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A Journal of Applied Mechanics and Mathematics by DrD
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#17  MDOF Vibrations / Part IV
A Journal of Applied Mechanics and Mathematics by DrD, # 17
© Machinery Dynamics Research, 2015 Vibrations  Part IV
Multiple Degrees of Freedom Introduction The first three parts of this series have involved only a single mass moving in a single direction. Such a system constitutes a single degree of freedom. In this part, a second mass is added which changes the required approach considerably. The new system is shown in Figure 1.
Fig. 1 Two Degree of Freedom System In the upper part of the figure, (a), the system is shown at equilibrium rest with no strain in either spring. In the lower figure, (b), the system is displaced x₁ and x₂ under the action of the applied external forces F₁(t) and F₂(t). The problem to be addressed in this part is the determination of the dynamic response of the system. VibsIVMDOF.pdf
#16  Vibration Part III/Effects of Damping
A Journal of Applied Mechanics and Mathematics by DrD, # 16
© Machinery Dynamics Research, 2015
Vibrations  Part III
Effects of Damping
Introduction In the previous discussions of the simple, displacement driven mechanical oscillator, damping has been explicitly omitted. In this part, viscous damping is taken into account. Viscous damping is the effect of forces that (1) always oppose existing relative motion, and (2) are proportional to the relative velocity between surfaces. It always removed energy from a system, converting mechanical energy into heat energy that is then lost by conduction, convection, or radiation. Note that, with no motion there is no viscous damping force. Viscous action is associated with surfaces separated by a film of oil, grease, or other viscous substance (such as animal fat, glycerin, or any other viscous fluid).
There are many other types of energy dissipation mechanisms, including dry friction (Coulomb damping), aerodynamic drag (proportional to the square of velocity), and various other nonlinear damping mechanisms. In many cases, these other models are closer to reality, but there is a reason why the viscous damping is often the preferred modeling technique instead; it is mathematically tractable, which is to say, that it lends itself to mathematical solution far more readily than most other models. (Tractable is simply a big word, popular with mathematicians, meaning that something is easy to work with for a solution.) In this part of the series, the same system considered previous is again presented in slightly modified form; a viscous damper is added as shown in Figure 1.
[Figure 1 Damped, Displacement Driven Mechanical Oscillator.] VibsIIIDampingRev.pdf
#15  SDOF Near & At Resonance, Part II
A Journal of Applied Mechanics and Mathematics by DrD, # 15
© Machinery Dynamics Research, 2015 Introduction & Review In Part I of this discussion, the single degree of freedom oscillator was introduced with displacement excitation. The system is shown in Figure 1.
The impressed support motion is s(t) and the response of the mass is measured by x(t). As shown in the upper figure, s(0)=0, x(0)=0, and there is no strain in the spring. The elongation (or compression) of the spring is s(t)x(t), so that the force acting to the right on the mass is F=K[s(t)x(t)]. Applying Newton's Second Law gives the system equation of motion Mx=K(sx) which may be rewritten as x+ω_{n}²x=ω_{n}²s where
ω_{n}²=K/M= square of the natural frequency of vibration. The quantity ω_{n} (or ω_{n}²) is a fundamental system property.
The driving displacement was specified as s(t)=Ssin(Ωt) and the solution was eventually determined, subject to the provision ω_{n}²≠Ω². In this part, the objective is to investigate what happens when the excitation is near or at the natural frequency, Ω≈ω_{n}. The frequency response plot, shown in Part I, leads us to think that the oscillations will be catastrophically large near resonance, but that is not the whole picture. VibsIIResonance.pdf
#14  SDOF Vibrations Part I
A Journal of Applied Mechanics and Mathematics by DrD, # 14
© Machinery Dynamics Research, LLC, 2015
SDOF Vibrations  Part I
Undamped System With
NonResonant Conditions Introduction The undamped single degree of freedom oscillator is a suitable model for countless physical systems. It might represent a wheel on a vehicle (without shock absorbers), a machine on an elastic foundation, a sensitive instrument mounted on a shaky support, or any number of other possible real situations. A simple schematic diagram for such a system is shown in Figure 1 below.
[Figure 1 Undamped SDOF Oscillator with Base Excitation.] The impressed support motion is s(t) and the response of the mass is measured by x(t). As shown in the upper figure, s(0)=0, x(0)=0, and there is no strain in the spring. SDOF Vibs1.pdf