tending to restore to its equilibrium position. This force, called the or return m restoring force force, is proportional to the displacement when is not large: x x = F x i^ –k where the constant of proportionality, is called the or and k, spring constant stiffness factor, is the unit vector in the positive The minus sign indicates that the restoring i^ x-direction. force is always opposite in direction to the displacement. By Newton’s second law Eqn. (1.2) can be written as where = return force per unit displacement per unit mass. called the angular k/m frequency of oscillation. 1.6 SIMPLE HARMONIC MOTION (SHM) If the restoring force of a vibrating or oscillatory system is proportional to the displacement of the body from its equilibrium position and is directed opposite to the direction of displacement, the motion of the system is simple harmonic and it is given by Eqn. (1.3). Let the initial conditions be = and = 0 at = 0, then integrating Eqn. (1.3), we get x A x t = cos x(t) A t where the maximum value of the displacement, is called the amplitude of the motion. If is A, T the time for one complete oscillation, then The general solution of Eqn. (1.3) is where and are determined from the initial conditions. Euqation (1.6) can be written as C D Where = = sin . The amplitude for the motion described by Eqn. (1.7) is now C A D A A=(C2D2) and the angular frequency is is uneffected by the initial conditions. The angle the is given by =tan-1 (D/C) where phase angle or phase constant or epoch chosen 0≤≤ 2π.
tending to restore m to its equilibrium position. This force, called the restoring force or return force, is proportional to the displacement x when x is not large: F = –k x i^ where k, the constant of proportionality, is called the spring constant or stiffness factor, and i^ is the unit vector in the positive x-direction. The minus sign indicates that the restoring force is always opposite in direction to the displacement. By Newton’s second law Eqn. (1.2) can be written as where k/m = return force per unit displacement per unit mass. called the angular frequency of oscillation. 1.6 SIMPLE HARMONIC MOTION (SHM) If the restoring force of a vibrating or oscillatory system is proportional to the displacement of the body from its equilibrium position and is directed opposite to the direction of displacement, the motion of the system is simple harmonic and it is given by Eqn. (1.3). Let the initial conditions be x = A and x = 0 at t = 0, then integrating Eqn. (1.3), we get x(t) = A cos t where A, the maximum value of the displacement, is called the amplitude of the motion. If T is the time for one complete oscillation, then The general solution of Eqn. (1.3) is where C and D are determined from the initial conditions. Euqation (1.6) can be written as Where C = A D = A sin . The amplitude for the motion described by Eqn. (1.7) is now A=(C2D2) and the angular frequency is is uneffected by the initial conditions. The angle the phase angle or phase constant or epoch is given by =tan-1 (D/C) where chosen 0≤≤ 2π.