PART 2a. PLANAR KINETICS OF LUMPED MASS SYSTEMS

Motion and Deformation of Mechanical Systems with 1 Degree of Freedom (1DOF)

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Textbook Chapter 3.1 thru 3.2d (Lectures 5-12)

Acronyms: M: mass, K: stiffness, C: damping, ODE: ordinary differential equation, EOM: equation of motion, SEP: static equilibrium position, DOF: degree of freedom, FBD: free body diagram, CE: characteristic equation, CME: Principle of Conservation of Mechanical Energy

 

 

Lecture

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Major Topics/WHAT YOU WILL LEARN

Recommended  homework problems

5

Newton’s Laws for particle kinetics and equivalence to change in linear momentum = linear impulse. Drawing a free body diagram (FBD) and identifying forces acting on system mass (M). Derivation of Mechanical Work = Conservation of Mechanical Energy (CME) Principle. Example of motion with constant acceleration. Learning use of energy integral substitution. Mechanical stiffness element (K): Selection of coordinate system. Derivation of fundamental EOM for mass-spring (M-K) system. Solution of ODE to arbitrary initial conditions: system free and constant force responses. Concept of natural frequency. Review of physical units.

Spring-mass models

3.2, 3.3, 3.4 (note: K has units of N/m), 3.5, 3.6

 

6

Derivation of EOM from CME: kinetic + potential energy = Work. Definition of static equilibrium position (SEP) and finding small amplitude motions (perturbation) about SEP. EOM including external (non-conservative) forces. Necessary condition for static stability (K>0).

Spring-mass models with constraints

3.30, 3.34, 3.48 & 3.41(without the damper)

7

Examples of simple M-K systems and interpretation of physical responses. Nonlinear EOM for simple pendulum and linearization for small amplitude motions: evaluation of natural frequency. Mechanical viscous damping element (C): constitutive relationship for energy dissipation element. Derivation of fundamental EOM for mass-spring-damper (M-K-C) system. Solution of ODE for arbitrary initial conditions: system free and constant force response. The concept of damped natural frequency and viscous damping ratio. Review of physical units. Necessary condition for dynamic stability (C>0). Types of system responses: underdamped, overdamped and critically damped

Viscous damping, transient response solutions, log dec

3.38, 3.41, 3.20

8

Transient (1) response of simple M-K-C systems.  Forced response including effect of initial conditions. Superposition of solutions: homogeneous + particular: formulas for forcing functions of the form = c + at + b t2.

Transient response solutions

3.09, 3.12, 3.13, 3.14

9

More Transient Response (2) Solutions - Base excitation

 

Base Excitation 3.15, 3.16,

 

10

Transient Response (3) Solutions - Base excitation: Ramp loading

 

3,17, 3.18

11

Harmonic Excitation: solution to equation of motion = transient + permanent or steady-state components. Definition of system frequency response. Effect of excitation frequency and damping ratio on amplitude (amplification factor) and phase lag of steady-state response. Regimes of operation, the concept of system resonance.
EOM for M-K-C system with harmonic base excitation. System response to harmonic base excitation. Effect of excitation frequency and damping ratio on amplitude (amplification factor) and phase lag of steady-state response. Regimes of operation. Example of road excitation on simple vehicle dynamics

Harmonic Excitation

3.19, 3.22, 3.23, 3.25 (note that the graph motion measures relative motion)

12

EOM for M-K-C system due to rotating imbalance. Physical nature of imbalance, magnitude of excitation force and its importance in rotating system dynamics. System response to mass imbalance. Effect of excitation frequency and damping ratio on amplitude (amplification factor) and phase lag of steady-state response. Interpretation of regimes of operation. Example of application to reduce vibration amplitude in a mechanical structure. Pulleys: EOMs and equations of constraint. Applications to M-K-C systems.

Harmonic Excitation & Rotating Imbalance

3.26, 3.27, 3.28

Appendix

Important design and operation issues/questions applying frequency response functions in simple M-K-C mechanical systems

 

 

Get all: Lectures 5-12

 

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