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WEEK 4 MECHANISMS

• References • (METU, Department of Mechanical Engineering)

Text Book: “Mechanisms” Web Page: http://www.me.metu.edu.tr/people/eres/ME301/Index.ht

• Analitik Çözümlü Örneklerle Mekanizma Tekniği, Prof. Dr. Mustafa SABUNCU, 2004

• Mekanizma Tekniği, Prof. Dr. Eres SÖYLEMEZ, 2007

• Theory of Machines and Mechanisms, J.J. Uicker, G.R.Pennock ve J.E. Shigley, 2003

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Cartesian polar.

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AB C A B= +

AB

C

Graphical Vector Addition

Adding two vectors A and B graphically can be visualized like two successive walks, with the vector sum being the vector distance from the beginning to the end point. Representing the vectors by arrows drawn to scale, the beginning of vector B is placed at the end of vector A. The vector sum C can be drawn as the vector from the beginning to the end point.

For many mechanisms, the sole purpose of analysis is to determine the location of all links as the driving link(s) of the mechanism is moved into another position.

POSITION ANALYSIS OF MECHANISMS

With motion analysis, we mean the analysis required to determine: a) Relative position of any link or a point on a link with respect to a reference frame which may be attached to a moving link or the fixed link; b) The path traced by a point on a link in plane which is attached to a moving link or the fixed link; c) The angular or linear displacement of a link or a point on a link relative to the fixed link. From the definition of the degree-freedom of a mechanism, in order to determine the position of each link, the number of independent parameters to be defined must be equal to the degree-of-freedom of the mechanism. The joint parameters will define the relative position of two links connected by a joint. The joint whose parameter is defined will be called the input joint. However, if one of the links connected by the joint is fixed, the word input link may also be used since the joint parameter defines the absolute position of the other link.

Vector Loops of a Mechanism The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints form closed polygons that we shall call a loop. The motion analysis of mechanisms is based on expressing these loops mathematically. In kinematic analysis we shall assume that all the necessary dimensions of each link is given and link length dimensions (i.e. the distance between the joints or the angles) can be determined from the given dimensions using the geometry of the link.

To determine the positions of the links 1-we must have a reference frame 2-we must define the angle (θ, in positive direction), which is related with the degree of freedom of the joint between links 3-We must obtain vector Loops of the mechanism

POSITION ANALYSIS

θ2

θ3

θ4

0 0 0 0 0A A AB BB B A+ + + =

2 2 2 2 3 3 3 3 4 4 4 4 1 0cos i sin j cos i sin j ( cos i sin j ) iθ + θ + θ + θ − θ + θ − =

y, j

x, i

Cartesian coordinates:

2 2 3 3 4 4 1 0i cos cos cos⇒ θ + θ − θ − =

2 2 3 3 4 4 0j sin sin sin⇒ θ + θ − θ =

Example 1.

Complex numbers: 0 0 0 0 0A A AB BB B A+ + + =

32 4

0 02 3 4 0ii iB Ar e r e r e rθθ θ+ − − =

2 2 2 2 3 3 3 3 4 4 4 4 1 0cos sin i cos sin i ( cos sin i )θ + θ + θ + θ − θ + θ − =

2 32 3 44 1 0real cos cos cos⇒ θ + − −θ =θ

2 2 3 3 4 4 0imag. sin sin sinθ⇒ θ + − =θ

2 known( input )θ →

Using Euler's Equation:

3 4, unknowns( outputs )θ θ →

To determine the positions of the links 1-we must have a reference frame 2-we must define the angle (θ, in positive direction), which is rela with the degree of freedom of the joint between links 3-We must obtain vector Loops of the mechanism

Example 2.

Cartesian coordinates : 0 0A A AB A H HB+ = +

θ2

θ3

X H

2 2 2 2 3 3 3 3cos i sin j cos i sin j Xi Cjθ + θ + θ + θ = +

2 2 3 3i cos cos X⇒ θ + θ =

2 2 3 3j sin sin C⇒ θ + θ =

Complex numbers: 0 0A A AB A H HB+ = +

322 3

iir e r e X Ciθθ + = +

2 2 3 3real cos os Xc⇒ θ + θ =

2 3 32imag. sin sin C⇒ θ =θ+

2 2 2 2 3 3 3 3cos sin i cos sin i X Ciθ + θ + θ + θ = +

2 known( input )θ →

3 ,X unknowns( outputs )θ →

Equations 1 and 2 are nonlinear equations. Therefore, a simple numerical iterative method known as Newton-Raphson method may be used to find a solution of such an equation set. There are also other numerical methods.

2 2 33co os Xs cθ + θ =

2 2 33sin sin Cθ + θ =

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Example 3:Shaping Machine (Quick Return Mechanism)

How will you determine the number of independent loops (vector loops) ?

Kinematic diagram:

Kinematic chain:

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******The number of vector loops of a mechanism is equal to the number of closed area in the kinematic chain.

Note: 1.The number of loops in a mechanism does not depend on the type of joints 2. The number of loops does not depend on the degree of freedom of space 3. The number of loops does not depend on the degree of freedom of the mechanism

Write the position equations using complex number:

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Sled element

Slider element

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Number of Independent Loops in a Mechanism

j = the number of joints in the open kinematic chain + the number of joints removed. j = (l-1) + L or L = j - l + 1

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