University:
California State University, Northridge
Course:
ME 584 | Modeling and Simulation of Dynamic Systems
Academic year:

2010
Views:

433

Pages:

39
Author:

Nathaly Davidson

chp3 1 Modeling Mechanical Systems chp3 2 Agenda • • • • Idealized Modeling Elements Modeling Method and Examples Lagrange’s Equation Case study: Feasibility Study of a Mobile Robot Design • Matlab Simulation Example • Active learning: Pair-share exercises, case study chp3 3 Idealized Modeling Elements chp3 4 Inductive storage Electrical inductance Translational spring Rotational spring Fluid inertia chp3 5 Capacitive Storage Electrical capacitance Translational mass Rotational mass chp3 6 Energy dissipators Electrical resistance Translational damper Rotational damper chp3 7 Springs - Stiffness Element - Stores potential energy x chp3 8 Actual Spring Behavior chp3 9 Spring Connections • Spring in series: KEQ=K1K2/(K1+K2) • Spring in parallel: KEQ=K1+K2 chp3 10 Dampers and Mass chp3 11 Dampers Connections • Dampers in series: BEQ=B1B2/(B1+B2) • Dampers in parallel: BEQ=B1+B2 chp3 12 Modeling Mechanical Systems chp3 13 Modeling Methods • State assumptions and their rationales • Establish inertial coordinate system • Identify and isolate discrete system elements (springs, dampers, masses) • Determine the minimum number of variables needed to uniquely define the configuration of system (subtract constraints from number of equations) • Free body diagram for each element • Write equations relating loading to deformation in system elements • Apply Newton’s 2nd Law: – F = ma for translation motion – T = Iα for rotational motion chp3 14 Example 1: Automobile Shock Absorber Spring-mass-damper Free-body diagram d 2 y(t ) dy(t ) M b  ky(t )  r(t ) 2 dt dt chp3 15 Example 2: Mechanical System • Draw a free body diagram, showing all forces and their directions • Write equation of motion and derive transfer function of response x to input u chp3 16 Example 2: Mechanical System chp3 17 Example 3: Two-Mass System • Derive the equation of motion for x2 as a function of Fa. The indicated damping is viscous. chp3 Example 3: Two-Mass System 18 chp3 19 Example 4: Three-Mass System • Draw the free-body-diagram for each mass and write the differential equations describing the system chp3 20 Example 4: Three-Mass System chp3 21 Example 5: Pair-Share Exercise • All springs are identical with constant K • Spring forces are zero when x1=x2=x3=0 • Draw FBDs and write equations of motion • Determine the constant elongation of each spring caused by gravitational forces when the masses are stationary in a position of static equilibrium and when fa(t) = 0. chp3 22 Example 5: Pair-Share Exercise: chp3 23 Example 5: Pair-Share Exercise: chp3 24 Example 6: Pair-Share Exercise • Assume that the pulley is ideal – No mass and no friction – No slippage between cable and surface of cylinder (i.e., both move with same velocity) – Cable is in tension but does not stretch • Draw FBDs and write equations of motion • If pulley is not ideal, discuss modeling modifications chp3 25 Example 6: Pair-Share Exercise • Pulley is not ideal – Add rotation mass and friction – Model the slippage behaviors – Add spring to model cable chp3 26 Example 7: Electric Motor • An electric motor is attached to a load inertia through a flexible shaft as shown. Develop a model and associated differential equations (in classical and state space forms) describing the motion of the two disks J1 and J2. • Torsional stiffness is given in Appendix B chp3 27 Example 7: Electric Motor chp3 28 Example 7: Electric Motor chp3 29 Example 7: Electric Motor chp3 30 Example 8: Pair-Share Exercise: Copy Machine • The device from a copying machine is shown. It moves in a horizontal plane. Develop the dynamic model, assuming that mass of bar is negligible compared to attached mass m2 and angular motions are small. The mass is subjected to a step input F, find an expression for the displacement of point B after the transient motions have died out. chp3 31 Example 8: Pair-Share Exercise: Copy Machine chp3 32 Example 9: Mass-Pulley System • A mechanical system with a rotating wheel of mass mw (uniform mass distribution). Springs and dampers are connected to wheel using a flexible cable without skip on wheel. • Write all the modeling equations for translational and rotational motion, and derive the translational motion of x as a function of input motion u • Find expression for natural frequency and damping ratio chp3 33 Example 9: Mass-Pulley System chp3 34 Example 9: Mass-Pulley System chp3 35 Example 9: Mass-Pulley System chp3 36 Example 10: Pair-Share Exercise: Double Pendulum • The disk shown in the figure rolls without slipping on a horizontal plane. Attached to the disk through a frictionless hinge is a massless pendulum of length L that carries another disk. The disk at the bottom of the pendulum cannot rotation relative to the pendulum arm. • Draw free-body diagrams and derive equations of motion for this system. chp3 Example 10: Pair-Share Exercise: Double Pendulum 37 chp3 38 Example 10: Pair-Share Exercise: Double Pendulum chp3 39 Example 10: Pair-Share Exercise: Double Pendulum

Modeling Mechanical Systems

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