Proceedings of RIT 2013 Vibrations Technical Conference RIT/VTC 2013 December 16, 2013, Rochester, NY, USA
VTC2013-1 ECCENTRIC MASS DYNAMIC VIBRATION ABSORBER Timothy G. Southerton, Brian T. Grosso, Kyle J. Lasher Kate Gleason College of Engineering Department of Mechanical Engineering Rochester Institute of Technology Rochester, New York, 14623 Email: tgs5800@ rit.edu,
[email protected],
[email protected]
ABSTRACT A physical model used to demonstrate the u se of a passive dynamic vibration absorber (DVA) to cancel out resonance of a classic mass spring damper system was developed for this project. It incorporates an eccentric mass as a rotational unbalance to generate movement with two degrees of freedom (DOF). The excitation frequency can be manually controlled to demonstrate discussed phenomena of a two mass system. In this paper, a vibratory model of a mass-spring-damper system is developed and compared with experimental results measured from the physical model. From this it is concluded that the theoretical model accurately reflects the real world results with minimal deviations. These differences are due to broad assumptions made in the theoretical development, so we can conclude that the model is valid.
INTRODUCTION Dynamic vibration absorbers are very fundamental devices when it comes to vibration management in systems, and their effects are interesting to even those outside of the field of engineering. The concept behind behind these passive passive components is simply to add a spring and mass that have a natural frequency tuned to that of the resonant excitation frequency of the system. Doing so transfers all of the resonance energy of the system to the DVA, leaving the original system undisturbed. However, the classic model of the vertical 2DOF springmass-damper system with a DVA is not widely available to demonstrate this phenomenon, so this project targeted creating this system with low-cost components as a demonstration for the Introduction to Engineering Vibrations class. Due to the the availability of electronic components from other coursework, it was decided that an electric motor with an unbalanced mass is the most effective way to generate a periodic applied force in the system so that the excitation frequency can be controlled. The biggest challenge in making a physical model of this 2DOF system is constraining the motion, for which there are a
variety of solutions ranging ranging widely in price. For this we chose an available middle ground that allowed for reasonable data collection, but significant improvements could be made to the experimental setup by using linear bearings and precision mounted slides. This paper focuses on comparing the theoretical results of the methods developed in class for a 2DOF DVA model to the data measured from the physical model to identify the validity of the development. Budget and time constrained fabrication and testing, but we were able to develop a reasonably robust first prototype of this system and to record relatively accurate displacement amplitudes amplitudes for comparison. comparison. There is much that could be done to this design in the future to further characterize or improve the system performance, but this is outside of the scope we have developed.
PHYSICAL MODEL For this project, a physical model of the system was developed using available materials, as can be seen in Fig. 1. The frame is 12" x 21.5" and is made from a 1x2x8 furring strip [1]. A Jameco ReliaPro ReliaPro 161382 geared motor that that was salvaged from a previous project is used to provide the periodic input force [2]. The eccentric and DVA masses are machined machined pieces of brass stock which we were given for free fro m the RIT machine shop. For the platform a machined machined aluminum bar is used that has four #8-32 clearance holes drilled at each corner, in which eye bolts are mounted [3]. A tapped #8-32 hole on the bottom in the center of the platform provides the mounting location for the eye bolt from which the DVA is suspended. Motor clamping to the platform is accomplished using two 2" #8-32 cap screws from the machine shop in tapped holes. A Lexan motor mount mount was made and sanded to give a frosted texture after being recycled from a previous project. The DVA mass is constrained similarly using a Lexan piece with two 5/16" holes for the motion constraining rods , and is connected to the DVA mass using a #8-32 eye bolt. 1/4" steel tubes that are mounted through the furring strips at the top
and bottom of the frame are used as constraining rods for system motion. To reduce pitching of the platform, 2" pieces of 3/8" copper tubing were secured through holes in the platform using adhesive. These work as minimal friction sli des on the steel constraining bars which do not not bind. Four extension springs are attached to eye hooks in the top of the frame and are used used to hang the platform [4, 5]. A spring that is slightly less stiff is used to hang the DVA from the platform. The electrical drive for the motor is a custom-built voltage regulator circuit connected to the motor with very light speaker wire to reduce unwanted constraints on the system motion. This unregulated DC power supply is 12VDC and 600mA, which feeds a LM317 regulator with heat sink through the circuit given in the datasheet. The regulator supplies from 1.25 to 11.75V to the motor with up to 1.5A of current [6].
Spring Stiffness 1.4
Spring
1.2
LB
)1.0 N ( e0.8 c r o0.6 F
LF RF RB
0.4
DVA
0.2 0.0 0.00
0.05
0.10
0.15
Spring Stretch (m)
FIGURE 3. SPRING STIFFNESS PLOTS MODEL DEVELOPMENT
FIGURE 1. DAMPER TEST STAND
FIGURE 4. SYSTEM SCHEMATIC
SYSTEM CHARACTERIZATION In order to characterize our model, data was taken on the system components using a triple beam balance from the Systems lab along with hanging masses. Every spring was measured individually using four mass increments so that trend lines could could be fit to calculate each stiffness value. Each component used for the platform and DVA assemblies was also weighed individually. The resulting system characterization values can be seen in Fig. 2. Spring stiffness graphs proved proved reasonably similar and linear, which can be seen in Fig. 3. Variable m1 m2 m0 e k LB LB k LF LF k RF RF k RB RB k 1 k 2
Item Platform mass Absorber mass Eccentric mass Eccentricity Left back spring stiffness Left front spring stiffness Right back spring stiffness Right back spring stiffness Eq. platform spring stiffness Absorber spring stiffness
Value 361.5 82.8 68.5 22.8 9.95 9.64 9.59 9.68 38.86 8.91
Units g g g mm N/m N/m N/m N/m N/m N/m
FIGURE 2. SYSTEM CHARACTERIZATION VALUES
Assumptions
1 Degree of motion, 2 degree of freedom system No platform pitch, roll, yaw, horizontal o motion etc... Frictionless, undamped motion Motor spins at a constant speed Ignore effect of gravity on spinning mass o Linear range of spring operation Not bottoming out or over-stretching o Frame is rigid and does not vibrate
From the system schematic in Fig. 4 and using the assumptions above, the system equation in Eq. 1 can be derived, which characterizes the vibratory system [7].
sin (1) 0 02 ̈̈ 2 22 22 2 = sin 0 Using the complex method, the global frequency response function (FRF) can be calculated directly, as in Eq. 2 and 3.
Η = [2Μ Κ]− = − 2 = 2 2 2 2 2 2 2 2 2 2 (2) 2 2 2 22 2 22 (3) Δ = = 2 2 The amplitude of the displacements of the platform and DVA predicted by this model can be found directly from the global FRF as in Eq. 4 and 5.
|2 = || = || +)∗ = −(− − +− ++− |2 2 = |2| = |2| 2 = −++∗− −+ −
(4)
(5)
(6)
(7)
For comparison purposes, the system can be modeled without the DVA, which is a simple 1DOF rotating unbalance system with no no damping. The resulting displacement amplitude response equation with respect to excitation frequency can be seen in Eq. 8, with r = ω / ω n.
∗ = − ∗
20 ) 15 s / d 10 a r ( ω
5
0 y = 1.7869x 1.7869x - 0.6166 R² = 0.9995
Also worth calculating directly are the system natural frequencies, which can also be established graphically from the global FRF. From Fig. 2, the platform and absorber natural frequencies are calculated using Eq. 6 and 7, respectively.
22 = √ = 10.4 rad/s
ω
0
These equations can be used to produce the displacement amplitude response with respect to excitation frequency for the vibratory system.
= √ = 10.4 rad/s
Motor Voltage Voltage vs. vs . 25
(8)
5
10
15
Applied Voltage (V)
FIGURE 5. MOTOR VOLTAGE VS. ROTATIONAL SPEED PLOT Static deflections of the platform alone, the platform with the DVA, and the DVA were first recorded as reference points. The voltage was then first swept over the range of available values (1.25-11.75V) in increments and the displacement amplitude of the platform alone were taken by holding a ruler parallel to the base and stopping it at the t he point where the mass stopped contacting, which was marked and measured on the frame. This provided the experimental reference data for the 1DOF system that we attempted to stop at resonance. The DVA was then connected and the process was repeated, recording both platform and DVA amplitudes. Multiple points where then taken later at voltage values around the points of resonance to provide more clarity for the system's real world response.
RESULTS COMPARISON Data collected was plotted in MATLAB along with theoretical model displacement amplitude response with respect to excitation frequency using the MATLAB code found in Appendix A of this paper and the recorded Excel data found in Appendix B. The resulting platform displacement amplitude amplitude plot of the experimental vs. theoretical results for the 1DOF system overlaid on the results for the 2DOF system can be seen in Fig. 6. Similarly, the resulting DVA displacement displacement amplitude plot of the experimental vs. theoretical results for the 2DOF system can be seen in Fig. 7.
DATA COLLECTION Using the access hatch on the regulator enclosure, the wires for the motor were disconnected from the power connector and a voltmeter was hooked up in parallel with the motor to record voltage values while testing. Rotational speeds in rev/s were calculated from time values taken by using a stopwatch to time 20 revolutions of the eccentric mass at lower rotational speeds. Videos were taken taken of the system at higher rotational speeds and the time taken for each revolution of the mass was established by stepping frame-by-frame through the video and and averaging three sample sample revolutions. These sample sample points were used to establish a voltage vs. rotational speed trend from which the excitation frequency was calculated for different voltage values. This can be seen in Fig. 5. FIGURE 6. DISPLACEMENT AMPLITUDE RESPONSE VS. FREQUENCY FOR SYSTEM PLATFORM
stiffnesses. The rig also exhibits some vibratory effects effects as it is not perfectly rigid, which is also a budgetary constraint and adds to error in the experimental values. However, we can see from the results comparison that the accumulation of errors due to the differences between the real world model and our assumptions in the theoretical model still produce favorable agreement. From this we can conclude t hat the physical model is an accurate representation of the system targeted, and that the theoretical model produces valid results.
FIGURE 7. DISPLACEMENT AMPLITUDE RESPONSE VS. FREQUENCY FOR SYSTEM DVA CONCLUSIONS As can be seen in Fig. 6, the experimental data for the platform matches very well with the t heoretical calculations for the 1DOF system. We were limited in the precision of the excitation frequency adjustment due to the potentiometer used, so it is reasonable to assume that the experimental and theoretical values are the same at the natural frequency of the platform (10.4 rad/s). For higher excitati on frequencies in the 1DOF system, the displacement measured is slightly lower than predicted, which we attribute to errors in our amplitude measurements. However, the amplitude values still trend as predicted by the model. Similarly, at low excitation frequencies, the motion was so small that we could not measure movement accurately, but it was very nearly zero as predicted. For the 2DOF results in Fig. 6, the same effects as noted in the 1DOF system system can be seen at the the new resonances. resonances. Notably here, at the second resonance the motion of the system was so chaotic that the DVA would sometimes hit the bottom of the rig and completely unload unload the DVA spring. These sort of effects led to the use of estimated values for the platform and DVA amplitudes at the second natural frequency, which can be seen as outliers in the figures. For the frequency at which which 1DOF resonance occurs, the 2DOF experimental setup showed the predicted attenuation. At the first 2DOF natural frequency, the resonance amplitude is small due mainly to the variable speed of the motor. This variable speed is due to the increased torque required to raise the eccentric mass vs. lowering it, which is more notable at lower excitation frequencies (when there is less rotational inertia). From Fig. 7, we can see that the discussed effects on the platform are translated directly to the DVA, which shows the same trends. The experimental results compare favorably with the model for the DVA amplitudes, which adds even more merit to the validity of the model. Overall, the experimental results were influenced by the unavoidable existence of friction (or dissipative losses) in the system and the ability of the platform to pitch from side to side. This pitching is due to wide clearances between the copper sleeves and the steel rods to prevent binding, which was a budgetary constraint for the project. The rig also seems to pitch more on the left side of the platform than the right, which could be due to positioning of the motor or slight differences in spring
REFERENCES [1] 1 x 2 x 8 Furring Strip. (n.d.). www.homedepot.com. www.homedepot.com . Retrieved December 4, 2013, from http://www.homedepot.com/p/Unbranded-1-x-2-x-8Furring-Strip-160954/100075477?cm_sp=BazVoice-_RLP-_-100075477-_-x#.Up_LrcRDuSo [2] Jameco Electronics. (n.d.). Jameco ReliaPro. ReliaPro. Retrieved December 4, 2013, from http://www.jameco.com/webapp/ wcs/stores/servlet/Product_10001_10001_161382_-1 [3] #8-32 x 1-5/8 in. Zinc-Plated Steel Eye Bolts with Nuts (2Pack). (n.d.). www.homedepot.com. www.homedepot.com. Retrieved December 4, 2013, from http://www.homedepot.com/p/Unbranded-832-x-1-5-8-in-Zinc-Plated-Steel-Eye-Bolts-with-Nuts-2Pack-14331/202704409#.Up_MocRDuSo [4] Everbilt 5/32 in. x 2-1/2 in. and 1/4 in. x 2-1/2 in. Zinc Plated Extension Springs (4-Pack). (n.d.). www.homedepot.com. www.homedepot.com . Retrieved December 4, 2013, from http://www.homedepot.com/p/Everbilt-5-32-in-x-2-1-2-inand-1-4-in-x-2-1-2-in-Zinc-Plated-Extension-Springs-4Pack-16080/202045471?keyword=760804# .Up_NnsRDuSo [5] #6 Zinc-Plated Screw Hooks (25-Pack). (n.d.). www.homedepot.com. www.homedepot.com . Retrieved December 4, 2013, from http://www.homedepot.com/p/Unbranded-6-Zinc-PlatedScrew-Hooks-25-Pack14092/100338097#.Up_OE8RDuSo [6] KA317 / LM317 3-Terminal Positive Adjustable Regulator. (n.d.). Fairchild Semiconductor . Retrieved December 4, 2013, from http://www.fairchildsemi.com/ds/LM/LM317.pdf [7] Inman, D. J. (2014). Engineering Vibration (4 ed.). Upper Saddle River, N.J.: Pearson Education, Inc.
Appendix A - MATLAB Code function UnbalancedDynamicAbsorber function UnbalancedDynamicAbsorber close all oneDOF = xlsread('C:\Users\Tim\Documents\Dropbox\Vibes xlsread('C:\Users\Tim\Documents\Dropbox\Vibes Project\vibsproject2.xlsx', Project\vibsproject2.xlsx' ,'1DOF' '1DOF'); ); twoDOF = xlsread('C:\Users\Tim\Documents\Dropbox\Vibes xlsread('C:\Users\Tim\Documents\Dropbox\Vibes Project\vibsproject2.xlsx', Project\vibsproject2.xlsx' ,'2DOF' '2DOF'); ); k1 = 38.86; %%(N/m) four spings in parallel k2 = 8.91; %%(N/m) m1 = 0.3615; %%(kg) m2 = 0.0828; %%(kg) M = [m1 0;0 m2]; K = [k1+k2 -k2;-k2 k2]; m0 = 0.0685; %%unbalanced mass(kg) ecc = 0.0228; %%eccentricity(m) w = linspace(0,30,1000); h(1000,3) = 0; x(1000,3) = 0; for i for i = 1:1000 H = (-w(i)^2*M+K)^-1;%%frequency (-w(i)^2*M+K)^-1;%%frequency response h(i,1) = H(1,1); h(i,2) = H(1,2); h(i,3) = H(2,2); f = [m0*ecc*w(i)^2;0];%%unbalanced [m0*ecc*w(i)^2;0];%%unbalanced force X = H*f; x(i,1) = X(1);%%x_1 X(1);%%x_1 x(i,2) = X(2);%%x_2 X(2);%%x_2 r = w(i)/sqrt(k1/m1); x(i,3) = r^2/abs(1-r^2)*m0*ecc/m1;%%1DOF r^2/abs(1-r^2)*m0*ecc/m1;%%1DOF end %%Platform displacement figure; hold on plot(w,abs(x(:,3)),'r-' plot(w,abs(x(:,3)), 'r-',oneDOF(:,1),oneDOF(:,2), ,oneDOF(:,1),oneDOF(:,2),'kO-' 'kO-',w,abs(x(:,1)), ,w,abs(x(:,1)),'b'b',twoDOF(:,1),twoDOF(:,2), ,twoDOF(:,1),twoDOF(:,2),'gO-' 'gO-') ) axis([0 20 -0.01 0.08])%%for 0.08])%%for x-axis line([0 20],[0 0],'Color' 0],'Color',[0 ,[0 0 0]) legend('Theo. legend('Theo. 1DOF', 1DOF','Exp. 1DOF', 1DOF','Theo. 2DOF', 2DOF','Exp. 2DOF', 2DOF' ,'Location' 'Location', ,'NorthWest' 'NorthWest') ) title('Platform title('Platform Displacement Amplitude (x_1_1)') (x_1_1)') xlabel('Excitation xlabel('Excitation Frequency (rad/s)'); (rad/s)'); ylabel('Displacement ylabel('Displacement Amplitude (m)'); (m)'); hold off %%Absorber displacement figure; hold on plot(w,abs(x(:,2)),twoDOF(:,1),twoDOF(:,3), 'O-' 'O-') ) axis([0 20 -0.01 0.08]) line([0 20],[0 0],'Color' 0],'Color',[0 ,[0 0 0])%%for 0])%%for x-axis legend('Theo.' legend('Theo.', ,'Exp.' 'Exp.', ,'Location' 'Location', ,'NorthWest' 'NorthWest') ) title('Absorber title('Absorber Displacement Amplitude (x_2_1)') (x_2_1)') xlabel('Excitation xlabel('Excitation Frequency (rad/s)'); (rad/s)'); ylabel('Displacement ylabel('Displacement Amplitude(m)'); Amplitude(m)'); hold off end
Appendix B - Excel Data
