Kinematics and dynamics of water Kinematics and dynamics of waterKinematics Dynamics.pptx

Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water
Kinematics and dynamics of water

Kinematics and dynamics of water

1. Kinematics and Dynamics Prof. Dr. Andreas Malcherek

2. A course in kinematics and dynamics This course in Mechanics is an integrated approch to kinematics,and dynamics . Based on the fundamental skill to solve ordinary differential equations (ODE) in MATLAB (will be shown in the course ) the fundamental equations of kinematice will be solved to follow the trajectory of a particle . Then the fundamental equations of dynamics will be solved to understand the forces of gravity , friction and elasticity . The next part of the course is dedicated to the ODE‘s describing rotation dynamics in pumps and turbines . Finally we will have a look on relativistic dynamics to get an understanding for the origin of magnetism .

3. Schedule: Kinematics and dynamics Foundation of kinematics Friction forces Newtonian law of motion Hookes‘s law and elasticty Center of mass Couple of forces Rotation kinematics and dynamics Angular momentum balance Moment of inertia Dynamics of a shaft in pumps and turbines Relativistic dynamics for Electrodynamics

4. Foundation of kinematics Prof. Dr.- Ing . Andreas Malcherek

5. The experiment

6. Trajektories or pathways of particles

7. Plot of a trajectory 7 t=[0:0.1:20]; x=0.1* t+cos (t); y=sin(t); plot( x,y )

8. Outflow trajectories   is unknown , but:

9. Definition of velocity 9 u   v   w  

10. Numerical Solution       function ParticleKinematics x0=0; y0=0 ; z0=0 ; dt =[0 10 ]; [T,F] = ode45(@ RHS,dt ,[x0 y0 z0]); x=F(:,1 ); y=F (:,2 ); z=F (:,3 ); plot ( T,x ) function dfdt = RHS( t,f ) x=f(1); y=f(2); z=f(3); u=2 ; v=0; w=0; dxdt =u ; dydt =v ; dzdt =w; dfdt = [ dxdt;dydt;dzdt ]; end end

11. The xt -diagram  

12. Acceleration Velocities can change . Change of velocity per time is acceleration . Brake is negative acceleration u   v   w        

13. Kinematics on earth       =0   =0   =-g   g = 9.81 m/s²

14. Horizontal motion   =0      

15. Vertical motion            

16. The trajectory z(x)    

17. Torricelli‘s law (1644) In general :

18. The program function ParticleKinematics x0=0; y0=0; z0=0; dt =[0 10 ]; [T,F] = ode45(@ RHS,dt ,[x0 y0 z0]); x=F(:,1 ); y=F (:,2 ); z=F (:,3 ); plot ( T,x ) function dfdt = RHS( t,f ) x=f(1); y=f(2); z=f(3); u=2; v=0; w=0; dxdt =u; dydt =v; dzdt =w; dfdt = [ dxdt;dydt;dzdt ]; end end Please change for yourself

19. Friction forces Prof. Dr.. Andreas Malcherek

20. 20 Linear friction

21. 21 Friction and gravitation

22. Quadratic friction

23. 1D Analytical solution

24. Where is the body ? Kinematics and dynamics

25. Programm function ballistics_linearfriction g=9.81; vecg =[0; -g]; k=2 ; vecx =[0; 0] ; vecv0=[10; 10 ]; [T,Y] = ode23(@RHS,[0 2],[ vecx vecv0 ]); plot (Y(:,1),Y(:,2),'LineWidth',2,'Color',[0 0 0]) ylim ([0 max (1.1*Y(:,2))]) xlabel ( 'x [m]','FontWeight','bold','FontSize',14); ylabel ( 'z [m]','FontWeight','bold','FontSize',14); function dfdt = RHS( t,f ) vecv =f(3:4); dvecxdt = vecv ; dvecvdt =-k* vecv+vecg ; dfdt = [ dvecxdt;dvecvdt ]; end end

26. Newtonian law of motion Prof. Dr. Andreas Malcherek

27. 27 Second Newtonian Law Is valid for obejcts with constant mass .

28. Newton’s gravitation law Vektorial formulation :

29. G ravitation force on the earth’s surface

30. 30 Gravitation in a earth surface co-ordinate system

31. 31 Gravitation potential

32. Elasticity and Hooke’ s Law Prof. Dr. Andreas Malcherek

33. 33 Newton‘s law of motion Such a body would always be accelerated Why stays the ball on the table ?

34. 34 Suspensions Elastic forces adapt automatically to the load Suspension forces come from contraction

35. 35 The ball on the table Equilibrium:

36. 36 … to pose something on the table

37. 37 Elasticity and friction

38. The program function elasticMovement dt =[0 0.05]; z0=0.1; w0=0; g=9.81; L0=0 ; M=1; D=1000000; r=100; [T,F] = ode45(@ RHS,dt ,[z0 w0]); plot(T,F(:,1),'LineWidth',2,'Color',[0 0 0]) xlabel ('Zeit [s]','FontWeight','bold','FontSize',12); ylabel ('Ort z [m]','FontWeight','bold','FontSize',12); function dfdt = RHS( t,f ) z=f(1); w=f(2); dzdt =w; dwdt=-g-D/M*(z-L0)-r*abs(w)*w; dfdt =[ dzdt ; dwdt ]; end end

39. The result

40. Center of mass movement Prof. Dr. Dipl.-Phys. Andreas Malcherek

41. Bodies as multiparticle -systems

42. Center- of - mass velocity

43. Internal forces Assumption

44. The movement of the center- of - mass COM movement does not refelct all the movements of a body : Rotation is the second movement Internal forces come back in continuum mechanics and thermodynamics

45. Couple of forces Prof. Dr. Andreas Malcherek

46. 46 From a point to a body

47. 47 The couple of forces Center of mass does not move but the body rotates .

48. Moment of two forces Moment with respect to reference location O

49. Moment of a couple of forces Moment eines Kräftepaars ist unabhängig vom Bezugspunkt

50. 50 Statics of a body

51. 51 Dynamics of a body Um welche Achse rotiert ein Körper? Wie schnell rotiert er um diese Achse?

52. Kinematics and dynamics of rotation Prof. Dr.- Ing . Andreas Malcherek

53. Translation and Rotation: A comparison Translation Linear motion Needs a lot of space velocity force Mass Kinetic Energy Rotation Movement on circles Spatially bounded Angular velocity torque Trägheitsmoment Rotational energy

54. Circular motion Movement on a full cycle :

55. Frequency and rounds per minute

56. Examples n = 3000 R/min => n = 50 R/s = 50 Hz

57. Angular velocity and particle velocity Vektoriell:

58. Centrifugal acceleration Centrifugal acceleration :

59. Centrifugal and centripetal force Eine Kreisbewegung entsteht dann, wenn der radiale Abstand (Radius) zum Rotationszentrum sich nicht ändert. Damit muss die Summe der radialen Kräfte Null sein:

60. Gravitation as centripetal force

61. Centripetal force in a shaft Hooke‘s law : Zentripetal force

62. A ngular momentum balance Prof. Dr.- Ing . Andreas Malcherek

63. Angular momentum The amount of rotation increases with the rotating mass with increasing velocity with increasing radius

64. Angular momentum balance Torques acting on a body :

65. Moment of inertia Prof. Dr.- Ing . Andreas Malcherek

66. Derivation

67. Moment of inertia of a shaft

68. Rules for the moment of inertia Two disjunct bodies : Parallel movement of the axis

69. Rotation dynamics of a shaft Prof. Dr.- Ing . Andreas Malcherek

70. Dynamics of a shaft

71. Estimation for the rotation friction

72. MATLAB-Program function shaftrotation rhoW =7850; L=4; R=1; theta =1/2* rhoW * pi *L*R^4 dR =0.001; LF=1; % Length of the shaft mu =1e-2; D=2* pi *R^3/ dR *LF* mu tau=1e4; [T,N]= ode45(@RHS,[ 0 10000],0); plot (T,N ); function domegadt =RHS( t,omega ) domegadt =tau/theta-D* omega / theta ; end end

73. Initiation of rotation

74. Stationary angular velocity Inertia of momentum is not decisive for the stationary angular velocity .

75. Shaft dynamics in pumps and turbines

76. Relativistic dynamics Prof. Dr.. Andreas Malcherek

77. Time and space

78. The so-called speed of light c = 300 000 000 m/s The speed of light is just a factor which transforms a spatial length to a time intervall . It‘s value comes from the anthropogenic SI-System.

79. Four-dimensional vectors Notation: An index i is related to a 4 dimensional vector in time and space . A vector array is a 3 dimensional spatial vector .

80. The imaginary time i i = -1 The time component has to be imaginary to describe distances between events correctly .

81. Gradients in 4D

82. Laplacian in 4D The Poisson equation becomes a wave equation !

83. Relativistic momentum The norm of the relativistic momentum of a particle remains always constant :

84. Rotations in 4 dimensions The norm remains constant if

85. The electrodynamic potential Potential: A particle is moving in the direction where a potential A decreases :

86. Reminder: Four dimensional gradients

87. Tensor of the electromagnetic field

88. Electrical Potential

89. Electric Field

90. Electric Field

91. Magnetic Field

92. Magnetic Field

93. The energy equation

94. The momentum equation

95. Relativistic Lorentz force

96. Relativistic momentum equation with Lorentz force Classical equation of the movement of an electrical particle :

97. Movement of an Electron in an EM-Field

98. Analytical Solution

99. Frequency and Period

100. Movement of an Elektrons in a EM field

101. Movement with/without magnetic field

102. function ElektronInEMField c=300000e3; qe =-1.6e-19; me =9.1093837015e-31; % Elektron E0=-1e-3; E=E0*[1 0 0]; vB =0.9*c; B0=E0/ vB ; B=B0*[0 1 0]; T0=2* pi * me / qe /B0; opts = odeset ('RelTol',1e-6); [T,Y] = ode45(@ lorentz ,[0 50*T0],[0 0 0 0 0 0], opts ); plot (T/T0,[Y(:,4)/ vB ],'Color','[0 0 0]','LineWidth',2,'DisplayName','u') hold on plot (T/T0,[Y(:,5)/ vB ],'Color','[1 0 0]','LineWidth',2,'DisplayName','v') plot (T/T0,[Y(:,6)/ vB ],'Color','[0 0 1]','LineWidth',2,'DisplayName','w') function dfdt = lorentz ( t,f ) x=f(1); y=f(2); z=f(3); u=f(4); v=f(5); w=f(6); vecv =[u v w]; gamma =1/ sqrt (1-vecv* vecv '/c^2); dvdt = qe / me / gamma *( E+cross ( vecv,B )-( vecv *E')* vecv /c^2); dfdt = [ vecv (1); vecv (2); vecv (3); dvdt (1); dvdt (2); dvdt (3)]; end end The program

103. What is magnetism? In a 4D world time and space are ( more or less ) equivalent . Forces only turn the momentum vector in such a space . Forces are described by a 4x4 antisymmetric Tensor. Electric forces are the time component of the forces . Magnetic forces are the spatial component of the forces .