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Oscillations
Published in William Bolton, Engineering Science, 2020
When the trolley, the mass of the system, is pulled to one side then one of the springs is compressed and the other stretched, this has the effect of providing a restoring force that is directed in such a direction as to endeavour to restore the trolley back to its rest position and that is always proportional to its displacement from the rest position. If the trolley is released from this deflected position, the restoring force causes the trolley to move back towards its original rest position and overshoot that position. The restoring force then reverses its direction to still be directed towards the rest position and so oscillations occur. If the displacement from the rest position is measured as a function of time then the result is as shown in Figure 28.3, the displacement variation with time being described by a cosine graph. This form of oscillation is termed simple harmonic motion (SHM). Simple harmonic motion (SHM) is said to occur when the motion is under the action of a restoring force which is always directed to a fixed point and has a magnitude which is proportional to the displacement from that point.
Free Fall and Harmonic Oscillators
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
We have already seen the simple problem of a mass on a spring as shown in Figure 2.4. Recall that the net force in this case is the restoring force of the spring given by Hooke’s Law, Fs=−kx, where k > 0 is the spring constant and x is the elongation of the spring. When the spring constant is positive, the spring force is negative, and when the spring constant is negative, the spring force is positive. The equation for simple harmonic motion for the mass-spring system was found to be given by mx¨+kx=0.
Motor Vibration and Acoustic Noise
Published in Wei Tong, Mechanical Design and Manufacturing of Electric Motors, 2022
A harmonic motion refers to the oscillations that are symmetrical about a position of equilibrium. The motion may have either one frequency or amplitude, defined as simple harmonic motion, or a combination of two or more components of harmonics, defined as complex harmonic motion. Simple harmonic motion oscillates with only the restoring force acting on the system. For this system, the restoring force is directly proportional to the displacement. As an example, a pendulum swings in a small arc, and in its periodic motion, the tangential component of gravity acts as the restoring force of the pendulum. It is the restoring force that interacts with the inertia property of mass (kinetic energy) to perpetuate the oscillation.
An accessible, justifiable and transferable pedagogical approach for the differential equations of simple harmonic motion
Published in International Journal of Mathematical Education in Science and Technology, 2019
The concept of harmonic oscillation is an important idea that is seen within curricula from courses on physics, classical mechanics, applied mathematics and engineering. Simple harmonic motion can act as a mathematical model to better understand the motion of dynamic phenomena, including: mass-spring systems; the oscillation of a pendulum; electrical systems; and molecular vibrations [1]. Simple harmonic motion can be learnt and taught at upper-high school levels, or more commonly, within the tertiary environment.
Feasibility study of iron ore fines beneficiation by shallow bed air fluidized separator
Published in Particulate Science and Technology, 2021
Ganesh Chalavadi, Ranjeet Kumar Singh
Assumptions:The feed rate effect is neglected.Deck vibration are approximated to simple harmonic motion without phase lag.Particle–particle collisions are neglected.
Research on topological configuration of lightweight high natural frequency plate structure
Published in Mechanics of Advanced Materials and Structures, 2023
Zichuang Guo, Fanchun Li, Yuan Zhang
In Eq. (3), is the amplitude of displacement vector and is the angular frequency of simple harmonic motion. By introducing Eq. (3) into Eq. (2), we can get the following results: