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ME451 Kinematics and Dynamics of Machine Systems. Introduction to Dynamics 6.1 October 09, 2013. Radu Serban University of Wisconsin-Madison. Before we get started…. Last Time: Concluded Kinematic Analysis Today: Towards the Newton-Euler equations for a single rigid body Assignments:
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ME451 Kinematics and Dynamics of Machine Systems Introduction to Dynamics 6.1October 09, 2013 Radu Serban University of Wisconsin-Madison
Before we get started… • Last Time: • Concluded Kinematic Analysis • Today: • Towards the Newton-Euler equations for a single rigid body • Assignments: • Matlab 4 – due today, Learn@UW (11:59pm) • Adams 2 – due today, Learn@UW (11:59pm) • Submit a single PDF with all required information • Make sure your name is printed in that file • Midterm Exam • Friday, October 11 at 12:00pm in ME1143 • Review session: today, 6:30pm in ME1152 • Midterm Feedback • Form emailed to you later today • Anonymous • Complete it and return on Friday
Kinematics vs. Dynamics • Kinematics • We include as many actuators as kinematic degrees of freedom – that is, we impose KDOF driver constraints • We end up with NDOF = 0 – that is, we have as many constraints as generalized coordinates • We find the (generalized) positions, velocities, and accelerations by solving algebraic problems (both nonlinear and linear) • We do not care about forces, only that certain motions are imposed on the mechanism. We do not care about body shape nor inertia properties • Dynamics • While we may impose some prescribed motions on the system, we assume that there are extra degrees of freedom – that is, NDOF > 0 • The time evolution of the system is dictated by the applied external forces • The governing equations are differential or differential-algebraic equations • We very much care about applied forces and inertia properties of the bodies in the mechanism
Dynamics M&S Dynamics Modeling • Formulate the system of equations that govern the time evolution of a system of interconnected bodies undergoing planar motion under the action of applied (external) forces • These are differential-algebraic equations • Called Equations of Motion (EOM) • Understand how to handle various types of applied forces and properly include them in the EOM • Understand how to compute reaction forces in any joint connecting any two bodies in the mechanism Dynamics Simulation • Understand under what conditions a solution to the EOM exists • Numerically solve the resulting (differential-algebraic) EOM
Roadmap to Deriving the EOM • Begin with deriving the variational EOM for a single rigid body • Principle of virtual work and D’Alembert’s principle • Consider the special case of centroidal reference frames • Centroid, polar moment of inertia, (Steiner’s) parallel axis theorem • Write the differential EOM for a single rigid body • Newton-Euler equations • Derive the variational EOM for constrained planar systems • Virtual work and generalized forces • Finally, write the mixed differential-algebraic EOM for constrained systems • Lagrange multiplier theorem (This roadmap will take several lectures, with some side trips)
What are EOM? • In classical mechanics, the EOM are equations that relate (generalized) accelerations to (generalized) forces • Why accelerations? • If we know the (generalized) accelerations as functions of time, they can be integrated once to obtain the (generalized) velocities and once more to obtain the (generalized) positions • Using absolute (Cartesian) coordinates, the acceleration of body i is the acceleration of the body’s LRF: • How do we relate accelerations and forces? • Newton’s laws of motion • In particular, Newton’s second law written as
Newton’s Laws of Motion • 1st LawEvery body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. • 2nd LawA change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. • 3rd LawTo any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction. • Newton’s laws • are applied to particles (idealized single point masses) • only hold in inertial frames • are valid only for non-relativistic speeds Isaac Newton (1642 – 1727)
6.1.1 Variational EOM for a Single Rigid Body
Body as a Collection of Particles • Our toolbox provides a relationship between forces and accelerations (Newton’s 2nd law) – but that applies for particles only • Idea: look at a body as a collection of infinitesimal particles • Consider a differential mass at each point on the body (located by ) • For each such particle, we can write • What forces should we include? • Distributed forces • Internal interaction forces, between any two points on the body • Concentrated (point) forces
Forces Acting on a Differential Mass dm(P) • External distributed forces • Described using a force per unit mass: • This type of force is not common in classical multibody dynamics; exception: gravitational forces for which • Applied (external) forces • Concentrated at point • For now, we ignore them (or assume they are folded into ) • Internal interaction forces • Act between point and any other point on the body, described using a force per units of mass at points and • Including the contribution at point of all points on the body
Newton’s EOM for a Differential Mass dm(P) • Apply Newton’s 2nd law to the differential mass located at point P, to get • This is a valid way of describing the motion of a body: describe the motion of every single particle that makes up that body • However • It involves explicitly the internal forces acting within the body (these are difficult to completely describe) • Their number is enormous • Idea: simplify these equations taking advantage of the rigid body assumption
A Model of a Rigid Body • We model a rigid body with distance constraints between any pair of differential elements (considered point masses) in the body. • Therefore the internal forces on due to the differential mass on due to the differential mass satisfy the following conditions: • They act along the line connectingpoints and • They are equal in magnitude, opposite in direction, and collinear
[Side Trip]Virtual Displacements A small displacement (translation or rotation) that is possible (but does not have to actually occur) at a given time • In other words, time is held fixed • A virtual displacement is virtual as in “virtual reality” • A virtual displacement is possible in that it satisfies any existing constraints on the system; in other words it is consistent with the constraints • Virtual displacement is a purelygeometric concept: • Does not depend on actual forces • Is a property of the particular constraint • The real (true) displacement coincideswitha virtual displacement only if theconstraint does not change with time Virtual displacements Actual trajectory
The Rigid Body Assumption:Consequences • The distance between any two points and on a rigid body is constant in time:and therefore • The internal force acts along the line between and and therefore is also orthogonal to :
[Side Trip]D’Alembert’s Principle Jean-Baptiste d’Alembert (1717– 1783)
[Side Trip]Principle of Virtual Work • Principle of Virtual Work • If a system is in (static) equilibrium, then the net work done by external forces during any virtual displacement is zero • The power of this method stems from the fact that it excludes from the analysis forces that do no work during a virtual displacement, in particular constraint forces • D’Alembert’s Principle • A system is in (dynamic) equilibrium when the virtual work of the sum of the applied (external) forces and the inertial forces is zero for any virtual displacement • “D’Alembert had reduced dynamics to statics by means of his principle” (Lagrange) • The underlying idea: we can say something about the direction of constraint forces, without worrying about their magnitude
