VIBRATION Copyright 2010 Pearson Education South Asia Pte

1 Undamped Free Vibration* n n A vibration is a periodic motion of a body or system of connected bodies displaced from a position of equilibrium Free vibration occurs when the motion is maintained by gravitational or elastic restoring forces Copyright © 2010 Pearson Education South Asia Pte Ltd

1 Undamped Free Vibration* n n Undamped vibrations can continue indefinitely as frictional effects are neglected The block has a mass m and is attached to a spring having a stiffness k Elastic restoring force F = kx is directed toward the equilibrium position Acceleration a is assumed to act in the direction of positive displacement Copyright © 2010 Pearson Education South Asia Pte Ltd

1 Undamped Free Vibration* n We have n Acceleration is proportional to displacement Motion is called simple harmonic motion Rearranging, we get ωn is the natural frequency (rad/s), is homogeneous, second-order, linear, differential equation with constant coefficients, and the general solution is n n Copyright © 2010 Pearson Education South Asia Pte Ltd

1 Undamped Free Vibration* n General solution is can expressed in terms of simple sinusoidal motion. Let We get Since sin (θ + Φ) = sin θ cos Φ + cos θ sin Φ, n When plotted on an x-versus-ωnt axis, n n Copyright © 2010 Pearson Education South Asia Pte Ltd

1 Undamped Free Vibration* n Amplitude is C and angle Φ is the phase angle n Sine curve completes one cycle in time when n Length of time is call a period, Copyright © 2010 Pearson Education South Asia Pte Ltd

1 Undamped Free Vibration* n Frequency f (hertz, Hz) is defined as the number

Example A 4 kg block is suspended from a spring, the spring is stretched a distance of 60 mm. Determine the frequency and the period of vibration for a 0. 5 kg block attached to the same spring. Copyright © 2010 Pearson Education South Asia Pte Ltd