VTU eNotes On Mechanical Vibrations (Mechanical Engineering)
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Introduction When an elastic body such as, a spring, a beam and a shaft are displaced from the equilibrium position by the application of external forces, and then released, they execute a vibratory motion, due to the elastic or strain energy present in the body. When the body reaches the equilibrium position, the whole of the elastic or stain energy is converted into kinetic energy due to which the body continues to move in the opposite direction. The entire KE is again converted into strain energy due to which the body again returns to the equilibrium position. Hence the vibratory motion is repeated indefinitely. Oscillatory motion is any pattern of motion where the system under observation moves back and forth across some equilibrium position, but dose not necessarily have any particular repeating pattern. Periodic motion is a specific form of oscillatory motion where the motion pattern repeats itself with a uniform time interval. This uniform time interval is referred to as the period and has units of seconds per cycle. The reciprocal of the period is referred to as the frequency and has units of cycles per second. This unit of combination has been given a special unit symbol and is referred to as Hertz Hz Harmonic motion is a specific form of periodic motion where the motion pattern can be describe by either a sine or cosine. This motion is also sometimes referred to as simple harmonic motion. Because the sine or cosine technically used angles in radians, the frequency term expressed in the units radians per seconds rad sec . This is sometimes referred to as the circular frequency. The relationship between the frequency in Hz cps and the frequency in rad sec is simply the relationship. 2 rad sec. Natural frequency is the frequency at which an undamped system will tend to oscillate due to initial conditions in the absence of any external excitation. Because there is no damping, the system will oscillate indefinitely. Damped natural frequency is frequency that a damped system will tend to oscillate due to initial conditions in the absence of any external excitation. Because there is damping in the system, the system response will eventually decay to rest. Resonance is the condition of having an external excitation at the natural frequency of the system. In general, this is undesirable, potentially producing extremely large system response. Degrees of freedom The numbers of degrees of freedom that a body possesses are those necessary to completely define its position and orientation in space. This is useful in several fields of study such as robotics and vibrations. Consider a spherical object that can only be positioned somewhere on the x axis.
This needs only one dimension, x to define the position to the centre of gravity so it has one degree of freedom. If the object was a cylinder, we also need an angle to define the orientation so it has two degrees of freedom. Now consider a sphere that can be positioned in Cartesian coordinates anywhere on the z plane. This needs two coordinates x and y to define the position of the centre of gravity so it has two degrees of freedom. A cylinder, however, needs the angle also to define its orientation in that plane so it has three degrees of freedom. In order to completely specify the position and orientation of a cylinder in Cartesian space, we would need three coordinates x, y and z and three