lectrieal Analo
udying R. B. STAMBAUGH Goodyear Tire & Rubber Co., Akron, Ohio
N ORDER to explain the complicated elastic and plastic behavior of rubber and other long-chain polymers, i t is often helpful t o design a mechanical model of the material, composed of a number of purely elastic springs and purely viscous hydraulic damping devices or “dashpots” which can be analyzed mathematically. Thus, a mathematical expression can be derived which will describe the motion of t8hematerial under given conditions.
FORCE RUBBER
A simple way to avoid these mechanical difficulties is to make use of the analogy between electrical and mechanical systems. The model can be built u p using the analogous electrical components instead of the mechanical ones, and the performance observed when appropriate electrical “forces” are applied. Not only can the electrical model be assembled easily, but the values of the components can be changed by simply throwing switches or turning knobs. In addition, the electrical system offers a very flexible time scale. Thus, either very slow or very fast deformations, by adjustment of the time scale, can be made to occur for the model in time intervals convenient for observation. The possible range of time scales is large without undue difficulty in selecting model components. THE ELECTRICAL ANALOGY
An analogy exists between mechanical and electrical quantities because there is mathematical similarity between the relations of the quantities of each system. This is immediately apparent from the equations of motion given in Equations 1 and 2 which describe the behavior of the mechanical and electrical oscillators shown in Figure 2. For the mechanical case, the equation of motion is
S2
D
E
d2X dx x M y + R - + - = F dt dt c
Figure 1. Mechanical Models of Rubber
(1)
and for the elect’ricalcase, it is The simplest model of a piece of rubber is a spring as drawn in Figure 1, B. The spring deflects when the force, F , is applied just as the rubber sample is compressed when the weight, TP, is placed upon it. To account for the plastic flow occurring in the compressed rubber, a dashpot, such as a loosely fitting piston in a cylinder containing a viscous oil, is added t o the spring model (Figure 1, C). To illustrate the recovery of the rubber after removal of the compressing force, the spring and dashpot are paralleled with another spring (Figure 1, D). The bar connecting the two branches of the model is rigid and remains horizontal. The irrecoverable flow, or permanent set, can be approximated by a very viscous dashpot in series v i t h the second spring, as in Figure 1, E . The spring and dashpot combination in series, commonly called a blaxwell unit, has been used bymany authors ( 1 ) todesign complex models which give force-deflection curves similar t o those obtained with rubber, plastirs, or textiles, as the case may be. T h i l e a fair representation of the true behavior of a material may often be given by a simple model, a more accurate represenbation requires the addition of further elements t o the model until i t becomes rather difficult to handle. The assigning of definite values t o the springs and dashpots also grows more tedious as the complexity of the model increases. The testing of a model may be done by substituting definite values in the mathematical equation of motion for the model and plotting a curve of the deflection versus time. This is time consuming and the forces applied must be limited t o simple, ideal cases. I t would be more satisfactory t o actually build the model and subject it to the same forces as the original material undergoes, but this presents mechanical diffimlties which may become as serious as the mathematical ones.
L, d2Q + R dQ - f - Q dt
dt
C
=
E
(2)
The equations are mathematically identical, and a solution of one becomes a solution of the other by simply changing to the corresponding units. B
A
s
D
Figure 2.
Mechanical Oscillator A . Spring B . Condenser
The analogy means that, just as the force deflects the spring in Figure 2, A , a voltage puts a charge on the condenser of Figure 2, B . The inductance provides electrical inertia as the mass does mechanically. If the force is removed from the mechanical system, the spring continues to oscillate a t the resonant frequency of the system until the energy is dissipated by the damping action of 1590
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INDUSTRIAL AND ENGINEERING CHEMISTRY
the dashpot. Similarly, when the applied voltage is reduced t o zero, the charge in t h e slectrical system continues t o flow from one side of the capacitor through the inductance t o the other side of the capacitor and back until the energy is dissipated as heat in the resistor. [CALIBRATING
1
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Ordinarily, this has no practical significance and the power, time, and impedance ratios may be adjusted t o the most convenient values as long as circuit Q remains the same. Thus, in the third column, ratios have been adjusted to give electrical components which are easily available. The consistency of the selected scale values may be checked by substituting in simple formulas which must hold for all conditions, such as Ohm’s law, etc. APPARATUS
tI I.ODELCJ U
AMPLlF I ER
/ ,
I I / DISPLAY
Figure 3.
1 SCREEN
Block Diagram of Apparatus
Thus, any mechanical model designed to reproduce the physical characteristics of a material can be duplicated by an electrical model in which forces or stresses are replaced by voltages, and deflections or strains by charges. There are other mechanicalelectrical analogies (8). For example, in one system, mechanical force is replaced by electrical current and displacement is the integral of the voltage over the time interval. There are a number of arguments in favor of this second analogy, a t least from the mathematical point of view, but the first has been selected as preferable for the present work because of the ease in representing forces by voltages on an oscilloscope screen, and because capacitors of low loss are easier t o obtain than inductors for representing springs. As an illustrative example of the design of an electrical model, consider the Goodyear Vibrotester ( 7 ) where a mass is held in position by two small rubber pellets and forced t o vibrate by means of a dynamic loudspeaker coil. This would be approximated by a mechanical model like the one of Figure 2, A . The compliance of the spring is the reciprocal of the modulus of the rubber stock corrected for the size and shape of the samples in the vibrotester. If a shape factor, p, is defined as the thickness divided by the area, compliance, c, equals plmodulus. Similarly, the resistance of t h e dashpot is obtained by dividing the internal friction, q, in poises by the shape factor: R = q / q . The mass, of course, is t h e same as actually used in the tester. T o change the model t o the analogous electrical one, it is necessary t o pick a convenient ratio of values. In Table I, the components of the mechanical system are listed in the first column for a typical rubber stock having a modulus of 60 kg. per square centimeter and an internal friction of 15 kilopoises, the shape factor being 0.5. The second column gives the corresponding components of an electrical model where 1 volt was arbitrarily given the value of 3160 dynes. This model has the same time and power scales as the mechanical one, so that lo7 ergs per second in the mechanical case correspond t o 1 watt in the electrical model.
Time ratio Power ratio Impedance ratio Mass Resonant frequency Compliance Resistance Force Resonant amplitude Resonant velocity
The equipment necessary t o study electrical models of polymers consists of a “force” generator which applies the desired voltage t o the model and an integrating amplifier which provides a signal proportional t o the charge in the model by integrating the current flow into it. This output voltage, being the “displacement” of the model, can be observed on a cathode ray oscilloscope screen or recorded graphically with a suitable instrument. The apparatus will vary in cost and complexity depending upon the precision desired and variety of conditions to be studied.
Figure 4.
Photograph of Apparatus
A block diagram of the electrical model analyzer used tn the present study is shown in Figure 3 and a photograph in Figure 4. The force generator consists of a number of motor-driven camactuated microswitches which can be arranged to produce step functions or a series of square waves according t o the type of force desired. A constantly rising force function may be generated by charging a capacitor through a constant-current pentode. Sinusoidal forces are obtained with a commercial, resistancetuned, variable frequency oscillator. The magnitude of the force voltage is adjusted t o the desired value by means of a direct current amplifier and this signal is applied to the model through a cathode-follower circuit employing a Type 3C33 tube.
The model is set up on a panel containing capacitors, inductors, and resistors in decade-switching arrangements. Continuously adjustable rheostats are also provided, the resistance of any setting being checked by a built-in Wheatstone bridge. External components may be plugged into the panel when necessary. The current which flows into the model upon application of the force voltacre uasses through a small. calibrated resistance causing a voltage drop proportional to the “velocity of motion” of the model. This voltage is applied to the input terminals of an TABLE I. MODELCOMPONENTS integrating amplifier which produces a voltage Electrical Analog Models Mechanical Model proportional to the time integral of the current 1:l 1:1 flowing into the model, which is, in turn, pro1:l 1:O.l .. 1:l 1 :o. 001 portional t o the electrical charge or mechanical 827 grams 827 henries 0.827 henrie displacement of the model. The amplifier is simBO cps. BOO s 60 cps. 0.0085 X 10-6 cm./dyne 0 . O&SPf. 8.5~f. ilar to analog computer integrating units which 30,000dyne-sec./cm. 30,000 ohms 30 ohms have been described in the literature (3). The 106 dynes 316 volts 3.16 volts voltage gain is adjustable from 1 t o 10 in unit 0.0885 om. 2 7 . 9 7 ~coulomb 279.79 coulomb 33.3 cm./sec. 0.01054amp. 0.1084amp. steps and can be used as either an integrator or multiplier by switching from capacitive to
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resistive feedback so that either the displacement or the velocity of the model can be studied. The voltage output of this amplifier is observed by means of a cathode ray oscilloscope with a 12-inch, long persistance-type (P7).screen. St the same time bhe amplifier signal is producing a vertical deflection on the screen, a “sweep” voltage, rising linearly with time, is applied to the horizontal deflection plates of the oscilloscope so that the display observed is a plot of the motion of the model as a function of time. The sweep generator also produces a voltage rising as the logarithm of the time (over the range of 0.01 t o 1 second) which is useful for observing creep curves.
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of IO7 so that the 1000 hours required only 0.36 second to display. The initial application of the voltage charges the condensers in series, the effective capacity of the system being equivalent to 2.28fif. The current f l o through ~ Rz gradually discharges CZuntil at the end of the cycle, the charge is held on C1 with a nearly constant leakage of current through R1. h 90.5-ohm resistor mas used to generate a voltage for the integiating amplifier. A photograph of the oscilloscope screen when the model was excited by a step function of 341 volts is shown in Figure 6. The beam intensity of the oscilloscope was modulated so that when the logarithmic sweep was used, a nearly constant light output was maintained to prevent overevposure at the slowly moving part of the trace. A plot on semilogarithmic paper of the actual deflection and creep curves of the GR-S stock, together n-ith the performance of the model taken from the photograph of Figure 6, is shoivn in Figure 7. The model deviates from the actual performance of the stock a t first because the model contains only t n o sections. Better results can be obtained by adding a third or fourth section. EQUIViLENT MODELS
Figure 5.
1Mechanical Model of GR-S in Creep with Electrical Analog
The force voltage may be applied to the horizontal deflection system instead of the s a eep voltage, thus giving the stress-strain characteristics of the model directly. In order t o synchronize the force with the sweep and to provide a continuous display on the oscilloscope, a timing cam is provided with a variable speed motor drive. The switches driven by the cams “zero” the integrating amplifier, discharge the model condensers, and start the force and sweep generators in the proper sequence. After the display, they shut off the devices and bring the circuits back t o their original condition and then repeat the cycle. The cycle is usually about ‘/2 second long although it may be second to several seconds. varied from The direct current deflection amplifiers for the oscilloscope have a variable gain control n-hich is adjusted t o give a satisfactory scale f6r the display. This gain setting is held fixed for any series of displays to be conipared with one another, and changes in gain nhich are necessary are made by means of the calibrated gain switch in the integrating amplifier. Over-all calibration of the system is made by switching in a generator which provides a series of square waves of accurately known voltages. The time base of the display is calibrated by a time-marker generator. A 5-inch oscilloscope is included in the assembly which may be switched into the important points of the various circuits as a check on their operation. I t is normally used to monitor the force gmerator output. TYPICAL MODEL
A GR-S stock having a modulus of 31.6 kg. per square centimeter, when placed in a creep-test apparatus, elongated 195% when a force of 1107 grams was applied ( 4 ) . The bench marks, originally 1.27 cm. apart, moved to 3.75 em. apart, showing a deflection of 2.48 cm. %hen the weight was added. After 1000 hours, the marks were 4.69 em. apart, showing a 3.42 em. deflection and 38% creep. An approximate model of this stock is shown in Figure 5 with the electrical analog. For the original deflection, the two springs, SI and Sz, act as a single spring of compliance equal to 2.285 X 10-6 cm. per dyne because of the high viscosity of the dashpots. A certain rate of creep occurs immediately, primarily because of the more fluid dashpot, Dz,but as Ss relaxes, the deflection is determined mainly by S1 with a final slow creep caused by DL. The electrical model used had the time scale reduced by a factor
The model of GR-S described was made up of two Maxxeell units in parallel. A generalized mechanical model of this type is shown in Figure 8, A with the electrical analog having the resistance and capacity values as shown. An equivalent model can be made using sections consisting of a spring in parallel with a dashpot (T‘oigt units), the sections being arranged in series, as shown in Figure 8 , B with the electrical analog. The mathematical equivalence of these models has been shown by several authors (3$6) and the values of the components of one type of model bear
‘100 H O U R S
TIME Figure 6.
Creep of GR-S 3Iodef
an algebraic relation to the components in the other. Thus the T’oigt model of the GR-S stock described above appears as shown in Figure 8, C. When this model is set up on the analyzer, the same curve as shown in Figure 6 is obtained. NONLIREAR MODELS
The modulus and internal friction of rubber and similar materials are functions of displacement and time because of temperature, thixotropic, and molecular orientation effects. The linear mechanical models proposed for these substances are therefore useful only for limited ranges of deformation and power input,. Models containing nonlinear elements have been proposed ( 5 ) which describe the performance of certain matrrials for very high deformations. While there are a number of difficulties involved
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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I 4r’
I
OR-S TREAD
I
I - 40
DEFLECTION
0
ACTUAL STOCK3 lo MODEL---
I Figure 7 .
“TIME-HOURS
loo
I
1000
Deflection and Creep Curves for GR-S Stock and Model
in building and testing nonlinear models, either mechanical or electrical, there are some nonlinear devices available for such an electrical model, such as thermistors, rectifiers, ferro-electric capacitors, etc. Vacuum tubes can sometimes be used if their rectifying action is not undesirable. Figure 9 shows the stressstrain curve of an electrical model reproducing the familiar increase in modulus with elongation observed in rubber. T o get this effect, a vacuum tube is arranged so that the plate current is biased off as the voltage rises, reducing the capacity or compliance. A linear model can be used to demonstrate the increme in modulus with rate of application of force observed id rubber, such as the two-time constant model of Figure 10, A . The stress-strain curve obtained slowly is shown in Figure 10, B, curve 1. Curves 2 and 3 are obtained by raising the force more rapidly.
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If the mechanical model is a single Maxwell element, the spring and dashpot being in series, t h e analog requires the resistor t o be in parallel with the capacitor and the losses are of the constant leakage type. R. is replaced by 1/Rpwo2C2, and &a becomes ECzR,wo, which is proportional t o the resonant frequency. The resonant current will rise as the square of the resonant frequency. These cases are pictured in Figure 11 The photograph marked velocity at the top of Figure 11was obtained using a 5-pf. capacitor with a 250-ohm resistor in series. Four different inductors (2.4, 1.2, 0.6, and 0.3 henries) were used t o give resonant frequencies a t 46,65,92, and 130 cycles per second, respectively. The photograph a t the t o p marked deflection also shows current resonance curves but the series resistance was varied to give resonant amplitudes inversely proportional to the resonant frequency as would be obtained if the actual displacement were observed in the case of constant series resistance. Similarly, the photograph marked velocity at the bottom of Figure 11 was obtained using a constant parallel resistance of 960 ohms, and this shows a resonant current proportional t o W O ~ . The resonance values in the photograph marked deflection, although they are actually velocity resonance curves for a different model, are directly proportional t o wg as would be observed for displacement resonance in the constant leakage case.
A . MAXWELL ELEMENTS
DYNAMIC MODELS
*
Performance of models under vibratory conditions has been observed by applying an alternating current force voltage and sweeping the frequency from 20 to 200 cps. The force generator amplifies the output of a Jackson resistance-tuned oscillator and supplies about 4 volts root mean square to the model at a 60-cycle impedance of about 5 ohms. The direct current series resistance added t o the circuit is less than 0.1 ohm. A synchronous motor turns the tuning condenser of the oscillator through its full range in about 1 second. Automatic switches reverse the motor at the beginning of each sweep of the cathode ray beam across the display screen. A linear time base provides a 1to 1correspondence between the position of the spot on the screen and the pointer on the Jackson oscillator scale. Since this scale is logarithmic with frequency, the screen displays the vertical deflection as a function of the logarithm of the frequency from 20 to 200 cps. It is difficult to make an integrating amplifier which will integrate properly over this frequency range and remain stable for the duration of the slow sweep through the frequencies. For this reason the amplifier is used as a simple multiplier and the current or velocity is observed instead of electrical charge or displacement. The envelope of these sine waves is the velocity resonance curve. The equations of motion given in Equations 1 and 2 describe the models drawn in Figure 2. When F is a sinusoidal force, the amplitude a t resonance, XU,equals F/Rw, where wois the resonant frequency, or QO = E/R,wu for the electrical case with the resistance in series with the capacitor. If the resonant frequency is changed by reducing the mass or inductance, XOand QOare reduced since they are inversely proportional to 00. The resonant velocity or current remains constant with resonant frequency. This condition is equivalent t o dielectric loss in the electrical capacitor.
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C. EQUIVALENT G R - S M O D E L Figure 8.
Equivalent Models
Over a limited range of temperatures and frequencies, the internal friction of rubber is nearly inversely proportional t o the frequency (7). The resonant displacement for such a material is independent of the resonant frequency. I n a model with this characteristic, the electrical capacitor must have some dielectric and some leakage losses. Over a small range of frequencies, this condition can be approximated by letting half the losses at a certain frequency, wl,be in a series resistor, and half in a parallel resistor. Then (3) and (4)
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Vol. 44, No. 7
--pF 1
I [(IL)J / I
I
LINEAR SWEEP
I
L
ELONGATION
-
GENERATOR
Figure 9.
B
A
ELONGATION
ELONGATION
Nonlinear Moidel and Stress-Strain Curve
Figure 10.
Sti-ess-Strairi Curbes a t Different Rates of Stress 1. Elcctrical model : c u r v e i 2 and 3 ohtainpd by rnizing force niorc rapidl?
B. Curve 1 obtained sSowl\
n i t h an analog computei, thr simplicity arid dirc~ctne~s of the electric analog are coiisidried important advantages ovei the more expensive and elaborate analog computei. The apparatus is useful in drsigning and testing models of all types of polymeric materials. I'rrformance of modela can be 011served under complex forces not easily repioduced for testing the actual material. I t should prove particularly valuahlr in dewloping nonlinear models which are difficult t o build and te-t i n the mechanical system. 4CKROWLEDGRZEVT
20
CPS
Figure 11.
200
20
c
PS
200
Effect of Resonant Frequency on Amplitude and Velocity
Top, internal friction of constant series type Bottom, internal friction of conitant leakage t y p e
The photograph marked deflection in lorn-er Figure 11 shoa-s the velocity resonance obtained v i t h such a model, the series reeistance being 125 ohms and the parallel resistor 1920 ohms. The displacement resonance values would change only slightly with the resonant frequency over this range. As a rule, a different model is used t o demonstrate the results of different types of test on a particular mateiial in addition to the obvious changes in scale and component values u hich correspond to the changes in size and shape of the test pieces used in different mechanical tests. It is desirable to use as simple a model as will adequately duplicate the characteristics being studied, and thus those sections of a theoretically complete model nhich do not appreciably affect the behavior characteristic- being studied may be omitted from a practical model. For instance, vhen studying the dynamic performance of a mateiial a t 60 cps., it is not necessary to include components in the model which account for the slow creep occurring aftei hours of stress. However, if the material being investigated were highly plastic, such as an uncured rubber gum stock, the model would be designed to show the effect of plastic flow on the dynamic properties. CONCLUSIONS
Equipment for studying mechanical models of polymers through the use of the electrical analog has been described and some examples of its application have been given. While the equations of motion for such models could be solved
The author Tvishes to express his thanks to the Gootlyeai Tiie Az Rubber Co. and to H. J . Osterhof for permission t o publish thib work. This investigation IT as carried out under the uponsoiship of the Office of Rubber Reseive, Reconstruction Finance Coip., in connection with the government synthetic rubher program. NO3IENC LA1'tiHE
Mechanical Case LII = mass, grams R = resistance, dyne-seconds per centimet,er c = compliance, centimeters per dyne x = displacement, centimeters t = t,ime, seconds F = force, dynes
dx
-
dt
=
velocity, centimeters per second
Electrical Case L = inductance, henries R = resistance, ohms C = capacitance, coulombs per volt or farads Q = charge, coulombs t = time, seconds E = electromotive force, volts
9 df
=
current, coulombs per second or amperes LITER.4TURE CITED
(1) Alfrey, T ., "Mechanical Behavior of High Polymers," New York, Interscience Publishers, 1048. (2) Alfrey, T., and Doty, P., J . Applied P h w , 1 6 , 7 0 0 (1915). (3) Frost, S., Electronics, 21, N o . 7 , 116 (1948). (4) Gehman, S. D., J . d p p l i e d Phw., 19, 456 (1948). (5) Halsey, G., White, H . J.. Jr., and Eyring, H., Teztile Resenicli J . , 15, KO.9 (1945). ( 6 ) Roscoe, R., Brit. J . Applied Phm., 1, 171 (1950) ( 7 ) Stambaugh, R. B., IND. ENQ.CHEM., 34,1358 (1942). (8) Whitehead, S., J . Sci. Instruments, 21, 73 (1944). ACCEPTED JIarch 2 1 , 1952. RECEIVED for review August 7 , 1951. Presented a t the XIIth International Congress of Pure and Applied Chemistry, New York. September 1951, Contribution 184 from the research lahoratory of the Goodyear Tire & Rubber Co.
