An electric motor is mounted on two inclined beam springs on each side, Figure a

Question

An electric motor is mounted on two inclined beam springs on each side, Figure a. A spring-mass model is shown in Figure b. The mass of the motor is 300 kg and the material for the beam spring is steel. Find the natural frequency w_n of the system. What is the maximum permissible amplitude of vibration, A_max , so that the maximum stress in each beam spring is less than the yield stress? Let initial displacement and initial velocity of the motor be and -0.3w_n A_max, respectively. Determine and sketch the vibratory motion of motor. Determine and sketch the safe region for initial displacement and velocity of the motor in vertical directions.

Explanation / Answer

(a) Change in KE = 0.5 * m * A^2 wn^2

      Change in PE = 2 * 0.5 * k * (A/sqrt(2))^2

      Change in KE = Change in PE

      m * wn^2 = k

      wn = sqrt(k/m)

(b) Energy stored in spring = 0.5 k * (Amax / sqrt(2) ) ^2 = 0.25 k * Amax ^2

     0.25 k * Amax ^2 = 0.5 * sigma ^2 / E

     Amax    =   sigma * sqrt( 2 / Ek )       where sigma is the yield stress and E is the youngs modulus of steel

(c)   y = C1 sin wn*t + C2 cos wn*t

       at t = 0   , y = -0.5 Amax

       =>    C2    = -0.5 Amax

      at t = 0 , dy/dt = -0.3 * wn * Amax

      =>    C1    = -0.3 Amax

      y = - ( 0.3 Amax * sin wn*t + 0.5 Amax * cos wn*t )

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