Consider an industrial machine mounted to a factory floor. The machine is modeled as a mass m₁ supported by two mounting springs each have stiffness k₁/2. This mass-spring system is excited by the motion of the vibrating floor. (Fig 1). At the frequency of the input velocity, the vibration amplitude of m1 is excessive. To minimize this vibration, an additional mass-spring system (m₂ and k₂) is attached to mass m_1. (See fig. 2). It is suggested that if this additional mass spring system is properly designed, it will serve as a vibration absorber and tend to cancel the vibration of the machine. A) Develop a set of state variable equations governing the motion of the machine with the vibration absorber systems. B) Develop a single differential equation for the velocity v₁ in terms of system parameters and input v_in. C) Solve for velocity v₁(t) assuming steady-state conditions with a sinusoidal input v_in(t)=Asin(omega)t. D) Explain how to design the vibration absorber. In other words, what choice of input frequency and parameters makes sense? E) For the rest of this problem, take m₁=1000kg, m₂=500kg, k₁=400,000 N/m, and k₂=312,500 N/m. Plot the amplifications factor (the amplitude ratio of v₁/v_in) vs. frequency. What input frequency minimizes the machine vibration? F) Assuming input v_in(t)=50 sin 25t mm/s, plot the steady state response v₁(t) with and without the vibration absorber. 
  

Transcribed Image Text:

V1 Industrial Machine m1 Vin Vibrating Floor (vertical motion only) Figure 1 for Problem 2 V1 Industrial Machine m1 Vin Vibrating Floor (vertical motion only) Figure 1 for Problem 2

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