PART 2a. PLANAR KINETICS OF LUMPED MASS SYSTEMS
Motion and
Deformation of Mechanical Systems with 1 Degree of Freedom (1DOF)
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 (get me) |
Major Topics/WHAT
YOU WILL LEARN |
Recommended homework
problems |
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Spring-mass models 3.2, 3.3, 3.4 (note: K has units of N/m), 3.5, 3.6 |
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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) |
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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 |
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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 |
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More Transient Response
(2) Solutions - Base excitation |
Base
Excitation 3.15, 3.16, |
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Transient
Response (3) Solutions - Base excitation: Ramp loading |
3,17, 3.18 |
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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. |
Harmonic Excitation 3.19, 3.22, 3.23, 3.25 (note that the graph motion
measures relative motion) |
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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 |
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Appendix |
Important design and operation
issues/questions applying frequency response functions in simple M-K-C
mechanical systems |
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Get all: Lectures 5-12
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