Question

A machine has a mass of 100 kg and unbalanced reciprocating parts of mass 2 kg which move through a vertical stroke of 80 mm with simple harmonic motion. The machine is mounted on four springs, symmetrically arranged with respect to centre of mass, in such a way that the machine has one degree of freedom and can undergo vertical displacements only. Neglecting damping, calculate the combined stiffness of the spring in order that the force transmitted to the foundation is 1 / 25 th of the applied force, when the speed of rotation of machine crank shaft is 1000 r.p.m. When the machine is actually supported on the springs, it is found that the damping reduces the amplitude of successive free vibrations by 25%. 

Find :

1.  the force transmitted to foundation at 1000 r.p.m., 

2.  the force transmitted to the foundation at resonance, and 

3.  the amplitude of the forced vibration of the machine at resonance.

Answer 1

Given m₁ = 100 kg; m₂ = 2 kg; l = 80 mm = 0.08 m; ε = 1/25; N = 1000 r. p.m. or ω = 2π x 1000/60 = 104.7 rad/s

Combined stiffness of springs

Let s = Combined stiffness of Springs in N/m, and

ω_n = Natural circular frequency of vibration of the machine in rad/s.

We know that transmissibility ratio (ε),

1. Force transmitted to foundation at 1000 r.p.m., 

Let F_T = Force transmitted, and

x₁ = Initial amplitude of vibration.

Since the damping reduces the amplitude of Successive free vibration by 25%, therefore final amplitude of vibration,

x₂ = 0.75x₁

_{We know that}

_{We know that damping coefficient or damping force per unit velocity,}

c = a x 2m₁ = 0.94 x 2 x 100 = 188 N/m/s

and critical damping coefficient,

c_c = 2m.ω_n = 2 x 100 x 20.5 = 4100 N/m/s

Actual value of transmissibility ratio,

 We know that the maximum unbalanced force on the machine due to reciprocating parts,

 F = m₂. ω². r = 2(104.7)²(0.08/2) = 877 N ..(r = l/2)

Force transmitted to the foundation,

F_T = ε.F = 0.44 x 877 = 38.6 N (ε = F_T/F)

2.  Force transmitted to the foundation at resonance,

Since at resonance, ω = ω_n , therefore tranmsissibility ratio,

and maximum unbalanced force on the machine due to reciprocating parts at resonance speed ω_n,

F = m₂(ω_n)²r = 2(20.5)² (0.08/2) = 33.6 N (r = l/2)

Force transmitted to the foundation at resonance,

F_T = ε.F = 10.92 x 33.6 = 367 N

3. Amplitude of the forced vibration of the machine at resonance

we know that amplitude of the forced vibration at resonance

= force transmitted at resonance/Combined stiffness = 367/42160 = 8.7 x 10^-3 m

= 8.7 mm