Room Number: SF 12
Phone: +91 80 2293 3250
Fax: +91 80 2360 8659
Department of Mechanical Engineering (Professor)
- MEMS and NEMS Sensors
- Transduction Targeted Material Development
- Energy Dissipation in Oscillations of Micro & Nano Scale Structures
- Vibratory Mechanobiology
- PhD, Cornell University, Ithaca, NY, USA, 1993
- MS, University of Arizona, Tucson, USA, 1987
- BTech (Hons), I.I.T. Kharagpur, India, 1985
- Professor, IISc., Bangalore 2008 – Present
- Associate Professor, IISc, Bangalore 2002 – 2008
- Assistant Professor, IISc, Bangalore 1996 – 2002
- Lecturer, Cornell University, USA, 1994 – 1996
Prof. Rudra Pratap’s research group is engaged in design and realization of small scale transducers and sensors using MEMS and NEMS technology. Their search for better and more efficient designs has led them to look at small scale sensors and transducers used by nature. Nature’s designs have been optimized over a very long period and are usually so elegant that man made systems for similar functions look so rudiemntary. Irrespective of whether the natural designs can be copied or not, their study and the revelation of their design is often a thing of sheer beauty and their understanding a source of immense joy. Prof. Pratap’s group has been recently engaged in two particular studies that have led to remarkable insights into nature’s design. A very brief description of these studies follows.
Bioacoustics of Field Crickets:
Field crickets produce incredibly loud songs using just their wings. The wings contain everything that the cricket needs to convert a low frequency (about 30 Hz) inaudible stridulation of the wing into a loud (~80 dB at 10 cm), high frequency (3 – 6 KHz) song. Prof. Pratap’s group, in collaboration with Prof. Rohini Balakrishnan’s group from the Centre for Ecological Sciences, has worked on unraveling the entire song production mechanism by studying and modelling all parts of the wing involved in the song production, and with careful finite element modelling, demonstrated how the song is produced by the cricket. Their study has also unravelled the basic design template used by nature for allocating distinct frequency bands to different species of field crickets and the ‘costs’ involved in moving away from the overall frequency band occupied by the most common field crickets. The underlying design principle is already bening exploited by Prof. Pratap’s group in designing new MEMS speakers.
Figure 1: (a) The forewing of a field cricket—G. bimaculatus, showing the different features of the forewing involved in the sound production. (b) The digitised coordinates along the boundary of the sound producing structure, called the harp. (c) The finite element model of the harp meshed using shell elements. (d) The computed first mode shape of the harp. (e) The recorded song of a cricket. (f) The song of the cricket recreated using the simulated response of the model to a train of impulses.
Soldier Fly Halteres as Gyroscopes:
Gyroscopes are sensors that sense the rate of rotation of a body. Gyrospoes are used for motion stabilization in ships, automobiles, aeroplanes, rockets, satellites, etc. Gyroscopes are now increasingly being used in cell phones, tablets, laptops, and gaming devices. Naturally, the demand for very small gyroscopes, particularly those using MEMS technology, has been growing exponentially. While the first two generations of MEMS gyroscopes are already under commercial production and use, the basic design of MEMS gyroscopes has not changed over the last two decades. It is this search for a new design that has led Prof. Pratap’s research group to look at nature for inspiration. They have been studying how certain dipteran insects seem to use a pair of very small organs called halteres for their flight stabilization, presumably using halteres as gyroscopes. Prof. Pratap’s research group, in collaboration with Prof. Sanjay Sane’s group at the National Centre for Biological Sciences, has studied the structure and motion of the haltere in great details, creating a model of the haltere that is used to simulate and show how the haltere is used for sensing the body rotations of the insect. This study shows that the design of the haltere is radically different from all existing MEMS gyroscopes. It is hoped that the understanding of the haltere design will lead to a new, elegant design of MEMS gyroscopes.
Figure 2 (a): Haltere of a Soldier fly, (b) Magnified view of the haltere, (c) Camapaniform sensilla at the base that encodes the rate of rotations, and (d)Proposed model of the haltere—it behaves like a rigid body in the actuation plane and like an elastic structure in the sensing plane.
We design and develop dynamics MEMS and NEMS devices with applications in inertial sensing (MEMS gyros, MEMS resonators), acoustics, (MEMS microphones) and ultrasonics (capacitive and piezo electric ultrasound transducers, also known as CMUTs and PMUTs). Our focus is on exploiting appropriate transduction mechanisms that are favourable at these small scales and designing low power and high sensitivity sensors that can impart intelligence to manmade objects. All the sensors we design and develop use extremely small structures that oscillate in response to some mechanical stimulus. Such structures necessarily require elastic and inertial elements that are suspended over a substrate with appropriate clearance to enable oscillations.
The dimensions involved and the compatibility of the fabrication process with CMOS make it ideal to combine these structures with electronics devices such as FETs. We are working on suspended Gate FETs where the transistor gate is a micromechanical structure capable of responding to a mechanical stimulus and causing substantial change in the drain current of the transistor. This work is in collaboration with Prof. Navakanta Bhat and the goal is to develop a platform technology for extremely sensitive dynamic displacement and pressure sensing.
Piezoelectric and piezoresistive materials are likely to play a major role in the development of micro and nano scale sensors and actuators. The transduction mechanism they provide for coupling mechanical and electrical energy domains is ideal for small scale transducers. A piezoresistor with low impedence, high gauge factor, and CMOS process compatability can lead to the development of a host of self-sensing structures. Similarly, CMOS process compatible piezoelectric materials with high coupling coefficients can help tremendously in the development of energy harvesters that can, in turn, make MEMS and NEMS sensor systems autonomous.
It is this theme that guides our work on the development of such materials. We are pursuing controlled electromigration induced nanoscale perturbations in structural topology of metallic films in order to enhance their piezoresistive sensitivity. This work has opened new frontiers in the dynamics of phase transformation and material transport at very small scales. Prelimiary studies indicate exciting possibilities in electric field mediated nanoscale material transport. We are also developing ZnO nanostructures with very high piezoelectric coupling coefficients for energy harvesting as the intended application.
Most of the devices and structures we design and develop exploit the dynamic response of some structural element for their essential operation. In studying these devices, one particular aspect of their response that has fascinated us the evaluation of energy dissipation in their micro and nano scale motions. Any dissipation of energy from the oscillating structure directly impacts its quality factor, Q. With so much interest in developing high-Q MEMS and NEMS devices, it becomes imperative to have a good understanding of energy transport from such oscillating structures.
It turns out that in most cases, the fluid- structure interaction with the surrounding fluid (mostly air) is responsible for maximum energy dissipation. In particular, we have been investigating squeeze film damping and acoustic radiation losses, and their effect on the Q of MEMS devices. Squeeze film damping refers to energy dissipation in fluid flow caused by “squeezing” of a very thin film of air or some other gas trapped between an oscillating mechanical structure and a fixed substrate.
We have been interested in understanding squeeze film flow and associated energy dissipation in all kinds of flow conditions – from continuum to molecular flow. We are also working on understanding acoustic radiation from these tiny structures and the associated energy losses particularly relevant to 2-D structures.
Our studies of MEMS and NEMS devices have led our attention to some of the small scale sensors and transducers that nature exploits. Nature’s design is particularly very elegant and most of the times awe inspiring.
We, in collaboration with colleagues from biology, are studying the mechanics of insect hateres as gyroscopic sensors, cricket harp as an efficient sound radiator and cell oscillations for signals of cell pathology. These are all incredibly complex and fascinating “devices” whose understanding has a huge bearing on our design of micro and nano scale systems.
S. D. Vishwakarma, A. K. Pandey, J. M. Parpia, D. R. Southworth, H. G. Craighead, and Rudra Pratap, “Evaluation of Mode Dependent Fluid Damping in a High Frequency Drumhead Microresonator”, (to appear),Journal of Microelectromechanical Systems, 2013.
S. Talukdar, P. Kumar, and Rudra Pratap, “Electric Current Induced Mass Flow in Very Thin Infinite Metallic Films”, IEEE Transactions on Electron Devices, Vol 60, No. 9, pp 2877-2883, September 2013.
Thejas, Navakanta Bhat,Rudra Pratap and K. N. Bhat,“Fringe Field Junctionless FET as a Sensitive Displacment Sensor”, Journal of ISSS, Vol. 2, No. 1, April 2013.
F. Jiang, A. Keating, M. Martyniuk, Rudra Pratap, L. Faraone and J. M. Dell, “Process control of cantilever deflection for sensor application based on optical waveguides”, Journal of Microelectromechanical Systems, Vol. 22, No. 3, pp 569-579, 2013.
S. M. Mohanasundaram, R. Pratap and Arindam Ghosh, “A cantilever resonator with integrated actuation and sensing fabricated using a single step lithography”, IEEE Sensors Journal, Vol. 13, No. 2, February 2013.
M. Mohanasundaram, Rudra Pratap and Arindam Ghosh, “Two orders of magnitude increase in metal piezoresistor sensitivity through nanoscale inhomogenization”, Journal of Applied Physics, Vol. 112, Issue 8, pp 084332-9, 2012.
1. Kaveri Rajaraman, Vamsy Godthi, Rudra Pratap, and Rohini Balakrishnan, “A novel acoustic-vibratory multimodal duet”, Journal of Experimental Biology, (accepted) to appear, 2015.
2. Charanjeet Kaur Malhi and Rudra Pratap, “On the Equivalence of Acoustic Impedance and Squeeze Film Impedance in Micromechanical Resonators”, Journal of Vibration and Acoustics, ASME, (accepted) to appear, 2015.
3. Charanjeet Kaur Malhi and Rudra Pratap, “On the method of extraction of lumped parameters for the radiation impedance of complex radiator geometries”, Microsystem Technologies, DOI 10.1007/s00542-015-2614-4, 2015.
4. Santanu Talukder, Praveen Kumar and Rudra Pratap, “Controlled material transport and multidimensional patterning at small length scales using electromigration”, Current Science, Vol. 108, No. 12, 25 June 2015.
5. Vamsy Godthi, K. Jayapraksh Reddy, and Rudra Pratap, “A Study of Pressure Dependent Squeeze Film Stiffness as a Resonance Modulator using Static and Dynamic Measurements”, Journal of Microelectromechanical Systems, DOI 10.1109/JMEMS.2015.2431633 2015.
6. Vamsy Godthi and Rudra Pratap, “Dynamics of the Cricket Sound Production”, in press, Journal of Vibration and Acoustics, ASME, doi:10.1115/1.4030090, 2015.
7. Kumar, Shishir; Kaushik, Swati; Pratap, Rudra; Raghavan, Srinivasan, “Graphene on Paper: A simple, low-cost chemical sensing platform”, ACS Applied Materials & Interfaces, Vol. 7(4), pp 2189-2194, 2015.
8. A. Roychowdhury. A. Nandy, C. S. Jog, and R. Pratap, “Hybrid elements for modeling squeeze film effects coupled with structural interactions in vibratory MEMS devices”, CMES: Computer Modeling in Engineering and Sciences, Vol. 103, No. 2, pp 91-110, 2014.
9. S. Rammohan, Sanketh Chiplunkar, C. M. Ramya, and Rudra Pratap, “Performance Enhancement of Piezoelectric Energy Harvesters using Multilayer and Multistep Beam Configurations”, IEEE Sensors Journal, Vol.15, No. 6, pp 3338-3348, June 2015.
10. Parween R. and Pratap R., “Modelling of Soldier Fly Halteres for Gyroscopic Oscillations”, BiologyOpen, Vol. 3, No. 13, Dec. 2014.
11. A. Roychowdhury, S. Patra, A. Nandy, and R. Pratap, “Analytical and numerical modeling of the effects of variable flow boundaries on the squeeze film behaviour in MEMS, Journal of ISSS, Vol. 3, No. 2, pp 26-38, Sept. 2014.
12. Shishir Kumar, George S. Duesberg, Rudra Pratap, Srinivasan Raghavan, “Graphene Field Emission Devices”, Applied Physics Letters, Vol. 105, 103107, 2014.
13. Rizuwana Parween, Rudra Pratap, Tanvi Deora, and Sanjay P. Sane, “Modeling Strain Sensing by the Gyroscopic Halteres in the Dipteran Soldier Fly”, Mechanics Based Design of Structures and Machines, Vol. 42, No. 3, 2014.
14. Santanu Talukder, Praveen Kumar, and Rudra Pratap, “Film Thickness Mediated Transition in the Kinetics of Electric Current Induced Flow of Thin Liquid Metal Films”, Applied Physics Letters, Vol. 104, 214102, May 2014.
15. S. D. Vishwakarma, A. K. Pandey, J. M. Parpia, D. R. Southworth, H. G. Craighead, and R. Pratap, “Evaluation of Mode Dependent Fluid Damping in a High Frequency Drumhead Microresonator”, Journal of Microelectromechanical Systems, Vol. 23, No. 2, pp 234-246, 2014.
16. B. Krishna Bharadwaj, Rudra Pratap and Srinivasan Raghavan, “Transfer free suspended graphene devices on silicon using electrodeposited copper”, Journal of Vacuum Science and Technology B, Vol. 32, 010603, 2014.
17. S. Rammohan, C. M. Ramya, S. Jayanth Kumar, Anjana Jain and Rudra Pratap, “Low frequency vibration energy harvesting using arrays of PVDF piezoelectric bimorphs”, Journal of ISSS, Vol. 3, No. 1, pp 18-27, March 2014.
18. A. Roychowdhury, A. Nandy, C. S. Jog, and R. Pratap, “A Monolithic FEM-Based Approach for the Coupled Squeeze Film Problem of an Oscillating Elastic Micro-Plate Using 3D 27-Node Elements”, Journal of Applied Mathematics and Physics, Vol 1, No 6, pp 20-25, 2013.
19. S. Talukdar, P. Kumar, and R. Pratap, “Electric Current Induced Mass Flow in Very Thin Infinite Metallic Films”, IEEE Transactions on Electron Devices, Vol 60, No. 9, pp 2877-2883, September 2013.
20. Thejas, Navakanta Bhat, Rudra Pratap, and K. N. Bhat, “Fringe Field Junctionless FET as a Sensitive Displacment Sensor”, Journal of ISSS, Vol. 2, No. 1, March 2013.
21. F. Jiang, A. Keating, M. Martyniuk, R. Pratap, L. Faraone and J. M. Dell, “Process control of cantilever deflection for sensor application based on optical waveguides”, Journal of Microelectromechanical Systems, Vol. 22, No. 3, pp 569-579, 2013.
22. S. M. Mohanasundaram, Rudra Pratap, and Arindam Ghosh, “A cantilever resonator with integrated actuation and sensing fabricated using a single step lithography”, IEEE Sensors Journal, Vol. 13, No. 2, February 2013.
23. S. M. Mohanasundaram, Rudra Pratap, and Arindam Ghosh, “Two orders of magnitude increase in metal piezoresistor sensitivity through nanoscale inhomogenization”, Journal of Applied Physics, Vol. 112, Issue 8, pp 084332-9, 2012.
24. Santanu Talukder, Arindam Ghosh and Rudra Pratap, “Nanoscale Control of Electro-migration for Resistance Tuning of Metal Lines”, Journal of ISSS, Vol. 1, No. 1, pp 16–22, 2012.
25. K. Jayaprakash Reddy, K. N. Bhat and Rudra Pratap, “Stiction Free Fabrication of MEMS Devices with Shallow Cavities Using a Two-Wafer Anodic Bonding Process”, Journal of ISSS, Vol. 1, No. 1, pp 1–9, 2012.
26. S. M. Mohanasundaram, Rudra Pratap, and Arindam Ghosh, “Tuning the sensitivity of a metal-based piezoresistive sensor using electromigration”, Journal of Microelectromechanical Systems, Vol. 21, No. 6, pp 1276-1278, December 2012.
27. S. M. Mohanasundaram, Rudra Pratap, and Arindam Ghosh, “Electromigration: A Unique Tool for Microstructure Engineering in Metal Films”, International Journal of Applied Physics and Mathematics, Vol. 2, No. 3, pp 2426–2431, 2012.
28. Babar Ahmad and Rudra Pratap, “Analytical Evaluation of Squeeze Film Forces in a CMUT with Sealed Air-filled Cavity “, IEEE Sensors Journal, Vol. 11, No. 10, pp 2426–2431, 2011.
29. Babar Ahmad and Rudra Pratap, “The effect of evacuated backside cavity on the dynamic characteristics of a capacitive micromachined ultrasound transducer”, International Journal of Advances in Engineering Sciences and Applied Mathematics, Vol. 2 (1-2), pp 50-54, 2010.
30. Babar Ahmad and Rudra Pratap, “Elasto-Electrostatic Analysis of Circular Microplates Used in Capacitive Micromachined Ultrasonic Transducers”, IEEE Sensors Journal, Vol. 10, Issue 11, pp 1767–1773, Nov.~2010.
31. Jaya Thakur, Rudra Pratap, Yannick Fournier, Thomas maeder and Peter Ryser, “Realization of a Solid Propellent based Microthruster Using Low Temperature Co-fired Ceramics”, Sensors & Transducers Journal, Vol. 117, Issue 6, pp 29–40, 2010.
32. Balaji Jayaramana, Navakanta Bhat and Rudra Pratap, “Thermal characterization of microheaters from the dynamic response”, Journal of Micromechanics and Microengineering, Vol. 19, No. 8, (085006), 2009.
33. K P Venkatesh, Nishad Patil, Ashok Kumar Pandey and Rudra Pratap, “Design and characterization of an in-plane MEMS yaw rate sensor”, Sadhana: Academy Proceedings in Engineering Sciences, Vol. 34, Part 4, pp 633–642, Aug 2009.
34. Balaji Jayaramana, Navakanta Bhat and Rudra Pratap, “Thermal Analysis of Microheaters using Mechanical Dynamic Response”, International Journal of Micro and Nano Systems, Vol. 1, No. 1, pp 15–20, June 2009.
35. A K Pandey, K P Venkatesh, and Rudra Pratap, “Effect of metal coating and residual stress on the resonant frequency of MEMS resonators”, Sadhana: Academy Proceedings in Engineering Sciences, Vol. 34, Part 4, pp 651–662, Aug 2009.
36. K P Venkatesh and Rudra Pratap, “Capturing Higher Modes of Vibrations of Micromachined Resonators “, Journal of Physics: Conference Series, Vol. 181, No. 1, pp 012079, 2009.
37. K A Lohar, Venkatesh K P and Rudra Pratap, “Effect of Process Induced Variations on Performance Characteristics of a Dual Mass Vibratory MEMS Gyroscope”, International Journal of Micro and Nano Systems, Vol. 1, No. 1, pp 57–63, June 2009.
38. Ashok Kumar Pandey and Rudra Pratap, “Modelling the Effect of Residual Stress and Perforations on the Dynamic Characteristics of MEMS Devices”, Advances in Vibration Engineering, Vol. 8(1), pp 17–26, 2009.
39. Ashok Kumar Pandey and Rudra Pratap, “A Semi-Analytical Model for Squeeze-Film Damping Including Rarefaction in a MEMS Torsion Mirror with Complex Geometry”, IOP Journal of Micromechanics and Microengineering, Vol. 18, No. 10, (105003), 2008.
40. Suhas S. Mohite, Venkata R. Sonti, and Rudra Pratap, “A Compact Squeeze-film Model including Inertia, Compressibility and Rarefaction Effects for Perforated 3D MEMS Structures”, Journal of Microelectromechanical Systems, Vol. 17, No. 3, pp. 709-723, June 2008.
41. Anil Arora, Ram Gopal, V.K. Dwivedi, Chandra Shekar, Babar Ahmad, Rudra Pratap, and P.J. George, “ Fabricating Capacitive Micromachined Ultrasonic Transducers with Wafer Bonding Technique”, Journal of Sensors & Transducers, Vol. 93, Issue 6, pp. 15-20, June 2008.
42. Ashok Kumar Pandey and Rudra Pratap, “Effect of Flexural Modes on Squeeze Film Damping In MEMS Cantilever Resonators”, Journal of Micromechanics and Microengineering, Vol. 17(12): pp. 2475-2484, 2008.
43. Ashok Kumar Pandey and Rudra Pratap, “Modelling the Effect of Residual Stress and Perforations on the Dynamic Characteristics of MEMS Devices”, Advances in Vibration Engineering Vol. 8(1), pp 17-26, 2009.
44. A. K. Pandey, Rudra Pratap and Fook Siong Chau, “Effect of Pressure on Fluid Damping In MEMS Torsional Resonators with Flow Ranging from Continuum to Molecular Regime”, Experimental Mechanics, Vol. 48, No. 1, pp 91-106, February 2008.
45. A. K. Pandey and R. Pratap, “A Comparative Study of Analytical Squeeze-Film Damping Models in Perforated MEMS Structures with Experimental Results”, Microfluidics and Nanofluidics, Vol. 4, No. 3, pp 205-218, March 2008.
46. Ashok Kumar Pandey, Rudra Pratap, and Fook Siong Chau, “Influence of Boundary Conditions on the Dynamic Characteristics of Squeeze Films in MEMS Devices”, IEEE/ASME Journal of MEMS, Vol. 16, Issue 4, pp. 893-903, August 2007.
47. R. Pratap and A. Arunkumar, “Material Selection for MEMS Devices”, Indian Journal of Pure and Applied Physics, Vol. 45, No. 4, pp 358-367, 2007.
48. R. Pratap, S. Mohite and A. K. Pandey, “Squeeze-Film Effects in MEMS Devices”, review paper, Journal of Indian Institute of Science, Vol. 87:1, pp 75-94, 2007.
49. A. K. Pandey, R. Pratap and Fook Siong Chau, “Analytical Solution of Modified Reynolds Equation for Squeeze Film Damping in Perforated MEMS Structures,” Sensors and Actuators A, Vol. 135, pp 839-848, 2007.
50. S. S. Mohite, Nishad Patil and R. Pratap, “Design, Modelling and Simulation of Vibratory Micromachined Gyroscopes”, Journal of Physics, Vol. 34, pp 757-763, 2006.
51. S. K. Jalan and Rudra Pratap, “Design of LTCC Micromachined Vibratory Rate Gyroscope Through Finite Element Analysis”, Advances in Vibration Engineering, Vol. 5(3), pp 233-240, 2006.
52. S. Kumar and R. Pratap, “Partitioning Design Space for Linear Tuning of Natural Frequencies in Planar Dynamic MEMS Structures”, Sensors and Actuators A: Physical, Vol. 125, Issue 2, pp 304-312, 10 January 2006.
53. S. S. Mohite, V. Haneesh Kesari, R. Sonti and R. Pratap, “Analytical Solutions for the Stiffness and Damping Co-efficients of Squeeze Films in MEMS Devices Having Perforated Back Plate”, Journal of Micromechanics and Microengineering, Vol. 15, pp 2083-2092, 2005.
54. S. S. Mohite, V. R. Sonti and R. Pratap, “A Comparative Study of the Equivalent Circuits for MEMS Capacitive Microphones and a Critical Evaluation of the Equivalent Parameters”, Advances in Vibration Engineering, Vol. 4(4), pp 337-349, 2005.
55. C. Venkatesh, S. Patil, N. Bhat, and R. Pratap, “A Torsional MEMS Varactor with Wide Dynamic Range and Low Actuation Voltage”, Sensors and Actuators A: Physical, Vol. 121(2), pp 480-487, 2005.
56. A. Sarkar, V. R. Sonti and R. Pratap, “A Coupled FEM-BEM Formulation in Structural Acoustics for Imaging a Material Inclusion”, Journal of Acoustics and Vibration, Vol. 10(1), pp 3-16, 2005.
57. A. K. Pandey, and R. Pratap, “Coupled Nonlinear Effects of Surface Roughness and Rarefaction on Squeeze Film Damping in MEMS Structures”, Journal of Micromech. Microeng. Vol. 14, pp 1430-1437, 2004.
58. S. Jagan Mohan, and R. Pratap, “A Natural Classifications of Vibration Models of Polygonal Ducts Based on Group Theoretic Analysis”, Journal of Sound and Vibration, Vol. 269, pp 745-764, 2004.
59. A. K. Pandey, and R. Pratap, “Studies in Nonlinear Effects of Squeeze Film Damping in MEMS Structures”, International Journal of Computational Engineering Science, Vol. 4, No. 3, pp 477-480, 2003.
60. S. Patil, C. Venkatesh, N. Bhat, and R. Pratap, “Voltage Controlled Oscillator using Tunable MEMS Resonator”, International Journal of Computational Engineering Science, Vol. 4, No. 3, pp 675-678, 2003.
61. Reddy, C. K and R. Pratap, “Multimodal Map and Complex Basin of Attraction of a Simple Hopper”, Physical Review E, Vol. 68(1), pp 16220-8, 2003.
62. S J Mohan, and R Pratap, “A Group Theoretic Approach to the Linear Free Vibration Analysis of Shells with Dihedral Symmetry”, Journal of Sound and Vibration, Vol. 252, No. 2, pp 317-341, 2002.
63. A Chatterjee, R Pratap, C K Reddy and A Ruina, “Persistent Passive Hopping and Juggling is Possible Even with Plastic Collisions”, International Journal of Robotics Research, Vol. 21, No.7, July 2002.
64. C K Reddy and R Pratap, “Can a Hopper Hop for Ever?”, Current Science, Vol. 79 No. 5, pp 639-645, 2000.
65. C K Reddy and R Pratap, “Equivalent Viscous Damping for a Bilinear Hysteretic Oscillator”, ASEE Journal of Engineering Mechanics, Vol. 126, No. 11, pp 1189-1196, 2000.
66. S Kulkarni and R Pratap, “Studies on the Dynamics of a Supercavitating Projectile”, Applied Mathematics Modelling, Vol. 24, pp 113-119, 2000.
67. J Judge and R. Pratap, “Asymptotic States of a Bilinear Hysteretic Oscillator in a Fully Dissipative Phase Space”, Journal of Sound and Vibration , Vol. 218(3), 1998, pp 548-557.
68. R Pratap and P J Holmes, “Chaos in a Mapping Describing Elastoplastic Oscillations”, Nonlinear Dynamics, Vol.18, pp 111-139, 1995.
69. R Pratap, S Mukherjee, and F C Moon, “Dynamics Behaviour of a Bilinear Hysteretic Elastoplastic Oscillator, Part I: Free Oscillations”, Journal of Sound and Vibration, Vol. 172(3), pp 321-338, 1994.
70. R Pratap, S Mukherjee and F C Moon, “Dynamics Behaviour of a Bilinear Hysteretic Elastoplastic Oscillator, Part II: Free Oscillations”, Journal of Sound and Vibration, Vol. 172(3), pp 321-338, 1994.
71. R Pratap, S Mukherjee and F C Moon, “Limit Cycles in an Elastoplastic Oscillator, Part I: Free Oscillations”, Physics Letters A, Vol.170, No. 5,pp 384-392, 1992.
72. R Pratap and T Kundu, “A Least Square Finite Element Formulation for Elastodynamic Problems”, International Journal of Numerical Methods in Engineering, Vol. 26, pp. 1883-1891, 1988.