compliant mounts to the supporting structure is proportional to the mount modal stiffness. The forces decrease rapidly at frequencies higher than the mode's natural frequency. When an excitation frequency is near one of the natural frequencies of the rigid body, both the rigid body displacement and the forces transmitted through the mounts can be large. The vibration isolation design goal emphasized here is the reduction of large transmitted forces by moving the system natural frequencies away from common operating excitation frequencies.
The rigid body spectrum of natural frequencies is depen-dent on the mount system design parameters (i.e., mount stiffness, attachment location, and orientation). A simulation of the rigid body dynamics, which calculates natural frequencies as a function of the design parameters without requiring a prototype system to be built for each case, is useful during the design process. This paper discusses a rigid body model which predicts the effects of mount design changes on both the rigid body vibration modes and the normalized forces transmitted to the supporting structure. Laboratory testing results are presented which verify its accuracy.
Contributed by the Design Automation Committee and presented at the Design Engineering Technical Conference, Cambridge, Mass, October 7-10, 1984 of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at ASME Headquarters, July 6,1984. Paper No. 84-DET-90.
The large number of combinations of design parameters which can be changed to yield an acceptable frequency spectrum makes the design process difficult when done manually. A computer optimization incorporating the abovementioned simulation increases the effectiveness of the design process. A penalty function optimization technique will be discussed which seeks to remove natural frequencies from an undesirable frequency range while avoiding large design changes. The simulation and optimization were combined into an interactive computer program which allows the analysis of many more potential mount design strategies than is possible with only laboratory testing. An example is discussed which uses this simulation and optimization to predict the natural frequencies of a V-6 diesel engine or rubber engine mounts.
Rigid Body Simulation A rigid body model [1, 2] is suitable for structures whose
geometry points remain fixed relative to one another. Figure 1 shows a rigid body on compliant mounts. The rigid body model consists of six degrees of freedom, which include a translation along and a rotation around each of the three orthogonal coordinate axes. This right-hand coordinate system has its origin at the center of gravity of the rigid body. Three assumptions were made in the derivation of the
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Undesirable Band
Frequency, a Fig. 2 Frequency-content penalty function, F(w)
equations of rigid body motion: (a) the rigid body will have "small" motion, (b) the compliant mounts have linear stiffness, and (c) the mounts have a combination of viscous and structural damping. These assumptions yield linear, three-dimensional equations of motion with six degrees of freedom for a rigid body on compliant mounts.
Mx + Cx + [K + /D]x = f (1) where xT = [xyzdx6ydz] is the displacement/rotation vector, M is the inertia matrix, C is the viscous damping matrix, K is the stiffness matrix, D is the structural damping matrix, f is the operating forces and moments, and / represents the imaginary part. This equation is useful for frequency and transient response calculations. However, in the vibration isolation systems considered here, the effect of damping on natural frequencies is assumed negligible. The homogeneous form of the undamped linear equations is
Mx + Kx = 0 (2) The solution to equation (2) may be assumed to be of the form
x = ue(/u" (3) Substituting this into equation (2) results in an eigenvalue problem [3, 4] of the form
Ku = XMu (4) Solution of equation (4) leads to a set of eigenvalues, X, which are the square of the natural frequencies of this system. Associated with each eigenvalue is an eigenvector, u, the mode shape at that natural frequency. Each mode shape is
made up of six components, three coordinate translations of the center of gravity and three coordinate rotations about the center of gravity. The forces in each mount are calculated as
F = Ku (5) Optimization. In the optimization to be described, a set of
mount design parameters is sought which satisfies the design criteria: Remove natural frequencies from an undesirable frequency range while avoiding costly large design changes. The procedure re-solves the eigenvalue problem each time the design parameters are changed in order to relate system natural frequencies to changes in the design parameters. This was done rather than the first-order Taylor series ap-proximation used by Bernard and Starkey [5, 6] because the computation required to solve a six-degree-of-freedom eigenvalue problem is relatively small. The design parameters are changed to minimize a penalty function that becomes smaller as the design criteria are met. The penalty function, P(E), is of the form
P(E) = aF(o>(E)) + bS(E) (6) where E is an (Nx 1) vector of normalized design parameters, co is a (6 x 1) vector of natural frequencies of the modified system, F is a scalar frequency-spectrum penalty function which is large when natural frequencies are in an undesirable range, and S(E) is a scalar size-of-change penalty function which becomes large when design changes begin to exceed prescribed limits. The scalars a and b indicate the relative importance of the size-of-change penalty as compared to the frequency-spectrum penalty. This allows the user to explore
a,b --
C = D = E = f = F =
= frequency and design change penalty weight values
vector = eigenvalue = frequency = natural frequency
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S(E)
- A A Size-of-Change, E
Fig. 3 Size-of-change penalty function, S(E)
the tradeoffs between a more desirable frequency spectrum and larger design parameter changes.
The frequency-content penalty, F, associated with a given design is a summation of the penalties assigned to the natural frequencies of the system. The natural frequencies are related to the design parameters by the eigenvalue problem, equation (4), which describes the dynamics of a rigid body on com-pliant mounts. Thus, the natural frequencies are a function of the design parameter changes. Equation (7) and Fig. 2 present a useful frequency-content penalty function.
F(co) = LltR[1 - cos[2*{«, - co,)/(«2 - « , ) ] ] (7) where R is the set of all natural frequencies that lie between co, and o>2, the undesirable frequency range.
Natural frequencies which lie inside the undesirable frequency range are assigned a penalty while those outside the undesirable range are assigned zero penalty. Natural frequencies near the center of the undesirable range are penalized more than those near the edges. An improved design is indicated when natural frequencies are moved away from the most critical frequency at the center of the range, which reduces the frequency-content penalty. A natural frequency which is removed from the undesirable frequency range no longer contributes to the penalty. This is reasonable because once a natural frequency is away from an excitation frequency, moving it twice as far away does not reduce transmitted forces proportionally.
This design technique is different in that the designer does not specify the new natural frequencies, but rather an un-desirable frequency range for the new design. The designer typically knows the excitation frequency, which defines an undesirable range. Specifying natural frequencies outside the undesirable range is generally difficult and not useful. The frequency penalty function above quantifies this design practice.
The size-of-change penalty used is shown in Fig. 3 and is expressed by
S(£) = ZNSj(Ej) = Ef (£} -A)2 for WEj II >A (8) The optimization penalizes large design parameter changes
much more than smaller design parameter changes. Thus, at some point, the increase in cost associated with the large design changes required to move the natural frequencies further will not be justified by the corresponding im-provement in the frequency spectrum. This penalty conforms with common design situations, where large design changes correspond to increased real cost to produce the vibration
isolation system. In some design situations, small design changes are possible which have no associated real cost. The size-of-change penalty function includes this situation through a zero penalty for small design changes less than some valued, with larger changes penalized.
Interative Computer Program The simulation of the rigid body on compliant mounts and
the natural frequency optimization were combined into an interactive computer program entitled ENGSIM. ENGSIM is an acronym for engine simulation, the initial design ap-plication. The input data needed for the simulation include {a) the mass and moments of inertia of the rigid body (b) the X,Y,Z coordinates of the rigid body C.G. and the mounts, and (c) the mount stiffness and orientation. The results of ENGSIM include (a) natural frequencies, (b) six degree-of-freedom mode shapes, (c) optimization of mount design parameters, (d) animation of engine modes, (e) static deflection, and (J) normalized mount forces for each mode.
The simulation forms the mass and stiffness matrices based upon the input data. These matrices are used to form an eigenvalue problem which can be solved and the results displayed on a mini-computer, such as a Prime 750, in less than three seconds. The speed of this process allows the user to interactively change the mount design parameters, view the resulting natural frequencies and mode shapes, and try many more design strategies than is possible experimentally. ENGSIM will also animate the mode shapes on selected display terminals. The animation of engine vibration modes provides direct visual information on the deflections of specific engine mounts. Because the forces transmitted through the mounts are proportional to the mount deflection, the relative sizes of these forces are available from the animation and also in tabular output. The simulation provides a technique for evaluating specific vibration isolation designs, but an optimization technique is needed to find the most effective design.
The optimization procedure begins with the mass and stiffness matrices and the natural frequencies obtained from the simulation. Within the design parameters allowed to be optimized by the user, the optimization procedure changes the design parameters (i.e., mount locations, stiffnesses, and/or orientations), reformulates the mass and stiffness matrices, and re-solves the eigenvalue problem to obtain new natural frequencies. The frequency and size-of-change penalty functions are then calculated to determine the penalty func-tion, P(E). Using the IMSL subroutine ZXM1N [7, 8], a locally optimal set of design changes, Ej, is found which define a local minimum of P(E). The £} may not yield a satisfactory design because the frequency spectrum may still be undesirable or the changes unacceptable. If this is the case, the user can redefine the allowable contraints and reinitiate the optimization process.
Example Problem The recent increased interest in smaller, more efficient
automobiles has brought about shorter automobile design cycles and has made the design of engine vibration isolation systems more difficult than in the past. The engine com-partments of new automobile designs are not more fully utilized which limits the possible engine mount locations and other changes to mount systems once the initial compartment layout has been established. Smaller engine designs with fewer cylinders and proportionally higher torque pulses at the same speed and power output have increased the level of vibration excitation. Because vibration excitation is more severe and engine mount configurations are more difficult to change, the method discussed here was developed to investigate the ef-fectiveness of engine mounting systems.
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Table 1 Simulation input, original engine system MODE 5 Mass (kg) 276.70 Inertia matrix (N-M-S')
X' Y' Z'
Coordinates (m)
C.G. Mount 1 Mount 2 Mount 3 Mount 4
X 15.80 0.80
-0 .90
X 0.000
-0.225 0.361
-0.195 0.293
Mount stiffness (N/m)
Mount 1 Mount 2 Mount 3 Mount 4
223667. 170167. 217167. 232167.
Mount orientation (deg)
Mount 1 Mount 2 Mount 3 Mount 4
Theta X 0.0 0.0 0.0 0.0
Y 0.80
11.64 3.20
Y 0.000
-0.309 -0.282
0.141 0.167
Z -0.90 -3.20 15.69
Z 0.000
-0.199 -0.251
0.229 -0.245
Compression Lateral fore/aft 44733. 44733.
126050. 48619. 434334. 108583. 464334. 116083.
Theta Y -45.0 -39.0 -75.0 -45.0
Theta Z 0.0
180.0 0.0
180.0
Table 2 Natural frequencies of original engine system Mode Shapes
X 0.080 0.414 -0.137 -0.093 0.078 0.018 Y -0.300 0.107 0.058 -0.018 -0.117 0.180 Z 0.029 0.050 1.000 -0.004 0.029 0.005
Simulation. A simulation of a V-6 diesel engine and its mounts was performed, with the engine inertia and mount parameters (Table 1) provided through laboratory measurements performed by Oldsmobile Division of General Motors Corp. The mass of the engine and the inertia matrix consisting of the principal moments and cross-products of inertia are shown (Table 1) along with coordinates, stiff-nesses, and orientations of the four engine mounts used for vibration isolation. Mount coordinates were measured from the engine center of gravity to each mount attachment point. The mount's compression, lateral, and fore/aft axes define a right-hand Cartesian system that is originally parallel to the engine's ATZ-coordinate system. Mount orientation is ob-tained by rotating about the engine X-axis, the y-axis, and the Z-axis, in that order. Simulation results (Table 2) show that the predicted natural frequencies span the range from 4.47 to 16.46 Hz.
Laboratory Testing. Independent laboratory modal measurements [2] were conducted on the V-6 diesel engine for
COMPUTER SIMULATION
z X-L-Y
EXPERIMENTAL TESTING
MODE 5
FREO (HZ) 12.47
DAMP (%) 2.68 1 S Z
X — ' — Y
Fig. 4 Predicted and measured mode shapes for Engine Mode 5
comparison with the simulation. The frequencies predicted (Table 3) are within 2 to 16 percent of those measured. This is good comparison in light of the difficulty in measuring the mount stiffness and orientation angles, which limited the accuracy of these input parameters. The sixth mode was not found in the laboratory data. The simulation later indicated that the sixth mode had a node near to the excitation shaker's attachment point, which explained why this mode was not excited in the laboratory test.
Figure 4 shows a comparison of the fifth mode shape predicted by analysis and the shape measured for that mode in the abovementioned laboratory test. This mode and all other modes calculated had mode shapes similar to those measured in the tests. The similarity between predicted and measured responses confirmed the reliability of the analysis.
Modal tests with the cooling and exhaust hoses attached to the engine (Table 4) further indicate the need to have accurate simulation input stiffness parameters. Although the hoses are much less stiff than the mounts, they do have a significant effect on the natural frequencies because they have a large moment arm. The differences in natural frequencies when the hoses are present is twice as large as the comparison men-tioned above of the differences between the natural frequencies of the engine without hoses and of the simulation in Table 3.
Optimization. Vibration isolation is critical at engine idle speeds. Typical idle speeds are between 600 and 750 rpm, which results in fundamental excitation frequencies between 10 and 12.5 Hz. Two of the predicted natural frequencies (Table 2) of the example engine and mounts system are at 9.97 and 12.26 Hz. The vibration isolation would be improved if all natural frequencies were removed from the idle excitation range.
An optimization of all design parameters which penalized natural frequencies between 9 and 13.5 Hz resulted in the values in Table 5. Both the new vibration isolation design parameters and the changes from the original design are shown. Nearly all the design parameters were changed by the optimization. Coordinates were changed from 4 to 43 mm, stiffnesses from 2.55 to 9.01 percent, and orientation from 0 to 140 deg. These changes resulted in natural frequencies (Table 6) that are moved away from the center of the
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Table 5 change
New design parameters optimization allowing all parameters to
penalized frequency range. Only one natural frequency, at 13.25 Hz, remains in the penalized frequency range. All natural frequencies are outside of the engine idle excitation frequency range. This set of design parameters should im-prove the engine idle vibration isolation.
An optimization of a limited set of design parameters may be more practical when some parameters are already fixed by other design constraints. Table 7 shows the results of an optimization in which only the mount orientations were allowed to change. In this case, mount orientations were changed by the optimization from 0 to 151 deg. The natural
frequencies from this optimization (Table 8) were again moved away from the center of the penalized frequency range. The fifth mode's natural frequency, 9.6 Hz, remains in the penalized frequency range. All frequencies were again removed from the engine idle excitation frequency range. The limited set of design parameter changes resulting from this optimization should again improve the vibration isolation system. Experience has shown that limiting further op-timization iterations to only the parameters with the largest changes in previous optimization results often will give ac-ceptable designs.
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Conclusions A simulation and optimization for solving a common
vibration isolation problem involving a rigid body on com-pliant mounts has been demonstrated. These were combined into an interactive computer program, ENGSIM. It is especially useful when there are natural frequencies near the excitation frequency of the external force acting on the rigid body. In this situation, one would like to remove any natural frequencies from around the excitation frequency because large mount deflections result which are proportional to the mount forces transmitted to the supporting structure.
The rigid body and mounts simulation calculates predicted natural frequencies and mode shapes. Comparisons of laboratory modal analysis and engine/mount simulation results verified the accuracy of the rigid body model. The ENGSIM animation of the modes provides direct visual information on the specific mounts which pass the largest forces to the structure. The rigid body and mounts simulation allows preliminary vibration isolation systems to be developed before the design of the supporting structure is fixed.
The optimization finds an effective design faster than trial-and- error methods. Frequencies in an undesirable frequency range are penalized along with large design parameter changes. The design parameters are changed until the penalties are minimized. This set of design parameters results in a vibration isolation system with natural frequencies removed from the center of the penalized frequency range. Final evaluation of vibration isolation designs will almost always require laboratory and/or field tests; however, the computer simulation and optimization discussed allow the
analysis of many more potential design strategies than is possible with prototype testing.
The simulation and optimization discussed does not automate the design process, but offers assistance to design engineers. It does allow the engineer to focus on the properties of the vibration isolation system and the effect that the design parameters have on those properties. Together with the engineer's insight and experience, the computer becomes a more valuable design tool. Acknowledgments
The authors are grateful for the support received from the Oidsmobile Division of General Motors Corp. and the Chrysler Corp. during the development of ENGSIM and for the use of laboratory facilities at Oidsmobile. References
1 Spiekermann, C. E., "Simulating Rigid Body Engine Dynamics," M.S. thesis, Michigan State University, 1982.
2 Radcliffe, C. J., Pickelmann, M., Spiekermann, C. E., and Hine, D., ' 'Simulation of Engine Idle Shake Vibration," SAE 830259, 1983.
3 Fox, G. L., "Matrix Methods for the Analysis of Elastically Supported Isolation Systems," Shock and Vibration Bulletin, Bulletin 46, Part 5, 1976, pp.135-146.
4 Meirovitch, L., Analytical Methods in Vibrations, Macmillan, New York, 1967, pp. 410-420.
5 Bernard, J. B., and Starkey, J. M., "Engine Mount Optimization," SAE 830257, 1983.
6 Starkey, J. M., "Redesign Techniques for Improved Structural Dynamics," Ph.D. thesis, Michigan State University, 1982.
7 Fletcher, R., "Fortran Subroutines for Minimization by Quasi-Newton Methods," Report R7125 AERE, Harwell, England, June 1972.
8 International Mathematics and Statistical Library Inc., Reference Manual, 8th Ed., Vol. 1, Houston, June 1980.
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