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# ENGR MECH DYNAMICS ME 16

UCSB

GPA 3.94

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This 13 page Class Notes was uploaded by Daren Beatty Jr. on Thursday October 22, 2015. The Class Notes belongs to ME 16 at University of California Santa Barbara taught by B. Bamieh in Fall. Since its upload, it has received 12 views. For similar materials see /class/227080/me-16-university-of-california-santa-barbara in Mechanical Engineering at University of California Santa Barbara.

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Date Created: 10/22/15

3 39w Wow 5W EM 013 sf We 4 4 KAN W3 CW0 Walt Mg ha 2 a a 5 19943 9 lgAIM VJ ya A mmv xy 1L IL 5 L2 w 3 90 meg margin Q2 007 V 32 634 WV 3 4 LC 1 4139 n 1 33 53133o39 Ml IN 393 En 39 3030 39 B N 2x w gar 3 wJ3039 m a A a 2ng N ma I quot A w g ask 61 can 4A 0WD quot 13 W183 O L e 0 3 53439 x kegs S wome 381M a g Mums M Vector Mechanics for Engineers Sample Problem 124 Dynamics The 12lb block B starts from rest and slides on the 30lb wedge A which is supported by a horizontal surface Neglecting friction determine a the acceleration of the wedge and b the acceleration of the block relative to the wedge SOLUTION The block is constrained to slide down the wedge Therefore their motions are dependent Express the acceleration of block as the acceleration of wedge plus the acceleration of the block relative to the wedge Write the equations of motion for the wedge and block Solve for the accelerations Vector Mechanics for Engineers Dynamics Sample Problem 124 SOLUTION The block is constrained to slide down the wedge Therefore their motions are dependent a3 Q aBA Write equations of motion for wedge and block 217x mAaA N1 sin30o mAaA 05N1WAgaA 217x mBax mg gA cos30 aBA WB sin30o WEgsz cos30 aBA aBA aA cos30 gsin30 EFy mBay mB aA sin30 N1 WB cos30 WBgaA sin30 m BaA Vector Mechanics for Engineers Dynamics Sample Problem 124 Solve for the accelerations 05N1 WA014 N1 WB cos 30 WB gaA sin 30 2WAgaA W3 cos30 WBgaA sin 30 gWB cos30 W 0A M 2301b121bsin30 aA 507fts2 LIBA aA cos30 gsin30 aBA 607ft52 zos30 622fts2m30 22u Q ij 5 ulna Mb A WSW 99 vwxn k 6 a 6 5W 9 13 M W 7543 1 yawg Cam WW W thwwn 39 Development of Terminology I m3 is central to the modeling y process but it is only one element contributing a problem39s solution 0 To form an appreciation for all of the equations we use to model a system it is useful to develop some terminology to more easily identify 1c 1000 these equations L0 1392 0 We will do this through an example of a crate sliding down a rough incline D and v0 are the init39al position and speed respectively The crate has been modeled as a particle Let39s determine the equations that govern motion of the crate The Newton Euler Equations The crate39s motion is rectilinear so we choose a Cartesian coordinate system and using the crate39s FBD we can write the crate39s Newton Euler equations a Z Fx 2 Fy max mgsin g 7 4 FS max7 may N 4 mgcos may 0 Newton39s second law is the basic postulate for a particle 9 However rotational motion is not defined for a particle 9 Therefore Newton39s second law is not sufficient for predicting the motion rigid bodies a An addi onal postulate or law is required which concerns the moments acting on the system a This realization is due to Leonard Euler 0 Therefore we w use the term Newton Euler N E equations The N E equations aren39t enough we still need to describe the kinematics as well as the forces acting on the system I l he Materials EquationsModels We begin with the description of the spring and friction forces the last set of equations come from the kinematics hese are what we call material equations in that they express the behavior of the spring and the behavior of the SI ing contact between e crate and the incline or the crate these equations are FS X 4 L0 and f ukN7 here k is the constant of the linear spring L0 is the unstretched length of the spring and uk is the coefficient of kinetic friction between the crate and the inc ine spring is linear elastic and its force lineary depends on the relative position between the two ends of the spring and therefore its form depends on the coordinate system chosen The force due to fr I l ion comes from the Coulomb model for fri The Kinematic Equations We know these well since we have been studying them almost 4 weeks 0 The nal set of equations comes from writing the accelerations in terms of the chosen component system these are the kinematic equations For our crate we know that the crate39s motion is parallel to the X direction and so y constant This means of course that ay O We now have all the equations we need since we have written the o NewtonEuler Equations These three sets of equations will always give us all the equations we Material Equations need to solve a problem Kinematic Equations I l Governing Equations Taken together the Newton Euler equations kinematic equations as given by material equationsand mgsin g f4 FS maX7 N 4 mgcos may7 4 kx 4 L07 P39kN ax39 i in H 7 7 ayOv are referred to as the governing equations of the system since they are the equations that govern the motion of the system

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