me451 kinematics and dynamics of machine systems introduction to dynamics 6.1 october 09, 2013 radu...
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ME451 Kinematics and Dynamics
of Machine Systems
Introduction to Dynamics6.1
October 09, 2013
Radu SerbanUniversity of Wisconsin-Madison
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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
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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
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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
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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)
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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
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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)
Variational EOM for a Single Rigid Body6.1.1
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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
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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
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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
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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 connecting
points and They are equal in magnitude,
opposite in direction, and collinear
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[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 coincideswith a virtual displacement only if theconstraint does not change with time
Actualtrajectory
Virtualdisplacements
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[Side Trip]
Calculus of Variations (1/3)
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[Side Trip]
Calculus of Variations (2/3)
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[Side Trip]
Calculus of Variations (3/3)
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Virtual Displacement of a Point Attached to a Rigid Body
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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 :
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[Side Trip]
An Orthogonality Theorem
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Variational EOM for a Rigid Body (1)
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Variational EOM for a Rigid Body (2)
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[Side Trip]
D’Alembert’s Principle
Jean-Baptiste d’Alembert(1717– 1783)
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[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
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[Side Trip]
PVW: Simple Statics Example