Chapter 5 Dynamics of Vibrations; Attenuation PDF

CHAPTER 5 DYNAMICS OF VIBRATIONS; ATTENUATION Damping damping, in physics, restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipation of energy. Unless a child keeps pumping a swing, its motion dies down b...

CHAPTER 5 DYNAMICS OF VIBRATIONS; ATTENUATION Damping damping, in physics, restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipation of energy. Unless a child keeps pumping a swing, its motion dies down because of damping. Shock absorbers in automobiles and carpet pads are examples of damping devices. A system may be so damped that it cannot vibrate. Critical damping just prevents vibration or is just su cient to allow the object to return to its rest position in the shortest period of time. The automobile shock absorber is an example of a critically damped device. Additional damping causes the system to be overdamped, which may be desirable, as in some door closers. The vibrations of an underdamped system gradually taper o to zero. There are many types of mechanical damping. Friction, also called in this context dry, or Coulomb, damping, arises chie y from the electrostatic forces of attraction between the sliding surfaces and converts mechanical energy of motion, or kinetic energy, into heat. Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a uid forced through a small opening by a piston, as in automobile shock absorbers. The viscous-damping force is directly proportional to the relative velocity between the two ends of the damping device. The motion of a vibrating body is also checked by its friction with the gas or liquid through which it moves. The damping force of the uid in this case is directly proportional to a quantity slightly less than the square of the body’s velocity and, hence, is referred to as velocity- squared damping. Besides these external kinds of damping, there is energy loss within the moving structure itself that is called hysteresis damping or, sometimes, structural damping. In hysteresis damping, some of the energy involved in the repetitive internal deformation and restoration to original shape is dissipated in the form of random vibrations of the crystal lattice in solids and random kinetic energy of the molecules in a uid. Source: h t t p s : / / w w w. b r i t a n n i c a. c o m / s c i e n c e / vibration#:~:text=vibration%2C%20periodic%20back%2Dand%2D,that%20tend%20to%20re store%20equilibrium. Inertia inertia, property of a body by virtue of which it opposes any agency that attempts to put it in motion or, if it is moving, to change the magnitude or direction of its velocity. Inertia is a passive property and does not enable a body to do anything except oppose such active agents ffi fl fl fl fl ff as forces and torques. A moving body keeps moving not because of its inertia but only because of the absence of a force to slow it down, change its course, or speed it up. There are two numerical measures of the inertia of a body: its mass, which governs its resistance to the action of a force, and its moment of inertia about a speci ed axis, which measures its resistance to the action of a torque about the same axis. See Newton’s laws of motion. Source: https://www.britannica.com/science/inertia Building Resonance: Structural stability during earthquakes All buildings have a natural period, or resonance, which is the number of seconds it takes for the building to naturally vibrate back and forth. The ground also has a speci c resonant frequency. Hard bedrock has higher frequencies softer sediments. If the period of ground motion matches the natural resonance of a building, it will undergo the largest oscillations possible and su er the greatest damage. Keypoints: Frequency of a wave refers to the number of waves that pass through a point in one second Period is the amount of time it takes one wave cycle to pass the given point Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others Resonant frequency of any given system is the frequency at which the maximum- amplitude oscillation occurs. All buildings have a natural period, or resonance, which is the number of seconds it takes for the building to naturally vibrate back and forth. Watch Video 1- Frequency, amplitude, period Watch Video 2- Read seismograph Source: h t t p s : / / w w w. i r i s. e d u / h q / i n c l a s s / a n i m a t i o n / building_resonance_the_resonant_frequency_of_di erent_seismic_waves Theory of Vibration by Ralph E. Blake Source: https://engfac.cooper.edu/pages/tzavelis/uploads/Vibration%20Theory.pdf Additional topics Source: https://www.brown.edu/Departments/Engineering/Courses/En4/Notes/ vibrations_forced/vibrations_forced.htm ff ff fi fi  Seismic Waves Seismic waves travel fastest through materials with high densities and elasticities, such as steel, which is a key structural element of modern infrastructure. As such, earthquake- resistant technology has been employed in recent years to reduce damage when buildings undergo violent vibrations. Base Isolation Devices are used to isolate buildings from the ground so that seismic waves are not transmitted directly upwards through them. These devices consist of exible pads which are placed between the ground and the foundation, and which act to resist lateral movement and absorb the earthquake force. By introducing exibility in the structure, the isolators also add damping to the system. Seismic Dampers are often introduced into buildings in place of structural elements, such as braces. Viscous, friction and yielding dampers all act as shock absorbers. By absorbing part of the earthquake energy, they work to damp the motion of the building. Given such technology, it is important to consider damping when modelling earthquake-building interactions. Through analysis of the motion of buildings undergoing seismic forces, it was found that the nature of the results was highly dependent on the frequency of the earthquake and the way the damping term had been de ned. It was important to consider the natural frequencies and periods at which the buildings oscillated to understand how certain earthquake frequencies would a ect the building’s motion. The proximity of the earthquake frequency to the natural frequency of the building greatly a ected the amplitudes of oscillation for the levels. When the earthquake frequency was close enough to the natural frequency, resonance was seen to occur. The duration of the earthquake e ected the magnitude of the displacement as well as the decay time for the system, however the key factor e ecting change was the frequency. Source: https://srs.amsi.org.au/wp-content/uploads/sites/92/2019/06/wilkins- researchpaper.pdf Seismic Dampers in Buildings Building safety and seismic protection are important considerations for civil engineers when designing structures. Seismic dampers in buildings, a relatively new technology, can signi cantly reduce the risk of damage and destruction that may occur during an earthquake or other seismic event. In this article, we will explore the basic principles of seismic dampers, the role they play in building construction, and their potential bene ts for engineers. Seismic dampers are devices that are installed in buildings to help protect them from damage during an earthquake or other seismic event. The damper is attached to the building at two points and is designed to absorb the energy of the earthquake, which reduces the amount of damage that the building sustains. Seismic dampers can be used in both new construction and retro t applications. Seismic dampers are an important part of any earthquake-resistant design. When used in conjunction with other earthquake-resistant measures, such as base isolation and shock- fl fi fi ff fi ff ff ff fi fl absorbing systems, they can signi cantly reduce the amount of damage that a building sustains during an earthquake. Seismic dampers in buildings are an important tool for civil engineers that can help to improve the safety and performance of buildings during an earthquake. As the use of seismic dampers grows, we can expect to see a reduction in the amount of damage that occurs to buildings during these events. What are the di erent types of seismic dampers? Seismic dampers are devices that are installed in buildings to help protect them from damage during an earthquake or other seismic event. There are two main types of seismic dampers: Viscous Dampers Viscoelastic Dampers Friction Dampers Tuned Mass Damper Yielding Dampers Magnetic Damper Viscous Seismic Dampers Viscous Dampers are used to reduce the shock waves that pass through the ground during earthquakes. Dampers are usually constructed of a heavy uid contained in cylinders or tanks and are placed along the length of an oil or gas pipeline. The uid inside each cylinder is undisturbed until it is subjected to a force. When an earthquake strikes, the uid moves from one end of the cylinder to another and absorbs energy from the seismic wave. The more viscous the liquid, the slower this energy will be absorbed. Viscous Seismic Dampers have been used for over 30 years. ff fi fl fl fl  Viscoelastic Dampers A viscoelastic damper is a device that absorbs shock by converting mechanical energy into another form of energy. The dampers are made from polyurethane and are placed between the oor and the upholstery. The material is able to absorb repetitive impacts through its structure, rather than just at the surface, which is how conventional suspension works. Viscoelastic dampers are used to eliminate the vibrations caused by any structure or machine. Vibration is a problem for many structures because it can cause damage over time. These damped structures have rubber elements which absorb some of the vibrations in order to ensure that they don’t cause any problems. There are two main types of viscoelastic dampers. Friction Dampers Friction dampers are commonly used in industrial machinery and material handling equipment to reduce the amount of friction generated between two moving objects. These friction dampers are made from a variety of materials ranging from steel to urethane as well as metals such as aluminum, stainless steel, brass and more. The kinetic energy of moving parts is converted into thermal energy by friction dampers, which lessen abrupt stops or prevent excessively high vibration amplitudes. These types of seismic dampers in buildings more cost e ective when compared some of other methods. fl ff Tuned Mass Damper A tuned mass damper is a device that’s installed in buildings to protect them against vibrations. Tuned mass dampers are essentially giant pendulums that slow down swaying as a building vibrates.  A seismic damper or harmonic absorber are other names for a tuned mass damper. It is a mechanical vibration-dampening device that is xed to buildings and consists of a mass mounted on one or more damped springs. Further, reading refer to article tuned mass damper. Tuned mass damper Yielding Dampers Yield dampers, commonly referred to as metallic dampers, are typically constructed of steel. In order to absorb the energy when a building vibrates during an earthquake, they are made to deform excessively. After a seismic incident, yield dampers typically cannot revert to their previous con guration. fi fi  Source: https://www.structuralguide.com/seismic-dampers/ Critical Damping D. Inman, in Encyclopedia of Vibration, 200 De nition of Critical Dampin Critical damping is de ned for a single-degree-of-freedom, spring-mass-damper arrangement, as illustrated in Figure 1. The equation of motion for this system is found from Newton's law and the free-body diagram to be: fi fi g 1  Figure 1. A single-degree-of-freedom system and free-body diagram m (t) + c (t) + kx(t)= 0 Equation 1 Here x(t) is the displacement in meters, (t)is the velocity in meters per second, (t) is the acceleration in meters per second per second, m is the mass in kilograms, k is thesti ness in Newtons per meter and c is the damping coe cient in Newton second per meter or kilograms per second. Eqn (1) is written in dimensionless form by dividing the expression by the mass. This yields: (t) + 2 ωn (t) + ω2n x (t) = 0 Equation 2 Here the undamped natural frequency is de ned to be: (in radians per second) Equation 3 And the damping ratio is de ned to be: which is dimensionless. Source: https://www.sciencedirect.com/topics/engineering/critical-damping ẍ ẍ 𝛇 ẋ ẋ fi ẋ fi ffi. ẍ ff