Vibrating mass-spring system

Fig. 1.9 Vibrating mass-spring system with damping...

A nuclear spin system behaves much like a damped harmonic oscillator. Suppose a mass is attached to one end of a spring and this mass-spring system is lowered into a liquid which provides damping. Attach the upper end of the spring to a motor which will vibrate the mass-spring system. As the frequency of the motor is varied through the natural resonance frequency of the system the response is a resonance curve as shown. 

Equation (4.48) is useful in correcting difficulties due to vibration. If a motor rests upon a base that has some flexibility which may be expressed in terms of a spring constant (k) and the mass of the motor and base (m), then the natural frequency of the system will be given by Eq. (4.48). When the rotational frequency of the motor corresponds to the natural frequency of the mass-spring system, then the amplitude of vibration will quickly rise to a high level. Not only will the motor move violently, but also the vibration will be transmitted to the floor. There are several remedies for this situation ... 

This may be defined as the oscillatory movement of a mechanical system, and it may be sinusoidal or non-sinusoidal (also known as complex). Vibration can occur in many modes, and the simplest is the single freedom-of-movement system. A mass/spring diagram (Figure 42.7) can explain the vibration of a system. 

To make a suspension-spring system with a natural frequency of 1 Hz, the weight of the mass should stretched the spring by 25 cm. Notice that Eq. (10.23) is exactly the formula for the natural frequency of a simple pendulum with length AL. To isolate the horizontal vibration, a pendulum is the... 

Figure 2 Energy and vibrational frequency of a classical harmonic oscillator. The sohd fine is the harmonic potential function. The dashed line represents the motion of the mass-and-spring system, displaced by Ar = 5 units then released...

Calculate the natural frequency of vibration for a system with two balls, one with mass 1 g and the other with mass 2 g, held together by a spring of length 10 cm and force constant 0.1 N m. ... 

A diatomic molecule can be considered as two vibrating masses connected by a spring. The bond distance continually changes, but an equilibrium or average bond distance can be definedi"When-ever the spring is stretched or compressed beyond this equilibrium distance, the potential energy of the system increases. 

Predictably, most systems that produce sound are more complex than the ideal mass/spring/damper system. And of course, most sounds are more complex than a simple damped exponential sinusoid. Mathematical expressions of the physical forces (thus the accelerations) can be written for nearly any system, but solving such equations is often difftcult or impossible. Some systems have simple enough properties and geometries to allow an exact solution to be written out for their vibrational behavior. A string under tension is one such system, and it is evaluated in great detail in Chapter 12 and Appendix A. For... 

We will learn more about the plucked string and higher-dimensional vibrating systems such as bars, plates, membranes, etc., in later chapters. The point of introducing the example of the plucked-string system here was to motivate the notion that sinusoids can occur in systems more complex than just the simple mass/spring/damper. 

Conventional dynamic vibration absorbers are composed of a mass, spring, and damper. Although dynamic vibration absorbers do not have sensors or controllers, they can provide vibration mitigation similar to that of actively controlled systems with a complicated sensor, control, and actuator system. Since an absorbers mass/spring/damper forms a single degree of freedom (DOF) vibration system, it consequently has a single resonant frequency and can exhibit an amplified response at this frequency. Dynamic absorbers behave similar to a system with a sensor to detect the specific frequency and a controller to amplify the vibration. Therefore, the absorbers natural frequency should be carefully tuned to a specific frequency for which the vibration amplitude of the host structure is to be reduced. The tuned frequency usually corresponds to natural modes and harmonically excited vibrations of a system. 

An important application of Hamilton s classical equations of motion is the vibrational mechanics of systems of particles. We shall begin with application to an idealized, simple problem, the harmonic oscillator. The harmonic oscillator is a special model problem consisting of a mass able to move in one direction and connected by a spring to an infinitely heavy wall, as shown in Figure 7.1. The spring is special because it is harmonic, which means that the restoring force is linearly proportional to the value of the coordinate that... 

If we think about two masses connected by a spring, each vibrating with respect to a stationary center of mass Xc of the system, we should expect the situation to be vei similar in form to one mass oscillating from a fixed point. Indeed it is, with only the substitution of the reduced mass p for the mass m... 

Materials and Reactions. Candle systems vary in mechanical design and shape but contain the same genetic components (Fig. 1). The candle mass contains a cone of material high in iron which initiates reaction of the soHd chlorate composite. Reaction of the cone material is started by a flash powder train fired by a spring-actuated hammer against a primer. An electrically heated wire has also been used. The candle is wrapped in insulation and held in an outer housing that is equipped with a gas exit port and rehef valve. Other elements of the assembly include gas-conditioning filters and chemicals and supports for vibration and shock resistance (4). 

Example The differential equation My" + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2 VkM. the roots of the characteristic equation... 

This system is the simplest of all vibration systems and consists of a mass suspended on a spring of negligible mass. Figure 5-5 shows this simple, single... 

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