Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation, angular velocity vector, description of velocity and acceleration in relatively moving frames. Euler angles, Review of methods of momentum and angular momentum of system of particles, inertia tensor of rigid body. Dynamics of rigid bodies - Euler's equation, application to motion of symmetric tops and gyroscopes and problems of system of bodies. Kinetic energy of a rigid body, virtual displacement and classification of constraints. D Alembert s principle. Introduction to generalized coordinates, derivation of Lagrange's equation from D Alembert s principle. Small oscillations, matrix formulation, Eigen value problem and numerical solutions. Modelling mechanical systems, Introduction to MATLAB, computer generation and solution of equations of motion. Introduction to complex analytic functions, Laplace and Fourier transform. PID controllers, Phase lag and Phase lead compensation. Analysis of Control systems in state space, pole placement, computer simulation through MATLAB. 1 Purpose: Preview of some fundamentals Focus on 4 Dimensions and Units 4 Vector algebra 4 Fundamentals of mechanics: statics and dynamics 4 Conversion of coordinates
4 Dimensions and Units: greatly facilitate formulating equations, checking dimensional homogeneity of an equation, and converting units 4 A dimension: the measure by which the magnitude of a physical quantity is expressed. 4 Four dimensions in dynamics: Mass, Length, Time, and Force. 4 Let s see Newton s second law of motion: Force = mass x acceleration 4 How many dimensions are involved here? 3 4 A unit is a determinate quantity adopted as a standard of measurement of dimensions. 4 Newton s equation of motion contains 4 units for the four dimensions: three are fundamental and one is derived. 4 Which ones are fundamentals? ML F = = MLT T 4 The answer to this question defines two unit systems: The International System of Units (SI) ML (kilogram)(meter) kg. m F = = = = N T (Second) s - The British Gravitational System (BG), M = FT L ( Pund )( Sekund) = Foot 4 lbf. s = ft
4 Examples of basic and derived units in the SI sys. Dimension Unit Symbol Length (L) Meter m Mass (M) kilogram kg Time (T) Second s Electric current (I) ampere A Temperature kelvin K Amount of material mole mol Light strength candela cd Dimension Unit Symbol Formula Force newton N kg.m.s - Pressure, stress pascal Pa N.m - Energy oule J N.m Power watt W J.s -1 el. N.m. s -1 Torsion newton-meter N.m 5 The principle of dimensional homogeneity: to prevent algebraic errors from occurring in complicated manipulations of equations. each term in an equation must have the same dimension, and the dimensions on both sides of the equal sign must be the same. 6
Vector analysis in mechanics 4 Expression of physical quantities in mechanics Scalars and vectors. Scalar: A quantity characterized by magnitude only. Examples: Mass, length, time, and volume Vector: A quantity that has both a magnitude and direction and obeys the parallelagram law of addition. Examples: Force, velocity, acceleration, and position of a particle in space A vector can be broken down into several components according to convenience. In the Cartesian coordinate system, for example, a vector a can be expressed in its components as a = a x i + a y + a z k (i, and k are unit vectors; a x, a y, and a z are components) Note: vector analysis plays an important role in dynamics 7 : Vector algebra Vector addition: c = a + b = a x i + a y + a z k + b x i + b y + b z k = (a x +b x )i+(a y +b y ) + (a z + b z )k Vector subtraction: c = a - b = (a x - b x )i + (a y - b y ) + (a z - b z )k Scalar product of two vectors: a b = abcosq Cosq = where a x, a y a z, b x, b y and b z are components of vectors a and b, and a is the magnitude of vector a and b is the magnitude of vector b. 8 a b ab ax bx + ayby + = ab a b z z
: Vector algebra Cross product of two vectors: a x b = (ab sin )e where is the angle between vectors a and b, and e is a unit vector perpendicular to the plane containing vectors a and b, and in the direction according to right-hand rule. The mathematical operation of cross product can be performed in matrix form as follows Triple scalar product Triple vector product 9 : Vector algebra Differentiation: the derivative of a vector, which is a function of time is defined as Other derivatives For a vector expressed with its components 10
: Vector algebra Gradient, Divergence and Curl Operations The gradient of a scalar f is defined as The divergence of a vector F The curl of a vector F is defined as 11 : Vector algebra Example: computation of curl of the vector V Where This is called Vorticity in fluid mechanics à Related with rotational velocity 1
Fundamentals of Mechanics a repetition 4 Study of motions, time and forces (imcluding interaction among these) 4 Study of a body or bodies and the forces that caused the motion 4 Study of the interaction between forces and the motions they cause 4 Study of obects at rest or in equilibrium under the action of forces and/or torques 4 Study of motion of bodies (only the space-time relation of bodies, with no concern about the forces causing the motion) 13 Fundamentals of Mechanics a repetition Construction Structure Mechanism Machine 4 Machine an arrangement of parts for doing work, a device for applying power or changing its direction. differs from a mechanism in its purpose. 4 Mechanism transmits power or force, but the dominant idea for a designer is to achieve a desired motion. Concept analogy 4 Structure ßà Statics 4 Mechanism ßà Kinematics 4 Machine ßà Kinetics 14
: Statics & dynamics 4 Equilibrium equations of statics for different dimensions of space 15 Conversion of Coordinates 4 Transformation matrices: Conversion from one coordinate system to the other is essential for diverse applications including 4... Computer graphics, Computer aided design GPS systems Initial position: WCS coincides with model coordinate system 16 Current position: Obect has undergone translation and rotation, wrt to WCS
Conversion of Coordinates 4 Translation of obect by a, b and c in x, y and z resp.: Relation between WCS and MCS x w = x m + a, y w = y m + b, z w = z m + c, Formulated in matix form Program command: Trans(a, b,c) Homogeneous transformation matrix; A convention used in CAD systems for example in OpenGL Exercise: Show the matrix form of the translation relation with the coordinates as row vectors 17 Conversion of Coordinates 4 Rotation of the obect by angle θ (about x-axis) from initial location at PQ to its current position at P Q Proection of obect in yz plane θ x w = x m y w = lcos(α + θ) = l(cosα cosθ - sinα sinθ) = (lcosα) cosθ - (lsinα) sinθ = y m cosθ - z m sinθ z w = lsin(α + θ) = l(cosα sinθ + sinα cosθ) = (lcosα) sinθ + (lsinα) cosθ = y m sinθ + z m cosθ 18
Conversion of Coordinates 4 Rotation of the obect by angle θ (about x-axis)... in matrix form Rot(x, θ) Homogeneous transformation matrix for rotation about x-axis Similarly, the homogeneous transformation matrices for rotation about y- and z-axis are: 19 Coordinate transformation 4 Example Find the homogeneous transformation matrix T for the following operations q Rotation α about X axis q Translation of a along X-axis q Translation of d along Z-axis q Rotation θ about Z-axis 0
Summary and Questions In this lecture we covered repetitions/basics of q Dimensions and units in SI system q Vector algebra: scalar and vector products q Classifications of mechanics: statics and dynamics q Coordinate transformation: Translation and rotation (read also about mapping and scaling)? Next: Kinematics of rigid bodies - coordinate transformation, angular velocity vector, description of velocity and acceleration in relatively moving frames 1