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From Rileys Dynamics Chapter 16 Kinetics of Rigid Bodies: Newtons Laws

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(Q) What are the Eulers Equations of Motion? Newtons Law applies only to the motion of a single particle translation G G R R only translationtranslation + rotation Newtons 2 nd Law Eulers Equations of Motion

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Rotation of a Rigid Body moment Starting Point Moment of F & f about A Newtons 2 nd Law Substitution yields

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Whats this? After integration, we can get the general form of the Eulers equations of motion. Very general equation about rotation. Need to unify the coordinate systems to {Axyz}.

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(Q) Simplified Version Plane Motion Mass center G lies in the xy-plane. r dm Now, After the similar calculation, we have

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product of inertia moment of inertia Using 0

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(Note) The 1 st 2 equations are required to maintain the plane motion about z-axis, especially for non-symmetrical geometry case. If the body is symmetric about the plane of motion,

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If (symmetry) + (acceleration of the point A = 0) If (symmetry) + (A = G)

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(Q) More about the Moment of Inertia For the particle dm For the entire body It uses the information about its geometry. THE SAME MASS BUT DIFFERENT GEOMETRY DIFFERENT MOMENT OF INERTIA IF widely distributed THEN larger moment of inertia

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There are various ways of choosing this small mass element for integration. A specific mass element may be easier to use than other elements.

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You may treat the rigid body as a system of particles.

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0 0 2 nd moment of area

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If the density of the body is uniform,

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Practical approach a rigid body summation of several simple shape rigid bodies composite body

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(Q) What is the radius of gyration? gyration [ ʤ aiəréi ʃ ən] n. U,C,, ( ); ( ). al [- ʃ ənəl] a.,. k m m I : moment of inertia about the axis (the moment of inertia about the axis) = mk 2 = NO useful physical interpretation!! Maybe baseball Home Run !!!!!

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(Q) What is the Parallel-Axis Theorem for Moments of Inertia? 0 0 measurement of the location of the mass center from the mass center = m

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z

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(Q) More about the Product of Inertia dm x y RzRz x y In 2-D space

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(Q) What is the effect of symmetry on the product of inertia? x y z x y y z x z

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x y z x z z y x y

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(Q) What is the Parallel-Axis Theorem for Product of Inertia? From definition or But, mass center from the mass center 0 0 and Therefore,

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(Q) What is the Rotation Transformation of Inertia Properties? Consider x y z x y z We know that We can represent i, j, and k w.r.t. i, j, and k. Substitution yields

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or a vector in the new frame a vector in the old frame [R] rotation transformation matrix from old to new frame old new

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(Example) x y x y Θ Θ It means that [R] is an orthonormal matrix. Rotation about z-axis

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y z y z Θ Θ z x z x Θ Θ Rotation about x-axis Rotation about y-axis

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Now, the rotational kinetic energy is Since energy is invariant Let : known old frame Let : unknown new frame newfrom old to new old This term will be derived in the next chapter.

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(Example) a = 240 mm b = 120 mm c = 90 mm x y z m = 60 kg Claim: [I] = ? (Idea) x y z x z y G

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G b c a

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G b c a x y z By using the parallel axis theorem,

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x y z x y z Θ Θ a b c

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Slender rod

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Thin rectangular plate

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Thin circular plate

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Quiz #1 X Y Z {xyz} Inertia matrix.

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(Q) How to analyze the General Plane Motion of NonSymmetric Bodies? For Plane Motion

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120 mm dia. m = 7.5 kg 30 mm dia. m = 1.2 kg l = 220 mm = /2-40/2 40 mm dia. 8.5 kg 600 rpm ccw increasing in speed at the rate of 60 rpm per second Bearing A resists any motion in the z-direction. Claim: 5 reactions & T ?

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120 mm dia. m = 7.5 kg 30 mm dia. m = 1.2 kg l = 220 mm = /2-40/2 40 mm dia. 8.5 kg The same result for this sphere since z G and x G are minus sign. The same result for this bar since z G and x G are minus sign. For the entire system

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Or next page

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z x z x Sym.

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(Q) How to analyze the 3-D Motion of a Rigid Body? Recall How? X Y Z O x z y A G dm All vectors are represented w.r.t. the body-fixed {xyz}.

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Eulers Equations of Motion Rotation of a Rigid Body moment Starting Point Moment of F & f about A Newtons 2 nd Law Substitution yields

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Whats this? After integration, we can get the general form of the Eulers equations of motion. Very general equation about rotation. Need to unify the coordinate systems to {Axyz}.

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If we use the Cartesian coordinate system, In vector-matrix form,

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Or

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= 75 rad/s constant = 25 rad/s constant

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= 75 rad/s constant = 25 rad/s constant or more mathematically

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= 75 rad/s constant = 25 rad/s constant Solvable!

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Therefore,

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