Question

(25) A grinding machine is supported on an isolator that has two springs, each with stiffness of k and one viscous damper with damping constant of c=1.8 kNs/m. The floor on which the machine is mounted is subjected to a harmonic disturbance due to the operation of an unbalanced engine in the vicinity of the grinding machine. The floor oscillates with amplitude Y=3 mm and frequency of 18 Hz. Because of other design constraints, the stiffness of each spring must be greater than 3.25 MN/m. What is the minimum required stiffness of each of the two springs to limit the grinding machine’s steady-state amplitude of oscillation to at most 10 mm? Assume that the grinding machine and the wheel are a rigid body of weight 4200 N and can move in only the vertical direction (the springs deflect the same amount).

Answer 1

Answer:

k = 15.62 MN/m

Explanation:

Given:-

- The viscous damping constant, c = 1.8 KNs/m

- The floor oscillation magnitude, Yo = 3 mm

- The frequency of floor oscillation, f = 18 Hz.

- The combined weight of the grinding machine and the wheel, W = 4200 N

- Two springs of identical stiffness k are attached in parallel arrangement.

Constraints:-

- The stiffness k > 3.25 MN/m

- The grinding machine’s steady-state amplitude of oscillation to at most 10 mm. ( Xo ≤ 10 mm )

Find:-

What is the minimum required stiffness of each of the two springs as per the constraints given.

Solution:-

- The floor experiences some harmonic excitation due to the unbalanced engine running in the vicinity of the grinding wheel. The amplitude "Yo" and the frequency "f" of the floor excitation is given

- The floor is excited with a harmonic displacement of the form:

                         $y ( t ) = Y_o*sin ( w*t )$

Where,

           Yo : The amplitude of excitation = 3 mm

           w : The excited frequency = 2*π*f = 2*π*18 = 36π

- The harmonic excitation of the floor takes the form:

                       $y ( t ) = 3*sin ( 36\pi *t )$                          

- The equation of motion for the floor excitation of mass-spring-damper system is given as follows:

                      $m*\frac{d^2x}{dt^2} + c*\frac{dx}{dt} + k_e_q*x = k_e_q*y(t) + c*\frac{dy}{dt}\\\\\frac{m}{k_e_q}*\frac{d^2x}{dt^2} + \frac{c}{k_e_q}*\frac{dx}{dt} + x = y(t) + \frac{c}{k_e_q}*\frac{dy}{dt}$

Where,

     m: The combined mass of the rigid body ( wheel + grinding wheel body)        c : The viscous damping coefficient

     k_eq: The equivalent spring stiffness of the system ( parallel )

     x : The absolute motion of mass ( free vibration + excitation )

- We will use the following substitutions to determine the general form of the equation of motion:

                                   $w_n = \sqrt{\frac{k_e_q}{m} } , \\\\p = \frac{c}{2\sqrt{k_e_q*m} } = \frac{1800}{2\sqrt{k_e_q*428.135} } = \frac{43.49628}{\sqrt{k_e_q} }$

Where,

               w_n: The natural frequency

               p = ζ = damping ratio = c / cc , damping constant/critical constant

                    

- The Equation of motion becomes:

                         $\frac{1}{w^2_n}*\frac{d^2x}{dt^2} + \frac{2*p}{w_n}*\frac{dx}{dt} + x = y(t) + \frac{2*p}{w_n}*\frac{dy}{dt}$

                     

- The steady solution of a damped mass-spring system is assumed to be take the form of harmonic excitation of floor i.e:

                         $X_s_s = X_o*sin ( wt + \alpha )$       

Where,

              X_o : The amplitude of the steady-state vibration.

              α: The phase angle ( α )

- The steady state solution is independent from system's initial conditions and only depends on the system parameters and the base excitation conditions.

- The general amplitude ( X_o ) for a damped system is given by the relation:

                        $X_o = Y_o*\sqrt{\frac{1+ ( 2*p*r)^2}{( 1 - r^2)^2 + ( 2*p*r)^2} }$              

Where,

                r = Frequency ratio =  $\frac{w}{w_n} = \frac{36*\pi }{\sqrt{\frac{k_e_q*g}{W} } } = \frac{36*\pi }{\sqrt{\frac{k_e_q}{428.135} } } = \frac{36*\pi*\sqrt{428.135} }{\sqrt{k_e_q} }$

   

- We will use the one of the constraints given to limit the amplitude of steady state oscillation ( Xo ≤ 10 mm ):

- We will use the expression for steady state amplitude of oscillation ( Xo ) and determine a function of frequency ratio ( r ) and damping ratio ( ζ ):

              

                    $(\frac{X_o }{Y_o})^2 \geq \frac{1+ ( 2*p*r)^2}{( 1 - r^2)^2 + ( 2*p*r)^2}\\\\(\frac{X_o }{Y_o})^2 \geq \frac{1+ ( 2*\frac{43.49628}{\sqrt{k_e_q} }*\frac{36*\pi*\sqrt{428.135} }{\sqrt{k_e_q} })^2}{( 1 - (\frac{36*\pi*\sqrt{428.135} }{\sqrt{k_e_q} })^2)^2 + ( 2*\frac{43.49628}{\sqrt{k_e_q} }*\frac{36*\pi*\sqrt{428.135} }{\sqrt{k_e_q} })^2}$

                    $(\frac{X_o }{Y_o})^2 \geq \frac{ 1 + \frac{41442858448.85813}{k_e_q^2 }}{[ 1 - (\frac{5476277.91201 }{k_e_q} )]^2 + \frac{41442858448.85813}{k_e_q^2 } }}\\\\(\frac{X_o }{Y_o})^2 \geq \frac{ \frac{k_e_q^2 + 41442858448.85813}{k^2_e_q }}{[ \frac{(k_e_q - 5476277.91201)^2 }{k_e_q^2} ] + \frac{41442858448.85813}{k_e_q^2 } }}$

                   $(\frac{X_o }{Y_o})^2 \geq \frac{ k_e_q^2 + 41442858448.85813}{ (k_e_q - 5476277.91201)^2 +41442858448.85813 }}\\\\(\frac{10 }{3})^2 \geq \frac{ k_e_q^2 + 41442858448.85813}{ k^2_e_q -10952555.82402*k_e_q +3.00311*10^1^3 }}\\\\\\10.11111*k^2_e_q -121695064.71133*k_e_q +3.33637*10^1^4 \geq 0$

- Solve the inequality (  quadratic ):

       $k1_e_q \geq 7811740.790197058 \frac{N}{m} \\\\k2_e_q \leq 4224034.972855095 \frac{N}{m}$

- The equivalent stiffness of the system is due to the parallel arrangement of the identical springs:

                $k_e_q = \frac{k^2}{2k} = \frac{k}{2}$

- Therefore,

                  $k1 \geq 7811740.790197058*2 = 15.62 \frac{MN}{m} \\\\k2 \leq 4224034.972855095*2 = 8.448 \frac{MN}{m}$  

- The minimum stiffness of spring is minimum of the two values:

                k = 15.62 MN/m