Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships -Physical Laws Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections Derive Equation of Motion (EOM) - MDOF ME375 Translation - 1 Variables x : displacement [m] v : velocity [m/sec] a : acceleration [m/sec ] f : force [N] p : power [Nm/sec] w : work ( energy ) [Nm] 1 [Nm] = 1 [J] (Joule) d x = x = v dt d d d d v= v= x = x= x= a dt dt dt dt d p = f v= f x = w dt t1 ( 1) = ( 0) + ( ) t0 wt wt pt dt t1 ( 0) ( ) t0 = wt + f x dt ME375 Translation - 1
Idealized Modeling Elements Inertia (mass) Stiffness (spring) Dissipation (damper) ME375 Translation - 3 Basic (Idealized) Modeling Elements Spring Stiffness Element K ( ) f = K x x S 1 Idealization Massless No Damping Linear Stores Energy x Reality 1/3 of the spring mass may be considered into the lumped model. In large displacement operation springs are nonlinear. (x ) ME375 Translation - 4
Practical Nonlinear Spring Engine Mount: Experimental #T06 VERTICAL Analytical 4000 000 LOAD (N) 0 1-000 -4000-0 -15-10 -5 0 5 DISP (mm) Restoring force = ( K + μδx ) Δx 1 Small motions K for isolation Large motions = K + μδx for static loads ME375 Translation - 5 Basic (Idealized) Modeling Elements Damper Friction Element Mass Mass Inertia Element x x D ( ) ( ) f = B x x = B v v 1 1 Dissipate Energy f f 3 i Stores Kinetic Energy i M f 1 M x= f = f f f 1 3 ( x x ) 1 ME375 Translation - 6 3
Interconnection Laws Newton s s Second Law d dt ( ) = = M v M x f Linear Momentum i EXTi Newton s s Third Law Action & Reaction Forces K M x K M Displacement Law ME375 Translation - 7 Modeling Steps Understand System Function, Define Problem, and Identify Input/Output Variables Draw Simplified Schematics Using Basic Elements Develop Mathematical Model (Diff. Eq.) Identify reference point and positive direction. Draw Free-Body Body-Diagram (FBD) for each basic element. Write Elemental Equations as well as Interconnecting Equations by applying physical laws. (Check:( # eq = # unk) Combine Equations by eliminating intermediate variables. Validate Model by Comparing Simulation Results with Physical Measurements ME375 Translation - 8 4
In Class Exercise ( (Blood Sampler) Schematic: Already done in Introduction; see those Filled Slides. ME375 Translation - 9 Energy Distribution EOM of a simple Mass-Spring Spring-Damper System M x + Bx + Kx = f() t Contribution Contribution Contribution of Inertia of the Damper of the Spring Total Applied Force We want to look at the energy distribution of the system. How should we start? K B M x f Multiply the above equation by the velocity term v : What have we done? Integrate the second equation w.r.t. time: What are we doing now? t1 t1 t1 t1 M x xdt + () t Bx xdt + Kx xdt = f t vdt 0 t 0 t 0 t0 ΔE 1 t1 1 Δ KE = M x t Bx dt 0 0 Δ PE = K x Total work done by the applied force f ( t) from time t0 to t1 ME375 Translation - 10 5
Example -- SDOF Suspension Suspension System Minimize the effect of the surface roughness of the road on the drivers comfort. Simplified Schematic (neglecting tire model) Define the reference position for the displacement of the car as the position when the spring does not have any deflection (i.e., the neutral position) ME375 Translation - 11 SDOF Suspension Draw FBD Apply Interconnection Laws Q: Since gravity is always present, is there a way to represent the suspension system by subtracting the effect of gravity? ME375 Translation - 1 6
SDOF Suspension (II) Relative Displacement Approach Define the reference position as the position of the car when the system is at rest in the gravity field, i.e., the spring force balances the car s s weight. FBD ME375 Translation - 13 SDOF Suspension (II) Interconnection Laws & Simplification Q: What are the differences between the two models? Q: Do the two models represent the same physical system? If they do, why are they different? ME375 Translation - 14 7
Series Connection Springs in Series x x K 1 K K EQ ME375 Translation - 15 Series Connection Dampers in Series x x B 1 B B EQ ME375 Translation - 16 8
Parallel Connection Springs in Parallel x x K 1 K EQ K ME375 Translation - 17 Parallel Connection Dampers in Parallel x x B 1 B EQ B ME375 Translation - 18 9
MDOF Suspension Suspension System Simplified Schematic (with tire model) ME375 Translation - 19 MDOF Suspension Draw FBD Apply Interconnection Laws ME375 Translation - 0 10
MDOF Suspension Matrix Form ME375 Translation - 1 11