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Published byJonathan Overbeck Modified over 2 years ago

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Example: Derive EOM for simple 1DOF mechanical system The figure shows a mechanical system comprised of two blocks connected by a pulley and (inextensible) cable system. The cable does NOT slip on the pulley. The block of mass M1 slides on a smooth surface and is connected to a fixed rigid wall through an elastomeric cable represented by stiffness K and viscous damping coefficient C. The system is initially at rest at its static equilibrium position. The external force F(t), applied to block of mass M2 for time t0 s, drives the system into motion. a)Select suitable coordinates for motion of the two blocks, show them on the Figure and explain rationale for your choice. b) Identify the kinematical constraint relating motions of the two blocks. c) Find the static deflection ( s) of the elastomeric cable attached to fixed wall d) Draw free body diagrams for motion of blocks and applicable for time t>0 s. Label all forces and define their constitutive relation in terms of the motion coordinates. e)Using Netwons Laws, state a EOM for each block (t0 s ), combine them to obtain a single EOM Luis San Andres©

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Parameters: M: mass K : stiffness coefficient Forces: W: weight N: normal to wall Fs: elastic force from top cable T cable tension (inextensible) Spring force (static) : F se = K s M2M2 W2W2 T e (cable tension) Wall reaction force M1M1 TeTe W1W1 W 1 sin N=W 1 cos Free Body Diagram – STATIC EQUILIBRIUM 2Te2Te TeTe Definition of SEP (Static equilibrium position): System is NOT in MOTION + there are NO external forces applied on the system. BALANCE OF STATIC FORCES s is the static deflection of spring needed to hold the system together at the SEP (3)

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Parameters: M: mass K : stiffness coefficient C: viscous damping coefficient Forces: W: weight N: normal to wall Fs: elastic force from top cable F D : viscous damper force top cable T cable tension (inextensible) F: external force Variables: X, Y : coordinates for motion of block 2 and block 1, resp. ( absolute coords., with origin at equilibrium state) DEFINITIONS: Y X Dashpot force: F D =CdY/dt Spring force F s = K ( s +Y) M2M2 2T W2W2 T T (cable tension) Wall reaction force M1M1 T W1W1 W 1 sin N=W 1 cos F(t), applied at t=0 Free Body Diagram System moving: Static equilibrium position (SEP) defines origin of coordinates X, Y describing the motion of blocks 2 and 1, respectively. Assumed state of motion to draw FBD : X>0, Y>0 Kinematic constraint – inextensible cable 2 T X = T Y, hence 2 X = Y 2 X = Y SEP (1)

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DERIVE EOMs Free Body Diagram System moving X>0, Y>0 Y X Dashpot force: F D =CdY/dt Spring force : F s = K ( s +Y) M2M2 2T W2W2 T T (cable tension) M1M1 T W1W1 W 1 sin F(t), applied at t=0 2 X = Y SEP Block 1 (top) Block 2 In Eq. (4), isolate the tension (T) and substitute constraint Y=2 X (4) (5) Substitute Eq. (6) into Eq. (5) to obtain (6)

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DERIVE final EOM Final EOM: From SEP: balance of forces for static equilibrium (6) (3) Cancel forces from SEP to obtain: Move to LHS terms related to motion: If using Y as the independent coordinate: Y=2 X

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Assume a state of motion: X > 0, Y >0 ENERGIES for system components (1a) Potential energy V = strain energy in cables + gravitational potential energy change (1b) Viscous dissipated power = External power (1c) (1d) Kinetic energy T Includes static deflection from spring. Datum for potential energy is SEP Substitute above constraint relating motion of blocks: Y=2 X

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ENERGIES for system components (1a) Potential energy V = strain energy in cables + gravitational potential energy change (1b) Viscous dissipated power External power (1c) (1d) Kinetic energy T Datum is SEP Substitute constraint relating motion of blocks: Y=2 X

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Derive EOM from PCME (3a) (3b) (2) Note how static SEP forces cancel Substitute Eqs. (3) into Eq. (2) to obtain

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Derive EOM from PCME (5) (2) Cancel velocity dX/dt to obtain final EOM:

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