Validating Automotive Frame Torsion Stiffness Measurement Techniques

Home , Torsion (mechanics)

A thesis submitted to the

Division of Research and Advanced Studies

of the

University of Cincinnati

in partial fulfillment of the requirements of the degree of

Master of Science

from the

Department of Mechanical and Materials Engineering

of the

College of Engineering and Applied Science

Alexander Young

Bachelor of Science, University of Cincinnati, 2011

13 July 2016

Committee:

Randall Allemang, Ph.D., Chair

Allyn W. Phillips, Ph.D.

David Brown, Ph.D.

Abstract

The stiffness, particularly torsion stiffness, of a frame, body or chassis is of paramount concern to the automotive structural engineer. An accurate measurement of torsion stiffness has several useful applications, among which are the characterization of construction quality, model calibration and suspension tuning. Static techniques for measuring torsion stiffness are perhaps as old as the automotive industry itself, and each manufacturer has its own version of the classical static torsion stiffness test. It is also common for a manufacturer to quote a new model’s torsion stiffness in industry publications, the values of which continually increase with improving material and construction technologies.

In recent years, dynamic methods have emerged which approach the problem of measuring torsion stiffness using experimental frequency response function (FRF) and modal analysis techniques. The primary purpose of this thesis is to further determine the utility of one such dynamic method, namely, the enhanced Rotational Compliance Function (eRCF), in terms of its accuracy in relation to an implementation of the classical static torsion stiffness test. To this end, two automotive frames made by the University of Cincinnati Formula SAE® (UC-FSAE)

Team are the test structures. When compared to typical commercial automotive structures, which contain embedded frames and surface shear panels, these steel triangulated frames are considered simple structures. In order to produce quality static values, the UC-FSAE static torsion stiffness measurement apparatus is to be updated and improved.

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Acknowledgements

I would like to first acknowledge my colleagues in the automotive industry, Jeff Poland of BMW

Manufacturing Company, as well as Luc Cremers, Martin Kolbe and Helmut Schneeweiss of

BMW-AG, who provided training and years of invaluable access to their expertise in this field.

Secondly, I’d like to thank the members of this committee, who have been generous of their time, not only during my graduate work, but also during my work in the automotive industry and as an undergraduate student.

Thirdly, I’m indebted to the 2016 UC-FSAE team members who assisted me in the course of this work, particularly Greg Claire, Kristine Huizing, Thomas “Woody” Neumann Stone and Oge

Okoh.

Fourthly, this thesis would not be possible without Thomas Steed and Hasan Pasha whose work at UC laid its foundation.

Lastly, I would like to express my gratitude to my family who were constant sources of encouragement during the course of my graduate work.

Table of Contents

1 Introduction: Torsion Stiffness and Its Importance to the Automotive Structure 1

2 Background and Theory 9

2.1 The Static Torsion Stiffness Test 9

2.2 The Enhanced Rotational Compliance Function 15

3 Results and Discussion 22

3.1 Results: Static 22

3.2 Results: eRCF 27

4 Calibration of a Finite Element Model 34

5 Conclusions and Recommendations 40

References 43

Appendix 45

A1 Quarter-car Model Parameters 45

A2 TSMM Parts 46

A3 eRCF Formulation 53

A4 eRCF MPE 55

List of Figures

1.1 Hooke’s Law 1

1.2 Springs in Series 5

1.3a Quarter-car Model with Rigid Body 6

1.3b Quarter-car Model with Compliant Body 6

1.4 Acceleration due to Road Input and the Effect of Body Stiffness 7

2.1.1 Typical Static Torsion Test Configuration 9

2.1.2 UC-FSAE 2016 Frame Static Torsion Stiffness Testing 11

2.1.3a UC-FSAE 2016 Frame Node Block Attachment to I-beam 12

2.1.3b UC-FSAE 2016 Frame Node Block Model 12

2.1.4a One Corner of TSMM with PCB® Load Cell 12

2.1.4b One Corner of TSMM Exploded View 12

2.1.5 UC-FSAE Static Torsion Stiffness Test Configuration 13

2.2.1 eRCF Configuration 16

2.2.2 FRFs to eRCF: Enhancement and Scaling 18

2.2.3 Portion of eRCF Used in MPE Part 1 19

2.2.4 Portion of eRCF Used in MPE Part 2 19

2.2.5 eRCF Model Elements 20

3.1.1 UC-FSAE 2014 Frame Static Torsion Stiffness Hysteresis Loops 22

3.1.2 UC-FSAE 2016 Frame Static Torsion Stiffness Hysteresis Loops 23

3.1.3 UC-FSAE 2016 Hub-to-hub Static Torsion Test 24

3.1.4 UC-FSAE 2016 Hub-to-hub Static Torsion Stiffness Hysteresis Loops 25

3.2.1 PCB® Modal Impact Hammers 27

3.2.2 UC-FSAE 2014 eRCF Model and Static Stiffness Estimate 28

3.2.3 UC-FSAE 2016 eRCF Model and Static Stiffness Estimate 30

3.2.4 UC-FSAE 2014 Frame as Springs in Series 32

4.1 UC-FSAE 2016 ANSYS® Model Mesh 36

4.2 MAC for FEA vs EMA Mode Shapes 38

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List of Tables

3.1.1 UC-FSAE 2014 Frame Static Torsion Stiffness Results 23

3.1.2 UC-FSAE 2016 Static Torsion Stiffness Results 24

3.1.3 UC-FSAE 2016 Hub-to-hub Static Torsion Stiffness Results 25

3.2.1 eRCF MRIT DSP Parameters 28

3.2.2 UC-FSAE 2014 Frame eRCF Static Estimates Based on Varying 29 Frequency Bands

3.2.3 UC-FSAE 2016 Frame eRCF Static Estimates Based on Varying 30 Frequency Bands

3.2.4 UC-FSAE 2016 Frame Two-sided SDOF Static Compliance 31 (x105)

3.2.5 UC-FSAE 2016 Frame Residual Flexibility Static Compliance 31 (x105)

3.2.6 UC-FSAE 2014 Frame Springs in Series Calculated vs. 32 Measured

4.1 FE Model Calibration Steps 34

4.2 First Iteration of FE Model Material Properties 37

4.3 Final Iteration of FE Model Material Properties 37

4.4 Hub-to-hub Stiffness Comparison 39

5.1 Comparing Section Rigidity 42

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1 Introduction: Torsion Stiffness and Its Importance to the Automotive Structure

In his seminal paper “Of Springs” (“De Potentiâ Restitutiva”), published in 1678, Robert Hooke summarized his experiments using various elastic wires, springs and weights, observing,

“It is very evident that the Rule or Law of Nature in every springing body is, that the force

or power thereof to restore itself to its natural position is always proportionate to the

distance or space it is removed therefrom, whether it be by rarefaction…of its parts the

one from the other, or by a Condensation…of those parts nearer together [1].”

Hooke’s Law, as it is commonly known, is the first order relationship between force and deformation, stated mathematically,

= 1.1

Figure 1.1: Hooke’s Law

The proportionality constant, K, is known as stiffness and is equivalent to the force (tension or compression) required to produce a given deformation (elongation or shortening); thus, its units are force per length [F/L]:

1.1a =

Hooke’s Law applies to torsional springs as well, where the torsion stiffness units are moment

(or torque) per angle of twist ([F-L/RAD] or [F-L/DEG]):

1.2 =

Hooke also concluded that linear springs are important components of harmonic motion, and his work in this field laid the foundation for the mechanics of elastic bodies.

In mechanics of materials, the angle of twist due to an applied torque on a shaft is a function of its length, shear modulus, G, and polar moment of inertia, J:

1.3 =

The equation can be rearranged in terms of torque per stiffness yields the familiar torsion stiffness constant. It is then clear that the stiffness decreases as the length of the section increases [2].

1.3a =

1.3b = = =

With the advent of the industrial revolution, the concept of stiffness went from a subject of academic inquiry to myriad applications, an enduring example being the automobile structure.

The horse-drawn carriage may seem the obvious structural forerunner to the modern automobile, however, the development of the bicycle played an equal, if not more important,

2 role. The bicycle was the first mass produced mode of self-transportation, and immediate attention was given to producing a lightweight structure in order to reduce the energy expended by the rider. In the 1870s alone, the bicycle industry saw such innovations as the tangential- spoked wheel, differential gearing (for tricycles) and triangulated tubular frame construction. In addition, better paved roads and a demand for motorized self-transportation were direct consequences of the popularity of the bicycle.

In 1895, the Englishman, F. W. Lanchester, made the first attempt to design a chassis, made of brazed steel tubes, which could carry engine and powertrain while still providing a solid floor for the passenger compartment. In the following two decades, as the popularity of the horseless carriage grew, innovations in mass production, steel and the oil and gas industry were seen. By the end of the 1920s, comfort was becoming important to the average car buyer who now demanded an enclosed passenger compartment. Such comforts challenged the automotive structural designer who needed to keep weight down, and the answer came in the form of improved sheet metal quality and forming techniques. Similar challenges were addressed in the

1930s as the customer began to desire aesthetically pleasing vehicles which meant long and low bodies with flowing lines.

In the 1930s, the unit-body or unibody design became an alternative to the frame-over-chassis design. This process saved weight and improved the overall stiffness by combining body and chassis as one structure through embedded frames and stamped body panels which were designed to resist shear. Frame-over-chassis is still a common construction form and has the advantages of interchangeability of frame and chassis, separate design and validation processes for the two sections and more options for isolating the body from undesirable chassis vibration. Increasing fuel standards have meant still further demand for keeping the weight of the structure to a minimum, thus, the advantage of the unibody construction is clear [3].

However, manufacturers of frame-over-chassis vehicles, such as pickup trucks, have responded by moving to lighter materials such as aluminum [4].

The stiffness of a structure is commonly measured in two cases, torsion and bending about the wheelbase. In terms of loading, the torsion case is more straightforward making testing easier and more uniform. In addition, torsion stiffness is considered more important because static bending does not affect the distribution of lateral wheel loads, for example, in the case of roll.

Also, generally speaking, when a structure achieves sufficient torsion stiffness, it has acceptable bending stiffness. Higher stiffness in an automotive body is desirable for several reasons. As a component of the complete suspension system, a stiff body is desirous for predictable suspension tuning and ride comfort. Fatigue is also a factor, as over the lifetime of an automobile, weakening of structural components, such as spot welds breaking or structural adhesive losing effectiveness, can lead to reduced performance [5].

When the UC-FSAE team performs suspension design, the chassis is assumed rigid for the simplicity of calculations, however, adjusting the suspension stiffness to address handling issues is not effective unless the chassis is much stiffer than the suspension. In the following equation, the relationship between the total system stiffness (Kt) of the vehicle is simplified as two springs in series, one for the suspension (Ks) and one for the body (Kb).

1 1 1 1.4 = +

Shown graphically in the following figure, when the body stiffness is much greater than that of the suspension, the total system stiffness approaches the suspension stiffness and all of the deflection due to an applied force occurs in the suspension spring. On the other hand, when suspension and chassis are nearly as stiff as one another, the deflection is shared equally.

Figure 1.2: Springs in Series

For practical reasons, the UC-FSAE team targets the chassis to be at least twice as stiff as the suspension.

In the case of passenger vehicles, especially luxury vehicles, ride comfort is of prime concern.

Ride comfort is tied to the amount of acceleration felt by a passenger. The wheel and suspension system, which functions as a low-pass filter of road input, has a resonance typically between 10 and 20 Hz. In the following quarter-car models, two cases are considered: a rigid body mass on a wheel (unsprung mass) and suspension system, and a compliant body on a wheel and suspension whose modal mass, stiffness and damping approximate the first torsion mode resonance of a car body at about 35 Hz (models based on Zeller [6], for parameters used, see Appendix A1).

Figure 1.3a: Quarter-car Model with Rigid Figure 1.3b: Quarter-car Model with Body Compliant Body

As illustrated in the following figure, the acceleration of the modal mass compared to the rigid body is much greater at the wheel resonance due to the first torsion mode. However, as seen by the effect of changing body stiffness in 10 % increments, the amplification is greatly reduced as the first torsion mode natural frequency increases due to an increasing body stiffness. Note also, the tuned mass damper effect of the modal mass in the frequency range of about 12 Hz.

Figure 1.4: Acceleration of Body due to Road Input and the Effect of Body Stiffness

The quarter-car model concept can be further expanded to include a seat-passenger system to gain more intuition about potential system responses which could affect ride comfort.

One metric which uses torsion stiffness to determine the quality of construction is Leichtbaugüte or the lightweight construction quality [6].

= 10 1.5 ∙

Here, the mass of the body is divided by its torsion stiffness and the product of the track and wheelbase of the vehicle.

The goal of this thesis is to show the ability of the eRCF method to determine static torsion stiffness in a way that corresponds to results from an implementation of the typical static torsion stiffness test. In order to show this in a step-by-step process, the thesis is divided into the following four chapters: first, descriptions of the background and theory for both the methods of determining static torsion stiffness; second, results from both methods; third, how the results

7 can be used to calibrate a finite element model of a frame; and, finally, conclusions and recommendations based on the results.

2 Background and Theory

2.1 The Static Torsion Stiffness Test

There are many ways in which to measure a structure’s torsion stiffness. A typical implementation of the test has the structure mounted to a test rig, typically at four locations associated with the suspension attachment, and a force couple is applied at the front of the structure while the rear is held fixed.

Figure 2.1.1: Typical Static Torsion Test Configuration

Milliken and Milliken [5] discuss one basic implementation of the typical static test wherein screw jacks are used to produce equal and opposite displacements (x1, x2), and the forces required to produce these displacements (f1, f2) are measured via scales under the screw jacks.

Displacements at other locations along the body are measured by dial indicators which are fixed to a rigid reference frame which hangs from one point on the front centerline of the body and two rear points where the twist is assumed to be zero. Calculating torsion stiffness is as follows:

( + ) 2.1 = 2

+ 2.2 = tan

1.2 =

Note that the length from front, where the force couple is applied, to the rear, where the structure is constrained, is not part of the equation as that case of a material test specimen.

Also, the assumption of zero motion at the constraint may not be valid, especially for stiff structures where large moments must be applied.

At the University of Cincinnati, the static torsion stiffness test uses a MTS® 320 Tire-Coupled

Road Simulator, commonly known as a four-poster. While its specialty is simulating various road inputs to a full-sized automobile in a laboratory setting, here it has the advantage of closed control loop hydraulic systems at each post which are used to maintain desired static displacements. Linear variable differential transformers (LVDTs) internal to the posts measure the displacements. Thus, using the MTS® system is similar to the jack screw approach in that forces are commensurate to desired displacements at each corner, rather than displacements corresponding to defined force inputs, a common implementation of the static test.

Between the four-poster wheel pans and the frame-to-be-tested is an updated version of the

UC-FSAE Torsion Stiffness Measurement Machine (TSMM) developed by Thomas Steed [7].

Three major improvements have taken place in the implementation of the TSMM since its original development. First, two 60-inch I-beams (W4X13) were modified with quarter-inch slots on the top flanges to accommodate the attachment of frames of various width, and four holes in the bottom flanges to mate with the TSMM load cell assembly. The I-beams serve as a rigid platform for the frame-to-be-tested and allow the TSMM corners to be set square in place prior to frame attachment. Because the I-beams are longer than FSAE frames are wide, the moment arm is also longer so that both the force and deflection at each post required to achieve a

10 certain torque and angle of twist are reduced. Secondly, PCB® model 1380-03A rod end load cells which use Wheatstone bridge circuitry were purchased and incorporated into the design to ensure more reliable force measurement versus the student-made half bridge strain gages used in preceding years. Aluminum parts of the TSMM were redesigned to mate with these new load cells [A2]. Lastly, the 2016 UC-FSAE team incorporated node blocks into their frame design which had threaded holes for ease of mating to the I-beams and provided massive joints which were thought less likely to result in local (rather than global) deflections.

The TSMM allows for small rotations of the I-beams via clevises between the load cell and I- beam. Also, a 1/8-inch mechanical fuse between the load cell and four-poster wheel pan is designed to prevent large loads which could result in permanent deformations in the frame.

Figure 2.1.2: UC-FSAE 2016 Frame Static Torsion Stiffness Testing

Figure 2.1.3a: UC-FSAE 2016 Frame Node Figure 2.1.3b: UC-FSAE 2016 Frame Node Block Attachment to I-beam Block Model

Figure 2.1.4a: One Corner of TSMM with Figure 2.1.4b: One Corner of TSMM PCB® Load Cell Exploded View

Using load cells at each of the four corners provides the key difference between the typical static torsion test and the UC-FSAE test: the additional knowledge of the moment needed to keep the rear points fixed.

Figure 2.1.5: UC-FSAE Static Torsion Stiffness Test Configuration

Because the I-beams are the same size, the front length and rear lengths are equivalent, so the stiffness calculations are nearly the same, except for an average of four moments as opposed to two:

( + ) + ( + ) ( + + + ) 2.1a = = 4 4

+ 2.2 = tan

1.2 =

The load cells are connected to a Vishay Measurements Group 2100 System Strain Gage

Conditioner and Amplifier which is used to power the bridge circuits and balance the strain gages prior to loading. The outputs from the amplifier and the MTS system are fed to VXI data acquisition cards controlled by UC-SDRL X-Modal research software. The Time Capture feature is used to collect the time histories. Each channel of data is viewed to ensure that the signal at each step is relatively flat, and that the displacements designated to be zero are, in fact, zero. Some minor noise is present in the data due to the hydraulic system. Initial displacements in increments of 0.02 inches (x1 and x2) are taken to verify that torque versus angle of twist is progressing linearly and smoothly, and after three or four such steps, increments are increased to 0.04 inches until a roughly 1000 lbf-ft couple is produced. A

MATLAB® code was written to post-process the data producing a mean value for each force and displacement channel and then performing the above calculations in 2.1a and 2.2.

Eventually a full load-deflection curve at each post can be converted into a torque-angular deflection curve, the hysteresis of which can be evaluated.

Another advantage of the four-poster setup is ease of producing displacements and angles of twist in both a positive sense (Fig. 2.6) and then in the opposite direction. The same concept is applied to rear points where both positive and negative angles are produced while the front points are held fixed. Before this second test, the body is released from its attachment and then re-attached as a repeatability check. The resulting data set is similar to Fig. 1.1 with two torque- angular deflection curves. By comparing these loops, the presence of non-linearity or possible measurement error may be detected at some level of load. An average is taken of linear regressions on all four pieces of data—front twist, positive; front twist, negative; rear twist, positive; rear twist, negative—so that four stiffness constants, KT, are averaged into one torsion stiffness value for the frame.

2.2 The Enhanced Rotational Compliance Function (eRCF) Method

The eRCF method was a joint development between the UC-SDRL and BMW Manufacturing

Company (BMW-MC) of Spartanburg, South Carolina. Because BMW-MC is a production facility, as opposed to a research and development facility, it has an interest in testing large numbers of bodies either for pre-series production validation or series production control purposes. A method was therefore desired which would require less technical training, setup and test time, and equipment investment compared to BMW’s long-standing version of the static torsion stiffness test. It was not necessarily a requirement that the new method produce exactly the same result as the historical method, but that it have a consistent relationship with the static method and be as useful in tracking various body stiffnesses. Another interest of BMW was an alternative bending stiffness test method, however, this was also not the primary objective of the project [8].

The UC-SDRL approached the problem using its tradition of experimental frequency response function (FRF), modal analysis and modal parameter estimation, producing the eRCF, which met the aforementioned requirements. For the doctoral dissertation which provided the theory and basis for practical implementation, a rectangular steel plate was used as the initial test structure, and a high fidelity finite element (FE) model was used to estimate static torsion stiffness. The eRCF method produced results within typical measurement error to the FE model

[9,10]

In the eRCF testing phase, raw data is captured using a standard multiple reference impact test

(MRIT) technique with a free-free boundary condition (air rides for a full-sized automobile body and foam for the UC-FSAE frame). Input force data is collected by roving a modal impact hammer to each of the four locations associated with the static test, while integrated electronic piezoelectric (IEPE) accelerometers placed at the same locations collect response data. These responses must be converted into displacement values via synthetic integration in order to

15 compare to static testing values measured by displacement sensors. One full cycle of testing consists of a four-by-four FRF matrix (four driving point and twelve cross measurements).

Calibrated data is of the utmost importance as various portions of the FRF data will be used to find real values.

The most common impact test error is the susceptibility of overloading channels. This error is more prevalent in the case of lightly damped steel frames, especially the driving point measurements. The overload case could either be the result of the response signal exceeding its predetermined range or the consequence of high frequency overloading, a phenomena associated with modern data acquisition systems. Users should pay particular attention to driving point phase measurements for evidence of contamination. One way to avoid the error is to increase the range of the overloaded channel [11].

Figure 2.2.1: eRCF Configuration

In the second phase, an FRF enhancement technique [12], which takes advantage of the principle of FRF superposition to keep, or “enhance,” desired modes or mode types while subtracting undesired modes. Also, as the raw FRFs are added or subtracted, their compliance units ([L/F]) must be scaled by the moment arm length to find angle and torque. A full derivation of the eRCF calculation can be found in the Appendix A3, however the final equation is:

Δ 2.3 () = = {} [()] {}

where,

1 2.4 ⎧+ ⎫ ⎪ ⎪ ⎪ 1 ⎪ ⎪− ⎪ {} = ⎨ 1 ⎬ − ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ + ⎩ ⎭

The enhancement process is a weighted-averaging of the torsion information (the positive- negative-negative-positive signs in {V} which mirror the assumed torsion configuration in Figure

2.2.1) in the raw FRFs. All torsion modes appear in the eRCF while any non-torsion mode is eliminated of greatly reduced. The most obvious effect of this operation can be seen in Figure

2.2.2 where the rigid body modes from 5-10 Hz are removed from the data.

Take note of the units of the FRF matrix, [H]. For example, data acquisition was taken in terms of British Imperial System units with acceleration in units of g. Therefore, [H] has compliance units of [g-s2/lbf]. This requires a conversion first to match the units of the vector, {V}. Thus, if

{V} has units of inches, a factor of 386 converts [H] from [g-s2/lbf] to [in/lbf]. The pre- and post-

17 multiplication by {V} in the eRCF operation effectively scales [H] to [RAD/F-L]. Therefore, a further conversion factor converts the eRCF from [rad/lbf-in] to [deg/lbf-ft].

Figure 2.2.2: FRFs to eRCF: Enhancement and Scaling

Modal parameter estimation (MPE) takes place in the third phase and is in two parts. The first part uses a Rational Fraction Polynomial algorithm which estimates the first torsion mode natural frequency and damping, λ, and its conjugate, λ* [13]. Only eRCF information near the peak is used for the natural frequency estimation (see Fig. 2.2.3). Second, a LS process simultaneously solves for residues, A and A*, and residual flexibility, RF. The residual flexibility accounts the stiffness contribution from all the modes outside the frequency range of the first torsion mode and its conjugate [14]. Because IEPE accelerometers are not very sensitive at low frequencies where the signal is already low, the information used for this part of the MPE process is based on the portion of the eRCF where the magnitude is flat and phase is nearly zero and relatively free of noise, beyond about 10 Hz, all the way up to the peak of the first torsion mode.

Figure 2.2.3: Portion of eRCF Used in MPE Part 1

Figure 2.2.4: Portion of eRCF Used in MPE Part 2

In the eRCF model (eRCFM) which will be used to find static stiffness, the three key elements: the SDOF, SDOF-conjugate and residual flexibility are added together to estimate the total rotational compliance; thus, torsion stiffness near zero frequency:

∗ 2.5 () = () + ∗ () + = + + − − ∗

Figure 2.2.5: eRCF Model Elements

The initial BMW dataset consisted of data from the historical static method and eRCF method for several car models and variants which constituted a large range in stiffness values. The eRCF method proved capable of tracking the static method, however, a delta between methods was seen wherein the eRCF result consistently underestimated the static result by a certain percentage. Explanations were put forward to explain the delta, such as the possibility of local deformations of the unibody sheet metal panels due to shearing (e.g., “oil-canning”) which could produce non-linear phenomena, as well as the large difference in force levels between static and dynamic processes.

Using the current, 2016 UC-FSAE car frame in addition to the 2014 frame as simple, intermediate automotive structures, somewhere between the rectangular plate and the BMW body, the primary goal of this thesis is to verify the ability of the eRCF method to produce accurate results in comparison to the improved TSMM method. The advantage over the

20 rectangular plate case is the experimental static test method as opposed to only a FE model, and the advantage over the BMW case is that the simple nature of the space frames were thought less likely to have the complexities of full-size automotive unibodies.

3 Results and Discussion

3.1 Results: Static

As mentioned in discussing the implementation of the UC-FSAE static torsion test, two cycles of twisting occurred for each test structure. The 2016 team suggested 0.75 degrees was the limit of elastic deformation, so extreme angles of twist were kept under 0.70 degrees. Initially, nylon mechanical fuses were used, also to guard against yield, however, it was quickly seen that loading above about 75 lbf at a corner would result in deformation and eventual failure of the nylon pins. This yielding would manifest itself in the torque-angular deflection curve: a change in angle without corresponding change in torque. Brass replacements were used and quality measurements, meaning tight, linear torque-angular deflection curves consisting of 36 steps, cycling from zero to maximum positive angle, through zero to maximum negative angle, back to zero, were taken.

Figure 3.1.1: UC-FSAE 2014 Frame Static Torsion Stiffness Hysteresis Loops

From the preceding figure, four stiffness constants were found from four linear regressions: the first quadrant data for the front twist, the third quadrant data from the front and similarly for the rear twist data.

Twist KT (lbf-ft/deg) Front, Positive 1308 Front, Negative 1284 Rear, Positive 1296 Rear, Negative 1255

Average 1286 Percent Difference 4.10

Table 3.1.1: UC-FSAE 2014 Frame Static Torsion Stiffness Results

The confidence in the data based on appearance was confirmed by the results showing the percent difference in absolute range divided by mean to be within ± 3 %, a commonly excepted value for measurement error. That the values are different positive to negative or front to rear are likely due to small asymmetries in placement over the four-poster, and averaging the values should compensate for these asymmetries.

Figure 3.1.2: UC-FSAE 2016 Frame Static Torsion Stiffness Hysteresis Loops

The 2016 frame results showed a similar quality as the 2014 frame and an obvious increase in stiffness: just under 1000 lbf-ft of torque was needed to twist the 2014 frame to the maximum angle of twist, whereas just over 1000 lbf-ft was needed for the same angle of twist of the 2016 frame.

Twist KT (lbf-ft/deg) Front, Positive 1509 Front, Negative 1548 Rear, Positive 1592 Rear, Negative 1516

Average 1541 Percent Difference 5.38

Table 3.1.2: UC-FSAE 2016 Static Torsion Stiffness Results

The percent difference was slightly higher, though still within ± 3 %.

After further construction of the 2016 frame and its suspension components, such as the A-arms and wheel hubs, the complete chassis, or hub-to-hub stiffness was measured.

Figure 3.1.3: UC-FSAE 2016 Hub-to-hub Static Torsion Test

The main difference in TSMM configuration consisted in different plates atop each corner which were designed to mate directly to the hubs, rather than I-beams. This proved a challenge because the four-poster would have to be adjusted while an engine hoist held the chassis above the hydraulic posts. Also, time constraints did not allow for removal and re-attachment, so the front and rear twisting occurred directly after one another. The larger variation in the four parts of the hysteresis loops is likely due to a less symmetrical placement.

Figure 3.1.4: UC-FSAE 2016 Hub-to-hub Static Torsion Stiffness Hysteresis Loops

Twist KT (lbf-ft/deg) Front, Positive 941 Front, Negative 848 Rear, Positive 885 Rear, Negative 1059

Average 933 Percent Difference 22.63

Table 3.1.3: UC-FSAE 2016 Hub-to-hub Static Torsion Stiffness Results

As mentioned in the introduction (Eq. 1.3b), the drop in stiffness from frame testing to hub-to- hub testing is expected as the distance between front moment and rear moment applications increased. In conventional automotive frame stiffness literature, the different sizes of vehicles is not taken into account making direct comparisons between stiffness values of various vehicles dubious.

3.2 Results: eRCF

In the testing phase of the eRCF method, two different modal impact hammers were used to ensure the method was robust. For the 2014 frame, a PCB® Model E086C40 hammer with a nominal sensitivity of 10 mV/lbf was used with a large, soft rubber tip. For the 2016 frame, a

PCB® Model 086C05 hammer with a nominal sensitivity of 1 mV/lbf was used with a small, rubber cap on the hard plastic tip. PCB® Model 352A56 teardrop accelerometers with nominal sensitivities of 100 mV/g were used as responses. The challenge of testing lightly damped structures, such as these space frames, is avoiding overloads which, if not removed from the data, can cause distortions in both phase and magnitude at low frequency. Careful attention should be paid to especially the driving point measurements to monitor the nature of the phase.

Figure 3.2.1: PCB® Modal Impact Hammers

The digital signal processing (DSP) used is summarized in the following table. Lengths used in the vector, {V}, were measured with tape measure. Because the accelerometer cannot be impacted directly, small markers were placed on the frame, and the accelerometer was located

27 on one side of the marker, and the spot of impact was on the other side of the marker, both response and impact as close to the marker as possible. Ten averages were taken in order to reduce noise.

Parameter fmax 125 Hz No. frequency lines 3200 FRF Estimator H1 Window Uniform No. averages 10 (Power Spectrum)

Table 3.2.1: eRCF MRIT DSP Parameters

Figure 3.2.2: UC-FSAE 2014 eRCF Model and Static Stiffness Estimate

Also investigated was the effect of changes in the frequency band used to estimated residues and residuals in the eRCF model. In Fig. 3.2.2, it is clear that the noise seen in the phase plot begins to reduce after 10-15 Hz. The following table summarized the static stiffness result as a function of the start frequency, from 10-30 Hz in steps of 5 Hz, and end frequency which is a function of natural frequency, as this will change from structure-to-structure. At 97 % of the natural frequency (ωn), the phase is still flat, while at 101 %, the phase is 180 degrees down

28 from zero. Green values are within 3 % the static result, while red is over and blue is under the

3 % threshold.

End Frequency (% of ωn) Range 97 98 99 100 101 Average (% Avg.) 10 1335 1328 1322 1306 1302 1319 2.5

15 1301 1296 1292 1279 1276 1289 2.0 20 1287 1282 1279 1266 1263 1275 1.8 (Hz) Start 25 1289 1283 1279 1263 1261 1275 2.2 Frequency Frequency 30 1270 1266 1264 1248 1245 1259 2.0 Average 1296 1291 1287 1272 1269 Range 5.0 4.8 4.5 4.6 4.5 (% Avg.)

Table 3.2.2: UC-FSAE 2014 Frame eRCF Static Estimates Based on Varying Frequency Bands

The table suggests that the stiffness estimate is more sensitive to the starting frequency; 10 Hz is either too low or too noisy for an accurate fit. Also, 30 Hz seems to underestimate the static result. At the other end, the end frequency does not affect the result as greatly, and when averaged over all the start frequencies, each column produces an average very close to the static result. To estimate the error due to measuring the lengths in {V}, the same calculations were done at ± 1/8-inch on each measurement which results in ± 4.3 % error.

Figure 3.2.3: UC-FSAE 2016 eRCF Model and Static Stiffness Estimate

End Frequency (% of ωn) Range 97 98 99 100 101 Average (% Avg.) 10 1623 1611 1596 1564 1559 1591 4.0

15 1565 1556 1546 1519 1516 1540 3.2 20 1539 1531 1522 1498 1495 1517 2.9 (Hz) Start 25 1524 1517 1508 1484 1481 1503 2.8 Frequency Frequency 30 1516 1508 1500 1474 1472 1494 3.0 Average 1553 1545 1534 1508 1504 Range 6.9 6.6 6.3 5.9 5.8 (% Avg.)

Table 3.2.3: UC-FSAE 2016 Frame eRCF Static Estimates Based on Varying Frequency Bands

The summarized results for the 2016 frame show more variance than the 2014 frame. The results most near the static test occur when the band is roughly 15-20 Hz to 97-100 % ωn. In order to study further the source of the variance, the following tables display the two static compliance results: the compliance due to the two-sided SDOF model and the compliance due to residual flexibility.

End Frequency (% of ωn) Range 97 98 99 100 101 Average (% Avg.) 10 8.46 8.02 7.59 6.94 6.90 7.58 20.5

15 8.04 7.73 7.43 6.93 6.90 7.40 15.4 20 7.84 7.59 7.35 6.92 6.90 7.32 12.9 (Hz) Start 25 7.73 7.51 7.31 6.91 6.90 7.27 11.5 Frequency Frequency 30 7.67 7.47 7.28 6.91 6.90 7.25 10.7 Average 7.95 7.67 7.39 6.92 6.90 Range 9.9 7.2 4.1 0.5 0.1 (% Avg.)

Table 3.2.4: UC-FSAE 2016 Frame Two-sided SDOF Static Compliance (x105)

End Frequency (% of ωn) Range 97 98 99 100 101 Average (% Avg.) 10 53.1 54.1 55.1 57.0 57.3 55.3 7.5

15 55.9 56.5 57.3 58.9 59.1 57.5 5.6 20 57.1 57.7 58.4 59.9 60.0 58.6 4.9 (Hz) Start 25 57.9 58.4 59.0 60.5 60.6 59.3 4.6 Frequency Frequency 30 58.3 58.8 59.4 60.9 61.1 59.7 4.6 Average 56.5 57.1 57.8 59.5 59.6 Range 9.1 8.4 7.5 6.6 6.4 (% Avg.)

Table 3.2.5: UC-FSAE 2016 Frame Residual Flexibility Static Compliance (x105)

From these tables, it can be seen that the SDOF model compliance varies more than the residual flexibility compliance based on the frequency band used in MPE. Essentially, when more information is used in the MPE near the peak, the residue estimate becomes more precise, and lower, which results in a lower static compliance.

A further use of the eRCF was used to estimate stiffnesses of various sections of the 2014 frame. This is useful in an automotive design process to determine which sections of the body

31 are most rigid. It was hypothesized that the result would be similar to springs in series, because when the test is done statically, displacement sensors are placed along the body to measure the different slope of the stiffness curve from section to section. Because the frame is designed for only four points of measurement at the suspension, static values for each section are not available. The eRCF method is versatile enough to take measurements anywhere on the frame where driving point measurements can be taken.

Figure 3.2.4: UC-FSAE 2014 Frame as Springs in Series

Section KT (lbf-ft/deg) Front 1292 Middle 1961 Rear 3024 Full (Calculated) 619 Full (Measured) 687 Percent Difference -9.89 100*(Calculated/Measured-1)

Table 3.2.6: UC-FSAE 2014 Frame Springs in Series Calculated vs. Measured

The table above summarizes the eRCF method of estimating torsion stiffness for each section individually, calculating the resulting full frame torsion stiffness using Eq. 1.3, and comparing this value with the measured full frame stiffness. The difference is about 10 %, however, it is clear that the sections are not perfectly springs in series. As expected, the rear section is by far the stiffest, as the section contains many members and the roll hoop. The benefit of this rigid cross section is also evident in the higher stiffness of the middle section than front.

4 Calibration of a Finite Element Model

A basic tool of modern automotive design is finite element analysis (FEA). The purpose of a FE model is to accurately predict a vehicle’s or component’s behavior in order to reduce the expense of testing. Many static and dynamic phenomena can now be modeled due to both improvements in computational methods and a long history of field tests. Experimental analyses are used in verification and validation (V&V) processes so that a particular model can be improved for future FEA or so that modeling techniques can be improved for the use of future designs [15].

The UC-FSAE team, specifically the frame design group, also desired to calibrate a FE model based on static stiffness testing and EMA in order to predict structural behavior and to continue to improve the UC-FSAE knowledge of FEA. Using ANSYS®, nodes are created at each joint location and are considered perfect joints, as opposed to modeling welds, for example.

Geometric properties for each member include cross section information (e.g., circular tube with specified outer and inner diameter) which is used to calculate the area and second moment of area of a member. These geometric properties along with the nominal material properties of

AISI 1020, cold drawn (CD) carbon steel, Young’s Modulus (or elastic modulus), E, density, ρ, and Poisson’s ratio, ν, are incorporated into the mass and stiffness matrices used to model static and dynamic deflections due to various load cases.

The basic philosophy of the calibration and validation process is carried out in the following steps:

Step Process Parameters Affected 1 Geometric Calibration (x,y,z), L, A, I 2 Mass Calibration ρ, A, L 3 Modal Calibration ρ, A, L, E, I 4 Stiffness Calibration A, L, E, I

Table 4.1: FE Model Calibration/Validation Steps

During geometric calibration, a coordinate measuring machine (CMM) is used to verify the location of each node in (x, y, z), and a process such as ultrasonic thickness measurement

(UTM) verifies the gauge thickness of member cross sections. These parameters control the length of all members as well as the area and second moment of area which, in turn, control mass and stiffness. In the case of UC-FSAE, such tools are not readily available, and the nominal values of cross section thickness as identified by the manufacturer of the steel tubes are used. The node locations are also unchanged from the initial design despite the inevitable changes due to the laser cutting and welding process.

In the mass calibration step, the structure is weighed. Because the geometric process was not calibrated via experimental measurement, only the material density can be changed to calibrate the mass. It was important in this step to include the four node blocks. ANSYS® models the weight of the structure by applying an acceleration of 1 g down on the structure and calculating and summing reaction forces at four nodes. Despite the lack of geometric calibration, the modeled mass was very close to the experimental value, as determined by weighing the frame, using the nominal density of the material, and ρ was modified only slightly.

During the third step, a modal analysis is performed in ANSYS® for comparison to EMA. In this step, only the first six deformation mode modal frequencies are used for comparison which are adequately defined by both the mass and stiffness matrices. Beyond frequency, it is important that modal vectors (mode shapes) match the experimental modal vectors in the correct mode order. Density can be altered further, but also Young’s Modulus. A further philosophy of the calibration process was to keep material properties within 3 % of nominal if possible.

Finally, as a measure of model validation, the results of the static torsion stiffness testing were compared to modeling the torsion stiffness in ANSYS® to modify the Young’s Modulus. In the model, the rear points were constrained in (x, y, z), but were free to rotate while 0.1-inch displacements in opposite directions were applied the front points. The stiffness validation was

35 considered the final step because the confidence in the precision and capability to measure the structure’s weight and modal frequencies and shapes were considered higher than the static stiffness. Also, the ability of the model to accurately predict static deflections was considered low because connections between members were modeled as perfect, rather than welded joints.

Figure 4.1: UC-FSAE 2016 ANSYS® Model Mesh

Fig. 4.1 shows the meshed model which consists of BEAM188 elements of 1/8-inch mesh size.

The following tables show the results following the calibration process.

First Iteration Material ρ (slug/in2) 8.88 x 10-3 Properties E (psi) 29.0 x 106 Experimental FE Model % Difference MASS Weight (lbf) 71.2 72.1 1.26 1. Torsion 70.4 68.5 -2.70 1. Bending (Lat.) 108.1 108.8 0.65 MODAL 2. Torsion 114.4 113.8 -0.52 (Hz) 1. Bending (Long.) 139.0 129.9 -6.55 2. Bending (Lat.) 145.1 144.4 -0.48 3. Torsion 156.5 160.8 2.75

STIFFNESS KT (lbf-ft/deg) 1541 1653 7.27

Table 4.2: First Iteration of FE Model Material Properties

Final Iteration Material ρ (slug/in2) 8.62 x 10-3 Properties E (psi) 29.3 x 106 Experimental FE Model % Difference MASS Weight (lbf) 71.2 69.9 -1.83 1. Torsion 70.4 69.9 -0.71 1. Bending (Lat.) 108.1 111.0 2.68 MODAL 2. Torsion 114.4 116.1 1.49 (Hz) 1. Bending (Long.) 139.0 132.6 -4.60 2. Bending (Lat.) 145.1 147.4 1.59 3. Torsion 156.5 164.0 4.79

STIFFNESS KT (lbf-ft/deg) 1541 1670 8.37

Table 4.3: Final Iteration of FE Model Material Properties

The material properties were modified so that density was 2.97 % lower than the first iteration, while the elastic modulus was increased 1.03 %. Poisson’s ratio remained unchanged from

0.29. The weight as modeled was within 2 % during the iteration process, however, density was further lowered and modulus increased in order to attempt to keep all modes within 5 % of their

37 measured frequencies. This resulted in a slightly larger error in static stiffness in which the model overestimates the measured stiffness by more than 8 %. Note that torsion modal frequencies were not treated as particularly important when changing model properties. This could be addressed in future model calibration attempts to model specifically torsion phenomena.

A further evaluation that the modal frequencies were appearing their correct order was to match mode shapes via the Modal Assurance Criterion (MAC) [16]. Because the EMA used only a subset of nodes, and only input force and measured response in vertical and lateral directions, the FE model was reduced to vectors matching those used in EMA. A MAC calculation was then performed for each mode, experimental and model, which essentially determines the linear relationship between the mode shape vectors. A MAC of unity means identical shapes.

Figure 4.2: MAC for FEA vs EMA Mode Shapes

It is clear from Fig. 4.2 that the modes are in correct order. An important step in the modal calibration was placing the node blocks, which are essentially point masses, in their proper location on the structure so that frequencies and shapes would be correct.

Even though the stiffness estimated by the model was relatively high, it was thought reasonable considering the simplicity of the model and lack of team knowledge of weld modeling issues.

The FE model was then updated to include suspension components, such as A-arms in, order to test the calibration of the model.

Experimental FE Model % Difference

KT (lbf-ft/deg) 933 1139 22.1

Table 4.4: Hub-to-hub Stiffness Comparison

The hub-to-hub stiffness prediction, even considering the 8 % error in the calibrated frame model, still has an error of nearly 14 %. Not only was the model expanded to include new parts, including the modeling of the A-arm spherical joints, but the experimental stiffness test result likely included a higher measurement error than the frame testing.

5 Conclusions and Recommendations

Compared with previous UC-FSAE team’s static stiffness results, the updated TSMM shows an increased ability to accurately measure torsion stiffness of such automotive space frames as future teams will build. The linear quality of the torque-angular deflection curves shown in this thesis should be used as the latest benchmark for the team. The updated aluminum parts should serve their capacity for the foreseeable future, but if the parts were to be made of steel instead, they could potentially last longer and possibly be able to accommodate larger structures for stiffness testing. New parts should be made to accommodate the I-beam in hub- to-hub testing, the results of which could be improved. Brass mechanical fuses should be used at a minimum, but steel pins are also included in the TSMM set. The best fuse would be a very brittle steel which would not yield during loading but shear at 250-300 lbf which would be enough force to reach 1000 lbf-ft. This may require a redesign of the size of the fuse.

All future UC-FSAE teams should incorporate the node block concept into their frame designs.

The node blocks served as very convenient test locations in TSMM mating and modal analysis.

If possible, smaller node blocks could be designed at the bottom of each major section in order to perform static stiffness testing of individual sections instead of only those concerning suspension locations.

The ability of the eRCF to accurately produce a static result within measurement error of the

TSMM is demonstrated in the case of the steel space frame. The method is robust in that it produced accurate results using different frames and impact hammers. Even considering measurement error in length, the result would be roughly ± 4 %. This suggests that the method is valid and that the BMW experience is a different case, perhaps due to the nature of the unibody construction. Future eRCF testing and analysis should attempt to determine the lowest frequency bound for useable data. A more sensitive study that focuses specifically on the lower frequency effect on the results or a modeling technique to account for the very low frequency (0-

10 Hz) characteristics of the data could further inform an automated eRCF static stiffness estimation. Data up to the natural frequency should be included for the most accurate residue estimation. Such studies could perhaps lead to a semi-autonomous method of analyzing the eRCF. Other future testing of the eRCF method should take place on automotive structures such as a truck frame and body system to determine whether it is more accurate for one or the other.

The eRCF is useful in analyzing the stiffness of different sections of the space frame, and if future testing allows for static testing of sections, comparisons may be made to determine how accurate this utility is.

Using FE modeling to predict static stiffness, even in the case of relatively simple frames, is still challenging. Future teams should make efforts to model welds and study the sensitivity of the process to determine whether a more accurate stiffness result may be achieved while still accurately modeling mass and modal characteristics. A similar statement can be made about hub-to-hub stiffness modeling.

In terms of a metric which might best describe the structural rigidity of a given frame or a section of a given frame, the following metric, which will be referred to as section rigidity (SR), is suggested which is based on Eq. 1.3b:

= 5.1

In this metric, the torsion stiffness is multiplied by the length between front to rear points used for analysis. This is as if one were to solve for the product of shear modulus (G) and polar moment of inertia (J) of the given frame section. It also effectively normalizes the torsion stiffness by the length dimension, which usually goes unaccounted. The following table summarizes the results for the frames tested in this paper.

2 Frame KT (lbf-ft/deg) L (ft) SR (lbf-ft /deg) 2014 1286 3.73 4797 2016 1541 3.54 5455

Table 5.1: Comparing Frame Section Rigidity

The SR for the 2016 frame results in a 13.7 % increase from its 2014 counterpart. On a final note, it should be kept in mind that a fundamental measure of the quality of a frame’s construction are its first few natural frequencies. The basic definition of a SDOF system’s natural frequency being the square root of stiffness divided by mass, a greater natural frequency is the result of a higher stiffness and less mass, both of which are desirable. In the case of the two frames in question, the first torsion mode natural frequencies for the 2014 and 2016 frames are 54.1 and 70.4 Hz, respectively. Thus, the 2016 frame’s first torsion mode natural frequency is 30.1 % higher than the 2014 frame.

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Appendix

A1 Quarter-car Model Parameters

Parameter Rigid Body Compliant Body Ktire 2250 lbf/in 2250 lbf/in Munsprung 115 lbm 115 lbm Cshock 15 lbf-s/in 15 lbf-s/in Kshock 1750 lbf/in 1750 lbf/in Ksuspension 200 lbf/in 200 lbf/in Mmodal -- 100 lbm Cbody -- 5 lbf-s/in Kbody -- 8750 ±20% lbf/in Mbody 750 lbm 650 lbm

A2 TSMM Updated Parts

All dimensions in inches.

A2.1: TSMM, Frame Configuration, Exploded View

A2.2: TSMM, Hub-to-hub Configuration, Exploded View

A2.3: TSMM, I-beam Attachment Plate (Steel)

A2.4: TSMM, Hub Attachment Plate (Steel)

A2.5: TSMM, Clevis Part (Steel)

A2.6: TSMM, Load Cell-to-Clevis Part (Aluminum)

A2.7: TSMM, Wheel Pan Plate to Load Cell Part (Aluminum)

A3 eRCF Formulation

Figure 2.2.1: eRCF Configuration

Moment balance:

↺ = 0

Transfer function definition:

{()} = [()]{()}

+ + ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪− ⎪ ⎪− ⎪ {} = = = = {} ⎨− ⎬ ⎨− ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + + ⎩ ⎭ ⎩ ⎭

{} = []{}

{}{} = {}[]{}

− − {}{} = − − + = −

Small angle assumption:

− − = tan ≈

{}{} = − = Δ

Δ = {}[]{}

Δ ( ) = = {} [( )] {}

A4 eRCF MPE

Part 1

From a basic second-order frequency domain MPE model:

( ) ( ) ( ) = = () () + () + ()

The characteristic equation:

+ + = 0

Normalizing for α2 (α2 =1):

∗ ∗ () ⋯ () ⋯ ∗ ∗ { } ()() ⋯ () () ⋯ {1}

∗ ∗ = −() () ⋯ () ⋯

Part 2

1 1 1 1 ⎡ ⋯ ⋯ ⎤ ⎢ − − − − − − ⎥ ∗ { } ⎢ 1 1 1 1 ⎥ ⎢ ∗ ⋯ ∗ ∗ ⋯ ∗⎥ ⎢ − − − − − − ⎥ ⎣ {1} ⎦

∗ ∗ = () ⋯ () ⋯