IMPROVEMENT OF VIBRATION TEST - CONVERTING A SINGLE-AXIS VIBRATION TABLE INTO A TWO-AXIS TABLE  By  Yanzhe Wu     A THESIS Submitted to  Michigan State University in partial fulfillment of the requirements for the degree of  Packaging  Master of Science 2016  ABSTRACT IMPROVEMENT OF VIBRATION TEST - CONVERTING A SINGLE-AXIS VIBRATION TABLE INTO A TWO-AXIS TABLE  By  Yanzhe Wu  An increasing number of companies find that their products pass standard vibration tests but are damaged during transportation. The main reason for this is that the vibration tables used in these tests only move up and down, meaning they lack 5 of the 6 motions that occur in real transportation. Converting a single-axis table to a six-axis table is almost impossible to do. Therefore this research investigated an alternative solution to this problem by adding the second most severe motion, roll. The concept of adding roll to a vertical shaker was to place a rocking platform on the table to act as the new vibration plane. When the table is vibrating, the platform will move both up and down and rock. Theoretically, the rocking motion can be made to match that in a trailer by adjusting two variables of the platform system. The theoretical RMS G could not be verified using test results due to unwanted noise and vibrations produced by the platform flexing and the axle wobbling. However, good agreement between the predicted and experimental rocking natural frequency showed that the concept has some merit. After fixing the problems with the structure of the platform, the next step for this research will be to test actual packages on a trailer and on the platform.    iii  ACKNOWLEDGMENTS    First and foremost I wish to express my gratitude to my thesis advisor, Dr. Gary Burgess, who is the one professor who truly made a difference in my life. He is very knowledgeable and experienced in packaging distribution; I would not have considered a graduate research in this area if he were not my major professor. He provided me with direction, technical support and became more of a mentor and friend than a professor. He helped me come up with the thesis topic, build the test prototype and figure out the problem I met in the research and the writing. Dr. Burgess is very good at inspiring and guiding people to think on their own, my critical and analytical thinking was developed during this process. I would also like to thank the other members of my committee, Dr. Robert Clarke and Dr. Brian Feeny. They shared their thoughts and concerns at the beginning and pointed out the shortcoming of the thesis and the better direction to revise it at the end. I am gratefully indebted to their very valuable comments on this research.    Finally, I must express my very profound gratitude to my beloved grandparents (Jiqin Chen and Zhenmei Wang), parents (Qikun Wu and Dongmei Chen) and my dear friends (Zhengyang Yang and Jin Zhang) for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. I would not make it complete without them.    iv  TABLE OF CONTENTS   LIST OF  .vii KEY TO  KEY TO ABBREVIATIONS CHAPTER 11 INTRODUCTION AND LITERATURE REVIEW.........1  1.1. Vibration test.............1 1.2. Problems with the vibration test.................3 1.3. Research on single-axis vibration versus multi-axis vibration 1.4. Alternative solution  combining the 2 most important motions 1.5. Objective and hypotheses CHAPTE METHODS AND MATERI  2.1. Vibration table simulation....13  2.2. Materials  2.3. Matching the trailer motion18   2.3.1. Target RMS G from trailer22   2.3.2. Simulated RMS .23   2.3.3. Predicted motion...26  RESULTS AND DISCUSSION..32 3.1. Predicted vs. Experimental rocking natural frequency 3.2. Predicted RMS G    4.1.   4.2. Limitations of this research  4.3.   APPENDIX A:  to Table 5........................................................................................49 APPENDIX B:  Table 6....52 APPENDIX C:  v     to Table 755 APPENDIX D:     8 .....59  vi  LIST OF TABLES   .3  Table 2 .18  Table 3 32  Table 4 Prediction vs. experimental rocking natural frequency from the sweep test.34  Table 5 Predicted combinations m35  Table 6 mx,P35  Table 7 .36  Table 8 and the spring41  vii  LIST OF FIGURES   Figure 1 Single-axis vibration table.1  .2  Figure 3 Stacked bags after t3  Figure 4 4  Figure 5 Lansmont mechanica  Figure 6 Circular-30° out-of-..6  Figure 7 MAST being us7  Figure 8 Multi-axis table being used to test auto parts in racks7  Figure 9 .8  Figure 10 Lansmont - CUBETM vibration t..8  Figure 11   Figure 12 Uneven road causes roll vs. single-axis vibration table.11  Figure 13 Initial idea (versi  Figure 14 A14  Figure 15 .15  Figure 16 15  Figure 17 ....16  Figure 18 .17  Figure 19 19  Figure 20 Decaying sine wave from   Figure 21 Sampled accelerat.22 viii  Figure 22   Figure 23 Relationship between vertical PSD plots for positi  Figure 24 Platform with SAVER 3X925  Figure 25 In.27  Figure 26 .27  Figure 27 28  Figure 28 Piecewise linear vertica30  Figure 29  33  Figure 30 Recorded acceleration in lat.37  Figure 31 38  Figure 32 .38  Figure 33 40  Figure 34 Plot of the relationship between RMS 42 Figure 35 Location of 45  ix  KEY TO SYMBOLS   a   Distance from the axle of the platform to the spring (in)  A and B  Constants  c   Viscous damping coefficient for the platform system (lb-sec/in)  C1, C2 and C3  Constants that depend on the platform construction  C4 to C9  Arbitrary constants  d   Horizontal distance from the axle of the platform to the center of  gravity (in)   D   Distance from the trailer centerline to the floor position to be simulated (in)     Rocking natural frequency of the platform (Hz)  F   Upward force exerted by the axle on the platform (lbs.)  g   Acceleration of gravity (386.4 in/sec2)       Acceleration (g     Lateral acceleration (g     Lateral acceleration of the platform (g     Lateral acceleration on the centerline of the trailer floor (g     Vertical acceleration (g     Vertical acceleration of the trailer floor at the position to be simulated, or vertical acceleration of the platform axle (g     Vertical acceleration on the centerline of the trailer floor (g     Height of the SAVER above the platform (in)  x     Height of the SAVER above the trailer floor (in)     System moment of inertia, an axis through the center of gravity on the  platform system (lb. in2)     Spring constant on one side of the platform (lb/in)  mg   Combined weight of mass, platform and test package (lbs.)  Mm   Weight of the mass on the platform (lbs.)     Counterclockwise rotation of the platform from the static position (rad)     Angle of rotation of the platform from the horizontal plane at rest (rad)     Starting value for the angle of the platform relative to the static position when vibrating (rad)     Angular velocity of the platform (rad/s)     Starting value for the angular velocity of the platform when vibrating  (rad/s)     Angular acceleration of the platform (rad/s2)     Angular acceleration of the trailer floor (rad/s2)  N   Total number of samples  P   Position on the trailer floor to be simulated  R    Damping ratio    Lateral RMS G of the platform (g    Lateral RMS G on the centerline of the trailer floor (g    Vertical RMS G of the trailer floor at position P     Vertical RMS G on the centerline of the trailer floor (g  Xm   Distance from the platform axle to the mass (in) xi      Time (msec)     Difference between and  (g     Time interval (msec)     Angular frequency (Hz)       xii  KEY TO ABBREVIATIONS   PSD    Power Spectral Density  ASTM    American Society of the International Association for Testing and  Materials  MAST    Multi-Axis Simulation Table  RMS G   Root Mean Square G  OD    Outside Diameter  ID    Inside Diameter  TTV    Touch Test Vibration Controller  TP3    Test Partner®  CG    Center of Gravity  std.dev    Standard deviation  1  CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW 1.1. Vibration test Mechanical vibrations and shocks that happen during transportation cause most of the damage to packages and products. In order to avoid insufficient or excessive packaging, a valid and economic testing method is necessary. The vibration test is the most common method used to simulate the transportation environment in labs.  The vibration test is performed with vibration tables. The most common vibration table is the single-axis shaker, which reproduces the most severe vibration - vertical motion. Test packages are mounted on the vibration table, which then moves only up and down. Figure 1 is an example of the single-axis vibration table (Lansmont, 2010).  Figure 1 Single-axis vibration table   The vibration table is driven by a power spectral density (PSD) plot. This is a plot of power density versus frequency (see Figure 2 as an example). These power densities are related to accelerations collected by vibration recorders mounted on the floor of a truck trailer or railcar. The transportation environment is simulated in the lab by using the specific PSD plot with the same frequencies and power densities as those that were recorded. In order to simplify the procedure, the standard ASTM D4728 (ASTM, 2012) 2  provides representative PSD plots that simulate typical random vibration environments. Figure 2 shows separate PSD plots for vertical motion, lateral (side-to-side) motion, and longitudinal (front to back) motion (Burgess, 2013). In the frequency range of interest (2-8 Hz), vertical vibration is normally 1 to 2 orders of magnitude higher than lateral and longitudinal vibrations. This is the main reason that vibration tables are constructed as single-axis shakers.  Figure 2 PSD plot for vertical, lateral and longitudinal vibrations of a trailer   This research focused on reproducing road transportation, which means tractor trailers. that vibrate at different frequencies. They are listed in Table 1.    3  Table 1 Different vibration parts of the trailer Suspension 2 Hz (fully loaded) to 8 Hz (empty trailer) Tires 15 Hz (low pressure) to 20 Hz (high pressure) Floor 50 Hz (fully loaded) to 100 Hz (empty trailer) 1.2. Problems with the vibration test An increasing number of companies find that their products pass the standard vibration test but are damaged during transportation. The images shown in Figure 3 are examples (D. Leinberger/ABF Freight, e-mail, 2010). As the vertical vibration cannot cause this damage, these two pallet loads should have experienced large lateral displacements.  Figure 3 Stacked bags after truck transportation  In order to solve this problem, it is necessary to figure out all of the movements that occur to a trailer during transportation. As shown in Figure 4, the trailer can move in 3 linear directions and rotate about 3 axes. The 3 linear movements are surge (front to back), sway (side to side) and heave (up and down). These are the same as the longitudinal, lateral and vertical directions in Figure 2. They happen when trailers change speed, change lanes and go over bumps in the road, respectively. The 3 rotations are roll (rotation about surge axis), pitch (rotation about sway axis) and yaw (rotation about 4  heave axis). Roll happens when the wheels on one side of the trailer go over potholes or bumps. Figure 4 Movements of a trailer  In Figure 3, the pallet-loads may have gone through roll and pitch motions. These are two movements that cannot be simulated on the vertical shaker. Real road transportation is a much more complex vibration environment. Current tables lack 5 of the 6 motions that occur in real transportation. This is why the test ASTM D4728 (ASTM, 2012) cannot be used to evaluate package integrity. Better vibration testing is badly needed.   1.3. Research on single-axis vibration versus multi-axis vibration  Singh (Singh, Antle, & Burgess, 1992) noticed that the power density level under 10 Hz could be as severe as that of vertical vibrations for heavily loaded truck trailers. It was because the rocking motion of the top load contributed to lateral vibration at low frequencies. A vibration test for stacked corrugated packaging was conducted by Bernad in 2010. The result showed stackable packaging could resist more load from the vertical direction 5  (compression) than other load directions (roll and pitch). Even though lateral and longitudinal excitations had less energy than vertical, they could contribute to sliding between layers in the stack and provoke the failure of the shipping unit (Bernad, Laspalas, González, Liarte, & Jiménez, 2010). Bernad expanded the research to demonstrate the need for multi-axis testing in the lab. The test showed that the three linear motions combined only slightly increase the power density while the addition of rotational movements made the most significant change in the PSD plot. These energies are neglected in single-axis vibration test (Bernad, Laspalas, González, Núñez, & Buil, 2011). Rouillard found that pitch and roll motions could be as damaging for shipments as vertical vibrations, even though they are relatively less severe (Rouillard, 2013). From experiments conducted to find the correlation between these three motions, Rouillard found that the type of road affects the overall intensity of the PSD plot, but not its shape. Also, the power density levels for all three motions generally increased linearly with vehicle speed. Peterson used single-axis and multi-axis excitations to do time-to-failure tests on 10 digital clocks. The result showed six-degree of freedom motion caused failure in roughly half the time compared to vertical excitations alone. Additionally, the author noticed that if one clock was mounted in the center of the table and another was mounted slightly off to the side, the time to failure of the one off to the side was about two-thirds of the time to failure for the one in the center because one off to the side absorbed more energy from rotations (Peterson, 2013). 6  Lansmont provides a low-cost mechanical shaker for package testing (See Figure 5). The shaker can perform vertical-linear, circular-synchronous and 30° out-of-phase motions (refer to Figure 6). In the vertical direction, the shaker works like a single-axis vibration table without PSD control. When it does circular-synchronous motion, the table moves in small vertical circles at constant rotational speed. Consequently, the vertical and horizontal motions are sinusoidal. When the motion is 30° out-of-phase, the shaker simulates vertical, longitudinal and pitch movements at the same time (Lansmont, 2012). Figure 5 Lansmont mechanical shakers  Figure 6 Circular-30° out-of-phase motion FedEx uses rotary vibration as one of its test procedures for packaged products weighing up to 150 lbs. The rotary vibration tester works similar to Lansmont mechanical shaker using circular-synchronous motion (FedEx, 2011). 7  A six degree of freedom multi-axis simulation table (MAST) is the one of the best simulators of the real environment (see Figure 7). The automotive industry was one of the first to employ this multi-axis vibration table. Figure 7 - 9 show a six-axis shaker used for automotive testing (Control Power-Reliance, personal communication, October 2013). Figure 7 MAST being used to test finished cars for rattles and squeaks  Figure 8 Multi-axis table being used to test auto parts in racks   8  Figure 9 Six hydraulic pumps need to drive the table  Lansmont provides a multi-axis vibration test system, called the CUBETM (see Figure 10). It claims to be able to simulate real world 6-degree of freedom motion. The top mounting surface is 32 × 32 in (Lansmont, 2014). At the present time, the price of the CUBETM is about $1,000,000 while their single-axis shaker (refer to Figure 1) is about $200,000. Lansmont believes that the value of multi-degree of freedom vibration testing will increase significantly in the next several years, especially for testing pharmaceutical and electronic products (J. Breault/Lansmont, e-mail, September 2015). Figure 10 Lansmont - CUBETM vibration tester  9  Lansmont conducted a test to verify the necessity of the multi-axis shaker. This experiment also showed how the truck really moves during transportation (Root, 2014). The truck was driven on California roads. Lansmont vibration recorder, the SAVER 9X30, was mounted on the trailer floor with two external triaxial accelerometers to record synchronous acceleration versus time measurements in 3 directions. At first, the CUBETM vibration tester was driven with only vertical input. Even though the tester reproduced the vertical motion very well, the test item did not respond the same way on the trailer. Other inputs (pitch, roll and yaw) were then added to the CUBETM. The motion of the test item was finally consistent with what happened on the truck. It showed that only multi-axis tables like the CUBETM could reproduce actual vibration.  1.4. Alternative solution  combining the 2 most important motions High-technology always comes with a high price. In addition to the shaker, there are also maintenance and space costs. 6-axis shakers usually require more lab space and more powerful drive mechanisms than the single-axis table. Therefore, these shakers are not used by most companies in the packaging industry at the present time.  Single-axis vibration tables are currently owned by a great many institutions and companies. This was not a small investment, even though the single-axis table is not nearly as expensive as a multi-axis shakercurrent vibration test standards are all written for single-axis shakers.  What improvements can be made for these vertical vibration tables? If the existing single-axis shakers could be made to function like a multi-axis vibration table at low-cost, this problem would be solved. However, this is almost impossible to do using the current single-axis table. It is possible however to add one of the five remaining motions, roll. 10  A truck trailer-ride or leaf-spring. They can both be modeled as a spring system. The end view of the trailer is shown in Figure 11. Figure 11 Simplified trailer suspension system  In addition to heave motion, roll is another motion that occurs all the time during transportation. Roll happens when the wheels on one side of the trailer go over potholes or bumps. Therefore, as long as the trailer is driving on an uneven road, there will be a roll motion. Uneven roads are what we have in the real world. Only when the wheels on both sides of the trailer hit a step bump at the same time, like raised pavement, will the trailers execute vertical motion. A comparison of real road conditions with road conditions that current vibration tables simulate is shown in Figure 12.   Stacked Packages 11  Figure 12 Uneven road causes roll vs. single-axis vibration table  When trailers drive over uneven roads, the axle is always inclined. This excitation is transferred to the trailer body by the suspension system and causes side to side movement of packages, especially tall stacks. For stacked packages, roll may be the primary risk. Packages are designed to support top load, so they have better resistance to compression than bending. Misalignment between stacked packages and layers could amplify the problem (Bernad et al., 2011). The remaining movements, surge, sway and yaw, only occur when the trailer changes speed, changes lanes and makes turns. These three excitations are long duration events that can last for several seconds. They require large displacements relative to vertical and roll motions that the vibration table would have to reproduce. Therefore, they cannot be replicated by single-axis shakers in the lab. The same is true for 6-axis shakers.   Stacked Packages 12  Roll and pitch are the only two motions that can be added to the single-axis vibration table. Pitch angles are much smaller than roll angles, so adding the second most severe motion, roll, to a single-axis vibration table was investigated in this research as an alternative to a six-axis shaker. 1.5. Objective and hypotheses The goal of this research was to reproduce the rolling motion of the trailer using an add-on to a single-axis vibration table. This add-on is a rocking platform.   In order to achieve this goal in a more controlled way, a prediction model was developed. This model was used to predict the rocking natural frequency of the rocking platform (, where  is the angle of rotation from the horizontal plane) and the overall severity of the vibration  Root Mean Square G (RMS G). A trustworthy prediction model should show good agreements between the predicted values and experimental results. Therefore, the hypotheses of this research were that predicted  and RMS G would be consistent with experimental results.  13  CHAPTER 2 METHODS AND MATERIALS 2.1. Vibration table simulation The original concept of adding rocking motion to a vertical shaker was to place a rocking platform on the table to act as the new vibration plane. A steel pipe is used as the axle. It passes through the platform and allows it to rotate. A spring is used to support the other side (refer to Figure 11). When the table is vibrating, the platform will heave and rock at the same time. Several prototypes of the platform have been explored based on this initial concept. a. The initial concept, shown in Figure 13, will be called version 1 (k is the spring constant). This system has a vertical frequency, which can be adjusted to simulate real transportation. The rocking frequency and amplitude can be set to match target values by adjusting the spring stiffness. Because the package is not directly over the pivot, the vertical motion will not be the same as that of the vibration table. In addition, the vertical amplitude could be magnified. This will require the PSD plot that drives the table to be modified. Additionally, the spring end might lift off the table, especially if resonance is created.   Figure 13 Initial idea (version 1)  Vibration table Platform Spring Test  Package 14  b. Figure 14 shows two more prototypes based on the initial concept. Both will be called version 2. Since the test package is over the pivot, the vertical frequency and amplitude of the package will exactly match the vertical frequency and amplitude of the vibration table. This is what we want because the table is being driven by a PSD plot that is supposed to recreate the vertical vibration of the floor of a truck trailer. But both of these designs have problems:  For the platform with springs (left), the rocking frequency and amplitude could be matched to that of the trailer floor by adjusting the spring stiffness. However, the spring end might lift off the floor, which poses a safety risk.  For the platform with a second axle (right), the rocking amplitude could be matched to that of the trailer floor by adjusting the position of this axle. For both, the rocking frequency of the platform is related to the vertical frequency of the vibration table while they should be independent.  Figure 14 Alternative platforms (version 2)    c. The prototype in Figure 15, called version 3, keeps all of the virtues of version 2 but mitigates their faults. Version 3 can simulate the rocking frequency and amplitude by adjusting the different spring stiffnesses on each side. Additionally, the rocking amplitude is self-limiting, which ensures that the platform will not go 15  into resonance. This is closer to a real trailer. However, since the spring stiffnesses on each side are different, the platform might not respond the same as the floor of the trailer, where the stiffnesses are the same on each side.  Figure 15 Better platform (version 3)  d. After further refining, the final prototype (version 4) of the platform is shown in Figure 16. The axle passes through the center of the platform and springs with the same spring constant are used on both sides. In addition to imitating the vertical motion, rocking motion is introduced by adding a solid mass on one side. When the table is vibrating, the platform will move up and down in synch with it and rotate, like a seesaw. This prototype can also account for the crown in the road by initially tilting the platform. Additionally, if the springs are replaced by blocks, the platform will revert back to a regular single-axis shaker (Figure 16, right).  Figure 16 Final prototype of the platform (version 4)  16  2.2. Materials a. -platform to provide a mounting surface for a vibration recorder, the SAVER. The actual platform is shown in Figure 17.   Figure 17 The actual platform  Weight: 31.33 lbs. Deck length x width x thickness:  Supporting frame underneath (thickness x width x length): 5 ribs along the length 8 ribs Two end pieces that can be removed to install up to 4 springs in parallel on each end  b. Axle Weight: 6.67 lbs. Steel pipe: 3 Inside Diameter (ID) c. Springs (see Figure 18) Compression Spring - P/N C48-187-192 (W.B. Jones Spring Co.) 17  1.470 OD, 0.187 music wire, 6-inch overall length Spring constant (k)  60 lbs. /in. Figure 18 Specifications for the spring   d. Bearings  Two wooden blocks with square holes for the axle.  Two screws in each block, one horizontal and one vertical, to adjust the position of too much during vibration.  e. Steel Masses Different weights were used to get the platform rocking. Their specifications are shown in Table 2.  18  Table 2 Steel Masses Number Weight (lbs.) Dimensions (length x width x thickness) 5 5  1 12.8  2 32  f. Vibration system  Lansmont vertical vibration table - Model 7000 and program TTV (TouchTest Vibration Controller) g. Vibration recorder Lansmont SAVER 3X 90 and program SaverXWare. h. External accelerometer Kistler 10 mV/g single-axis piezoelectric accelerometer and program TestPartner® (TP3) 2.3. Matching the trailer motion Since the test package is placed directly over the axle, its vertical motion is the same as that of the vibration table. Therefore, how to reproduce the rocking motion was the main objective that needed to be researched. Rocking frequency and rocking amplitude are two independent parameters that need to be considered. The natural frequency in the rocking mode can be determined from either a bump test or a frequency sweep looking for resonance.  For the bump test, an accelerometer was mounted on the platform and connected to a computer with TestPartner® (TP3) (see Figure 19).   19  Figure 19 Platform in the bump test  After the platform was pushed down and released, TP3 recorded the vibration. It showed an exponential decay curve as in Figure 20. The time interval between peaks, such as  to , is the period of vibration. The reciprocal of the period is the rocking frequency:  Figure 20 Decaying sine wave from the bump test  20  For a damped spring-mass system responding to a bump test, Equation (1) describes the behavior in Figure 20 (Shabana, 1995):     ( 1 ) The solution to Equation (1) is   ( 2 ) where A and B are constants. The natural frequency is:      ( 3 ) and the damping ratio is:     ( 4 ) The constants C1 and C2 in Equation (1) are relaFigure 20 by      ( 5 )     ( 6 ) where the accelerations  and  are the first and last peaks at times  and .  The sweep test is another way to find the natural frequency. It uses the vibration table. The standard procedure is ASTM D999 (ASTM, 2015). The table is driven so that it moves up and down sinusoidally, slowly increasing the frequency from 3 Hz to 100 Hz. The natural frequency can be determined by observing the platform movement during the sweep test. Maximum amplitude can be identified by attaching an accelerometer to the 21  platform. When the platform rocks wildly, it has reached resonance. It resonates whenever the table frequency matches its natural frequency.  The resonant frequency recorded during the sweep test should match the result from the bump test. Nevertheless, both tests give a natural frequency, which need not be the same as the rocking frequency during transportation because this frequency depends on the spacing of bumps on the road and the truck traile. Rocking natural frequency is not a variable that will be targeted but is nevertheless important because it does enter into calculations later.  The other parameter, rocking amplitude, is the maximum angle that the platform achieves during vibration. It cannot be measured directly because the SAVER only measures linear accelerations in three perpendicular directions. However, by mounting the SAVER above the platform, the horizontal acceleration that it measures during rocking can be related to angular acceleration. Angular acceleration can then be used to determine the rocking amplitude.  Since vibration in transit is normally random motion, rocking motion is also. Therefore, the RMS G (Root-Mean-Square G), which is a quantity that is measured by the SAVER, will be targeted. This is used to represent the overall severity of the motion. The goal will be to match this with what the trailer does.  Figure 21 shows a portion of a random vibration signal. The dots are acceleration samples recorded by the SAVER. The average of the recorded accelerations in Figure 21 will be zero because there are as many positive accelerations as negative ones. Positive accelerations result from the trailer floor moving up and negative accelerations from it moving down. The standard deviation, which is a measure of the variation in G values 22  around the mean, is not zero. If N is the number of samples, the standard deviation or  in the x (lateral) direction is:   ( 7 ) Figure 21 Sampled accelerations  2.3.1. Target RMS G from trailer Different locations on the trailer floor experience different RMS Gs. The motion at the center, midway between walls, is usually the smoothest and near the walls, it is the roughest. The data recorded by the SAVER only represents the vibration where the SAVER is located. There are several steps needed to get the RMS G that the platform is supposed to simulate.  First, a lightweight beam is attached to the trailer floor as shown in Figure 22. The SAVER is mounted on it midway between walls at height Ht above the trailer floor. The SAVERlateral () and vertical () accelerations are recorded at regular intervals, usually every 1 millisecond (refer to Figure 21). At every instant, the floorangular acceleration () is related to the lateral acceleration experienced by the SAVER through:      ( 8 ) 23  where g is the acceleration due to gravity, 386.4 in/sec2.  Figure 22 End view of a trailer with the SAVER on beam  Next, if the motion at location P (Figure 22) on the trailer floor is to be simulated, where P is at distance D from the center line, the vertical acceleration () at P will be:     ( 9 ) The platform should be set up to reproduce the angular and vertical accelerations in Equations (8) and (9) as closely as possible. Since it is highly unlikely that the platform will be able to reproduce them at every instant, only their RMS values will be targeted.  2.3.2. Simulated RMS G The vertical motion of the trailer floor at position P is . This signal should be used to drive the vibration table. This makes the vertical acceleration of the platform directly over the axle the same as that of the trailer at P.  24  Vibration tables are driven by PSD plots. In order to get the PSD plot for location P, the recorded SAVER data needs to be processed. Substituting  from Equation (8) into Equation (9) relates  to the lateral and vertical accelerations recorded by the SAVER:      ( 10 ) The vertical  s vertical  and lateral  by:    where the sums are over the N samples taken at regular intervals. The middle sum is zero because  and  are independent random oscillations, both with means of zero. The first and third sand lateral directions.      ( 11 ) Since the RMS G squared is the area under its PSD plot, Figure 23 shows the meaning of Equation (11). In order to get the vertical PSD plot for position P, start with the vertical PSD plot the SAVER on beam provided and raise it up until the crosshatched area in Figure 23 is . If the plot spans the usual frequency range of 3 Hz to 100 Hz, all power density (G2/Hz) values are raised the same amount , where . 25  Figure 23 Relationship between vertical PSD plots for position P and center of trailer  Now the rocking motion (angular acceleration) must be matched. Since the SAVER on the platform is mounted right over the axle to record the ) and vertical () accelerations (Figure 24), the relationship between the lateral acceleration of the platform and its angular acceleration () is:      ( 12 ) Figure 24 Platform with SAVER 3X90 mounted on it  26  To make the platform move like the trailer, the angular accelerations ( and ) should be the same:       ( 13 ) instant should therefore be made to relate at every instant by:     ( 14 )  Trying to get the platform to do this at every instant will not be possible. Instead, Equation (14) will be satisfied in an RMS sense. In view of the definition of RMS G, the lateral RMS G:     ( 15 )  The goal is therefore to make the lateral acceleration  measured by the SAVER on the platform be  times the lateral acceleration  measured by the SAVER on the beam attached to the trailer floor. This can be done by adjusting the locations of the mass attached to the platform and the number of springs used. In order to prove this, the next section analyzes the theoretical motion of the platform.  2.3.3. Predicted motion When the table is turned off, the platform is at rest (Figure 25). In this state, the platform rotates angle  relative to the horizontal plane (st means static).     27  Figure 25 Initial state of the platform  Since the platform is not moving, there is no damping force. The force diagram is shown in Figure 26.  Figure 26 Force diagram of the platform at rest  mg = combined weight of mass, platform and test package (lbs.) F = support force exerted by axle (lbs.) a = distance from the axle to the spring (in) k = spring constant (lb/in) Xm = distance from the axle to the mass (in) d = horizontal distance from axle  gs three are used, etc.). m m to be varied.  28  The static rotation  is obtained by summing moments about the axle:      ( 16 )      ( 17 ) When the vibration table is running, the platform is in a dynamic state. See the force diagram in Figure 27.  Figure 27 Force diagram of the platform in motion   Summing vertical forces and moments about the center of gravity requires that:     ( 18 )   ( 19 ) c = viscous damping coefficient for the system (lb-sec/in) = known vertical acceleration of the axle = counterclockwise rotation from the static position. = angular velocity of the platform = angular acceleration of the platform  = system moment of inertia about an axis through the CG 29  Equation (18) was multiplied by d and added to Equation (19), producing Equation (20):     ( 20 )      ( 21 )      ( 22 )     ( 23 ) The homogeneous solution to differential Equation (20) is:    ( 24 ) where C4 and C5 are arbitrary constants to be solved for later. The angular frequency is:    ( 25 ) where   Since the SAVER records vertical acceleration only at discrete times (every 1 millisecond),  in Equation (20) is only known at discrete times. In order to solve Equation (20) accurately, the vertical acceleration is assumed to be piecewise linear between samples. For the first time interval (), the vertical accelerations are shown in Figure 28.  and  are the sampled accelerations.    30  Figure 28 Piecewise linear vertical acceleration   Within this segment, the vertical acceleration is represented by:      ( 26 ) where C6 and C7 were related to the sampled accelerations at the start and end of the interval:       ( 27 )      ( 28 ) The particular solution to Equation (20) is:      ( 29 )       ( 30 )     ( 31 ) The complete solution to the differential Equation (20) for  is the sum of Equation (24) and Equation (29):   ( 32 ) Taking the derivative of Equation (32) with respect to time gives: 31   ( 33 ) In order to get C4 and C5, the starting values for the angle () and angular velocity () are needed.  At , if ,       ( 34 ) At , if ,      ( 35 ) This completes the solution to Equation (32 angle and angular velocity at the end of the first interval are obtained using  in Equations (32) and (33). These values are then used as starting values for the next interval (). In this way, the solution can be obtained in a recursive manner. The angular acceleration () and lateral acceleration () at each instant are therefore also known from Equations (20) and (12). The  can therefore be obtained using the  values calculated at each time step. The results will be presented in the next chapter.    32  CHAPTER 3 RESULTS AND DISCUSSION 3.1. Predicted vs. Experimental rocking natural frequency There were two tests conducted to get the rocking natural frequency  the sweep test and the bump test. Both results will be compared to the predictions.  In the bump test, there was no test package. The only weight on the platform was a 5 pound steel block. Also, the number of support springs on each side was 2. In order to eliminate noise on the recorded waveform produced by the accelerometer mounted on the platform, the filter frequency was set to 5 times as the predicted rocking natural frequency. This is common practice in this field. The results of the bump test are shown in Table 3. The rocking frequency was measured from the decaying sine wave (Figure 29) and R was calculated from Equation (4).  Table 3 Rocking natural frequency and damping ratio from the bump test Bumps Filter Frequency (Hz) Period (ms) Bump Test_f(Hz) G1 GN R 1 st 44 122 8.20 2.33 1.25 0.03 2 nd 44 124 8.06 2.18 1.02 0.04 3 rd 44 146 6.85 2.2 1.56 0.02 4 th 44 126 7.94 2 1.13 0.03 5 th 44 130 7.69 3.03 1.82 0.03    33  Figure 29  First bump test filtered at 44 Hz  Different bumps produced different results for and  even though they were conducted under the same experiment conditions. The accelerometer is too sensitive, requiring filtering to eliminate noise, and the platform probably flexed, which could be reasons for this result. Compare to the bump test, the sweep test results were much more repeatable. Table 4 shows the comparison between the predicted  and the sweep test value. The sweep test value is the vibration table frequency that caused resonance. The predicted natural frequency was obtained from Equation (25) using the platform specifications in section 2.2.       34  Table 4 Prediction vs. experimental rocking natural frequency from the sweep test  Predicted  (Hz) Sweep Test  (Hz) Predicted  (Hz) Sweep Test  (Hz) Predicted  (Hz) Sweep Test  (Hz) Springs on each  side Mass weight (lbs.) 2 3 4 5 8.73 8.65 10.69 11.6 12.34 12 37 4.98 5.05 6.10 6.2 7.04 6.9 69 3.85 4.2 4.71 4.85 5.44 5.5 101.8 3.24 3.5 3.96 4 4.58 4.65 The good agreement between the predicted and experimental  shows that Equations (25) appear to be trustworthy.  3.2. Predicted RMS G  A series of vertical acceleration () samples were generated at 1 ms intervals and used as the trailer input . The vertical accelerations  of a targeted position 30 inches from the centerline were then obtained to drive the vibration table.  For the prototype wooden platform, the spring constant (k) and the weight of the mass (Mm) are adjustable, while the locations of the springs and the mass are fixed (refer to  Mm, a program that uses the theoretical equations in Chapter 2 was written in Excel Macros (see Appendix A). For the targeted of the m are shown in Table 5. In this situation, the spring location (a) was 23 inches and mass location (Xm) was 20.5 inches.   35  Table 5 m Number of springs  Spring constant (lb/in) Weight of the mass (lbs.) Simulated RMS G's 2 120 5 0.0109 3 180 5 0.0115 4 240 5 0.0119 The platform used in this research was a simplified version. As the locations of the springs and the mass were fixed, the mass needed to be removed to adjust the weight and/or springs needed to be taken out or added. This way is not very convenient for actual use. An easier adjustment for a commercial version of the platform is recommended. In the commercial version, only one set of springs and mass are needed  Xmables. Adjustments can be made more easily by a crank handle as needed. Table 6 m as the simplified version based on the same input data. In this case, the spring constant (k) and the weight of the mass (Mm) were chosen to be 350 lb/in can be m  Table 6  Xmx,P Spring location away from axle (in) Mass location away from axle (in) 3 7.84 4 6.38 5 6.49 6 6.7 7 6.15 8 5.74 9 5.77 10 5.93 11 5.97  36  Table 6 (cont 12 6.11 13 6.31 14 6.47 15 6.63 16 6.84 17 7.12 18 7.47 20 8.29 21 8.74 22 9.21 23 9.7  The theoretical results in Table 5 show that it is possible to target a given lateral  by using different combinations of spring constant and weight of the mass. An actual test was conducted to check if the predicted results could be verified. The input data was downloaded from a SAVER that was mounted on the A-frame of the platform. There were 3 springs supporting each side of the platform and one 5 pound weight was placed on it. The recorded vertical accelerations were used as the input , and then the angular acceleration  and lateral acceleration  were obtained through Equations (20) and (12). The RMS G from the recorded lateral accelerations was then compared with the one from the predicted lateral acceleration . See Table 7 for the result.  Table 7  Predicted lateral  Recorded lateral  0.139 1.2881  Even though the predicted and actual RMS G 37  other most likely because of the unwanted noise and vibration produced by the platform flexing and the axle wobbling. They were inferred from the signals the SAVER recorded. Figure 30 shows the recorded lateral response. The SAVER recorded a constant It can be inferred from the beat pattern that the A-frame and/or the platform were resonating. Figure 30 Recorded acceleration in lateral direction  The recorded longitudinal accelerations (in the direction of the axle) are shown in Figure 31. Accelerations  were recorded. They  as there should be no vibration in this direction. The reason for this is that the platform was not stiff enoughA-.    -6.00E+00-4.00E+00-2.00E+000.00E+002.00E+004.00E+006.00E+000.00E+005.00E-021.00E-011.50E-012.00E-012.50E-013.00E-01Acceleration_G'sTime_s CH1 Acc (G's) (lateral)38  Figure 31 Recorded acceleration in longitudinal direction  Additionally, the ASTM 4169 Truck II PSD spectrum (ASTM, 2005) was used to drive the table, which means the vertical accelerations SAVER recorded should be about Figure 32 shows vertical accelerations as large as This was probably due to the axle being loose in the bearings, so unwanted noise and vibrations were produced.  Figure 32 Recorded acceleration in vertical direction    -3.00E+00-2.00E+00-1.00E+000.00E+001.00E+002.00E+003.00E+000.00E+005.00E-021.00E-011.50E-012.00E-012.50E-013.00E-01Acceleration_G'sTime_s CH3 Acc (G's)(longitudinal)-4.00E+00-3.00E+00-2.00E+00-1.00E+000.00E+001.00E+002.00E+003.00E+004.00E+000.00E+005.00E-021.00E-011.50E-012.00E-012.50E-013.00E-01Acceleration_G'sTime_s CH2 Acc (G's)(vertical)39  CHAPTER 4 CONCLUSIONS AND FUTURE WORK For the reasons mentioned in Chapter 3, the predicted RMS G could not be verified using the experimental test results due to flexing of the platform and A-frame and rattling of the axle in the bearings. However, the predicted rocking natural frequency was verified using the sweep test, because this property  depend on instantaneous accelerations. It is likely that we can trust the predicted RMS G even though the SAVER data does not confirm this.  If we want the vibration recorders to record noise free data, the platform should be built as stiff as possible with tight tolerances in the axle. This viewpoint is also a concern of the single-axis vibration table manufacture, Lansmont, in their instruction manual (Lansmont, 2011). Also, hard filtering acceleration data to remove unwanted platform vibrations will be helpful. This can be done by specifying some upper frequency cutoff and hard wiring a simple analog filter into the circuit.  In order to get the target  from the trailer, a multi-axis vibration recorder like the SAVER and its support frame will be needed. Lightweight beams as shown in Figure 22 should be used to build the frame.  In addition to rocking motion, this platform can also reproduce pitching motion, which happens when the front and back wheels of the trailer go over bumps at different times. As long as the beam attached to the trailer is parallel to the longitudinal direction, all the steps in section 4.1 can be repeated. As the length of a trailer is much larger than its width (40 feet vs. 8 feet), the angle of rotation caused by bumps or potholes will be about 5 times less than in the lateral direction, which is why pitching motion should not be considered.  40  4.1. Industry application of this research and matters that need attention The following steps must be followed to use the platform to add rocking motion to a single-axis vertical vibration table:   a) Construct a lightweight beam like the one in Figure 22, making it as stiff as possible. b) Mount a SAVER on the beam and attach the beam to the floor of a trailer. Mount this beam over one of the floor support beams if possible, refer to Figure 33 (Archer, June 2012). Make sure the SAVER is mounted on it midway between walls at a known height  (30 inches for example) above the trailer floor.  Figure 33 Bare chassis of a trailer showing support beams   c) Drive the trailer over the road that needs to be simulated and let the SAVER record the vertical and lateral accelerations ( and ) at 1 ms or 2 ms intervals.  41  d) Process the data from the SAVER through Equation (10) to get the vertical PSD plot () at the target position P to be simulated. Follow the procedure in section 2.3.2.  e) Construct a rigid platform (refer to Figure 17) with an axle that fits tightly into bearings. Attach the bearings to the vibration table and add one spring (500 lbs./in would be a good start) on each side under the platform to support it.  f) Put a solid block (20 lbs. would be a good start) on one side of the platform and mounted a SAVER on the top of the A-frame (refer to Figure 24).  g) Place the test package on the center of the platform and secure it if necessary.  h) Run the vibration table with the target vertical PSD plot for position P and let the SAVER record the vibration in the vertical and lateral directions.  i) Download the SAVER data and compare the lateral RMS G with the predicted  in Equation (15).  j) Adjust the spring location (a) and the mass location (Xm) until the test result matches the prediction. Use the fact that the lateral RMS G will increase if the mass is moved further from the axle. It may both increase and decrease if the spring is moved further from the axle. The Table 8 and Figure 34 show the predicted trends.   Figure 8  Location of spring from axle (in) Location of mass from axle (in) Simulated RMS G's 2 6 0.0171 2 10 0.0269 2 14 0.0347 2 18 0.0403 6 6 0.0245 42  Table 8 (cont 6 10 0.0372 6 14 0.0459 6 18 0.0517 10 6 0.0266 10 10 0.0421 10 14 0.0542 10 18 0.0622 14 6 0.0243 14 10 0.0386 14 14 0.0501 14 18 0.0584 18 6 0.0194 18 10 0.0315 18 14 0.0424 18 18 0.052 22 6 0.0156 22 10 0.0255 22 14 0.0345 22 18 0.0425 Figure 34 Plot of the relationship between RMS G and two variables  By trial and error, move the spring and the mass in small increments until the target  is reached, because the RMS G is sensitive to the locations of them, especially 43  for the mass. Therefore, if the difference between target RMS G and tested result is big, adjusting the location of mass will hit the target quicker. Figure 34 also shows the adjustment of the spring is harder than the mass because the lateral RMS G will increase and decrease if increase the distance between the axle and the spring, Another observation regarding the motion of platform deserves mentioning. At the moment when the vibration table was turned on, the vibration table surged upward. This is normal. When this happened, the platform rotated through a big angle. This suggests that vertical spikes are accompanied by rotational spikes. This also happens to the trailer floor when the trailer is over a pothole. This is confirmed by Equation (20). Wha large input vertical acceleration , the angular acceleration  will be large too.  There is always the possibility that the center of gravity (CG) of the system can be directly over the axle of the platform. This could happen if the test package has an irregular shape or is placed on the platform off center. In this case, the distance d from will be zero, which makes C3 in Equation (23) zero. Therefore, the angular acceleration in Equation (20) will be zero, which means the platform will only move up and down after the vibration table is turned on. In reality, the platform may rotate through a very small angle, but will be too small to be considered rocking vibration. When conducting the vibration test with the platform, people should pay attention to how to place the test package to avoid this situation CG should not be over the axle. Making sure that the platform rotates through some angle at rest prevents this.   44  4.2. Limitations of this research  a) Since the platform only has two parameters that can be adjusted to match rocking motion, the entire PSD plot cannot be reproduced by simply adjusting these limited variables. Instead, the area under the PSD plot (RMS G squared) was targeted to reproduce the lateral motion of the trailer. RMS G is the most important property of a PSD plot because it represents the overall severity of the ride.  b) This research focused on how to reproduce the rocking motion of the trailer floor. Both the test package and the platform are assumed to be solid masses. The predicted motion in section 2.3.3. was based on the assumption that the test package is a rigid mass, but actual packages do not always meet this criterion. If the package is flexible, its center of gravity will change because the product moves around inside the box. Therefore, the predicted RMS G will have some error. However, the space available for the product to move around inside the package is limited, so the location of the t change a lot. As a result, the predicted RMS G should not change very much.  c) Ideally, the platform should have the same decay rate as the trailers when bumped. This would require matching the damping motion and the RMS G at the same time. In order to match the damping motion of the trailer, the damping ratio R should be the same. Both can be obtained from a bump test. Mount a SAVER on the beam shown in Figure 35 (P. Singh, personal communication, 2006). Drive the trailer over a single bump (like over a curb) and get the decaying sine wave response from the SAVER. Analyze the response to get the rocking frequency and damping ratio through Equations (3)-(6) and then c will be obtained from Equation (21).    45  Figure 35 Location of the vibration recorder  The damping coefficient used for this research was obtained from the bump test on the platform, because this test is the only way to get it.  The damping coefficient c for the platform is determined by different kinds of friction, dry friction and viscous friction. There are two places where dry friction occurred. One is between the springs and the wooded pockets of the platform, because the spring is constantly rubbing against the wood during vibration. Another is between the axle and the bearings. on a commercial platform because it will be built with ball bearings . For viscous friction, there are two factors as well. One is the internal friction of the steel coils when the spring is compressed. The other is the air mass that must be moved during rocking. This provides resistance to the platform during vibration.  Since the adjustments to the platform are limited, the most important property of rocking motion should be match lateral RMS G. The damping 46  coefficient of the platform can be made to be adjustable by adding dampers to the platform. This will increase the damping coefficient.  d) The platform cannot simulate the motion of a trailer when it changes speed (surge), changes lanes (sway) and turns corners (yaw). These three motions are long duration events, lasting several seconds. The platform would have to move several feet to simulate this. Fortunately, these 3 motions only happen occasionally relative to vertical and rocking motion. In addition, they rarely provide enough power to affect the PSD plot, which is used to drive the vibration table. Multi-axis shakers have these same problems. e) Even if the platform and its A-frame can be built as stiff as the vibration table itself, and with no noise coming from the axles, the simulated rocking motion will only get close to the real motion. In a trailer, there are 2 random inputs (left wheels and right wheels), while on a vibration table there is only 1 input. Therefore, there is no way to make the platform motion agree with trailer motion exactly.  4.3. Future work First, a platform that is much stiffer must be built. The platform in this version should use the commercial design where the locations of the spring and the mass are variable. This new prototype should then be used to further verify the prediction model by comparing the lateral RMS G that SAVER measured with the one predicted by the model.  The next step would be to test actual packages on a trailer and on the platform. The actual package should be placed at whatever position on the trailer floor is to be simulated (position P). The vibration table should be driven with the PSD plot obtained through Equation (11). Either the program in Excel Macros or the trial and error approach 47  above can be used to get the required information on how to set up the platform. The package tested on the platform should then be compared to the one tested on the trailer based on the damage they get. Consistency in damage levels will indicate how well the trailer movements are reproduced.   48  APPENDICES 49  Appendix A  Table 5 Sub auto1() Dim I As Integer Dim Xm, Xg, Yg, Ip, Il, Im, MoI, Q, OMEGA As Variant Dim C1, C2, C3, C4, C5, C6, C7, C8, C9, B1, B2, B3 As Variant Dim SUMXT, SUMZT, SUMZP, DDX0, DDX1, DDXT, DDXT0, DDXT1, DDZT0, DDZT1, DDZT, DDZP, TH, TH1, DTH, DTH1, DDTH, AVG, DDTHETA, DDX, SUM As Variant 'trailer Gz & Gx and target platform rms N = 3500: G = 386.4: T = 0.001  HTR = 36 'height of trailer saver above floor SUMXT = 0: SUMZT = 0 : C = 5 For I = 1 To N DDXT = Sheet1.Cells(I, "A").Value : DDZT = Sheet1.Cells(I, "B").Value  SUMXT = (DDXT) ^ 2 + SUMXT : SUMZT = (DDZT) ^ 2 + SUMZT Next I RMSXT = Sqr(SUMXT / N) : RMSZT = Sqr(SUMZT / N) Sheet1.Cells(1, "D").Value = "trailer: lateral Grms= " & Round(RMSXT, 8) Sheet1.Cells(2, "D").Value = "trailer: vertical Grms= " & Round(RMSZT, 8) D = 30 'location on trailer floor to simulate - distance from centerline HPL = 15 ' height of saver above axle on platform SUMZP = 0 For I = 1 To N DDXT = Sheet1.Cells(I, "A").Value : DDZT = Sheet1.Cells(I, "B").Value  DDTHETA = -DDXT * G / HTR 'target position's angular acc's DDZP = DDZT + D * DDTHETA / G 'target position's vertical acc's/axle gz SUMZP = SUMZP + (DDZP) ^ 2 Next I RMSZP = Sqr(SUMZP / N) 50  RMSXP = (HPL / HTR) * RMSXT 'taget rms - rmsxp Sheet1.Cells(1, "E").Value = "platform: lateral Grms= " & Round(RMSXP, 8) Sheet1.Cells(2, "E").Value = "platform: vertical Grms= " & Round(RMSZP, 8) 'platform settings Mp = 40 / G: Lp = 48: Hp = 6: Xp = 0: Yp = 0 Ml = 48 / G: Ll = 12: Hl = 36: Xl = 0: Yl = 18                      Lm = 6.75: Hm = 2: Ym = 4: Xm = 20.5 IJ = 2 : A = 23  For K = 60 To 240 Step 60  For Mm = 5/G To 101.8/G Step 5/G  Mass = Mp + Ml + Mm     Xg = (Mp * Xp + Ml * Xl + Mm * Xm) / Mass     Yg = (Mp * Yp + Ml * Yl + Mm * Ym) / Mass     Ip = Mp * ((Lp) ^ 2 + (Hp) ^ 2) / 12 + Mp * ((Xp - Xg) ^ 2 + (Yp - Yg) ^ 2)     Il = Ml * ((Ll) ^ 2 + (Hl) ^ 2) / 12 + Ml * ((Xl - Xg) ^ 2 + (Yl - Yg) ^ 2)     Im = Mm * ((Lm) ^ 2 + (Hm) ^ 2) / 12 + Mm * ((Xm - Xg) ^ 2 + (Ym - Yg) ^ 2)     MoI = Ip + Il + Im : Q = MoI + Mass * (Xg) ^ 2     C1 = C * (A) ^ 2 / Q: C2 = K * (A) ^ 2 / Q: C3 = -Mass * Xg / Q     OMEGA = Sqr(C2 - (C1) ^ 2 / 4)     B1 = Exp(-C1 * T / 2): B2 = Sin(OMEGA * T): B3 = Cos(OMEGA * T) TH = 0: DTH = 0 DDXT00 = Sheet1.Cells(1, "A").Value : DDZT00 = Sheet1.Cells(1, "B").Value  DDTHETA00 = -DDXT00 * G / HTR  DDZP00 = DDZT00 + D * DDTHETA00 / G  SUM = (HPL * C3 * DDZP00) ^ 2 For J = 1 To N - 1         DDXT0 = Sheet1.Cells(J, "A").Value          DDZT0 = Sheet1.Cells(J, "B").Value          DDTHETA0 = -DDXT0 * G / HTR          DDZP0 = DDZT0 + D * DDTHETA0 / G  51                   DDXT1 = Sheet1.Cells(J + 1, "A").Value          DDZT1 = Sheet1.Cells(J + 1, "B").Value          DDTHETA1 = -DDXT1 * G / HTR          DDZP1 = DDZT1 + D * DDTHETA1 / G                  C6 = DDZP0 * G: C7 = (DDZP1 - DDZP0) * G / T         C9 = C3 * C7 / C2: C8 = (C3 * C6 - C1 * C9) / C2         C5 = TH - C8: C4 = (DTH + C1 * C5 / 2 - C9) / OMEGA         TH1 = B1 * (C4 * B2 + C5 * B3) + C8 + C9 * T         DTH1 = B1 * (-(OMEGA * C5 + C1 * C4 / 2) * B2 + (OMEGA * C4 - C1 * C5 / 2) * B3) + C9         DDTH = C3 * DDZP1 * G - C1 * DTH1 - C2 * TH1         DDX = HPL * DDTH / G         SUM = SUM + (DDX) ^ 2         DTH = DTH1 : TH = TH1 Next J     RMS = Sqr(SUM / N)     If Abs(RMS - RMSXP) < 0.001 Then     Sheet1.Cells(IJ, "F") = K : Sheet1.Cells(IJ, "G") = M     Sheet1.Cells(IJ, "H") = Round(RMS, 4)     IJ = IJ + 1     End If     Next M     Next K     Sheet1.Cells(1, "F") = "Spring constant on each side_lb/in"     Sheet1.Cells(1, "G") = "Weight of the mass_lbs."     Sheet1.Cells(1, "H") = "Simulated RMS G's"     Sheet1.Cells(1, "I") = "Difference between simulated and target RMS G's" End Sub  52  Appendix B   and Table 6 Sub auto2() Dim I As Integer Dim Xm, Xg, Yg, Ip, Il, Im, MoI, Q, OMEGA As Variant Dim C1, C2, C3, C4, C5, C6, C7, C8, C9, B1, B2, B3 As Variant Dim SUMXT, SUMZT, SUMZP, DDX0, DDX1, DDXT, DDXT0, DDXT1, DDZT0, DDZT1, DDZT, DDZP, TH, TH1, DTH, DTH1, DDTH, AVG, DDTHETA, DDX, SUM As Variant 'trailer Gz & Gx and target platform rms N = 3500: G = 386.4: T = 0.001 : K = 350: C = 5 HTR = 36 'height of trailer saver above floor SUMXT = 0: SUMZT = 0 For I = 1 To N DDXT = Sheet1.Cells(I, "A").Value : DDZT = Sheet1.Cells(I, "B").Value  SUMXT = (DDXT) ^ 2 + SUMXT : SUMZT = (DDZT) ^ 2 + SUMZT Next I RMSXT = Sqr(SUMXT / N) : RMSZT = Sqr(SUMZT / N) Sheet1.Cells(1, "D").Value = "trailer: lateral Grms= " & RMSXT Sheet1.Cells(2, "D").Value = "trailer: vertical Grms= " & RMSZT D = 30 'location on trailer floor to simulate - distance from centerline HPL = 15 'height of saver above axle on platform SUMZP = 0 For I = 1 To N DDXT = Sheet1.Cells(I, "A").Value : DDZT = Sheet1.Cells(I, "B").Value  DDTHETA = -DDXT * G / HTR 'target position's angular acc's DDZP = DDZT + D * DDTHETA / G 'target position's vertical acc's/axle gz SUMZP = SUMZP + (DDZP) ^ 2 Next I RMSZP = Sqr(SUMZP / N) 53  RMSXP = (HPL / HTR) * RMSXT 'taget rms - rmsxp Sheet1.Cells(1, "E").Value = "platform: lateral Grms= " & RMSXP Sheet1.Cells(2, "E").Value = "platform: vertical Grms= " & RMSZP 'default settings Mp = 40 / G: Lp = 48: Hp = 6: Xp = 0: Yp = 0 Ml = 48 / G: Ll = 12: Hl = 36: Xl = 0: Yl = 18 Mm = 10 / G: Lm = 8: Hm = 2: Ym = 4 Mass = Mp + Ml + Mm : IJ = 2 For A = 3 To 23 Step 1  LOW = 6: HIGH = 20: DXM = 1 3   For Xm = LOW To HIGH Step DXM      Xg = (Mp * Xp + Ml * Xl + Mm * Xm) / Mass     Yg = (Mp * Yp + Ml * Yl + Mm * Ym) / Mass     Ip = Mp * ((Lp) ^ 2 + (Hp) ^ 2) / 12 + Mp * ((Xp - Xg) ^ 2 + (Yp - Yg) ^ 2)     Il = Ml * ((Ll) ^ 2 + (Hl) ^ 2) / 12 + Ml * ((Xl - Xg) ^ 2 + (Yl - Yg) ^ 2)     Im = Mm * ((Lm) ^ 2 + (Hm) ^ 2) / 12 + Mm * ((Xm - Xg) ^ 2 + (Ym - Yg) ^ 2)     MoI = Ip + Il + Im : Q = MoI + Mass * (Xg) ^ 2     C1 = C * (A) ^ 2 / Q: C2 = K * (A) ^ 2 / Q: C3 = -Mass * Xg / Q     OMEGA = Sqr(C2 - (C1) ^ 2 / 4)     B1 = Exp(-C1 * T / 2): B2 = Sin(OMEGA * T): B3 = Cos(OMEGA * T) TH = 0: DTH = 0 DDXT00 = Sheet1.Cells(1, "A").Value : DDZT00 = Sheet1.Cells(1, "B").Value  DDTHETA00 = -DDXT00 * G / HTR  DDZP00 = DDZT00 + D * DDTHETA00 / G  SUM = (HPL * C3 * DDZP00) ^ 2 For J = 1 To N - 1         DDXT0 = Sheet1.Cells(J, "A").Value : DDZT0 = Sheet1.Cells(J, "B").Value          DDTHETA0 = -DDXT0 * G / HTR          DDZP0 = DDZT0 + D * DDTHETA0 / G          DDXT1 = Sheet1.Cells(J + 1, "A").Value  54          DDZT1 = Sheet1.Cells(J + 1, "B").Value          DDTHETA1 = -DDXT1 * G / HTR          DDZP1 = DDZT1 + D * DDTHETA1 / G          C6 = DDZP0 * G: C7 = (DDZP1 - DDZP0) * G / T         C9 = C3 * C7 / C2: C8 = (C3 * C6 - C1 * C9) / C2         C5 = TH - C8: C4 = (DTH + C1 * C5 / 2 - C9) / OMEGA         TH1 = B1 * (C4 * B2 + C5 * B3) + C8 + C9 * T         DTH1 = B1 * (-(OMEGA * C5 + C1 * C4 / 2) * B2 + (OMEGA * C4 - C1 * C5 / 2) * B3) + C9         DDTH = C3 * DDZP1 * G - C1 * DTH1 - C2 * TH1         DDX = HPL * DDTH / G        SUM = SUM + (DDX) ^ 2         DTH = DTH1 : TH = TH1 Next J     RMS = Sqr(SUM / N)     If RMS < RMSXP Then GoTo 1     LOW = Xm - DXM: HIGH = Xm: DXM = DXM / 10     If DXM < 0.00001 Then     Sheet1.Cells(IJ, "F") = A     Sheet1.Cells(IJ, "G") = Round(Xm, 2)      IJ = IJ + 1     GoTo 2     End If     GoTo 3 1   Next Xm 2 Next A     Sheet1.Cells(1, "F") = "SPRING'S LOCATION AWAY FROM AXLE_in"     Sheet1.Cells(1, "G") = "MASS'S LOCATION AWAY FROM AXLE_in"    End Sub   55  Appendix C  Excel Macros for prTable 7 Sub rmsmeasuredandcalculated() Dim Gz0, Gz1, Gr1, C4, C5, C6, C7, C8, C9, TH1, DTH1, DDTH, DTH, TH, SUMR1, SUMX1 As Variant N = 2048 K = 180: C = 5: T = 0.001: G = 386.4: HPL = 15: A = 23 SUMX1 = 0: SUMR1 = 0 Mp = 40 / G: Lp = 48: Hp = 6: Xp = 0: Yp = 0 Mm = 5 / G: Lm = 6: Hm = 0.5: Ym = 3.25: Xm = 20.5 Mass = Mp + Mm     Xg = (Mp * Xp + Mm * Xm) / Mass     Yg = (Mp * Yp + Mm * Ym) / Mass     Ip = Mp * ((Lp) ^ 2 + (Hp) ^ 2) / 12 + Mp * ((Xp - Xg) ^ 2 + (Yp - Yg) ^ 2)     Im = Mm * ((Lm) ^ 2 + (Hm) ^ 2) / 12 + Mm * ((Xm - Xg) ^ 2 + (Ym - Yg) ^ 2)     MoI = Ip + Im : Q = MoI + Mass * (Xg) ^ 2     C1 = C * (A) ^ 2 / Q: C2 = K * (A) ^ 2 / Q: C3 = -Mass * Xg / Q     OMEGA = Sqr(C2 - (C1) ^ 2 / 4)     B1 = Exp(-C1 * T / 2): B2 = Sin(OMEGA * T): B3 = Cos(OMEGA * T) TH = 0: DTH = 0 Gz010 = Sheet1.Cells(2, "C").Value SUMX2 = (HPL * C3 * Gz010) ^ 2 For I = 2 To N Gz01 = Sheet1.Cells(I, "C").Value Gz11 = Sheet1.Cells(I + 1, "C").Value C6 = Gz0 * G: C7 = (Gz11 - Gz01) * G / T C9 = C3 * C7 / C2: C8 = (C3 * C6 - C1 * C9) / C2 C5 = TH - C8: C4 = (DTH + C1 * C5 / 2 - C9) / OMEGA TH1 = B1 * (C4 * B2 + C5 * B3) + C8 + C9 * T 56  DTH1 = B1 * (-(OMEGA * C5 + C1 * C4 / 2) * B2 + (OMEGA * C4 - C1 * C5 / 2) * B3) + C9 DDTH = C3 * Gz11 * G - C1 * DTH1 - C2 * TH1 DDX = HPL * DDTH / G    SUMX1 = SUMX1 + (DDX) ^ 2  DTH = DTH1 TH = TH1 Next I For J = 2 To N + 1 Gr1 = Sheet1.Cells(J, "B").Value SUMR1 = SUMR1 + (Gr1) ^ 2  Next J RMSx1 = Sqr(SUMX1 / N) RMSr1 = Sqr(SUMR1 / N) Sheet1.Cells(2, "F").Value = "Predicted lateral RMS= " & Round(RMSx1, 4) Sheet1.Cells(3, "F").Value = "Recorded lateral RMS= " & Round(RMSr1, 4) End Sub   57  Appendix D  related to Table 8 Sub variousRMS3GB() Dim I As Integer Dim Gx, Gz, DDTHETA, Gzp As Variant Dim MM, MASS, XG, YG, IP, IM, IL, MOI, Q, C1, C2, C3, R, OMEG As Variant Dim TH, DTH, SUM, Gz0, Gx0, Gz1, Gx1, DDTHETA0, DDTHETA1, Gzp0, Gzp1 As Variant Dim C6, C7, C9, C8, C5, C4, Z1, Z2, Z3, DDTH, Gxp As Variant D = 30: HTR = 36: HPL = 15: N = 3500: T = 0.001: G = 386.4 C = 5: J = 2: K = 500  MP = 40 / G: LP = 48: HP = 6: XP = 0: YP = 0 MM = 20 / G: LM = 8: HM = 2: YM = 4: 'Xm = 20 ML = 48 / G: LL = 12: HL = 36: XL = 0: YL = 18 MASS = MP + MM + ML For A = 2 To 23 Step 4      For Xm = 6 To 20 Step 4      XG = (MP * XP + MM * Xm + ML * XL) / MASS     YG = (MP * YP + MM * YM + ML * YL) / MASS     IP = MP * ((LP) ^ 2 + (HP) ^ 2) / 12 + MP * ((XP - XG) ^ 2 + (YP - YG) ^ 2)     IM = MM * ((LM) ^ 2 + (HM) ^ 2) / 12 + MM * ((Xm - XG) ^ 2 + (YM - YG) ^ 2)     IL = ML * ((LL) ^ 2 + (HL) ^ 2) / 12 + ML * ((XL - XG) ^ 2 + (YL - YG) ^ 2)     MOI = IP + IM + IL: Q = MOI + MASS * (XG) ^ 2     C1 = C * (A) ^ 2 / Q: C2 = K * (A) ^ 2 / Q: C3 = -MASS * XG / Q     R = C1 / 2 / Sqr(C2): OMEG = Sqr(C2 * (1 - (R) ^ 2))     TH = 0: DTH = 0     Gx00 = Sheet3.Cells(1, "a").Value     Gz00 = Sheet3.Cells(1, "b").Value     DDTHETA00 = -Gx00 * G / HTR     Gzp00 = Gz00 + D * DDTHETA00 / G 58      SUM = (HPL * C3 * Gzp00) ^ 2         For I = 1 To N - 1         Gx0 = Sheet3.Cells(I, "a").Value: Gx1 = Sheet3.Cells(I + 1, "a").Value         Gz0 = Sheet3.Cells(I, "b").Value: Gz1 = Sheet3.Cells(I + 1, "b").Value         DDTHETA0 = -Gx0 * G / HTR: DDTHETA1 = -Gx1 * G / HTR         Gzp0 = Gz0 + D * DDTHETA0 / G: Gzp1 = Gz1 + D * DDTHETA1 / G         C6 = Gzp0 * G: C7 = (Gzp1 - Gzp0) * G / T         C9 = C3 * C7 / C2: C8 = (C3 * C6 - C1 * C9) / C2         C5 = TH - C8: C4 = (DTH + C1 * C5 / 2 - C9) / OMEG         Z1 = Exp(-C1 * T / 2): Z2 = Sin(OMEG * T): Z3 = Cos(OMEG * T)         TH = Z1 * (C4 * Z2 + C5 * Z3) + C8 + C9 * T         DTH = Z1 * (-(OMEG * C5 + C1 * C4 / 2) * Z2 + (OMEG * C4 - C1 * C5 / 2) * Z3) + C9         DDTH = C3 * Gzp1 * G - C1 * DTH - C2 * TH         Gxp = HPL * DDTH / G         SUM = SUM + (Gxp) ^ 2         Next I     RMS = Sqr(SUM / N)     Sheet3.Cells(J, "D").Value = A     Sheet3.Cells(J, "E").Value = Xm     Sheet3.Cells(J, "F").Value = Round(RMS, 4)     J = J + 1     Next Xm Next A End Sub59  REFERENCES60  REFERENCES  Archer, R. 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