Once more, increasing the value of the spring constants increases the frequency of the oscillations of the solutions. To explore the dependence of (2M-2S-HSMD) on the values of the masses, consider the exact same initial conditions as above. Let k1 =45 N/m , k2=42 N/m, m1=5 kg, m2= 4 kg, r1=0.5, r2=0.5. And observe the behavior of the solutions for the first ten seconds Let k1=45 N/m, k2=42 N/m, m1= 10 kg, m2=8 kg, r1=0.5, r2=0.5 (Change only the values of the masses.) Once again, observe the behavior of the solutions for the first ten seconds. As the values of the masses increased, the frequency of the oscillations of the solutions decreased. To explore how (2M-3S-HSMD) is affected by changes in the values of the masses, I used the same values as used above, with k3=40 in both cases. As the values of the masses increased, the frequency of the oscillations of the solutions decreased. Non-Homogenous Case 1. Consider (2M-2S-HSM) with forcing term mg in each ODE (g= -9.8 m/s). This is the natural forcing term for a vertically hanging spring, as the force of gravity acts upon each mass as it oscillates. m1 x1”(t) = -k1x1(t)-k2 (x1(t)-x2(t))+m1g m2 x2”(t)= -k2(x2(t)-x1(t))+m2g , t >0 (2M-2S-HSMD) with forcing term mg: m1 x1”(t) = -k1x1(t)-k2 (x1(t)-x2(t))-r1x1’(t)+ m1g m2 x2”(t)= -k2(x2(t)-x1(t))-r2x2’(t)+m2g , t >0
Once more, increasing the value of the spring constants increases the frequency of the oscillations of the solutions. To explore the dependence of (2M-2S-HSMD) on the values of the masses, consider the exact same initial conditions as above. Let k1 =45 N/m , k2=42 N/m, m1=5 kg, m2= 4 kg, r1=0.5, r2=0.5. And observe the behavior of the solutions for the first ten seconds Let k1=45 N/m, k2=42 N/m, m1= 10 kg, m2=8 kg, r1=0.5, r2=0.5 (Change only the values of the masses.) Once again, observe the behavior of the solutions for the first ten seconds. As the values of the masses increased, the frequency of the oscillations of the solutions decreased. To explore how (2M-3S-HSMD) is affected by changes in the values of the masses, I used the same values as used above, with k3=40 in both cases. As the values of the masses increased, the frequency of the oscillations of the solutions decreased. Non-Homogenous Case 1. Consider (2M-2S-HSM) with forcing term mg in each ODE (g= -9.8 m/s). This is the natural forcing term for a vertically hanging spring, as the force of gravity acts upon each mass as it oscillates. m1 x1”(t) = -k1x1(t)-k2 (x1(t)-x2(t))+m1g m2 x2”(t)= -k2(x2(t)-x1(t))+m2g , t >0 (2M-2S-HSMD) with forcing term mg: m1 x1”(t) = -k1x1(t)-k2 (x1(t)-x2(t))-r1x1’(t)+ m1g m2 x2”(t)= -k2(x2(t)-x1(t))-r2x2’(t)+m2g , t >0