! " # $ % ) * + ', +"!! & *,)+&,, $ ,+*, $$ ..


 Neil Osborne
 3 years ago
 Views:
Transcription
1 ! " # $ % " & ' %( ) * + ', +"!! & *,)+&,, $ ,+*, $$ ..
2
3 Dissertation for the Degree of Doctor of Philosophy in Engineering Science with Specialization in Solid Mechanics presented at Uppsala University in 2002 Abstract Widehammar, S A Method for Dispersive Split Hopkinson Pressure Bar Analysis Applied to High Strain Rate Testing of Spruce Wood. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala. ISBN The aim was to establish a method for studying the relationbetweenstressandstraininspruce wood at high strain rate. This was achieved by adapting and somewhat further developing the split Hopkinson pressure bar (SHPB) technique. Hopkinson bars usually have a circular crosssection and a diameter much smaller than the operative wavelengths. The wave propagation in the bar is then approximately nondispersive and a onedimensional (1D) wave propagation model can be used. When, as in this study, it is not certain that the transverse dimensions of the bars are small in relation to the wavelengths, a solely 1D wave propagation model is insufficient and the geometry of the crosssection, which was square in this study, must be taken into account. Therefore, an approximate 3D wave propagation model for bars with arbitrary crosssection was developed using Hamilton s principle. The model provides a dispersion relation (wavenumber vs. angular frequency) and average values for displacements and stresses over the bar/specimen interfaces. A calibration procedure was also developed. Tests on spruce wood specimens were carried out at a high strain rate (about 10 3 s 1 )using the adapted SHPB technique, and for comparison at low ( s 1 ) and medium (17 s 1 ) strain rates using a servohydraulic testing machine. The moisture contents of the wood specimens corresponded to oven dry, fibre saturated and fully saturated, and the testing was performed in the radial, tangential and axial directions relative to the stem of the tree. In each case, five tests were run at room temperature. The results show the strain rate dependence of the relation between stress and strain for spruce wood under all conditions studied. Keywords: split Hopkinson, wave propagation, dispersion, Hamilton s principle, strain rate, calibration, spruce, wood Svante Widehammar, Department of Materials Science, Uppsala University, Box 534, SE Uppsala, Sweden c Svante Widehammar 2002 ISSN X ISBN Printed in Sweden by Uppsala University, Tryck och medier, Uppsala 2002
4 Akademisk avhandling för avläggande av teknologie doktorsexamen i teknisk fysik med inriktning mot hållfasthetslära vid Uppsala universitet år 2002 Sammanfattning Widehammar, S En metod för dispersiv analys av försök med delad hopkinsonstång tillämpad på provning av granved vid hög töjningshastighet. Syftet var att etablera en metod för att studera sambandet mellan spänning och töjning för granved vid hög töjningshastighet. Detta åstadkoms genom att anpassa och något vidareutveckla tekniken med delad hopkinsonstång ( Split Hopkinson Pressure Bar, SHPB). Vanligtvis har hopkinsonstavar cirkulärt tvärsnitt och en diameter som är mycket mindre än de verksamma våglängderna. Under sådana förhållanden är vågutbredningen i stängerna approximativt ickedispersiv, och en endimensionell (1D) vågutbredningsmodell kan användas. När det, som är fallet i denna studie, däremot inte kan säkerställas att stängernas tvärdimensioner är små i förhållande till våglängderna, är en helt igenom 1D vågutbredningsmodell otillräcklig, och tvärsnittets geometri, vilken var kvadratisk i denna studie, måste beaktas. Därför utvecklades med hjälp av Hamiltons princip en approximativ 3D vågutbredningsmodell för stänger med godtyckligt tvärsnitt. Modellen ger ett dispersionssamband (vågtal som funktion av vinkelfrekvens) samt medelvärden för förskjutningar och spänningar över gränsytorna mellan stänger och provstav. En kalibreringsprocedur utvecklades också. Provning av granved genomfördes vid hög töjningshastighet (omkring 10 3 s 1 ) med den anpassade SHPBtekniken, samt för jämförelse vid låg ( s 1 ) och måttlig (17 s 1 ) töjningshastighet med en servohydraulisk provningsmaskin. Fukthalterna i veden motsvarade ugnstorr, fibermättnad och fullständig mättnad, och proven utfördes i radiell, tangentiell och axiell riktning i förhållande till trädets stam. För vart fall utfördes fem försök vid rumstemperatur. Resultaten visar töjningshastighetsberoendet för sambandet mellan spänning och töjning för granved under alla studerade förhållanden. Svante Widehammar, Institutionen för materialvetenskap, Uppsala universitet, Box 534, 75121Uppsala
5 Preface This work was initiated by the former Sunds Defibrator Industries AB (now Metso Paper AB) and the Mid Sweden University. The work was carried out at the Fibre Science and Communication Network at Mid Sweden University in Sundsvall, in cooperation with the Division of Solid Mechanics at Uppsala University. It was supported economically by the following external agencies: Foundation for Knowledge and Competence Development, the National Graduate School of Scientific ComputingandMetsoPaperAB. I would like to thank Professor Per Gradin, Mid Sweden University, and Professor Bengt Lundberg, Uppsala University, for their guidance and support during the course of this work. I would also like to thank all my colleagues in Sundsvall and Uppsala for providing a pleasant and creative atmosphere, in particular Staffan Nyström and Max Lundström for their assistance in the laboratory, including the superb coffee. Special thanks also to Göran Widehammar and Örjan Bäcklund who used their free time to find and cut a suitable tree and to manufacture the wood specimens. Finally, I would like to thank my children Ronja and Hugo for being there, even though they could not see the meaning of storing small cubes of wood in water and writing funny signs and curves on the computer, and my beloved wife, AnnaKarin, who, after all, managed to cope with me during these years. Sundsvall in December, 2002 Svante Widehammar
6 List of papers This thesis comprises a survey of the following four papers: I. Widehammar S, Gradin P.A. and Lundberg B. (2001). Approximate determination of dispersion relations and displacement fields associated with elastic waves in bars. Method based on matrix formulation of Hamilton s principle. Journal of Sound and Vibration 246(5): II. Widehammar S. Estimation of 3D field quantities and energy flux associated with elastic waves in a bar. Journal of Sound and Vibration (Accepted) III. Widehammar S. Split Hopkinson pressure bar testing: procedures for evaluation and calibration. Experimental Mechanics (Submitted) IV. Widehammar S. Stressstrain relationships for spruce wood: influences of strain rate, moisture content and loading direction. Experimental Mechanics (In progress)
7 Contents 1. Introduction 9 2. The refining process Fundamentals of SHPB testing D formulation Displacement, particle velocity, strain and stress in a SHPB Dispersionrelation EvaluatingSHPBtests Twopulsemethod Determinationofamplitudes Discussion D formulation Displacement, particle velocity, strain and stress in a SHPB Dispersionrelation EvaluatingSHPBtests Twopulsemethod Determinationofamplitudes Discussion Calibration Dispersionrelation Poisson sratio Preparatory tests Experimentalsetup EstimationofPoisson sratio EstimationofYoung smodulus Evaluation of redundant measurements Comparisonofevaluationoptions Discussion... 34
8 8. Testing of spruce wood Experimentalconditions SHPBtests Resultsanddiscussion Conclusions 38 References 39
9
10 DISPERSIVE SHPB ANALYSIS OF WOOD 9 1. Introduction The aim of this work was to establish and use a method for determining the relation between stress and strain in spruce wood under conditions which are similar to those which prevail in a refiner in the thermomechanical pulping process. Characteristic of these conditions are high temperature, a moisture content above fibre saturation and a high strain rate. The last condition excludes the use of a traditional servohydraulic testing machine for determining the stressstrain relations. In order to achieve a high strain rate, the split Hopkinson pressure bar (SHPB) technique has proven suitable. A general presentation of the technique is given by AlMousawi et al. [1]. Hopkinson bars usually have a circular crosssection and a diameter which is much smaller than the wavelengths associated with the measured pulses. Under such conditions, the wave propagation in the bars is approximately nondispersive, and a onedimensional (1D) wave propagation model can be used with good accuracy. This is the traditional method, described by, e.g., Graff [2]. Preliminary tests showed that dispersion could not be neglected in this work. Dispersion occurs when the transverse dimensions of the bars are of the same order of magnitude as the operative wavelengths. Compensation for dispersion is a standard procedure nowadays, and it has been used by many researchers, e.g., Follansbee and Frantz [3], Gong et al. [4], Zhao and Gary [5] and Li and Lambros [6]. Common for all these investigations is that a dispersion compensation procedure is applied to the standard 1D model of the bar, in which warping of the bar crosssection is neglected and the geometry of the crosssection is assumed to be insignificant. The dispersion relation (wavenumber vs. angular frequency) must then be known, and the exact solution for a bar with a circular crosssection, which was originally presented by Pochammer [7] and Chree [8] and further developed by, e.g., Kolsky [9], Davis [10] and Bancroft [11], is usually used. For practical reasons, a square crosssection was chosen for the spruce specimens as well as for the bars in this work. No exact solution seems to be available for such bars. For arbitrary crosssections, finite element methods have been used by, e.g., HladkyHennion [12], Volovoi et al. [13] andtaweel et al. [14]. InPaperI, an approximate 3D wave propagation model for bars with arbitrary (and in particular square) crosssection was developed using Hamilton s principle. The model uses Fourier series and leads to an eigenvalue problem which has to be solved for each angular frequency associated with the Fourier components. The eigenvalue problem provides a dispersion relation, and average values for displacements and stresses over the bar/specimen interfaces can be calculated, as is shown in Paper II. This means that warping can also be included in the evaluation of SHPB tests. The effects of warping have been studied for circular bars by Tyas and Watson [15 17], who used
11 10 SVANTE WIDEHAMMAR the PochammerChree equations. They showed that the effects of warping may not always be negligible. The formulations with Fourier series used in the wave propagation model are suitable for redundant measurements, which may increase the accuracy in the results. This is evaluated in Paper II. The technique with redundant measurements was previously used by Hillström et al. [18] to estimate the complex modulus of the material of a bar. A calibration procedure for the experimental setup using an ordinary SHPB projectile and an optimizing procedure has also been developed from the wave propagation model. The calibration procedure is developed in Papers II and III. The fully 3D model, including redundant measurements and parts of the calibration procedure, is validated on a bar with a free end in Paper II. A mixed evaluation procedure, using the dispersion relation from the 3D model and the 1D formulations for variables such as displacements and stresses in the bars, was applied to SHPB tests and validated in Paper III. Finally, high strain rate testing of spruce wood was performed in the work described in Paper IV. Low and medium strain rate testing was performed in a servohydraulic testing machine for comparison with the high strain rate SHPB tests. Oven dry, fibresaturatedandfullysaturatedwoodweretestedintheradial,tangential and axial directions with reference to the stem of the tree. 2. The refining process The thermomechanical pulping (TMP) process produces pulp for papermaking out of wood chips. The central part of the process takes place in a refiner, which consists of two concentric circular discs. Either one disc rotates and one is fixed, or both rotate in opposite directions. Preheated wood chips and water are fed into the centre of the refiner and are transported radially outwards between the discs due to inertia. Through contact with radial bars on the discs, the chips are defibrated into fibres. The discs have a coarse bar pattern near the centre, the breaker bar zone, and a much finer bar pattern at the periphery. The gap between the two discs decreases from the centre to the periphery, where it is of the order of less than one millimetre. The discs are about 1.5 meters in diameter and normally operate at 1500 or 1800 revolutions per minute. All aspects of mechanical pulping are treated in a book by Sundholm [19]. The TMP process is very energyconsuming and much effort has been spent on reducing the energy consumption without any loss of pulp quality. A better understanding of the refining process requires a better knowledge of the behaviour of wood
12 DISPERSIVE SHPB ANALYSIS OF WOOD 11 chips under realistic refining conditions. The initial breaking of the wood chips is performed at high temperature (about 125 C),atamoisturecontentabovefibre saturation, and at very high strain rate (about 10 3 s 1 ). It is generally recognized that species of the spruce family are the most favourable raw materials for mechanical pulping, and spruce is also the most widely used species for that purpose in Europe and Canada. Experiments aimed at quantifying the behaviour of wood under such conditions have been carried out, e.g., by Uhmeier and Salmén [20] and by Renaud et al. [21, 22]. Uhmeier and Salmén used servohydraulic testing equipment and reached 25 s 1 at 98 C for spruce wood, while Renaud et al. used the SHPB technique in order to achieve a high strain rate at room temperature. The latter authors used three different kinds of hardwood which are all of minor importance for the Swedish pulping industry. Other tests on wood at high strain rates have been carried out, e.g., by Bragov and Lomunov [23] and by Reid and Peng [24]. The latter investigators performed their experiments on dry wood. There is an evident lack in the literature of test data on wet spruce wood at strain rates similar to those which occur in a refiner. The SHPB technique is well suited to obtain such data. 3. Fundamentals of SHPB testing For compression testing with the SHPB technique, a specimen of the material to be tested is inserted between the ends of two bars with colinear axes, as shown in Figure 1. The material of the bars, which has known properties, is commonly linearly elastic and isotropic, and small deformations can be assumed in the bars. A projectile is fired onto the free end of one of the bars, the sender bar. This generates an incident pulse, which propagates towards the specimen where the pulse is partially reflected and partially transmitted into the second bar, the receiver bar. Measurement and analysis of the incident, reflected and transmitted pulses enables the mechanical Incident pulse Reflected pulse Transmitted pulse Projectile Sender bar Specimen Receiver bar Figure 1: Principle setup of SHPB experiments.
13 12 SVANTE WIDEHAMMAR behaviour of the material of the specimen to be estimated. The stress in the specimen at time is approximated by the mean value of the forces from the two bar ends in contact with the specimen, divided by the specimen crosssectional area, i.e., () =[ b A (0)+ b B (0)] 2 (1) where b is the crosssectional area of the bars, A ( ) and B ( ) are the axial stresses in the sender and receiver bars, respectively, and is the axial coordinate. The two bars have individual coordinate systems, where =0is the axial position of the bar/specimen interface in each system. An overbar denotes the mean value over the bar crosssection. The strain in the specimen is approximated by the difference between the averages of the axial displacements of the two bar ends divided by the initial length of the specimen, i.e., () =[ B (0) A (0)] (2) where A ( ) and B () are the axial displacements in the two bars. Differentiation with respect to time gives the strain rate () =[ B (0) A (0)] (3) where a dot denotes time derivative. From relations (1) and (2), a stressstrain curve can be established with time as a parameter. The strain rate (3) generally varies with time, but it is often approximately constant during a test. In order to be able to evaluate the conditions at the two sides of the specimen, axial strains in the bars are measured at some distances from the bar/specimen interfaces. A procedure will be formulated in which the specimen behaviour can be calculated from the measured strains. 4. 1D formulation 4.1. Displacement, particle velocity, strain and stress inashpb Using a discrete Fourier transform (DFT), a pulse can be expressed as a sum of harmonic waves. If the wavelength of such a harmonic wave is large compared to the crosssectional dimensions of the bar, it can be assumed that transverse displacements and stresses are small compared to the axial displacements and stresses.
14 DISPERSIVE SHPB ANALYSIS OF WOOD 13 If the transverse quantities are neglected, axial displacement, axial particle velocity and axial strain and stress in the SHPB:s depend only on and, and not on the crosssectional coordinates and. The mean value over the crosssection is then trivial, so the overbar is left out in this section. Under this assumption, the axial displacement in the sender bar can be approximated by 2 X A ( ) = 0A + 0A + 0A +Re h³ i i e i + r e e ii (4) where the first two terms on the righthand side represent rigid body motion with constant velocity 0A,and 0A is a constant axial strain. The sum represents harmonic waves travelling in positive and negative directions. These waves are obtained from the DFT:s of the incident (index i) andreflected (index r) pulses, respectively. The constants i and r are complexvalued amplitudes and and are the wavenumber and the angular frequency, respectively, associated with Fourier component number. The constant is recognized as the length of a measuring record, and Re denotes the real part. The axial particle velocity in the bar is obtained by differentiating equation (4) with respect to time, i.e., 2 X A ( ) = 0A +Re h³ i i e i + r e i e ii (5) and the axial strain is obtained by differentiating equation (4) with respect to, i.e., 2 X A ( ) = 0A +Re h³ i i e i + r e i e ii (6) With the assumptions that the bar material is linearly elastic and the stress is uniaxial, the axial stress is obtained by multiplying the strain by Young s modulus, i.e., 2 X A ( ) = 0A + Re h³ i i e i + r e i e ii (7) For the receiver bar, correspondingly, the axial displacement is 2 X B ( ) = 0B + 0B + 0B +Re h i t e i e i (8) where index t denotes the transmitted pulse. The axial particle velocity, strain and stress are obtained in the same way as for the sender bar.
15 14 SVANTE WIDEHAMMAR 4.2. Dispersion relation The angular frequencies are obtained from the definition of the DFT, but a dispersion relation = () is lacking in the previous expressions. Hamilton s principle can be utilized to find the dispersion relation. In the absence of external forces, Hamilton s principle states that µz 2 ( )d =0 (9) 1 where is the kinetic energy, is the elastic strain energy, and 1 and 2 are two arbitrary instants of time The kinetic energy is here given by ZZZ µ 1 = 2 2 d (10) where is the mass density and is the volume of the body. The elastic strain energy stored in the body is ZZZ µ 1 = 2 d (11) Here, the axial strain is = 0, where the prime denotes partial differentiation with respect to, and the axial stress is = 0. Inserting these relations and equations (10) and (11) into Hamilton s principle (9), gives µz 2 ZZZ µ ( ) d d =0 (12) By performing the variational operation, an integration by parts with respect to time of the first term and an integration by parts with respect to of the second term, it is obtained that + 00 =0 (13) together with initial and axial boundary conditions, which will not be treated here. Inserting a harmonic wave similar to those in equation (4), i.e., n =Re e i( )o (14) gives which leads to where the phase velocity is constant. 2 e i( ) 2 e i( ) =0 (15) = 0 (16) 0 = p (17)
16 DISPERSIVE SHPB ANALYSIS OF WOOD Evaluating SHPB tests Inserting expressions (4), (5) and (7) evaluated at =0, and similar expressions for the receiver bar, into relations (1)(3) gives () = b 2 Re X ( i + r t )i e i (18) and () = Re X (t i r )e i (19) () = Re X (t i r )i e i (20) where the constants 0, 0 and 0 are most easily determined directly from the conditions that () = () = () =0before the incident pulse reaches the specimen. This is done by plotting the results in the time domain using only the Fourier series in equations (18) and (19). In these plots, the initial parts of the curves are generally straight sloping lines. The constants are then determined so as to make these lines coincide with the horizontal axis. This evaluation method is called the threepulse method in Paper III, as all three pulses, incident, reflected and transmitted, are used in the evaluation Twopulse method If the SHPB test is well designed, it can be assumed that the stress is constant throughout the specimen, i.e., A (0)= B (0). Expression(7)andthecorresponding expression for the receiver bar then give i = r + t. Using this equality in equations (18)(20) gives () = 0 2 b Re X t i e i (21) and () = Re X r e i (22) () = Re X r i e i (23)
17 16 SVANTE WIDEHAMMAR Here, only the reflected and transmitted pulses are used in the evaluation Determination of amplitudes To be able to determine the amplitudes associated with harmonic waves, axial strains are measured at axial positions =1 2 on each bar. After a DFT of each strain record has been performed, the axial strain at = can be approximated by 2 X ( )= 1 +Re e i (24) where are complexvalued amplitudes obtained from the DFT. Comparison of this expression with the axial strain in the sender bar given by equation (6) yields = ³ i i e i + r e i (25) for =2 32. For each Fourier component, this provides a system of equations e i1 e i1 1 (i ) i.. = e i e i. (26) r (i ) for the two unknown amplitudes i and r. Thus, for 2 this system of equations is overdetermined and an appropriate approximation of i and r can be obtained by the leastsquares method. An overdetermined system of equations contains redundant information, and as a result, it may give more accurate results for i and r than a system which does not contain redundancies. If a measuring section is placed so far from the bar/specimen interface that the incident and reflected pulses are separated in time at this particular section, the measuring record can be split into two parts, viz., one part containing the incident pulse and one part containing the reflected pulse. The two split records should be padded with zeros at the end or at the beginning, respectively, to maintain the original record length (), and to let each of the two pulses appear at the same times in the split records as they did in the original record. This simplifies the analysis. Performing a DFT on each of the two split measuring records gives the constants i and r for the incident and reflected pulses, respectively. The equation corresponding to row in the system of equations (26) can then be separated into two new equations, viz., e i 0 i 0 e i r = i (i ) r (i ) (27)
18 DISPERSIVE SHPB ANALYSIS OF WOOD 17 The splitting of one equation into two equations adds information to the original system and therefore gives a greater accuracy in the determined amplitudes. In the receiver bar, only waves propagating in the positive direction are considered. The leastsquares method can be used together with redundant data also here, even though there is only one amplitude to determine for each Fourier component Discussion It is shown in Paper III that equations (21)(23) together with equation (16) can be formulated as () = b t (0)= µ b t t + t (28) 0 () = 2 Z 0 r (0)d = 2 Z µ 0 r r + r d (29) and () = 2 0 r (0)= 2 µ 0 r r + r (30) 0 where t ( t ) and r ( r ) are the measured axial strains in the measuring sections = t and r for the transmitted and reflected pulses, respectively, and t = t and r = r are the distances between the measuring sections and the bar/specimen interfaces. These expressions are traditionally often used for evaluation of SHPB experiments, see, e.g., AlMousawi et al. [1] and Graff [2]. It can be seen that if the distances t and r are equal, there is no need for the time shift. The method is then very easy to use and also very robust. Expressions (28)(30) may indicate that the formulations with Fourier series seem unnecessarily complicated here. Anyway, although not really consistent, correction for dispersion is often used in connection with the 1D formulation, see, e.g., [3 6]. That technique is also used in Paper III and Paper IV. The frequent use of a 1D formulation mixed with a dispersion correction justifies the use of Fourier series in the 1D case. Also, the obvious similarities with the 3D formulation which follow will make comparisons easier. 5. 3D formulation 5.1. Displacement, particle velocity, strain and stress inashpb If the wavelengths of the operative harmonic waves in the Fourier series in equations (4)(8) are comparable to the crosssectional dimensions of the bar, lateral displace
19 18 SVANTE WIDEHAMMAR ments cannot be neglected. An expression for a general displacement vector in the sender bar corresponding to equation (4) in the 1D formulation, is then u A ( )=( 0A + 0A )[0 0 1] T + 0A [ ] T 2 X + Φ ( )Re h³ i i d e i + r d e e ii (31) where is Poisson s ratio, Φ is a 3 matrix containing given coordinate functions of the crosssectional coordinates and, and d are normalized vectors of complexvalued constants. The amplitudes i and r have the physical dimension of length, while Φ and d are dimensionless. The coordinate functions in Φ allow for variations in the displacement vector over the crosssection, warping, and the constants in d are the weights for each coordinate function. The vectors d contain the weights for waves travelling in the negative direction, i.e., connected with the reflected pulse. It will later be shown that d are the complex conjugates of the vectors d. Differentiation of equation (31) with respect to time gives the particle velocity u A ( )= 0A [0 0 1] T 2 X + Φ ( )Re h³ ei i d e i + r d i e ii (32) The vector of strain components ε =[ ] T and the displacement vector u are related through the kinematic relation ε = u (33) where contains the partial differential operators and. For a study of wave propagation in a bar with its axis oriented in the direction, it is advantageous to split the operator matrix into two parts, i.e., = + (34) where = = (35)
20 DISPERSIVE SHPB ANALYSIS OF WOOD 19 The parameter is a reference length, such as the radius of a bar with circular crosssection or half the side of a square crosssection. It is introduced in order to obtain the same physical dimension of and. From equations (31) and (33)(35), the vector of strain components in the sender bar is ε A ( ) = 0A [ ] T 2 X + Φ ( )Re h³ ei i d e i + r d e ii 2 X + Φ ( )Re h³ i i d e i + r d e i e ii (36) For a linearly elastic and isotropic material, the vector of stress components τ = [ ] T is given by the generalized Hooke s law τ =Cε (37) where is Young s modulus and C = (38) is a symmetric matrix containing dimensionless elastic constants 1 = 1 (1 + )(1 2) 2 = (1 + )(1 2) 3 = 1 2(1+) (39) From equations (36)(39), the vector of stress components in the sender bar is obtained as τ A ( ) = 0A [ ] T 2 X + C Φ ( )Re h³ i d e i + r d i e i e i 2 X +C Φ ( )Re h³ i d e i + r d i ei i e i (40)
21 20 SVANTE WIDEHAMMAR 5.2. Dispersion relation As in the 1D case, Hamilton s principle will be used to obtain the dispersion relation. The displacement vector to be used here is similar to the harmonic waves in equation (31), viz., n u =Re Φ ( ) de i( )o (41) The kinetic energy is here given by ZZZ = and the elastic strain energy by ZZZ = µ 1 2 ut u d (42) µ 1 2 τ T ε d (43) It is shown in Paper I that equations (33), (34), (37) and (41)(43) together with Hamilton s principle (9) lead to the eigenvalue problem " K 0 +i µ 2 µ 2 ³K 1 K T 1 + K 2 M# d = 0 (44) where 0 = 1 are reference quantities, and ZZ K 0 = and 0 = 1 s (45) ( Φ) T C Φ d (46) ZZ K 1 = ( Φ) T C Φ d (47) ZZ K 2 = ( Φ) T C Φ d (48) M = 1 2 ZZ Φ T Φ d (49) are real and dimensionless matrices. The derivation in Paper I is more general since neither a cartesian coordinate system nor an isotropic material are assumed. The eigenvalue to solve for in the eigenvalue problem (44) is one of the quantities 0 or 0 with the other known. For a given angular frequency, equation
22 DISPERSIVE SHPB ANALYSIS OF WOOD 21 (44) is a quadratic eigenvalue problem with 2 solutions. The eigenvalue 0 is generally complexvalued, but for modes corresponding to propagating waves, 0 is real. Nonreal wavenumbers correspond to end modes which decay with increasing distance from the bar end. See, e.g., Taweel et al. [14]. Suppose that a solution ( d) with positive and real is found. From equation (41), it is evident that this corresponds to a wave travelling in the positive direction. Now, by forming the complex conjugate of equation (44), it can be shown that ( d ) where the star denotes a complex conjugate, is also a solution of the eigenvalue problem. This second solution corresponds to a wave of the same mode which travels in the negative direction, which has already been used in equation (31) and in the expressions derived from it. The 2 solutions of (44) approximate in general several modes of wave propagation such as flexural, torsional and longitudinal. In a SHPB, only the first longitudinal mode is of interest. The coordinate functions can then be constructed to include the symmetries related to this mode. Here, these coordinate functions are polynomials in the lateral coordinates and, given in detail in Papers I and II. For longitudinal modes, it is shown in Paper II that the matrix Φ can be partitioned according to Φ = φ T 0 φ T 0 0 φ T (50) The partitioning of Φ into subvectors also motivates a partition of d, accordingto d d = (51) It is also shown in Paper II that all the components of d have the same phase angle, and that this angle differs by 2 from the phase angle of all the components in d. If the vector d is an eigenvector, so is de i, which makes it possible to change the phase angle of the eigenvector d. If the phase angle is chosen so that d is real, the subvector d is purely imaginary. This implies that d d = (52) d The convergence rate of the method can be demonstrated by solving the eigenvalue problem (44) with different orders of the polynomials in Φ, seefigure2. In Paper I, solutions of the eigenvalue problem (44) are compared to other approximate solutions from the literature for bars with a square crosssection, and with the exact solution for bars with a circular crosssection. In Paper II, it is shown that the d
Dispersion diagrams of a waterloaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell
Dispersion diagrams of a waterloaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell B.K. Jung ; J. Ryue ; C.S. Hong 3 ; W.B. Jeong ; K.K. Shin
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationPlate waves in phononic crystals slabs
Acoustics 8 Paris Plate waves in phononic crystals slabs J.J. Chen and B. Bonello CNRS and Paris VI University, INSP  14 rue de Lourmel, 7515 Paris, France chen99nju@gmail.com 41 Acoustics 8 Paris We
More informationObjectives. Experimentally determine the yield strength, tensile strength, and modules of elasticity and ductility of given materials.
Lab 3 Tension Test Objectives Concepts Background Experimental Procedure Report Requirements Discussion Objectives Experimentally determine the yield strength, tensile strength, and modules of elasticity
More informationTorsion Testing. Objectives
Laboratory 4 Torsion Testing Objectives Students are required to understand the principles of torsion testing, practice their testing skills and interpreting the experimental results of the provided materials
More informationNumerical Analysis of Independent Wire Strand Core (IWSC) Wire Rope
Numerical Analysis of Independent Wire Strand Core (IWSC) Wire Rope Rakesh Sidharthan 1 Gnanavel B K 2 Assistant professor Mechanical, Department Professor, Mechanical Department, Gojan engineering college,
More informationIntroduction to acoustic imaging
Introduction to acoustic imaging Contents 1 Propagation of acoustic waves 3 1.1 Wave types.......................................... 3 1.2 Mathematical formulation.................................. 4 1.3
More informationFinite Element Formulation for Beams  Handout 2 
Finite Element Formulation for Beams  Handout 2  Dr Fehmi Cirak (fc286@) Completed Version Review of EulerBernoulli Beam Physical beam model midline Beam domain in threedimensions Midline, also called
More informationSoil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay
Soil Dynamics Prof. Deepankar Choudhury Department of Civil Engineering Indian Institute of Technology, Bombay Module  2 Vibration Theory Lecture  8 Forced Vibrations, Dynamic Magnification Factor Let
More informationSolid Mechanics. Stress. What you ll learn: Motivation
Solid Mechanics Stress What you ll learn: What is stress? Why stress is important? What are normal and shear stresses? What is strain? Hooke s law (relationship between stress and strain) Stress strain
More informationChapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations
Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.
More informationFatigue Testing. Objectives
Laboratory 8 Fatigue Testing Objectives Students are required to understand principle of fatigue testing as well as practice how to operate the fatigue testing machine in a reverse loading manner. Students
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
More informationsin ( sin(45 )) 70.5
Seismology and Global Waves Chap. 4 HW Answers 1. A swinging door is embedded in a N S oriented wall. What force direction would be required to make the door swing on its hinge?. A ray is travelling in
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationPSS 27.2 The Electric Field of a Continuous Distribution of Charge
Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight ProblemSolving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.
More informationDetailed simulation of mass spectra for quadrupole mass spectrometer systems
Detailed simulation of mass spectra for quadrupole mass spectrometer systems J. R. Gibson, a) S. Taylor, and J. H. Leck Department of Electrical Engineering and Electronics, The University of Liverpool,
More informationSolved with COMSOL Multiphysics 4.3
Vibrating String Introduction In the following example you compute the natural frequencies of a pretensioned string using the 2D Truss interface. This is an example of stress stiffening ; in fact the
More informationAUTOMATED THREEANTENNA POLARIZATION MEASUREMENTS USING DIGITAL SIGNAL PROCESSING
AUTOMATED THREEANTENNA POLARIZATION MEASUREMENTS USING DIGITAL SIGNAL PROCESSING John R. Jones and Doren W. Hess Abstract In this paper we present a threeantenna measurement procedure which yields the
More informationCenter of Mass/Momentum
Center of Mass/Momentum 1. 2. An Lshaped piece, represented by the shaded area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the Lshaped
More information1. Introduction. 2. Response of Pipelines
1. Introduction Blasting is common in the coal industry to remove rock overburden so that the exposed coal can be mechanically excavated. A portion of the blast energy released is converted to wave energy
More informationDesign of Cellular Composite for Vibration and Noise Control: Effects of Inclusions and Filling Patterns on Acoustic Bandgap
Design of Cellular Composite for Vibration and Noise Control: Effects of Inclusions and Filling Patterns on Acoustic Bandgap Vijay Kumar and D. Roy Mahapatra Department of Aerospace Engineering Indian
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiplechoice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationSIMPLE HARMONIC MOTION
SIMPLE HARMONIC MOTION PURPOSE The purpose of this experiment is to investigate one of the fundamental types of motion that exists in nature  simple harmonic motion. The importance of this kind of motion
More informationPrelab Exercises: Hooke's Law and the Behavior of Springs
59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationForce measurement. Forces VECTORIAL ISSUES ACTION ET RÉACTION ISOSTATISM
Force measurement Forces VECTORIAL ISSUES In classical mechanics, a force is defined as "an action capable of modifying the quantity of movement of a material point". Therefore, a force has the attributes
More informationPhysics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd
Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle
More informationSimple Harmonic Motion Concepts
Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called
More informationMETU DEPARTMENT OF METALLURGICAL AND MATERIALS ENGINEERING
METU DEPARTMENT OF METALLURGICAL AND MATERIALS ENGINEERING Met E 206 MATERIALS LABORATORY EXPERIMENT 1 Prof. Dr. Rıza GÜRBÜZ Res. Assist. Gül ÇEVİK (Room: B306) INTRODUCTION TENSION TEST Mechanical testing
More informationPhysics 53. Wave Motion 1
Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can
More informationChapter 18 4/14/11. Superposition Principle. Superposition and Interference. Superposition Example. Superposition and Standing Waves
Superposition Principle Chapter 18 Superposition and Standing Waves If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum
More informationMASTER DEGREE PROJECT
MASTER DEGREE PROJECT Finite Element Analysis of a Washing Machine Cylinder Thesis in Applied Mechanics one year Master Degree Program Performed : Spring term, 2010 Level Author Supervisor s Examiner :
More informationCalibration and Use of a StrainGageInstrumented Beam: Density Determination and WeightFlowRate Measurement
Chapter 2 Calibration and Use of a StrainGageInstrumented Beam: Density Determination and WeightFlowRate Measurement 2.1 Introduction and Objectives This laboratory exercise involves the static calibration
More informationStrain rate dependence of human body soft tissues using SHPB
Strain rate dependence of human body soft tissues using SHPB R. Marathe, S. Mukherjee, A. Chawla Department of Mechanical Engineering, IIT Delhi, India R. Malhotra Department of Orthopedics, All India
More informationEFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLESTAYED BRIDGES
EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLESTAYED BRIDGES YangCheng Wang Associate Professor & Chairman Department of Civil Engineering Chinese Military Academy FengShan 83000,Taiwan Republic
More informationFinite Element Analysis for Acoustic Behavior of a Refrigeration Compressor
Finite Element Analysis for Acoustic Behavior of a Refrigeration Compressor Swapan Kumar Nandi Tata Consultancy Services GEDC, 185 LR, Chennai 600086, India Abstract When structures in contact with a fluid
More informationSound Power Measurement
Sound Power Measurement A sound source will radiate different sound powers in different environments, especially at low frequencies when the wavelength is comparable to the size of the room 1. Fortunately
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationInternational Journal of Engineering ResearchOnline A Peer Reviewed International Journal Articles available online http://www.ijoer.
RESEARCH ARTICLE ISSN: 23217758 DESIGN AND DEVELOPMENT OF A DYNAMOMETER FOR MEASURING THRUST AND TORQUE IN DRILLING APPLICATION SREEJITH C 1,MANU RAJ K R 2 1 PG Scholar, M.Tech Machine Design, Nehru College
More informationσ = F / A o Chapter Outline Introduction Mechanical Properties of Metals How do metals respond to external loads?
Mechanical Properties of Metals How do metals respond to external loads? and Tension Compression Shear Torsion Elastic deformation Chapter Outline Introduction To understand and describe how materials
More informationChapter Outline. Mechanical Properties of Metals How do metals respond to external loads?
Mechanical Properties of Metals How do metals respond to external loads? Stress and Strain Tension Compression Shear Torsion Elastic deformation Plastic Deformation Yield Strength Tensile Strength Ductility
More informationMathematical Physics
Mathematical Physics MP205 Vibrations and Waves Lecturer: Office: Lecture 19 Dr. Jiří Vala Room 1.9, Mathema
More informationRock Bolt Condition Monitoring Using Ultrasonic Guided Waves
Rock Bolt Condition Monitoring Using Ultrasonic Guided Waves Bennie Buys Department of Mechanical and Aeronautical Engineering University of Pretoria Introduction Rock Bolts and their associated problems
More informationThe simulation of machine tools can be divided into two stages. In the first stage the mechanical behavior of a machine tool is simulated with FEM
1 The simulation of machine tools can be divided into two stages. In the first stage the mechanical behavior of a machine tool is simulated with FEM tools. The approach to this simulation is different
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More information7 Centrifugal loads and angular accelerations
7 Centrifugal loads and angular accelerations 7.1 Introduction This example will look at essentially planar objects subjected to centrifugal loads. That is, loads due to angular velocity and/or angular
More informationLesson 19: Mechanical Waves!!
Lesson 19: Mechanical Waves Mechanical Waves There are two basic ways to transmit or move energy from one place to another. First, one can move an object from one location to another via kinetic energy.
More informationFrequency response of a general purpose singlesided OpAmp amplifier
Frequency response of a general purpose singlesided OpAmp amplifier One configuration for a general purpose amplifier using an operational amplifier is the following. The circuit is characterized by:
More informationTorsion Tests. Subjects of interest
Chapter 10 Torsion Tests Subjects of interest Introduction/Objectives Mechanical properties in torsion Torsional stresses for large plastic strains Type of torsion failures Torsion test vs.tension test
More informationA Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle
A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle Page 1 of 15 Abstract: The wireless power transfer link between two coils is determined by the properties of the
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationImpact Load Factors for Static Analysis
Impact Load Factors for Static Analysis Often a designer has a mass, with a known velocity, hitting an object and thereby causing a suddenly applied impact load. Rather than conduct a dynamic analysis
More informationTypes of Elements
chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could
More information3D WAVEGUIDE MODELING AND SIMULATION USING SBFEM
3D WAVEGUIDE MODELING AND SIMULATION USING SBFEM Fabian Krome, Hauke Gravenkamp BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin, Germany email: Fabian.Krome@BAM.de
More information1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids
1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids  both liquids and gases.
More informationNew approaches in Eurocode 3 efficient global structural design
New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beamcolumn FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural
More informationLABORATORY 9. Simple Harmonic Motion
LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the massspring system and the simple pendulum. For the massspring system we
More informationNonlinear Models of Reinforced and Posttensioned Concrete Beams
111 Nonlinear Models of Reinforced and Posttensioned Concrete Beams ABSTRACT P. Fanning Lecturer, Department of Civil Engineering, University College Dublin Earlsfort Terrace, Dublin 2, Ireland. Email:
More informationPower Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.
Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture  1 Electric Drive Today, we will start with the topic on industrial drive
More informationStress and deformation of offshore piles under structural and wave loading
Stress and deformation of offshore piles under structural and wave loading J. A. Eicher, H. Guan, and D. S. Jeng # School of Engineering, Griffith University, Gold Coast Campus, PMB 50 Gold Coast Mail
More informationCosmosWorks Centrifugal Loads
CosmosWorks Centrifugal Loads (Draft 4, May 28, 2006) Introduction This example will look at essentially planar objects subjected to centrifugal loads. That is, loads due to angular velocity and/or angular
More informationStanding Waves on a String
1 of 6 Standing Waves on a String Summer 2004 Standing Waves on a String If a string is tied between two fixed supports, pulled tightly and sharply plucked at one end, a pulse will travel from one end
More informationMENG 302L Lab 4: Modulus of Elasticity and Poisson s Ratio
MENG 302L Lab 4: Modulus of Elasticity and Poisson s Ratio Introduction: In Lab 4 we will measure the two fundamental elastic constants relating stress to strain: Modulus of Elasticity and Poisson s Ratio.
More informationWelcome to the first lesson of third module which is on thinwalled pressure vessels part one which is on the application of stress and strain.
Strength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture 15 Application of Stress by Strain Thinwalled Pressure Vessels  I Welcome
More informationElectromagnetic Waves
Electromagnetic Waves Maxwell s equations predict the propagation of electromagnetic energy away from timevarying sources (current and charge) in the form of waves. Consider a linear, homogeneous, isotropic
More informationExperimental Question 1: Levitation of Conductors in an Oscillating Magnetic Field SOLUTION ( )
a. Using Faraday s law: Experimental Question 1: Levitation of Conductors in an Oscillating Magnetic Field SOLUTION The overall sign will not be graded. For the current, we use the extensive hints in the
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationEML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss
Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000
More informationStress Strain Relationships
Stress Strain Relationships Tensile Testing One basic ingredient in the study of the mechanics of deformable bodies is the resistive properties of materials. These properties relate the stresses to the
More informationSTANDING WAVES. Objective: To verify the relationship between wave velocity, wavelength, and frequency of a transverse wave.
STANDING WAVES Objective: To verify the relationship between wave velocity, wavelength, and frequency of a transverse wave. Apparatus: Magnetic oscillator, string, mass hanger and assorted masses, pulley,
More informationIntroduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1
Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1 In this tutorial, we will use the SolidWorks Simulation finite element analysis (FEA) program to analyze the response
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationDYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE
DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE ÖZDEMİR Y. I, AYVAZ Y. Posta Adresi: Department of Civil Engineering, Karadeniz Technical University, 68 Trabzon, TURKEY Eposta: yaprakozdemir@hotmail.com
More informationStructural Integrity Analysis
Structural Integrity Analysis 1. STRESS CONCENTRATION Igor Kokcharov 1.1 STRESSES AND CONCENTRATORS 1.1.1 Stress An applied external force F causes inner forces in the carrying structure. Inner forces
More informationSTRESSSTRAIN RELATIONS
STRSSSTRAIN RLATIONS Strain Strain is related to change in dimensions and shape of a material. The most elementar definition of strain is when the deformation is along one ais: change in length strain
More informationStochastic Doppler shift and encountered wave period distributions in Gaussian waves
Ocean Engineering 26 (1999) 507 518 Stochastic Doppler shift and encountered wave period distributions in Gaussian waves G. Lindgren a,*, I. Rychlik a, M. Prevosto b a Department of Mathematical Statistics,
More informationBlasius solution. Chapter 19. 19.1 Boundary layer over a semiinfinite flat plate
Chapter 19 Blasius solution 191 Boundary layer over a semiinfinite flat plate Let us consider a uniform and stationary flow impinging tangentially upon a vertical flat plate of semiinfinite length Fig
More informationFoundation Engineering Prof. Mahendra Singh Department of Civil Engineering Indian Institute of Technology, Roorkee
Foundation Engineering Prof. Mahendra Singh Department of Civil Engineering Indian Institute of Technology, Roorkee Module  03 Lecture  09 Stability of Slopes Welcome back to the classes of on this Stability
More informationMechanical Properties of Metals Mechanical Properties refers to the behavior of material when external forces are applied
Mechanical Properties of Metals Mechanical Properties refers to the behavior of material when external forces are applied Stress and strain fracture or engineering point of view: allows to predict the
More informationDIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION
1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding
More informationPOWER SCREWS (ACME THREAD) DESIGN
POWER SCREWS (ACME THREAD) DESIGN There are at least three types of power screw threads: the square thread, the Acme thread, and the buttress thread. Of these, the square and buttress threads are the most
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationA wave lab inside a coaxial cable
INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 25 (2004) 581 591 EUROPEAN JOURNAL OF PHYSICS PII: S01430807(04)76273X A wave lab inside a coaxial cable JoãoMSerra,MiguelCBrito,JMaiaAlves and A M Vallera
More information1 of 10 11/23/2009 6:37 PM
hapter 14 Homework Due: 9:00am on Thursday November 19 2009 Note: To understand how points are awarded read your instructor's Grading Policy. [Return to Standard Assignment View] Good Vibes: Introduction
More informationSubminiature Load Cell Model 8417
w Technical Product Information Subminiature Load Cell 1. Introduction... 2 2. Preparing for use... 2 2.1 Unpacking... 2 2.2 Using the instrument for the first time... 2 2.3 Grounding and potential connection...
More informationPHYSICAL QUANTITIES AND UNITS
1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them
More information3 Concepts of Stress Analysis
3 Concepts of Stress Analysis 3.1 Introduction Here the concepts of stress analysis will be stated in a finite element context. That means that the primary unknown will be the (generalized) displacements.
More informationMCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements
MCE380: Measurements and Instrumentation Lab Chapter 9: Force, Torque and Strain Measurements Topics: Elastic Elements for Force Measurement Dynamometers and Brakes Resistance Strain Gages Holman, Ch.
More informationLecture L19  Vibration, Normal Modes, Natural Frequencies, Instability
S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19  Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free
More informationCH 6: Fatigue Failure Resulting from Variable Loading
CH 6: Fatigue Failure Resulting from Variable Loading Some machine elements are subjected to static loads and for such elements static failure theories are used to predict failure (yielding or fracture).
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationCOMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS
International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, MarchApril 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2
More informationGas Dynamics Prof. T. M. Muruganandam Department of Aerospace Engineering Indian Institute of Technology, Madras. Module No  12 Lecture No  25
(Refer Slide Time: 00:22) Gas Dynamics Prof. T. M. Muruganandam Department of Aerospace Engineering Indian Institute of Technology, Madras Module No  12 Lecture No  25 PrandtlMeyer Function, Numerical
More informationREVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More information