Forced mass-spring oscillator

Fig. 1.10 Illustration of a forced mass-spring oscillator. The left end of the spring is wiggled back and forth with an angular frequency a and a maximum amplitude F0. Damping is proportional to friction, i.e., to negative velocity...

Fig. 29.2 (a) Schematic of the jet element movement along the trajectory with the direction of the forces and velocities involved (b) coordinates of the upper boundary (shown by u.b. ) and the center of the mass (shown by subscript c.m. ) of the liquid element (c) analogy between an oscillating 2D drop and a forced mass-spring system... 

Just as a mass/spring system oscillates due to the inteiplay of an inertial force associated with the mass with a restoring force from the spring, an elastic wave... 

Figure 11.4 Mass/spring/damped system driven by an external force. Here a periodic force is applied, and this will determine the frequency at which the system oscillates. As the driving frequency nears the natural resonant freqnency (determined by the mass and spring) the size of the oscillations will increase.

Based on an analogy between the oscillations of a two-dimensional (2D) droplet and a mass spring system (similar to the Taylor analogy breakup (TAB) model), we assume that the deformation of our 2D liquid droplet is dependent on the viscous (Fv), surface tension (Fj), and inertial (Fa) forces. So, performing a force balance in the X2-direction for the half element (shaded) in Fig. 29.2c, we can write... 

A 10,0-g mass connected by a spi itig to a statiotiaiy poitit executes exactly 4 complete cycles of harmonic oscillation in 1,00 s. What are the period of oscillation, the frequency, and the angular frequency What is the force constant of the spring ... 

Three 10,0-g masses are connected by springs to fixed points as harmonic oscillators showui in Fig, 3-12, The Hooke s law force constants of the springs ai e 2k. k, and k as showui, where k = 2.00 N m, What are the pei iods and frequencies of oscillation in hertz and radians per second in each of the three cases a, b, and e ... 

The relationship between the weight of mass M and the static deflection of the spring can be calculated using the equation W = ZZ,. If the spring is displaced downward some distance, Zo, from Zj and released, it will oscillate up and down. The force from the spring, F, can be written as follows, where a is the acceleration of the... 

The vertical spring and mass is an example of a stable system and by definition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a finite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail. 

The proportionality constant k is known as the force constant. The minus sign in equation (4.1) indicates that the force is in the opposite direction to the direction of the displacement. The typical experimental representation of the oscillator consists of a spring with one end stationary and with a mass m... 

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

Let us consider a diatomic molecule and assume that it behaves as a harmonic oscillator with two masses, nii and m2, connected by an ideal (constant-force) spring. At equilibrium, the two masses are at a distance Xq by extending or compressing the distance by an amount X, a force F will be generated between the two masses, described by Hooke s law (cf equation 1.14) ... 

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

The harmonic oscillator is used as a simple model to represent the vibrations in bonds. It includes two masses that can move on a plane without friction and that are joined by a spring (see Fig. 10.3). If the two masses are displaced by a value x0 relative to the equilibrium distance / , the system will start to oscillate with a period that is a function of the force constant k (N m ) and the masses involved. The frequency, which is independent of the elongation, can be approximated by equation (10.2) where n (kg) represents the reduced mass of the system. The term harmonic oscillator comes from the fact that the elongation is proportional to the exerted force while the frequency i/yib is independent of it. 

The sizer sections are joined together by yokes and the entire unit freely oscillates due to the action of a vibrator, the necessary support being provided by cylindrical springs. The vibrator shaft mounts two unbalanced masses whose resultant force causes vibrations in horizontal and vertical planes. The vibrator is operated from a motor through V-belt transmission. 

In the treatment of two atoms connected together, a simple harmonic oscillator model can be adopted involving the two masses connected with a spring having a force constant fk. Thus, the vibrational frequency in wavenumbers 2 depends from the reduced mass p, from fk with c being the velocity of light. 

Assignments for stretching frequencies can be approximated by the application of Hooke s law. In the application of the law, two atoms and their connecting bond are treated as a simple harmonic oscillator composed of two masses joined by a spring. The following equation, derived from Hooke s law, states the relationship between frequency of oscillation, atomic masses, and the force constant of the bond. 

Figure 2.1-1 Models of a harmonic oscillator, a mass m on a spring with the force constant/, b two masses, m and m2, connected by a spring with the same force constant.

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