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Conceptual basis of classical mechanics
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
The simple harmonic motion (SHM) is perhaps the simplest and most elegant of all dynamical systems. It considers the problem of a mass-spring system being attracted to a given fixed point by a force which by Hooke’s law is assumed to be directly proportional to the distance from the point. Obviously the SHM, which is also referred to as the simple harmonic oscillator, is a conservative system. In a one-dimensional setting, along the x-axis, the force is given by F (x) = −kx, where k > 0, which is linear and of a restoring type. F (x) being an odd function of x is negative when x > 0 and positive when x < 0. For such a force we run into the differential equation as given by Newton’s second law () mx¨+kx=0→x¨+ω02x=0,ω0=km
Nonlinear Vibrations
Published in William T. Thomson, Theory of Vibration with Applications, 2018
In a conservative system the total energy remains constant. Summing the kinetic and potential energies per unit mass, we have () 12x˙2+U(x)=E=constant
Basics of lumped parameter vibration
Published in Indrajit Chowdhury, Shambhu P. Dasgupta, Dynamics of Structure and Foundation – A Unified Approach, 2008
Indrajit Chowdhury, Shambhu P. Dasgupta
In any conservative system, sum of the kinetic and potential energy is constant. For free vibration of an undamped system, the energy is partly potential and partly kinetic. That is T+U=constant⇒ddt(T+U)=0
An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints
Published in Mechanics Based Design of Structures and Machines, 2018
A lane change maneuver to the left is simulated with a sinusoidal steer angle 𝜃 = −amp⋅sin(ωt) rad, for 0≤ωt≤2π, and 𝜃 = 0 thereafter. With an initial velocity of 15 m/s (≈35 min/h), amp = 0.004 rad, and ω = om = 1 rad/s, the conventional configuration with ϕ = 0.2 rad executes the approximately 4 m lane change shown at the left of Fig. 6. The chopper configuration with ϕ = π∕4 rad, the same initial velocity, amp = 0.0078 rad, and ω = om = 1 rad/s executes approximately the same lane change shown at the right of Fig. 6. Thus, the standard configuration is substantially more sensitive to steer input than the longer chopper, which might have been expected. Both simulations were carried out with h = 0.001 s, intol = 10−6, and utol = Btol = Htol = 10−7. Total energy was constant to six places for this essentially conservative system.