St_Mod6_F10

St_Mod6_F10 - Module 6: Moments and Equivalence...

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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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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 z-axis? 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.

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