`   equations:{a == b1 + c*x''[t], b2 == d*x''[t] + e*(x[t]-y[t]) + f*(x'[t]-y'[t]),
    g*y''[t] + e*(y[t] - x[t]) + f*(y'[t] - x'[t]) + m == 0,
    n*z''[t] == m - R, R == 0, b1 == b2, y[t] == 2*x[t],
    y'[t] == 2*x'[t], y''[t] == 2*x''[t]}

Reduce[{a == b1 + c*x''[t], b2 == d*x''[t] + e*(x[t]) + f*(x'[t]) ,
  g*y''[t] + e*(y[t] - x[t]) + f*(y'[t] - x'[t]) + m == 0,
  n*z''[t] == m - R, R == 0, b1 == b2, y[t] == 2*z[t],
  y'[t] == 2*z'[t], y''[t] == 2*z''[t]}, {x''[t], z''[t]}, {b1, b2, R,
   y[t], y'[t], y''[t],m}]