Chapt.2 (Model of 1DOF System)

Outline
This chapter introduces the model to derive the correction description of the dynamics of the system with one degree of freedom. The principles can be easily extended to obtain models of systems having two or more degrees of freedom (Chap. 6 and 8). Pukyong National University Intelligent Mechanics Lab.
Note how much simpler the velocity and acceleration expressions are when described by normal-tangential coordinates
Pukyong National University
Intelligent Mechanics Lab.
Pukyong National University
Intelligent Mechanics Lab.
Degree of Freedom
Degree of freedom (DOF) : the minimum number of coordinates required to completely describe the motion of the object Generalized coordinates : coordinates that reflect the existence of constraints and yet still provide a complete description of the motion of the object (For pendulum, angle θ, coordinate x and y are not) Constraint : restrict the motion of an object usually means that fewer coordinates are needed to describe the motion of the object (constant length L)
Curvilinear Translation
Rectilinear Translation
General Plane Motion
Fixed-axis Rot University
Intelligent Mechanics Lab.
Kinematics
Example : Single Pendulum
A system having 1 DOF
A multi-pulley system
Pukyong National University
Intelligent Mechanics Lab.
Plane Motion of a Rigid Body
Particle : A mass of negligible dimensions Motion can consist of translation only Rigid body : An object negligible deformations but considered dimensions Motion can involve both translation and rotation Newton’s second law : Acceleration of a mass particle is proportional to the vector resultant force acting on it and is in the direction of this force
Kinematics
Kinematics (운동학 운동학, 운동학 動 ):
• Study of motion only without regard for the masses that are in motion or the forces that produce the motion • Primarily concerned with relationship among displacement, velocity & acceleration • Provides the basis for describing the motion of the systems • Restrict our attention to motion in a plane • This means that the object can translate in 2-dimensions only and can rotate only about an axis that is perpendicular to the plane containing these 2-dimensions • The completely general motion involves translation in 3-dimensions and rotation about 3-axes : considerably more complex to analyze • Many practical engineering problems can be handled with the plane motion methods
∑ F = f = ma = m dt
dv dv
Vector sum of external forces acting on a body of mass m must equal the time rate of change of linear momentum L
f= d (mv ) dL = dt dt
String or cable
• Infinite DOF • y = f(x)
Pukyong National University
Intelligent Mechanics Lab.
Coordinates & Degree of Freedom
Examples : DOF & Kinematics
• If without constraint 3 translations 3 rotations 6 DOF
• If with constraint (only vertical motion) 1 translations (x) 1 DOF
Pukyong National University
Coordinates
Coordinate : An independent quantity used to specify position The proper choice of coordinates can simplify the resulting expression for position, velocity, or acceleration Categories of coordinates : Fixed or moving rectangular coordinate Path coordinate (normal-tangential coordinate) Polar coordinate (cylindrical coordinate)
• 2 DOF • Generalized coordinates (θ1, θ2)
• 2 DOF • Generalized coordinates (x1, x2)
• Pure rolling motion • 1 DOF (∆x = R∆θ) • Generalized coordinates θ or x
Normal and tangential coordinate system Fixed rectangular coordinate system
Velocy vecor : & v = v e = Lθ e
t
t
Acceleration vector : v2 & && & a = e n + v et = Lθ 2 e n + Lθ et L
Equilibrium
힘( ) (force)
Pukyong National University
물체
운동( 動) (motion)
Intelligent Mechanics Lab.
Kinematics
Kinematics : Categories of Plane Motion
Translation(¿ 動), Rotation(eo 動) 4 categories of plane motion • Rectilinear translation • Curvilinear translation • Rotation about a fixed axis • General plane motion
M=
d ( Iω) dH & = =H dt dt
I : Body’s mass moment of inertia about the axis & ω : Angular velocity about that axis (ω = θ ) Term torque T are often used instead of moment Calculating inertia : mass moment of inertia I about a specified reference axis
William J. Palm III : Mechanical Vibration