Understanding Forced Vibration in Spring-Mass Systems
This content explains forced vibration in spring-mass systems, deriving the differential equation of motion and discussing solutions for both the complementary and steady-state scenarios. It delves into concepts like resonance conditions and the behavior of structures under larger amplitude vibrations.
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Forced Vibration Aim: Derive the differential equation of motion of spring mass system subjected to external applied excitation force. x k F(t) M
Forced Vibration 1. First we define coordinate to specify position of mass M. We choose a coordinate fixed at natural length of spring. Suppose position of mass M is x at any instant of time t. x k F(t) M 3. Apply Newton second law of motion: 2. Draw free body diagram of mass M. Fx = m ax m = F (t) Fs Mg m + Fs = F(t) F(t) Fs m + k x = F(t) N Fs is spring force. F(t) is external applied time varying force. This equation is known as differential equation of motion (DEOM). N is normal contact force.
Forced Vibration m + k x = F(t) - - - - - (1) Solution of equation (1) is a) Case 1 : F(t) is harmonic force x(t) = complementary / transient solution +steady state solution. x(t) = xc (t) + xss (t). F(t) = F0 Sin ( t) For complementary solution: m c + k xc = 0 F0 = Force Amplitude. = exciting frequency For steady state solution: Equation (1) is second order Non homogenous differential Equation. m ss + k xss = F0 Sin ( t)
Forced Vibration For steady state solution: m ss + k xss = F0 Sin ( t) Steady state solution take a form xss (t) = X0 Sin( t ) X0 = Displacement amplitude. = Phase angle. Substitute in above equation we Get, - m X0 2 Sin ( t ) + k X0 Sin( t ) = F0 Sin ( t) ( k m 2 ) X0 Sin( t ) = F0 Sin ( t)
Steady state solution ( k m 2 ) X0 Sin( t ) = F0 Sin ( t) After equating above equation we get, (k m 2 ) X0 = F0 F0 (k m 2 ) X0 = = 00 Steady state solution: xss (t) = X0 Sin( t) Note : Phase angle is equal to zero.
Resonance condition Take excitation frequency is equal to natural frequency. If = n then displacement amplitude X0 is equal to infinity. So this condition is called as resonance. Characteristic of resonance: Spring mass system vibrate is larger amplitude then our structure will fail.
