An Introduction to Three-Dimensional, Rigid Body Dynamics. James W. Kamman, PhD. Volume II: Kinetics. Unit 3

Transcription

1 Summary An Introduction to hree-dimensional, igid ody Dynamics James W. Kamman, PhD Volume II: Kinetics Unit Degrees of Freedom, Partial Velocities and Generalized Forces his unit defines the concepts of degrees of freedom, generalized coordinates, partial elocities, partial angular elocities and generalized forces. hese concepts form an introduction to methods of treating systems with multiple bodies as systems rather than one body at a time as with the Newton/Euler equations of motion. Page ount Examples Suggested Exercises opyright James W. Kamman, 6

2 Degrees of Freedom of Mechanical Systems System onfiguration and Generalized oordinates he configuration of a mechanical system is defined as the position of each of the bodies within the system at a particular instant. In general, both translational and rotational coordinates are needed to describe the position of a rigid body. ogether the translation and rotation coordinates are called generalized coordinates. For the simple pendulum shown, x G Simple Pendulum and y G are translational coordinates that describe the location of the mass center of the bar, and is a rotational coordinate that describes the orientation of the bar. ypically, the generalized coordinates used to define the configuration of the mechanical system form a dependent set. hat is, the coordinates are not independent of each other. For example, for the simple pendulum x G, can be written y G, and are not independent, because the following two independent constraint equations x Lsin ( ) y Lcos( ) G G Gien the alue of one of the coordinates, these equations can be used to compute the alues of the other two coordinates, so only one of the coordinates is needed. Any pair of these coordinates forms a dependent set. Note: Generalized coordinates need not always directly represent translational or rotational ariables, but they will for the purpose of this text. Generalized oordinates and Degrees of Freedom he number of degrees of freedom (DOF) of a mechanical system is defined as the minimum number of generalized coordinates necessary to define the configuration of the system. For a set of generalized coordinates to be minimum in number, the coordinates must form an independent set. he figures to the right show examples of one, two, and three degree-of-freedom planar systems. For mechanical systems that consist of a series of interconnected bodies, it may not be obious how many opyright James W. Kamman, 6 Volume II, Unit : page /8

3 degrees of freedom the system possesses. For these systems, the number of degrees of freedom can be found by first calculating the number of degrees of freedom the system would possess if the motions of all the bodies were unrestricted and then subtracting the total number of constraints on the motion. For example, for the twodimensional slider-cran mechanism shown, the number of degrees of freedom can be calculated in the following ways: a) ounting bodies: (cran, slider, and connecting bar) DOF/body = pin joints = = 9 possible DOF = 6 constraints slider joint = = constraints Actual DOF = 96 = DOF b) ounting bodies: (cran and connecting bar) DOF/body = = 6 possible DOF pin joints = = 4 constraints pin/slider joint = Actual DOF = 64 = constraint = DOF he slider cran mechanism shown is a two-dimensional mechanism. Each pin joint in the mechanism restricts the X and Y motion of that point. herefore, each pin joint proides two constraint equations. he slider joint of part (a) restricts the rotational motion and the Y translational motion of the slider. Hence, it proides two constraint equations. In part (b), the slider is not counted as a body, so the joint at is considered to restrict the motion of the connecting rod A. he bar is free to translate in the X direction and to rotate at, but it is not free to translate in the Y direction. Hence, the pin/slider joint proides only one constraint equation. A similar approach could be used to erify that the two dimensional four-bar mechanism shown aboe also has only one degree of freedom. he system has three moing bars (with 9 possible degrees of freedom) and four pin joints proiding eight constraint equations. his approach can be applied to three dimensional systems as well. As with two dimensional systems, care must be taen to determine exactly how each joint restricts the motion of the system. opyright James W. Kamman, 6 Volume II, Unit : page /8

4 Example : hree-dimensional Slider-ran Mechanism he slider-cran mechanism shown has three members a cran, a slider, and a connecting rod. he cran (dis) is pinned to and rotates about its center in the xy plane. he slider (collar) at translates along and rotates about the fixed bar EF. he connecting rod A is connected to the cran using a ball and socet joint, and it is connected to the slider using a pin joint. Find: alculate the number of degrees of freedom (DOF) of the mechanism. Solution: he pin joint on the cran at proides fie constraints by restricting all but the rotational motion of the cran about the z direction relatie to the ground. he ball and socet joint at A proides three constraints by restricting the x, y and z translations of that point relatie to the cran. ar EF proides four constraints by restricting the x and z translations and the x and z rotations of the slider relatie to the ground. Finally, the pin joint at proides fie constraints by restricting all but one rotation of the bar relatie to the slider. Summary: 6 DOF/body = 6 pin joint at = 5 ball and socet joint at A = = 8 possible DOF = 5 constraints = constraints cylindrical joint on slider = 4 = 4 constraints pin joint at = 5 = 5 constraints Actual DOF = = DOF Partial Velocities If the elocity of some point P within a mechanical system can be written in terms of a set of generalized coordinates q (,, n) and their time deriaties q (,, n), then the partial elocities of P are defined to be the partial deriaties of with respect to the q (,, n). P q P (,, n) hese ectors represent the changes in the elocity of P resulting from changes in each of the q. It can be shown that they also represent the changes in r P the position of P resulting from changes in the opyright James W. Kamman, 6 Volume II, Unit : page /8

5 q (,, n). hey are a measure of the sensitiity of the elocity (or position) of P to changes in the q (or q ). In this regard, they can also proide a measure of the mechanical adantage or disadantage associated with changes in any of the generalized coordinates. If small changes in the generalized coordinates produce large changes in the position of P, then the proider has a mechanical disadantage. If large changes in the generalized coordinates produce small changes in the position of P, then the proider has a mechanical adantage. From the perspectie of multiariate calculus, if the position ector of P is a function of the generalized coordinates and time, that is, r r ( q, t), then P the elocity of P can be written as follows. P P P dr r r dt q t n P P P q From this result, it also follows that q P r q P he term r P t accounts for position ectors that depend explicitly on time (e.g. some specified motion). Note that since the elocity is linear in the q (,, n), the partial elocities are found by inspection of the elocity ector. Partial Angular Velocities Similarly, if the angular elocity of a body within a mechanical system can be written in terms of a set of generalized coordinates q (,, n) and their time deriaties q (,, n), then the partial angular elocities of are defined to be the partial deriaties of with respect to the q (,, n). q (,, n) hese ectors represent changes in the angular elocity of a body resulting from changes in the q (,, n). In general, the angular elocity can be written in terms of the partial angular elocities as n q ( ) t q where ( ) is that part of the angular elocity ector that depends explicitly on time. t opyright James W. Kamman, 6 Volume II, Unit : page 4/8

6 Note the aboe equation cannot generally be formulated using a differentiation process as there is no ector whose deriatie is the angular elocity. he angular elocity is formed using the summation rule for angular elocities, and as with the partial elocities, the partial angular elocities are found by inspection. Example : Planar, losed-loop, Slider ran Mechanism System onfiguration he figure shows a simple slider cran mechanism with no offset. Gien the physical dimensions of the lins (, L ), the configuration of the system at any instant of time can be described by one or all the generalized coordinates q,, x. As only one coordinate is required to define the system configuration at any instant, a set of two constraint equations can be written relating the three coordinates. For example, the ector loop-closure equation r / r / r / gies the following two scalar constraint equations. A A S LS L x (two constraint equations relating the three dependent generalized coordinates) Here, as preiously in this text, the symbols S, S, and hae been used to represent the sines and cosines of the angles. Partial Angular Velocities of the Lins Using the angles shown, the angular elocities of the cran and connecting bar can be written as A and. From these results, two obious partial angular elocities can be defined. A A second set of partial angular elocities can also be defined by first relating the angular rates and. his can be done by differentiating the first of the two constraint equations with respect to time (using the chain rule). L Using this result to express one of the angular rates in terms of the other, the following additional partial angular elocities can be defined opyright James W. Kamman, 6 Volume II, Unit : page 5/8

7 L L L L A Note that becomes zero when the cran angle 9 (deg). In this position, the connecting rod is translating and not rotating, so changing the angular elocity of the cran has no effect on the angular elocity of the connecting rod. onersely, A becomes undefined when the cran angle 9 (deg). Near this position, the angular elocity of the cran is ery sensitie to changes in the angular elocity of the connecting rod. Partial Velocities of the Slider he elocity of the slider can be written most simply as x i. From this result, the following partial elocity can be defined x i Additional partial elocities can be defined by relating x to the angular rates and. his can be done by differentiating the second of the constraint equations with respect to time and using the chain rule. x S LS Using this result, the elocity of the slider can be written as x i S LS i Using this result and the equation aboe relating the angular rates and, the following additional partial elocities can be defined. S L S i L L S LS i Note when the cran angle S S i L S S i (or 8) (deg), both of these partial elocities are zero, but when 9 (deg), i and is undefined. ecall that when (or 8) (deg), the elocity of opyright James W. Kamman, 6 Volume II, Unit : page 6/8

8 the slider is zero and both bars are rotating about their end points, and when 9 (deg) the connecting rod is translating. When the connecting rod is translating, the cran angular elocity has its largest influence on the elocity of the slider. Example : Six DOF Arm he system shown is a six DOF double pendulum or arm. he first lin is connected to ground and the second lin is connected to the first with ball and socet joints at O and A. he ground frame is :,, i i i frames are i N N N and the lin L : n, n, n ( i,). he orientation of each lin is defined relatie to using a -- body-fixed rotation sequence. he lengths of the lins are and. Find: a) Partial angular elocities of the lins associated with the rates of the six orientation angles. b) Partial elocities of point associated with the rates of the six orientation angles. Solution: Preious results: In Unit 7 of Volume I the angular elocities of the lins and the elocity of were found to be n n n with i i i Li i i i S S i i i i i i S S i i i i i i i i i i ( i,) n n n n a) Using these results, the following partial angular elocities can be identified. L SS n S n n L n S n L n L S S n S n n L n S n L n b) he following partial elocities can also be identified. opyright James W. Kamman, 6 Volume II, Unit : page 7/8

9 n S S n n n n S S n n n ecall that matrices and transform the lin-based components into the base frame. Hence, the results shown for the partial elocities and partial angular elocities could all be easily transformed into the base frame using the transformation matrices. Generalized Forces Gien a mechanical system whose configuration is defined by a set of generalized coordinates q (,..., n), the generalized forces associated with each of the n generalized coordinates as F F M i j q i j forces q torques q (,, n ) ( i) ( j) Here the index i represents each of the forces, the index j represents each of the torques, and the index represents each of the generalized coordinates. he forces contributions to this sum are said to be actie. onseratie and Nonconseratie Forces onsider a particle that moes from position to position along one path (forward path) and bac again to position along a second path (return path) as shown in the diagram. A force acting on the particle is said to be conseratie if the net wor it does oer the closed path is zero. Suppose, for example, that the wor done by the force as the particle moes from position to position is positie, then the force does the same amount of wor as the particle returns to position, except that this wor is negatie. F i and the torques M j that hae non-zero In this way, conseratie forces do not permanently add or remoe energy from a system. When the conseratie force is doing negatie wor, the system is said to be gaining potential energy that can later be transformed into inetic energy. It is also true of conseratie forces that the wor done in moing from one position to another is independent of the path of the particle. Examples include weight forces and spring forces and torques. opyright James W. Kamman, 6 Volume II, Unit : page 8/8

10 Forces whose net wor around a closed circuit is not zero are called nonconseratie forces. he wor done by these forces as a particle moes from one position to another is dependent on the path of the particle. Examples include friction and damping forces and torques. onseratie Forces and Potential Energy (V) he generalized force associated with conseratie forces and torques can be written in terms of a potential energy function, V. For weight forces and linear spring forces and torques, the potential energy functions are V W y mg y ( y is the height of the particle aboe some arbitrary datum) V V e ( e is the elongation or compression of the spring (units of length)) ( is the elongation or compression of the spring (radians or degrees)) For systems with multiple conseratie forces, the system potential energy is V V. If conseratie forces and torques do wor on a system (i.e. if they are actie), their contribution to the generalized forces can be calculated using the definition gien aboe or they can be calculated using the potential energy function V. i i F q V q (,, n) Viscous Damping and ayleigh s Dissipation Function () One type of nonconseratie force or torque is associated with iscous damping. One way of modeling this phenomenon is to assume that the forces or torques are proportional to the relatie elocity or relatie angular elocity of the end points of the element. Force: F crel orque: rel M c n Here, the direction of the unit ector n is defined by the right-hand rule. For these types of nonconseratie forces and torques, ayleigh s dissipation function is Force: c x orque: rel c rel Here, the symbols x rel and rel hae been used to represent the rates of translational and rotational motion between the ends of the damping element. For systems with multiple proportional damping elements, the system dissipation function is. i i opyright James W. Kamman, 6 Volume II, Unit : page 9/8

11 he contribution of these forces and/or torques to the generalized forces can be calculated using the definition gien aboe or they can be calculated using ayleigh s dissipation function. F q q (,, n) Example 4: Planar, One-Lin Pendulum he one lin pendulum shown is acted upon by graity and by the applied moment M. he Y axis is pointed in the direction of graity. he single coordinate describes the position of the pendulum. Find: F, the generalized force associated with the coordinate Solution: he mechanical system in this case is a single body the pendulum lin. Using the general definition of the generalized force acting on the system associated with the coordinate is defined to be F F M F W M i j O G i j O forces torques ( i) ( j) zero he angular elocity of the bar, the elocity of the mass center G can be written as follows. d G LS i L j L i S j dt Substituting into the generalized force equation gies G F W M W j L i S jm W LS M Note: he contribution of the weight force could hae been calculated using the potential energy function for graity. Assuming a horizontal datum along the X axis passing through O F W L W LS W opyright James W. Kamman, 6 Volume II, Unit : page /8

12 Example 5: Loaded, Planar, Slider-ran Mechanism he figure shows a slider cran mechanism under the action of an external torque acting on lin A, external forces P and Q acting at and, and a linear spring attached at. he spring has stiffness, an unstretched length of u, and it always remains horizontal. Neglect weight forces and friction. Find: F the generalized force associated with the coordinate Solution: he system in this case is the entire slider-cran mechanism. he actie forces and torques acting on the system are the forces P and Q, the torque, and the linear spring force. he pin forces at A, and and the wall force on the slider are not actie. Why? he pin at A has zero elocity and zero partial elocity. he pin at has nonzero elocity and partial elocity, but the internal forces on members A and are equal and opposite, so their net contribution is zero. he same is true for the pin forces at acting on the and the slider. Without friction, the wall force is perpendicular to the elocity of, and hence it is not actie. So, the generalized force F can be written as follows. F F M F F F F i j i j forces torques ( i) ( j) P Q spring A P i Q i Fs i he angular elocity of A and the elocity of can be written as A / A i S j S i j he elocity of was found in Example to be S S i Substituting these results into the definition of F gies A F P i Q i Fs i P Fs i S i j Q i S S i opyright James W. Kamman, 6 Volume II, Unit : page /8

13 s F F P S Q S S he spring force is equal to the product of the spring stiffness and elongation, or Fs u. he contribution of the spring force to F can also be calculated using the potential energy function of the spring. Specifically, spring u u u V F S S spring Example 6: Planar Arm with Driing orques he figure shows a two-lin arm in a ertical plane with driing torques at the joints. he two uniform slender lins are assumed to be identical with mass m and length. he system has two degrees of freedom described by the generalized coordinate set. Find: F and, F the generalized forces associated with and Solution: he actie forces in this system are the two weight forces and the two driing torques. he contribution of the weight forces can be found by using the potential energy function for graity. Assuming a horizontal datum passing through the fixed point O, the potential energy of the system can be written as V V V mg mg mg mg V V he contributions of the weight forces to the generalized forces can then be written as V F mg mg mg S weights V F mg mg mg S weights o calculate the contributions of the driing torques, first note that the torque M is applied to the first lin by the ground, and the torque M is applied to the second lin by the first lin. he torque M applied to the second lin is accompanied by a reaction torque M applied to the first lin. With this in mind, the generalized actie forces associated with the driing torques are opyright James W. Kamman, 6 Volume II, Unit : page /8

14 F M M M M M M torques F M M torques F M M M M M M torques F he generalized forces torques M F and F are the sums of the aboe results F F F M M mg S torques weights Example 7: Six DOF Arm with End Force A force F F N F N F N is applied to the end of the six degree of freedom arm described in Example. Find: F ( i,,) and i F ( i,,) the generalized forces i associated with F and the six orientation angles Solution: In Unit 7 of Volume I, the components of the elocity of in the base frame were found to be F F F M mg S torques weights N N N with S S i i i i i i S S i i i i i i i i i i Using these results, the six partial elocities of can be written as follows. N N SS N N N N N N N opyright James W. Kamman, 6 Volume II, Unit : page /8

15 N N SS N N N N N N N Using these results, the six generalized forces associated with the applied force F can be written as follows. F F F F F SS F F F F F F F F F F F F F F F SS F F F F F F F F F F Exercises. he system shown consists of three bodies with eighteen possible degrees of freedom. he collar at is connected to the fixed horizontal bar using a prismatic joint so it can translate along the bar but not rotate. he collar at A is connected to the fixed ertical bar with a cylindrical joint so it can translate along and rotate about the bar. ar A is connected to the collar at using a ball and socet joint, and it is connected to the collar at A using a pin joint. opyright James W. Kamman, 6 Volume II, Unit : page 4/8

16 Using the counting procedure discussed aboe, erify that the system has only one degree of freedom. How many degrees of freedom does the system hae if the joint at A is changed to a ball and socet joint? How many degrees of freedom does the system hae if the collar at is connected to the fixed horizontal bar with a cylindrical joint? Answers: DOF as is; DOF if ball and socet joint at A; DOF if collar at can rotate. he one degree of freedom system shown consists of slender bar A of mass m and length and a piston P of mass m P. he system is drien by the force Ft () and graity. A spring and damper are attached to the light slider at. he spring has stiffness and is unstretched when x. Find F the generalized force associated with the coordinate. Neglect friction. Answer: () P F F t S m g S mg S S c. he system shown is a two degree of freedom pendulum (or arm) that moes in a ertical plane. he external force P P i P j acts on the end of the second lin. Find x F and y F the generalized forces associated with the two coordinates and. Include the weight forces and the external force P. he lins are identical uniform lins with mass m and length midpoints of the lins. Answers:. he mass centers are at the F P P S mg S x y.4 he system shown consists of a slider of mass m and a uniform slender bar A of mass m and length. he slider is connected to the fixed horizontal bar with a prismatic joint it translates along the bar but does not rotate about it. he slider is attached to the ground by a spring of stiffness and linear iscous damper with damping coefficient c. he bar is pinned to the slider at A and rotates freely about that point. he system is drien by the forces F x y F P P S mg S F i and P P i P j. he spring is unstretched at x. Find x y F x and F the generalized forces associated with the coordinates x and. opyright James W. Kamman, 6 Volume II, Unit : page 5/8

17 Answers: Fx F Px x cx F Px Py S m g S.5 he two degree of freedom system shown consists of two bodies dis D and slender bar. he dis has radius and mass m D. he slender bar has length and mass m. he unit ector set D( i, j, ) are fixed in the dis along the X, Y and Z axes. he rotation of the dis about the Z axis is gien by the angle ( ), and the rotation of the bar relatie to the dis about the X axis is gien by the angle. A rotational spring-damper located at pin P restricts the motion between the bar and the dis. he spring has stiffness and the linear, iscous damper has damping coefficient c. An external force F F i F j F is applied to the end of the bar. X Y Z A motor torque M is applied by the ground to the dis about the Z axis, and a motor torque M is applied by the dis to the bar at P. Find F and F the generalized forces associated with the coordinates and. F M mg S c FY FZ S Answers: F M FX b S.6 he system shown is a three-dimensional double pendulum or arm. he first lin is connected to ground and the second lin is connected to the first with uniersal joints at O and A, respectiely. he ground frame is :,, i i i are i N N N and the lin frames L : n, n, n ( i,). he orientation of L is defined relatie to and the orientation of L is defined relatie to L each with a - body-fixed rotation sequence. Lin OA is oriented relatie to the ground frame by first rotating through an angle about the N direction, and then rotating about an angle about the lin OA by rotating first through an angle about the about the n direction. he lengths of the lins are and n direction. Lin A is oriented relatie to n direction, and then through an angle with mass centers are at their midpoints. he system is drien by graity and by four motor torques, one on each axis of the uniersal joints. he four motor torques can be written as follows opyright James W. Kamman, 6 Volume II, Unit : page 6/8

18 M M N M M n M M n M M n he two constraint torques transmitted through the joints can be written as Find O O A A n n N n F, F, F and. F the generalized forces associated with the four orientation angles. In the process, show that the contribution of the constraint torques O and A are zero. Assume the N direction is ertical. Answers: ecall that matrix transforms components from the base frame () to L, and the matrix transforms components from L to L. F M W S S W W S S M W S W S W S S S S S F M W W W S S M W S W S W S S F M W M W S S F M W M W S S S S S opyright James W. Kamman, 6 Volume II, Unit : page 7/8

19 eferences:. L. Meiroitch, Methods of Analytical Dynamics, McGraw-Hill, Kane, P.W. Liins, and D.A. Leinson, Spacecraft Dynamics, McGraw-Hill, Kane and D.A. Leinson, Dynamics: heory and Application, McGraw-Hill, L. Huston, Multibody Dynamics, utterworth-heinemann, H. aruh, Analytical Dynamics, McGraw-Hill, H. Josephs and.l. Huston, Dynamics of Mechanical Systems, Press, 7... Hibbeler, Engineering Mechanics: Dynamics, th Ed., Pearson Prentice Hall, 8. J.L. Meriam and L.G. raig, Engineering Mechanics: Dynamics, rd Ed, F.P. eer and E.. Johnston, Jr. Vector Mechanics for Engineers: Dynamics, 4 th Ed, 984 opyright James W. Kamman, 6 Volume II, Unit : page 8/8