Exact Solutions for Strings, Membranes, Beams, and Plates
Content
- About the authors
- Chapter 1: Introduction to Structural Vibrations
- Chapter 2: Vibration of Lines
- Chapter 3: Membrane Oscillations
- Chapter 4: Vibration of Trees
- Chapter 5: Vibrations of Isotropic Plates
- Chapter 6: Plate vibrations and critical effects
- Chapter 7: Vibrations of Stress-Free Plates
- Description
Preface
WHAT IS VIBRATION?
Vibrations can be defined as repeating motion over time or oscillations of a system around an equilibrium point.
Kelly (2007). Vibrations can occur in a variety of ways, such as the swinging of a pendulum, plucked guitar strings, tidal motion, chirping of a male cicada, flapping of airplane wings in turbulence, soothing motion of a massage chair, and swaying of a tall building due to wind or earthquake.
The main parameters for describing vibration are amplitude, period, and frequency.
The maximum displacement of a vibrating particle or body from its equilibrium position, proportional to the applied energy, is known as the amplitude of vibration. The period refers to the time it takes to complete one cycle of motion. The frequency refers to the number of cycles per unit time, or the reciprocal of the period. We calculate the angular (or circular) frequency in radians per unit time by multiplying the frequency by 2.
We can classify vibrations as either free or forced. Free vibration occurs when a system oscillates based on internal forces without external ones. Free vibration causes a system to vibrate at its natural frequencies, which vary based on mass, stiffness, and boundary conditions. Periodically applying an external force to a system causes forced vibration.
Undamped vibrations are those that do not have frictional effects. In reality, damping occurs to some extent for all vibrations. Only slight damping causes a free vibration to gradually decrease its amplitude until it stops after a predetermined time. Sufficient damping reduces vibration and allows the system to quickly return to its equilibrium position. As long as we apply the periodic force causing the vibration, we maintain a dampened forced vibration. The magnitude of damping forces affects the amplitude of vibrations.
Vibration is the alternating exchange of potential and kinetic energy, according to energy theory. Damped systems dissipate energy during each vibration cycle, leading to the eventual cessation of motion. We must compensate for the energy loss due to damping to maintain a consistent vibration motion.
A brief historical review of the vibration of strings, membranes, beams, and plates.
Rao (1986, 2005) suggests that interest in vibration stems from the discovery of early musical instruments like whistles, strings, and drums, which generate sound through vibration. Egyptian tombs dating back to 3000 BC contain drawings of stringed instruments on their walls.
Pythagoras (582-507 BC), a Greek mathematician and philosopher, conducted experiments on vibrating strings to understand why certain notes sounded better than others. He discovered that the tension and length of the string affected the pitch of the note, or the frequency of sound. Galileo (1638) was an Italian physicist and astronomer who measured the length and frequency of vibration for pendulums and strings, as well as observed resonance between two connecting bodies. Marinus Mersenne (1636), a French mathematician and theologian, investigated the behavior of vibrating strings. English scientist Robert Hooke (1635-1703) and French mathematician and physicist Joseph Sauveur (1653-1716) investigated the pitch-frequency relationship of a vibrating taut string. Scholars credit Sauveur for coining terms like nodes (stationary points), loops, fundamental frequency, and harmonics, and for being the first scientist to document beats.
Sir Isaac Newton (1687) was the first to formulate the laws of classical mechanics, while Gottfried Leibniz (1693) and Newton jointly developed calculus. Euler (1744) and Bernoulli (1751) discovered the differential equation for lateral vibration of prismatic bars and studied its solution for small deflections. Lagrange (1759) made significant contributions to the theory of vibrating strings. Euler (1766) developed equations for the vibrations of rectangular membranes under uniform tension and rings. Poisson (1829) developed the equation for vibrating circular membranes and provided solutions for axisymmetric vibration modes. Pagani (1829) developed the nonaxisymmetric vibration solution for circular membranes.
Coulomb (1784) studied the torsional oscillations of a metal cylinder suspended on a wire.
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