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Unformatted text preview: Module 6: Moments and Equivalence
September 10, 2010 Module Content:
1. Cross products are frequently used in sta=cs for calcula=ons involving moments.
2. The moment about a par=cular axis is calculated using the scalar triple product, which is a combina=on of the dot product and cross product.
3. Equivalence calcula=ons allow us to simplify a loading scenario on a structure. Module Reading, Problems, and Demo:
Reading: Chapter 4
Problems: Fund. Prob. 4.6, Fund. Prob. 4.12 (cross products)
Demo: none
Homework PlaQorm: hRp://www.masteringengineering.com
Course Blog: hRp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 6
1 Concept: Three Scenarios what do you no+ce?
MAE 2300 Sta=cs © E. J. Berger, 2010 6
2 Concept: Three Scenarios what do you no+ce?
MAE 2300 Sta=cs © E. J. Berger, 2010 6
2 Concept: Three Scenarios what do you no+ce?
MAE 2300 Sta=cs © E. J. Berger, 2010 6
2 Concept: Three Scenarios what do you no+ce?
MAE 2300 Sta=cs © E. J. Berger, 2010 6
2 Theory: Moments as Cross Products
• the moment of a force around an axis can be constructed by considering two vectors:
• the force vector (Cartesian) • a (rela=ve) posi=on vector which locates the line of ac=on of the force with respect to the axis • NOTE that this posi=on vector can point to anywhere on the line of ac=on of the force vector! • HOW? Let’s derive it together... MAE 2300 Sta=cs © E. J. Berger, 2010 6
3 Theory: Moments as Cross Products
• the moment of a force around an axis can be constructed by considering two vectors:
• the force vector (Cartesian) • a (rela=ve) posi=on vector which locates the line of ac=on of the force with respect to the axis • NOTE that this posi=on vector can point to anywhere on the line of ac=on of the force vector! • HOW? Let’s derive it together... MAE 2300 Sta=cs © E. J. Berger, 2010 6
3 Theory: Moments as Cross Products
• the moment of a force around an axis can be constructed by considering two vectors:
• the force vector (Cartesian) • a (rela=ve) posi=on vector which locates the line of ac=on of the force with respect to the axis • NOTE that this posi=on vector can point to anywhere on the line of ac=on of the force vector! • HOW? Let’s derive it together... MAE 2300 Sta=cs © E. J. Berger, 2010 6
3 Deriva=on: Cross Product MAE 2300 Sta=cs © E. J. Berger, 2010 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) MAE 2300 Sta=cs © E. J. Berger, 2010 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) MAE 2300 Sta=cs © E. J. Berger, 2010 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) r1 MAE 2300 Sta=cs θ1 © E. J. Berger, 2010 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) θ2 MAE 2300 Sta=cs θ1 © E. J. Berger, 2010 r2
r1 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) θ3
r3
θ2 MAE 2300 Sta=cs θ1 © E. J. Berger, 2010 r2
r1 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) d = r sinθ = r1 sinθ1 = r2 sinθ2 = r3 sinθ3
θ3
r3
θ2 MAE 2300 Sta=cs θ1 © E. J. Berger, 2010 r2
r1 6
4 Deriva=on: Cross Product
based upon a physical argument: the perpendicular distance d is the “right” length to use for the moment calcula+on (Mo = Fd) d = r sinθ = r1 sinθ1 = r2 sinθ2 = r3 sinθ3
Mo = r x F = Fr sinθ
(for any r which connects the axis O to the line of ac=on of F!)
θ2 MAE 2300 Sta=cs θ1 © E. J. Berger, 2010 θ3
r3
r2
r1 6
4 Theory: Cross Products as Moments
• the last slide (and our discussion in class on Friday 9/5) illustrated that calcula=ng the moment of a force about a par=cular point or axis is actually a cross product opera=on of the form: • IF YOU NEED A REMINDER ABOUT CROSS PRODUCTS: Sec=on 4.3 OR Google it • other useful moment rela=ons: • moment summa=on: • principle of moments: MAE 2300 Sta=cs © E. J. Berger, 2010 6
5 Theory: Cross Products as Moments
• the last slide (and our discussion in class on Friday 9/5) illustrated that calcula=ng the moment of a force about a par=cular point or axis is actually a cross product opera=on of the form: • IF YOU NEED A REMINDER ABOUT CROSS PRODUCTS: Sec=on 4.3 OR Google it • other useful moment rela=ons: • moment summa=on: • principle of moments: MAE 2300 Sta=cs © E. J. Berger, 2010 6
5 Theory: Cross Products as Moments
• the last slide (and our discussion in class on Friday 9/5) illustrated that calcula=ng the moment of a force about a par=cular point or axis is actually a cross product opera=on of the form: • IF YOU NEED A REMINDER ABOUT CROSS PRODUCTS: Sec=on 4.3 OR Google it • other useful moment rela=ons: • moment summa=on: • principle of moments: MAE 2300 Sta=cs © E. J. Berger, 2010 6
5 Theory: Cross Products as Moments
• the last slide (and our discussion in class on Friday 9/5) illustrated that calcula=ng the moment of a force about a par=cular point or axis is actually a cross product opera=on of the form: • IF YOU NEED A REMINDER ABOUT CROSS PRODUCTS: Sec=on 4.3 OR Google it • other useful moment rela=ons: • moment summa=on: • principle of moments: MAE 2300 Sta=cs © E. J. Berger, 2010 6
5 Theory: Cross Products as Moments
• the last slide (and our discussion in class on Friday 9/5) illustrated that calcula=ng the moment of a force about a par=cular point or axis is actually a cross product opera=on of the form: • IF YOU NEED A REMINDER ABOUT CROSS PRODUCTS: Sec=on 4.3 OR Google it • other useful moment rela=ons: • moment summa=on: • principle of moments: MAE 2300 Sta=cs perhaps i and j components of a vector © E. J. Berger, 2010 6
5 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Moment about a Speciﬁed Axis
• the convergence of cross and dot products comes about when we wish to determine the component of a moment about a speciﬁc axis • here’s an example: whatʼs the moment around the zaxis? z x
y MAE 2300 Sta=cs © E. J. Berger, 2010 6
6 Theory: Scalar Triple Product (STP)
• we can generalize the result from the last slide to include the moment around any speciﬁc axis: • and we note that the dot product part always uses a UNIT vector to indicate the desired axis MAE 2300 Sta=cs © E. J. Berger, 2010 6
7 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 6
8 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 6
8 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 6
8 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: M = Fd • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 6
8 Theory: Couple Moments
• couple moments are created when two equal and opposite forces exist oﬀset from each other: M = Fd
(d is the distance between the forces) • from an equilibrium standpoint, the result is a pure moment (the forces cancel out because they are equal and opposite) • couple moments are free vectors, which means that the moment they create is the same regardless of the point we choose around which to take the moment MAE 2300 Sta=cs © E. J. Berger, 2010 6
8 Derivation: Huh? MAE 230 Statics © E. J. Berger, 2008 7 9 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce iden=cal eﬀects on the structure, then they are equivalent • “iden=cal eﬀects” simply means that from the standpoint of equilibrium calcula=ons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 6
10 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce iden=cal eﬀects on the structure, then they are equivalent • “iden=cal eﬀects” simply means that from the standpoint of equilibrium calcula=ons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 6
10 Concept: Equivalence
• we can deﬁne this: if two loading scenarios produce iden=cal eﬀects on the structure, then they are equivalent • “iden=cal eﬀects” simply means that from the standpoint of equilibrium calcula=ons, the eﬀects are the same MAE 2300 Sta=cs © E. J. Berger, 2010 6
10 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calcula=on (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equa=ons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 6
11 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calcula=on (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equa=ons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 6
11 Theory: Equivalence Calcula=ons
• remember that we wish to calculate an equivalent set of loads whose eﬀect in the equilibrium calcula=on (i.e., it has the same “external eﬀects” on the body) • original two types:
•
• • force equivalence
moment equivalence
so we need to make sure that the original and the equivalent systems appear the same in the equilibrium equa=ons equivalent loading at O
MAE 2300 Sta=cs © E. J. Berger, 2010 6
11 ...
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This note was uploaded on 02/09/2012 for the course MAE 2300 taught by Professor Staff during the Fall '10 term at UVA.
 Fall '10
 Staff
 Statics

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