Strength of Materials Notes eBook

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INTRODUCTION When a mechanical system undergoes free vibrations, an initial force causing some displacement is impressed upon the system, and the system is allowed to vibrate under the influence of inherent elastic properties. The system however, comes to rest, depending upon the amount of damping in the system. In engineering situations, there are instances where in an external energy source causes vibrations continuously acting on the system. Then the system is said to undergo forced vibrations, as it vibrates due to the influence of external energy source. The external energy source may be an externally impressed force or displacement excitation impressed upon the system. The excitation may be periodic, impulsive or random in nature. Periodic excitations may be harmonic or non harmonic but periodic. The amplitude of vibrations remains almost constant. Machine tools, internal combustion engines, air compressors, etc are few examples that undergo forced vibration. 3.2 FORCED VIBRATIONS OF SINGLE DOF SYSTEMS UNDER HARMONIC EXCITATION Consider a spring mass damper system as shown in Figure 3.1 excited by a sinusoidal forcing function F Fo Sin t
k
Cx
kx
C
x
F m F F0 Sin t F Fo
Figure 3.1
Let the force acts vertically upwards as shown in FBD. Then the Governing Differential Equation GDE can be written as
mx - Kx - C x F
m x C x Kx F --------------- 3.1 is a linear non homogeneous II order differential equation whose solution is in two
parts. 1. Complementary Function or Transient Response