A FORCE BALANCE TECHNIQUE FOR MEASUREMENT OF YOUNG'S MODULUS Abhinav A. Kalamdani Dept. of Instrumentation Engineering, R. V. College of Engineering, Bangalore, India. kalamdani@ieee.org Abstract: A new scheme for measurement of one of the vital characteristic of a material is been proposed here. The proposed scheme, is the force balance technique utilizing the property of a permanent magnet DC motor and its inherent relationships between, speed, torque and current. The parameters to be measured are the rotor position and the current profile. With the position and current data the Young's modulus can be measured. This strategy is presented with analysis showing the possibility of implementing in the real world. Keywords: force-balance, current, torque, position, Young's modulus. 1 Introduction A wide variety of materials are used in the design and building of the structures and objects. The process of mechanical design is highly dependent on the strengths and bending properties of the materials being used. These factors are in turn dependent on the elastic properties like Youngs modulus. Hence the knowledge of Youngs modulus is very much vital for design. There have been many methods for measurement of the Youngs modulus. One such method is been devised here using the electric machines. The stress applied to a material causes the strain. Stress and strain are directly proportional to each other in the elastic zone of the complete behaviour of the material. The behaviour is as shown in figure 1. Stress Elastic Zone Figure 1. Elastic property of a material Strain
The relation between stress and strain in elastic zone is given by equation (1). stress =modulus of elasticity (1) strain Considering the stress applied along the longitudinal axis of an object, the strain occured would be change in the length L. L F ΔL Figure. Force applied to an object If force F is applied to the object transversely a deflection of ΔL is observed along the direction of the force applied. The cross section area of the object is A and the Youngs modulus Y is given by equation (). Y = FL A L () The knowledge of force applied, cross section area, length and deflection of the object will determine the Youngs modulus. Dynamic Mechanical Model The analogous model of the mechanical system is realised with essentially two components. The inertial aspect of the object given by mass and the elasticity property given by a spring. Figure 3. Analogous mechanical model The mass is M of the block, spring constant K, force applied F and the dynamic distance moved x(t). The Newton's law gives rise to the equation Force applied = Inertial load + Opposition due to spring (3) The dynamic form the equation (3) is given in terms of F and x as
F=M d x Kx (4) dt Rearranging the equation () for force and deflection equation (5) is obtained. F= AY L L= K L (5) The relation K = AY defines the value of the spring constant included in the L mechanical model. Here the parameters A and L are the cross section and the length, describing the geometry of the object. The parameter Y is the Youngs modulus which describes the elastic property of the object. The measurement of the spring constant K will be used to determine the Youngs modulus. Including the inertial part of the equation (4) in the equation (5), the dynamic model of the elasticity of the object is obtained in equation (6) F=M d dt L AY L L (6) The force F that is applied to the object is converted into torque T and the linear motion x is converted into rotational motion θ. This conversion is done using the rack and pinion arrangement. The arrangement is chosen such that the inertia and mass of the rack and pinion together are low. The radius(r) of the pinion is known. The figure 4 shows this arrangement. F, x T, θ r Pinion Rack Object Figure 4. Force to torque conversion One side of the object is fixed to a reference position and the other is fixed to the rack. The pinion wheel drives the rack forward and backward. The equations (7) and (8) give the conversion relation for the force and linear motion. The conversion factor is mainly the radius(r) of the pinion wheel. F= T r (7)
x=r (8) These conversion relations are used in equation (4) and the dynamic equation for linear motion is modified into dynamic equation for rotational motion. T =Mr d dt Kr (9) The equation (9) specifies the dynamic rotational motion when the torque is applied to the pinion wheel, and gives the information of the deflection that has taken place in the object. 3 Permanant Magnet DC Motor Modelling The permanant magnet DC motor has fixed magnetic field around the rotor, hence the modelling is easier and the dynamic electro-magnetic characteristics completely depend on the armature inductance, resistance and the motor constants. The simplified general model of the motor is as shown in the figure 5. L R i 1 Va Eb + A - T m, θ J B T L Figure 5. Electro-mechanical Model of DC motor Here the parameters L and R are inductance and resistance of the armature coils respectively. E b is the back emf of the motor, V a is the armature voltage and i is the armature current. The mechanical part consists of the shaft whose inertia is J, viscous coefficient is B and T L is the load torque. The torque generated by the motor is T m and the rotational motion θ. The motor constants are the back emf constant K b and the torque constant, the dynamic equation describing the motor is given in equation (10). V a =ir L di dt K d b dt (10) The torque generated by the motor is given by equation (11) in terms of current and torque constant. T m =i (11)
The generated torque is used to drive the shaft whose motion is opposed by inertia, friction and the load torque. The dynamic equation for the torque is given by equation (1). T m =J d dt B d dt T L (1) The block diagram of the model of the DC motor is as shown in the figure (6). Figure 6. Block diagram of the DC motor model 4 Force Balancing and System Integration The concept of force balance is used in the transduction techniques for measurement of pressures. This electrodynamic process, involves a feedback mechanism where in the amount of force applied is measured by measuring the quantity that is responsible for causing the force. The measurement also gives the effect of the force applied. Applying the concept here, the dc motor provides an in-built force balance mechanism, by the feedback of the speed through the back emf to the input voltage. Here the torque is applied to the pinion wheel. The armature current gives a measure of the torque being applied. The elastic property of the material, causes a deflection in the motor shaft. The feedback of speed increases or reduces the armature current when subjected to the load torque as shown in the figure. If the load torque increases, then the armature current also increases dynamically. System integration of the mechanical and the electrical systems, will perform the force balancing scheme. The defined models of the mechanical and the electrical systems can be integrated at the mechanical level at the shaft of the motor and the pinion wheel, along with the gear train. The direction of the motion of the motor is such that the force exterted on the object causes elongation. The motor is turned on, and the deflection profile and the current profiles are recorded. The angular deflection profile is sensed using a very high resolution optical encoder. The profile of the deflection will also provide the information of the angular speed, by performing the weighted backward differentiation on the angular deflection data. The weighted backward differentiation will
filter all the noise. The armature current is sensed using a very low value sense resistance. The voltage across the resistance will give the current information. Using this configuration the deflection and the current profiles are recorded in real time. This information can give the estimation of the required parameter. The motor shaft is coupled to the speed reducer gear train. The gear ratio being 'n', the output torque is magnified n times, and speed reduced n times. Refering the equation (9), the torque required to have an elongation in the object, is defined as the load torque (T L ) in the equation (1), hence giving equation of the integrated system in (13). T m = J Mr d dt B d dt Kr n (13) The equation (11) is used to modify the equation (13) to give the model in terms of the armature current and the rotational motion, as equation (14) i = J Mr d dt B d dt Kr n (14) Considering the variables as current i and deflection θ, and normalising the equation (14) as a general second order differential equation, i= J Mr d dt B d dt Kr (15) The equation (15) gives the response of the current with respect to the changing θ, the equation is differentiated in time domain, the derivative will give the response of the rate of change of current. The equation (16) shows the differentiation. di dt J Mr = d 3 dt B d Kr d 3 dt dt (16) The responses of the current and the rate of current are used in the motor equation (10), to provide the relation between the armature input voltage and the deflection θ, equation (17) gives the relation. V a =[ J Mr L ] d 3 dt 3 [ J Mr R B ] d dt [ Kr L K nk b BR ] d t dt [ Kr R ]... (17) The equation (17) gives the response of the deflection θ, when the input voltage to the motor V a is provided. The general model of the system is described the block diagram in figure (7).
The model can be used to see the behavior of the deflection when the motor is subjected to the deflection dependent load torque. However the system is assumed to have a few parameters to be very low and Figure 7. Block diagram of the system model can be neglected. Since it is a negligible acceleration system, the 3 rd and the nd order differentials can be neglected. This will make the models to be quite linear in behavior and hence also easier to analyse. Linearising the model equations (15) and (17) would provide the models, described by equations (18) and (19) respectively. i= B d Kr (18) dt V a =[ Kr L K nk b BR ] d t dt [ Kr R ] (19) The models are described with first order differentials and zero order terms. The term d is nothing but the angular speed ω. Transforming the differential equations into dt laplace domain with variable s, the relations, would be defined as in equations (0) and (1). i = B s Kr (0) nk = t V a Kr L nbr K b s Kr R (1)
The equations (0) and (1) will yield the relation between the current and the input armature voltage V a. The relation is described by equation () in the laplace domain. i nbs Kr = V a Kr L nbr K b s Kr R () The behavior of the current profile can be known completely using the equation (). The equation (18) is the prime equation for the system. The differential is replaced with the variable ω specifying the speed to give equation (3). i= B Kr (3) The constants containing in the equation (3) are B,, K and r. The constants B, and r are known. As described earlier, the determination of the constant K will yield the Youngs modulus Y using the relation Y = KL. Here the deflection(θ) and current(i) A profiles are recorded in real time using the sensors as specified earlier. The speed(ω) profile is computed using the backward differentiation. This available data is used to compute K. The method used to compute the constant K is the Least Squares method of curve fitting. The equation (3) is rearranged so as to give a linear equation of the form y = mx. The equation (4) gives the form. i B = Kr (4) The left hand side term is fully known, and in the right hand side the except the parameter K everything is known. Hence this is simplified by introducing two more terms, z and λ both defined as in equations (5) and (6). z=i B (5) = r (6) The equation (4) is modified to equation (7) using the terms z and λ as defined in (5) and (6). z=k (7) Considering the term K, a linear polynomial P(λ) can be stated by the equation (8) which will be equivalent to the equation (7). P = K (8)
The term K is the estimated value of the term K. This estimate is computed using the Least Squares Approximation. The cost function I is given as in equation (9). This function is the square of the estimation error between the value z and the estimated P term. I K = z j K j =minimum (9) j=0 Here N is the total number of samples of the data collected. The cost function I has to be minimized. This is done by performing partial differentiation on the equation (9) with respect to the term K. This is given by equation (30). I K = z j K j j =0 (30) j=0 The equation (30) can be simplified for the middle term as equation (31). j=0 z j j = K j j=0 (31) Solving the equation (31) for K as a function of the data available K = z j j = j=0 j j=0 i j B j=0 K j r t r j=0 nk j t j (3) The equation (3) is the ultimate equation for estimation of the parameter now applied in the relation with Youngs modulus in equation (33). K. This is Y = L A K (33) Using the relation of (33), the Youngs modulus is finally determined. This is the conceptualized algorithm for the determination the elastic property of a ductile material. 6 Simulations The performance of the system is simulated using MATLAB. The linear mathematical models which were built in the previous sections, are used here. The PMDC motor that is used here for simulation is Mavilor's MS- DC
servo motor, whose parameters are: Rated Voltage 180 V Rated Power.5 kw Maximum Torque 49.14 Nm Maximum Current - 68 A Back EMF Constant (K b ) 0.53 V/rad/s Torque Constant ( ) 0.53 Nm/A Damping Co-efficient (B) 0.478x10-3 Nm/rad/s Armature Resistance (R) 0.67 Ω Armature Inductance (L) 50 μh Inertia (J) 5.1x10-3 kg/m Radius of the pinion wheel (r) 0.05 m Gear Ratio (n) 100 Geometry of the Object: Cross Section Diameter 0.01 m Cross Section Area (A) 7.86x10-5 m Length (L) 0.3 m Geometry Ratio (A/L).6x10-4 m Youngs modulus used in simulation x10 10 N/m (steel) Computed Spring Constant (K) 5.4x10 6 N/m The equations (1) and () are used giving the dynamic variation of the deflection versus armature voltage and current versus voltage. These equations are simulated with the given parameters. The deflection profile obtained is shown in figure (8). Figure 8. Deflection profile
The speed profile is shown in figure (9). The current profile is shown in figure (10). Figure 9. Speed profile Figure 10. Current profile The collected data of deflection, speed, and current is used in the prime equation to compute the spring constant parameter K. The estimate of the parameter is computed from the 'z' versus 'λ' curve shown in the figure (11).
Figure 11. z versus lambda plot The program is written for computation of the parameter using least squares equation (3). The estimated value is found to be 5.3x10 6 quite close the actual value and hence the estimated Youngs modulus will be 5.3x10 6 /.6x10-4 that is 1.996x10 10 N/m. The estimated value of the Youngs modulus is very close to the actual value. 5 Conclusions The proposed concept for measurement of the Youngs modulus using an electrical based force balance technique is in the view of replacing the hydraulic systems for the purpose with smaller volume consumption for the apparatus. The motivation for the concept is the versatile behavior of the permanant magnet DC motor. It can be very well used for the force balance methods of transduction and measurement. However there are many challenges for the system to be developed. First of all, the coils of the motor should be able to handle the maximum current at stall torque. The next challenge is to make the whole rack and pinion arrangement with gearing to be very light using suitable composites. The deflection sensing of the shaft is another challenge, since the deflections are in the range of 10-3 degree. If these challenges are taken care of, the proposed system will be an efficient alternative for the measurement of elastic properties of the materials. References: 1. P. C. Sen, "Principles of Electric Machines and Power Electronics", /e John Wiley, 001.. Erwin Kreyszig, "Engineering Mathematics", Pearson Education, 00. 3. M.K.Jain, S.R.K Iyengar, R.K.Jain, "Numerical Methods for Scientific and Engineering Computation", 3/e New Age International, 001.
4. Thomas Young, Freedman, "University Physics", 5/e Addison Wesley, 1999. 5. C.S.Rangan, G.R.Sarma, V.S.V.Mani, "Instrumentation Devices and Systems", /e Tata McGrawhill, 001.