SOLVING DYNAMICS OF QUIET STANDING AS NONLINEAR POLYNOMIAL SYSTEMS
|
|
- Belinda Brown
- 5 years ago
- Views:
Transcription
1 SOLVING DYNAMICS OF QUIET STANDING AS NONLINEAR POLYNOMIAL SYSTEMS Zhiming Ji Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey 070 Abstract Many problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations, as is succinctly pointed out by Raghavan and Roth (995. Developing methods for solving sets of nonlinear polynomial equations represents one of Bernard Roth s many significant contributions towards establishment of theoretical basis for the mechanisms analysis and synthesis and the mechanical aspects of robotics. In this paper, results from studying the human postural stability during quiet standing are discussed to show that some problems in biomechanical systems may also lead to systems of nonlinear polynomial equations. Keywords: Systems of polynomial equations, Postural stability, Biomechanical systems.. Introduction Postural control is coordinated by the central nervous system with input from three systems: visual, vestibular, and somatosensory (collectively known as the proprioceptive system. Body sway is used as an indicator of postural stability. Falls due to impaired postural control present a serious health hazard to elderly as well as to persons with balance disorder (for examples, deficits of the proprioceptic system or muscle weakness, and diminish a person's ability to perform activities of daily living. The Sensory Organization Test (SOT, Motor Control Test (MCT and Adaptation Test (ADT form the core battery of tests, recognized as Computerized Dynamic Posturography (CDP, for diagnosing the functional impairments underlying balance disorders. Most Computerized Dynamic Posturography systems qualify postural stability based on force plate technology. These systems measure the ground reaction force with transducers on force plate to determine the center of pressure (COP. Then they use the upward projection of the COP as an estimate for the body center of mass (COM. Different low-pass filters are used on the COP time series to remove the high frequency content in the resulted COP (Benda, Riley and Krebs, 994; Caron, Faure and Breniere, 997, based on the assumption that postural sway is quasi-static.
2 Another approach is to estimate COM with the second integral of horizontal acceleration, which is assumed to be proportional to the horizontal ground reaction force (Shimba, 984. This method requires the initial constants of integration and several techniques were suggested for estimating these integration constants (Zatsiorsky and King, 998. Winter et al (998 estimate COM based on their 4-segment model and the measurements of markers. This approach works well for research purpose but less practical for clinical applications. Most of the studies on quiet standing use the single-joint inverted pendulum model (Winter, Patla, Rietdyk and Ishac, 00; Morasso and Schieppati, 999 and only the moment equilibrium is considered as the system equation. By including the force equilibrium into the system equation, we can determine the COM directly from the measured ground reaction force, without using integration or filtering. We show that the resulted nonlinear differential equation can be solved as a set of nonlinear polynomial equations, even for quiet standing on inclined surface.. Equations of Motion for Quiet Standing A complete set of dynamic equilibrium equations can be easily derived to establish the relationship between sway movement and the ground reaction forces. Free body diagrams in Figures and illustrate the human inverted pendulum for sway in the sagittal plane. Figures shows the entire body excluding feet as inverted pendulum rotating about the ankle joint A. M is the mass of body above ankle, F, and F V are horizontal and vertical force acting H A at ankle joint, τ is moment acting at ankle joint by muscles and θ is absolute sway angle with respect to fixed vertical reference. Figure. Free body diagram of body (above ankle
3 Figures shows the feet together with the force plate. m is the total mass of the feet and the force plate; F F and front and rear transducers respectively; F R are ground reaction forces perpendicular to the force plate, measured with F H is ground reaction force parallel to the force plate, measured with transducer at pin join; d is the distance between to the pin axis and transducers that measures forces perpendicular to the force plate; e is the distance between ankle join and transducer that measuring force parallel to force plate; a is distance from the line through ankle and pin joints to the center of mass of the feet, θ m is sway angle of the center of mass relative to the line perpendicular to force plate, and φ is incline angle of the force plate. Figure. Free body diagram of feet with force plate If h is the distance between the ankle joint and the center of mass of the body and I is the moment of inertia of the body about the ankle joint, then the equation of motion for the body can be written as F ( &, θ cosθ & H A = M h θ sinθ ( F ( & θ sinθ & V = Mg Mh + θ cosθ ( τ + Mgh sin θ = I & θ (3 The equilibrium of the feet can be written as 3 ( F F φ F = F cosφ + + sin (4 H, A H F R
4 F V = ( F + F cosφ F sinφ mg (5 F R H τ = ( F F d + F e mga cosφ (6 F R H After eliminating all the internal forces and moments F,, F V, and τ from eqs. (-6, we H A have the following three differential equations for the postural system: Mh ( & θ cosθ & θ sinθ = F cosφ + ( F + F sinφ (7 H F Mh ( & θ sinθ + & θ cosθ = ( M + m g ( F + F cosφ + F sinφ (8 F I & θ = Mghsinθ ( F F d F e mga cosφ (9 F R H + R R H The ground reaction forces F F, F R and F H are induced by the sway motion. Once they are measured by the transducers embedded in the force plate, the unknown state θ, & θ, and & θ of the sway for a fixed inclination φ can be obtained by solving eqs. (7-9. Although & θ and & θ are derivatives of the sway θ (t, they can be treated as independent to each other. Let x = sinθ, x = cosθ, & x 3 = θ, and x = & θ 4, we convert the set of differential eqs. (7-9 into a set of polynomial equations a ( = a x x4 xx3 xx4 x x3 4 x a5x4 a6 + x = (0 a ( = a a = + ( 3 + ( x (3 where a = Mh, a = FH cosφ + ( FF + FR sinφ, a 3 = ( M + m g ( FF + FR cosφ + FH sinφ, a 4 = Mgh, a5 = I, a6 = ( FF FR d + FH e mga cosφ. The resulted polynomial system can be solved with one of the three well-known methods (Raghavan and Roth, 995: Dialytic elimination, Polynomial Continuation, and Grobner bases. Applying Sylvester's Dialytic Elimination, we obtain a quadratic equation of x as: [( a + a a + a a ] x a a ( a a + a a x + a a a a 0 a ( = 4
5 From the solution of equation (4, we can obtain the rest by formulas [ aa6 ( aa4 a3a5 x ] aa5 x = + (5 x 3 a3x a x = ( a (6 4 a3x a x x = ( + a (7 The sway angle can now be solved as θ = ATAN ( x, x. Note that the process only yields & x 3 = θ. Thus only the magnitude θ &, not the direction, of the angular velocity θ & is found. Therefore, this direction information must be derived from the time history of angle θ. For small sway, an approximation sin θ θ and cosθ may be used to simplify the solution. The ankle moment can be evaluated with eq. (6. With the computed sway angle and ankle moment, muscle stiffness during quiet standing can now be evaluated. 3. Application Examples We have been using EquiTest system (NeuroCom International, 00 to evaluate patients and subjects for balance and postural stability. This system consists of a movable dual platform, which is surrounded by a visual scene that can also rotate about the ankle joints. It quantifies the ground reaction force using five force transducers. The two force plates are connected by a pin joint and supported by four force transducers mounted symmetrically on a supporting plate. These four transducers measure forces perpendicular to the force plate. A fifth transducer is mounted to the supporting plate directly beneath the pin joint for measuring shear force. The force transducers are sampled at 00Hz, with a detection threshold about one Newton. The system estimates COM from COP using a moving average filter. We applied the derived formulas with measured ground reaction forces from several groups of subjects. The subjects body segment lengths and inertial parameters are calculated using anthropometric data taken from the literature (Dempster, 955; Miller and Nelson, 973; Le Veau, William and Lissner, 977, based on the subjects height and weight. The results for one of the subjects are presented here. The subject s height and weight are H =. 67 m and W = kgf, respectively. We find M = kg, m =. 65 kg, I = 85.0 kg m, h = m, d = m, e = m and a = m. The following plots are generated based on one of the trials (each trial lasts 0 seconds. 5
6 Figure 3 shows the computed COM ( y = h sinθ and the moving average reported by the Equitest system for the same trial. We can see clearly the smoothing effect of the moving average by comparing the computed and the reported COM. The system estimates h as 0.557H or h = 0. 93m, which is slightly smaller than the value we used. As a result, the reported COM curve is lower than our computed COM. Figure 3. Example of COM displacement Fig. 4 shows the result of a linear regression of ankle moment versus sway angle for the trial presented in Fig. 3. The resulted line is τ = θ ( N m with the coefficient of determination R = This indicates that the ankle stiffness closely resembles an ideal spring, which is in agreement with (Winter, Patla, Rietdyk and Ishac, 00. This computed ankle stiffness is very close to the value of Mgh, which is In this particular trial, the difference between the stabilizing ankle moment and the destabilizing gravity moment is less than % of the ankle moment, as shown in Fig. 5. The discrete nature of the force measurement can be seen from the plot of the net moment Mgh θ τ. 6
7 Figure 4. Example of linear regression between ankle moment and sway angle Figure 5. Example of net moment at the ankle 4. Conclusions The equations of motion for human quiet standing, when modeled as a single-joint inverted pendulum, form a set of nonlinear differential equations. By using the measured ground reaction force, this set of equations can be solved as a set of nonlinear polynomial equations. The resulted polynomial equations can be solved based on the work of Raghavan and Roth to obtain the COM 7
8 displacement during quiet standing. Such a solution can be used to evaluate ankle muscle stiffness. 5. Acknowledgements The work reported here was inspired by the scholarship of Bernard Roth and discussions with Thomas Findley and Hans Chaudhry. References Benda B. J., Riley P. O., and Krebs D. E., 994, Biomechanical relationship between the center of gravity and center of pressure during standing, IEEE Transactions on Rehabilitation Engineering, Vol., pp Caron O., Faure B. and Breniere Y., 997, Estimating the centre of gravity of the body on the basis of the centre of pressure in standing posture, Journal of Biomechanics, Vol. 30(/, pp Dempster W. T., 955, Space Requirements of the seated operator, Aero Medical Lab., Wright- Patterson Air Force Base, WADC TR Miller D., and Nelson R., 973, Biomechanics of sport, Lea & Febiger, Philadelphia. NeuroCom International Inc., 00, EquiTest System operator s manual, ver B. Le Veau, William and Lissner, 977, Biomechanics of human motion, nd. Ed., W. B. Saunders Co., Philadelphia. Morasso P.G., and Schieppati M., 999, Can muscle stiffness alone stabilize upright standing? J. Neurophysiol. Vol. 8(3, pp Raghavan M., and Roth B., 995, Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators, Special 50th Anniversary Design Issue, Trans. ASME 7, pp Shimba T., 984, An estimation of center of gravity from force platform data, Journal of Biomechanics, Vol. 7, pp Winter D. A., Patla A. E., Prince F., Ishac M., and Gielo-Perczak K., 998, Stiffness control of balance in quiet standing, J. Neurophysiol. Vol. 80 (3, pp.-. Winter D. A., Patla A.E., Rietdyk, S., and Ishac, M. G., 00, Ankle muscle stiffness in the control of balance during quiet standing, J. Neurophysiol. Vol. 85 (6, pp Zatsiorsky V. M. And King D. L.,998, An algorithm for determining gravity line location from posturographic recordings, Journal of Biomechanics, Vol. 3(, pp
Computational Method To Evaluate Ankle Postural Stiffness. With Ground Reaction Forces. Zhiming Ji* Department of Mechanical Engineering
! #"$% & '( *)+, -/. 0 '1' 3 4 )5 67 8 9: ; )
More informationA Biomechanical Model That Confirms Human Ankle Angle Changes Seen During Short Anterior Perturbations of a Standing Platform
A Biomechanical Model That Confirms Human Ankle Angle Changes Seen During Short Anterior Perturbations of a Standing Platform Rakesh Pilkar 1, 2, John Moosbrugger 3, Viprali Bhatkar 1, 2, Robert Schilling
More informationAdaptive Control of Human Posture in a Specific Movement
Journal of Informatics and Computer Engineering (JICE) Vol. 2(5), Oct. 216, pp. 195-199 Adaptive Control of Human Posture in a Specific Movement Seyyed Arash Haghpanah School of Mechanical Engineering,
More informationThree-Dimensional Biomechanical Analysis of Human Movement
Three-Dimensional Biomechanical Analysis of Human Movement Anthropometric Measurements Motion Data Acquisition Force Platform Body Mass & Height Biomechanical Model Moments of Inertia and Locations of
More informationBIOMECHANICS AND MOTOR CONTROL OF HUMAN MOVEMENT
BIOMECHANICS AND MOTOR CONTROL OF HUMAN MOVEMENT Third Edition DAVID Α. WINTER University of Waterloo Waterloo, Ontario, Canada WILEY JOHN WILEY & SONS, INC. CONTENTS Preface to the Third Edition xv 1
More informationKIN Mechanics of posture by Stephen Robinovitch, Ph.D.
KIN 840 2006-1 Mechanics of posture 2006 by Stephen Robinovitch, Ph.D. Outline Base of support Effect of strength and body size on base of support Centre of pressure and centre of gravity Inverted pendulum
More informationEE Homework 3 Due Date: 03 / 30 / Spring 2015
EE 476 - Homework 3 Due Date: 03 / 30 / 2015 Spring 2015 Exercise 1 (10 points). Consider the problem of two pulleys and a mass discussed in class. We solved a version of the problem where the mass was
More informationPresenter: Siu Ho (4 th year, Doctor of Engineering) Other authors: Dr Andy Kerr, Dr Avril Thomson
The development and evaluation of a sensor-fusion and adaptive algorithm for detecting real-time upper-trunk kinematics, phases and timing of the sit-to-stand movements in stroke survivors Presenter: Siu
More informationA model of a human standing. by Richard Denker. January 7, 2013
A model of a human standing by Richard Denker January 7, 2013 Analytical Mechanics (FYGC04), HT 2012 Faculty of Technology and Science Department of Physics Contents 1 Introduction 3 2 Theory 3 2.1 The
More informationMethod for stabilogram characterization using angular-segment function
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 61, No. 2, 2013 DOI: 10.2478/bpasts-2013-0038 Method for stabilogram characterization using angular-segment function J. FIOŁKA and Z.
More informationExam 3 Practice Solutions
Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at
More informationFigure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody
Lecture 27. THE COMPOUND PENDULUM Figure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody diagram The term compound is used to distinguish the present
More informationFriction and Motion. Prof. Paul Eugenio 13 Sep Friction (cont.) Motion: kinetics and dynamics Vertical jump Energy conservation
Friction and Motion Friction (cont.) Motion: kinetics and dynamics Vertical jump Energy conservation Ukulele means jumping flea Prof. Paul Eugenio 13 Sep 2018 Lecture 5 Reactive, Normal, and Friction Forces
More informationLecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws
Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,
More informationModels and Anthropometry
Learning Objectives Models and Anthropometry Readings: some of Chapter 8 [in text] some of Chapter 11 [in text] By the end of this lecture, you should be able to: Describe common anthropometric measurements
More informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting
More informationChapter 8. Centripetal Force and The Law of Gravity
Chapter 8 Centripetal Force and The Law of Gravity Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = s = rφ = Frφ Fr = τ (torque) = τφ r φ s F to s θ = 0 DEFINITION
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = rφ = Frφ Fr = τ (torque) = τφ r φ s F to x θ = 0 DEFINITION OF
More informationIn-Class Problems 30-32: Moment of Inertia, Torque, and Pendulum: Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 TEAL Fall Term 004 In-Class Problems 30-3: Moment of Inertia, Torque, and Pendulum: Solutions Problem 30 Moment of Inertia of a
More informationHow a plantar pressure-based, tongue-placed tactile biofeedback modifies postural control mechanisms during quiet standing
Exp Brain Res (2007) 181:547 554 DOI 10.1007/s00221-007-0953-9 RESEARCH ARTICLE How a plantar pressure-based, tongue-placed tactile biofeedback modifies postural control mechanisms during quiet standing
More informationSports biomechanics explores the relationship between the body motion, internal forces and external forces to optimize the sport performance.
What is biomechanics? Biomechanics is the field of study that makes use of the laws of physics and engineering concepts to describe motion of body segments, and the internal and external forces, which
More informationI B.TECH EXAMINATIONS, JUNE ENGINEERING MECHANICS (COMMON TO CE, ME, CHEM, MCT, MMT, AE, AME, MIE, MIM)
Code.No: 09A1BS05 R09 SET-1 I B.TECH EXAMINATIONS, JUNE - 2011 ENGINEERING MECHANICS (COMMON TO CE, ME, CHEM, MCT, MMT, AE, AME, MIE, MIM) Time: 3 hours Max. Marks: 75 Answer any FIVE questions All questions
More informationForces and Newton s Laws Reading Notes. Give an example of a force you have experienced continuously all your life.
Forces and Newton s Laws Reading Notes Name: Section 4-1: Force What is force? Give an example of a force you have experienced continuously all your life. Give an example of a situation where an object
More informationMechanical energy transfer by internal force during the swing phase of running
Available online at www.sciencedirect.com Procedia Engineering 34 (2012 ) 772 777 9 th Conference of the International Sports Engineering Association (ISEA) Mechanical energy transfer by internal force
More informationCHAPTER 4 NEWTON S LAWS OF MOTION
62 CHAPTER 4 NEWTON S LAWS O MOTION CHAPTER 4 NEWTON S LAWS O MOTION 63 Up to now we have described the motion of particles using quantities like displacement, velocity and acceleration. These quantities
More informationStiffness Control of Balance in Quiet Standing
Stiffness Control of Balance in Quiet Standing DAVID A. WINTER, 1 AFTAB E. PATLA, 1 FRANCOIS PRINCE, 2 MILAD ISHAC, 1 AND KRYSTYNA GIELO-PERCZAK 1 1 Department of Kinesiology, University of Waterloo, Waterloo,
More information» APPLICATIONS OF CONTROL
» APPLICATIONS OF CONTROL A PID Model of Human Balance Keeping KIMURA HIDENORI and YIFA JIANG The basic requirement of biped (two-legged) motion is apparatus records what we know today as the center of
More informationInternal models in the control of posture
PERGAMON Neural Networks 12 (1999) 1173 1180 Neural Networks www.elsevier.com/locate/neunet Internal models in the control of posture P.G. Morasso a,b, *, L. Baratto b, R. Capra b, G. Spada b a Department
More informationCHAPTER 12 OSCILLATORY MOTION
CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time
More informationPhysics 4A Solutions to Chapter 10 Homework
Physics 4A Solutions to Chapter 0 Homework Chapter 0 Questions: 4, 6, 8 Exercises & Problems 6, 3, 6, 4, 45, 5, 5, 7, 8 Answers to Questions: Q 0-4 (a) positive (b) zero (c) negative (d) negative Q 0-6
More informationStabilization of Motion of the Segway 1
Stabilization of Motion of the Segway 1 Houtman P. Siregar, 2 Yuri G. Martynenko 1 Department of Mechatronics Engineering, Indonesia Institute of Technology, Jl. Raya Puspiptek-Serpong, Indonesia 15320,
More informationCoordinating Feet in Bipedal Balance
Coordinating Feet in Bipedal Balance S.O. Anderson, C.G. Atkeson, J.K. Hodgins Robotics Institute Carnegie Mellon University soa,cga,jkh@ri.cmu.edu Abstract Biomechanical models of human standing balance
More informationChapter 12 Static Equilibrium
Chapter Static Equilibrium. Analysis Model: Rigid Body in Equilibrium. More on the Center of Gravity. Examples of Rigid Objects in Static Equilibrium CHAPTER : STATIC EQUILIBRIUM AND ELASTICITY.) The Conditions
More informationPHYS 1114, Lecture 33, April 10 Contents:
PHYS 1114, Lecture 33, April 10 Contents: 1 This class is o cially cancelled, and has been replaced by the common exam Tuesday, April 11, 5:30 PM. A review and Q&A session is scheduled instead during class
More informationChapter 8. Rotational Motion
Chapter 8 Rotational Motion The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of
More informationResearch Article On the Dynamics of the Furuta Pendulum
Control Science and Engineering Volume, Article ID 583, 8 pages doi:.55//583 Research Article On the Dynamics of the Furuta Pendulum Benjamin Seth Cazzolato and Zebb Prime School of Mechanical Engineering,
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.6 The Action of Forces and Torques on Rigid Objects Chapter 8 developed the concepts of angular motion. θ : angles and radian measure for angular variables ω :
More informationτ = F d Angular Kinetics Components of Torque (review from Systems FBD lecture Muscles Create Torques Torque is a Vector Work versus Torque
Components of Torque (review from Systems FBD lecture Angular Kinetics Hamill & Knutzen (Ch 11) Hay (Ch. 6), Hay & Ried (Ch. 12), Kreighbaum & Barthels (Module I & J) or Hall (Ch. 13 & 14) axis of rotation
More informationChapter 5 Gravitation Chapter 6 Work and Energy
Chapter 5 Gravitation Chapter 6 Work and Energy Chapter 5 (5.6) Newton s Law of Universal Gravitation (5.7) Gravity Near the Earth s Surface Chapter 6 (today) Work Done by a Constant Force Kinetic Energy,
More informationGeneral Physics I Spring Applying Newton s Laws
General Physics I Spring 2011 Applying Newton s Laws 1 Equilibrium An object is in equilibrium if the net force acting on it is zero. According to Newton s first law, such an object will remain at rest
More informationKinematics, Kinetics, Amputee Gait (part 1)
Kinematics, Kinetics, Amputee Gait (part 1) MCE 493/593 & ECE 492/592 Prosthesis Design and Control October 16, 2014 Antonie J. (Ton) van den Bogert Mechanical Engineering Cleveland State University 1
More informationSection 6: 6: Kinematics Kinematics 6-1
6-1 Section 6: Kinematics Biomechanics - angular kinematics Same as linear kinematics, but There is one vector along the moment arm. There is one vector perpendicular to the moment arm. MA F RMA F RD F
More informationPhysics 41 HW Set 1 Chapter 15 Serway 8 th ( 7 th )
Conceptual Q: 4 (7), 7 (), 8 (6) Physics 4 HW Set Chapter 5 Serway 8 th ( 7 th ) Q4(7) Answer (c). The equilibrium position is 5 cm below the starting point. The motion is symmetric about the equilibrium
More informationSOLUTION di x = y2 dm. rdv. m = a 2 bdx. = 2 3 rpab2. I x = 1 2 rp L0. b 4 a1 - x2 a 2 b. = 4 15 rpab4. Thus, I x = 2 5 mb2. Ans.
17 4. Determine the moment of inertia of the semiellipsoid with respect to the x axis and express the result in terms of the mass m of the semiellipsoid. The material has a constant density r. y x y a
More informationAP Physics. Harmonic Motion. Multiple Choice. Test E
AP Physics Harmonic Motion Multiple Choice Test E A 0.10-Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation. 1) Torque Produces angular
More informationPhysics 2001/2051 The Compound Pendulum Experiment 4 and Helical Springs
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar
More informationLecture 8. Torque. and Equilibrium. Pre-reading: KJF 8.1 and 8.2
Lecture 8 Torque and Equilibrium Pre-reading: KJF 8.1 and 8.2 Archimedes Lever Rule At equilibrium (and with forces 90 to lever): r 1 F 1 = r 2 F 2 2 General Lever Rule For general angles r 1 F 1 sin θ
More informationUNITS AND DEFINITIONS RELATED TO BIOMECHANICAL AND ELECTROMYOGRAPHICAL MEASUREMENTS
APPENDIX B UNITS AND DEFINITIONS RELATED TO BIOMECHANICAL AND ELECTROMYOGRAPHICAL MEASUREMENTS All units used are SI (Système International d Unités). The system is based on seven well-defined base units
More informationChapter 4 Dynamics: Newton s Laws of Motion
Chapter 4 Dynamics: Newton s Laws of Motion Force Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the Normal Force Applications
More information!T = 2# T = 2! " The velocity and acceleration of the object are found by taking the first and second derivative of the position:
A pendulum swinging back and forth or a mass oscillating on a spring are two examples of (SHM.) SHM occurs any time the position of an object as a function of time can be represented by a sine wave. We
More informationAppendix W. Dynamic Models. W.2 4 Complex Mechanical Systems. Translational and Rotational Systems W.2.1
Appendix W Dynamic Models W.2 4 Complex Mechanical Systems W.2.1 Translational and Rotational Systems In some cases, mechanical systems contain both translational and rotational portions. The procedure
More informationDynamic Optimization of the Sit-to-Stand Movement
Journal of Applied Biomechanics, 011, 7, 306-313 011 Human Kinetics, Inc. Dynamic Optimization of the Sit-to-Stand Movement Hiroshi R. Yamasaki, 1 Hiroyuki Kambara, and Yasuharu Koike 1 Showa University;
More information7. FORCE ANALYSIS. Fundamentals F C
ME 352 ORE NLYSIS 7. ORE NLYSIS his chapter discusses some of the methodologies used to perform force analysis on mechanisms. he chapter begins with a review of some fundamentals of force analysis using
More informationQUESTION TWO: BUNGY JUMPING. Acceleration due to gravity = 9.81 m s 2
6 QUESTION TWO: BUNGY JUMPING Acceleration due to gravity = 9.81 m s Standing on a platform that is 5.0 m above a river, Emma, of height.00 m and mass m, is tied to one end of an elastic rope (the bungy)
More information( )( ) ( )( ) Fall 2017 PHYS 131 Week 9 Recitation: Chapter 9: 5, 10, 12, 13, 31, 34
Fall 07 PHYS 3 Chapter 9: 5, 0,, 3, 3, 34 5. ssm The drawing shows a jet engine suspended beneath the wing of an airplane. The weight W of the engine is 0 00 N and acts as shown in the drawing. In flight
More information24/06/13 Forces ( F.Robilliard) 1
R Fr F W 24/06/13 Forces ( F.Robilliard) 1 Mass: So far, in our studies of mechanics, we have considered the motion of idealised particles moving geometrically through space. Why a particular particle
More informationChap. 10: Rotational Motion
Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics - Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Newton s Laws for Rotation n e t I 3 rd part [N
More informationA consideration on position of center of ground reaction force in upright posture
sice02-0206 A consideration on position of center of ground reaction force in upright posture Satoshi Ito ),2) Yoshihisa Saka ) Haruhisa Kawasaki ) satoshi@robo.mech.gifu-u.ac.jp h33208@guedu.cc.gifu-u.ac.jp
More information2. a) Explain the equilibrium of i) Concurrent force system, and ii) General force system.
Code No: R21031 R10 SET - 1 II B. Tech I Semester Supplementary Examinations Dec 2013 ENGINEERING MECHANICS (Com to ME, AE, AME, MM) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions
More informationExercises on Newton s Laws of Motion
Exercises on Newton s Laws of Motion Problems created by: Raditya 1. A pendulum is hanging on a ceiling of a plane which is initially at rest. When the plane prepares to take off, it accelerates with a
More informationBiomechanical Modelling of Musculoskeletal Systems
Biomechanical Modelling of Musculoskeletal Systems Lecture 6 Presented by Phillip Tran AMME4981/9981 Semester 1, 2016 The University of Sydney Slide 1 The Musculoskeletal System The University of Sydney
More informationUnit 7: Oscillations
Text: Chapter 15 Unit 7: Oscillations NAME: Problems (p. 405-412) #1: 1, 7, 13, 17, 24, 26, 28, 32, 35 (simple harmonic motion, springs) #2: 45, 46, 49, 51, 75 (pendulums) Vocabulary: simple harmonic motion,
More informationOscillatory Motion. Solutions of Selected Problems
Chapter 15 Oscillatory Motion. Solutions of Selected Problems 15.1 Problem 15.18 (In the text book) A block-spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and
More informationChapters 10 & 11: Rotational Dynamics Thursday March 8 th
Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Review of rotational kinematics equations Review and more on rotational inertia Rolling motion as rotation and translation Rotational kinetic energy
More informationPhys101 Lecture 5 Dynamics: Newton s Laws of Motion
Phys101 Lecture 5 Dynamics: Newton s Laws of Motion Key points: Newton s second law is a vector equation Action and reaction are acting on different objects Free-Body Diagrams Ref: 4-1,2,3,4,5,6,7. Page
More informationTheory of Vibrations in Stewart Platforms
Theory of Vibrations in Stewart Platforms J.M. Selig and X. Ding School of Computing, Info. Sys. & Maths. South Bank University London SE1 0AA, U.K. (seligjm@sbu.ac.uk) Abstract This article develops a
More information6. Find the net torque on the wheel in Figure about the axle through O if a = 10.0 cm and b = 25.0 cm.
1. During a certain period of time, the angular position of a swinging door is described by θ = 5.00 + 10.0t + 2.00t 2, where θ is in radians and t is in seconds. Determine the angular position, angular
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination
More informationCh 6 Using Newton s Laws. Applications to mass, weight, friction, air resistance, and periodic motion
Ch 6 Using Newton s Laws Applications to mass, weight, friction, air resistance, and periodic motion Newton s 2 nd Law Applied Galileo hypothesized that all objects gain speed at the same rate (have the
More informationStatic Equilibrium, Gravitation, Periodic Motion
This test covers static equilibrium, universal gravitation, and simple harmonic motion, with some problems requiring a knowledge of basic calculus. Part I. Multiple Choice 1. 60 A B 10 kg A mass of 10
More informationPractice. Newton s 3 Laws of Motion. Recall. Forces a push or pull acting on an object; a vector quantity measured in Newtons (kg m/s²)
Practice A car starts from rest and travels upwards along a straight road inclined at an angle of 5 from the horizontal. The length of the road is 450 m and the mass of the car is 800 kg. The speed of
More informationTorque. Introduction. Torque. PHY torque - J. Hedberg
Torque PHY 207 - torque - J. Hedberg - 2017 1. Introduction 2. Torque 1. Lever arm changes 3. Net Torques 4. Moment of Rotational Inertia 1. Moment of Inertia for Arbitrary Shapes 2. Parallel Axis Theorem
More informationPOTENTIAL ENERGY AND ENERGY CONSERVATION
7 POTENTIAL ENERGY AND ENERGY CONSERVATION 7.. IDENTIFY: U grav = mgy so ΔU grav = mg( y y ) SET UP: + y is upward. EXECUTE: (a) ΔU = (75 kg)(9.8 m/s )(4 m 5 m) = +6.6 5 J (b) ΔU = (75 kg)(9.8 m/s )(35
More informationThe content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam.
WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system is always
More informationPH201 Chapter 5 Solutions
PH201 Chapter 5 Solutions 5.4. Set Up: For each object use coordinates where +y is upward. Each object has Call the objects 1 and 2, with and Solve: (a) The free-body diagrams for each object are shown
More informationChapter 6: Work and Kinetic Energy
Chapter 6: Work and Kinetic Energy Suppose you want to find the final velocity of an object being acted on by a variable force. Newton s 2 nd law gives the differential equation (for 1D motion) dv dt =
More informationWorksheet #05 Kinetic Energy-Work Theorem
Physics Summer 08 Worksheet #05 June. 8, 08. A 0-kg crate is pulled 5 m up along a frictionless incline as shown in the figure below. The crate starts at rest and has a final speed of 6.0 m/s. (a) Draw
More informationFr h mg rh h. h 2( m)( m) ( (0.800 kg)(9.80 m/s )
5. We consider the wheel as it leaves the lower floor. The floor no longer exerts a force on the wheel, and the only forces acting are the force F applied horizontally at the axle, the force of gravity
More informationEQUATIONS OF EQUILIBRIUM & TWO- AND THREE-FORCE MEMBERS
EQUATIONS OF EQUILIBRIUM & TWO- AND THREE-FORCE MEMBERS Today s Objectives: Students will be able to: a) Apply equations of equilibrium to solve for unknowns, and, b) Recognize two-force members. APPLICATIONS
More informationAP Pd 3 Rotational Dynamics.notebook. May 08, 2014
1 Rotational Dynamics Why do objects spin? Objects can travel in different ways: Translation all points on the body travel in parallel paths Rotation all points on the body move around a fixed point An
More informationRigid Body Dynamics, Constraints, and Inverses
Rigid Body Dynamics, Constraints, and Inverses Hooshang Hemami Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 Bostwick F. Wyman Department of Mathematics,
More informationA. B. C. D. E. v x. ΣF x
Q4.3 The graph to the right shows the velocity of an object as a function of time. Which of the graphs below best shows the net force versus time for this object? 0 v x t ΣF x ΣF x ΣF x ΣF x ΣF x 0 t 0
More informationApplication of Newton/GMRES Method to Nonlinear Model Predictive Control of Functional Electrical Stimulation
Proceedings of the 3 rd International Conference on Control, Dynamic Systems, and Robotics (CDSR 16) Ottawa, Canada May 9 10, 2016 Paper No. 121 DOI: 10.11159/cdsr16.121 Application of Newton/GMRES Method
More informationUnits are important anyway
Ch. 1 Units -- SI System (length m, Mass Kg and Time s). Dimensions -- First check of Mathematical relation. Trigonometry -- Cosine, Sine and Tangent functions. -- Pythagorean Theorem Scalar and Vector
More informationPreLab 2 - Simple Harmonic Motion: Pendulum (adapted from PASCO- PS-2826 Manual)
Musical Acoustics Lab, C. Bertulani, 2012 PreLab 2 - Simple Harmonic Motion: Pendulum (adapted from PASCO- PS-2826 Manual) A body is said to be in a position of stable equilibrium if, after displacement
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED SUBSIDIARY GCE UNIT 761/01 MATHEMATICS (MEI) Mechanics 1 MONDAY 1 MAY 007 Additional materials: Answer booklet (8 pages) Graph paper MEI Examination Formulae and Tables (MF) Morning Time: 1 hour
More informationAngular Kinetics. Learning Objectives: Learning Objectives: Properties of Torques (review from Models and Anthropometry) T = F d
Angular Kinetics Readings: Chapter 11 [course text] Hay, Chapter 6 [on reserve] Hall, Chapter 13 & 14 [on reserve] Kreighbaum & Barthels, Modules I & J [on reserve] 1 Learning Objectives: By the end of
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull
More information1 CHAPTER 8 IMPULSIVE FORCES
8.1 Introduction. 1 CHAPTER 8 IMPULSIVE FORCES As it goes about its business, a particle may experience many different sorts of forces. In Chapter 7, we looked at the effect of forces that depend only
More informationLevers of the Musculoskeletal System
Levers of the Musculoskeletal System Lever system consists of: lever fulcrum load force Three classes of levers 1. first class (a) - pry bars, crowbars 2. second class (b) - wheelbarrow 3. third class
More informationPhysics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium
Physics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium Strike (Day 10) Prelectures, checkpoints, lectures continue with no change. Take-home quizzes this week. See Elaine Schulte s email. HW
More informationCOMPUTATIONAL AND ROBOTIC MODELS OF HUMAN POSTURAL CONTROL. by Arash Mahboobin
COMPUTATIONAL AND ROBOTIC MODELS OF HUMAN POSTURAL CONTROL by Arash Mahboobin B.S., Azad University, 1998 M.S., University of Illinois at Urbana Champaign, 2002 Submitted to the Graduate Faculty of the
More informationChapter 4: Newton s Second Law F = m a. F = m a (4.2)
Lecture 7: Newton s Laws and Their Applications 1 Chapter 4: Newton s Second Law F = m a First Law: The Law of Inertia An object at rest will remain at rest unless, until acted upon by an external force.
More information11th Grade. Review for General Exam-3. decreases. smaller than. remains the same
1. An object is thrown horizontally with a speed of v from point M and hits point E on the vertical wall after t seconds as shown in the figure. (Ignore air friction.). Two objects M and S are thrown as
More informationChapter 9- Static Equilibrium
Chapter 9- Static Equilibrium Changes in Office-hours The following changes will take place until the end of the semester Office-hours: - Monday, 12:00-13:00h - Wednesday, 14:00-15:00h - Friday, 13:00-14:00h
More informationNotes on Torque. We ve seen that if we define torque as rfsinθ, and the N 2. i i
Notes on Torque We ve seen that if we define torque as rfsinθ, and the moment of inertia as N, we end up with an equation mr i= 1 that looks just like Newton s Second Law There is a crucial difference,
More information