Chapter 19.5, Problem 19.147P

To determine

Show that if a periodic force of magnitude $P=P_m\sin {\omega }_{f}t$ is applied to the element, the amplitude of the fluctuating force transmitted to the foundation is $F_m=P_m\sqrt{\frac{1+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}{{\left[1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right]}^2+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}}$.

Answer to Problem 19.147P

The amplitude of the fluctuating force transmitted to the foundation is $F_m=P_m\sqrt{\frac{1+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}{{\left[1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right]}^2+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}}$ and hence it is proved.

Explanation of Solution

Given information:

The magnitude of periodic force (P) is $P=P_m\sin {\omega }_{f}t$.

Calculation:

The expression for the motion of the machine (x) as follows:

$x=x_m\sin \left({\omega }_{f}t-\varphi \right)$

Differentiate the above equation with respect to time ‘t’.

$\dot{x}=x_m{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)$

Calculate the force transmitted $\left(F_s\right)$ to the foundation for the springs using the relation:

$F_s=kx$

Substitute $x_m\sin \left({\omega }_{f}t-\varphi \right)$ for x.

$F_s=k{x}_m\sin \left({\omega }_{f}t-\varphi \right)$

Calculate the force transmitted $\left(F_d\right)$ to the foundation for the dashpot using the relation:

$F_d=c\dot{x}$

Substitute $x_m{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)$ for $\dot{x}$.

$F_d=c{x}_m{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)$

Calculate the total force transmitted $\left(F_t\right)$ to the foundation using the relation:

$F_t=F_s+F_d$

Substitute $k{x}_m\sin \left({\omega }_{f}t-\varphi \right)$ for $F_s$ and $c{x}_m{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)$ for $F_d$.

$\begin{array}{c}{F}_t=k{x}_m\sin \left({\omega }_{f}t-\varphi \right)+c{x}_m{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)\\ =x_m\left[k\sin \left({\omega }_{f}t-\varphi \right)+c{\omega }_f\cos \left({\omega }_{f}t-\varphi \right)\right]\end{array}$

Since $A\sin y+B\cos y=\sqrt{A^2+B^2}\sin \left(y+\psi \right)$.

Write the total force transmitted $\left(F_t\right)$ to the foundation,

$F_t=\left[x_m\sqrt{k^2+{\left(c{\omega }_f\right)}^2}\right]\sin \left({\omega }_{f}t-\varphi +\psi \right)$

The expression for the natural circular frequency $\left({\omega }_n\right)$ as follows:

$\begin{array}{l}{\omega }_n=\sqrt{\frac{k}{m}}\\ {\omega }_n^2=\frac{k}{m}\\ k={\omega }_n^{2}m\end{array}$

The expression for the critical damping coefficient $\left(c_c\right)$ as follows:

$\begin{array}{c}{c}_c=2m{\omega }_n\\ m=\frac{c_c}{2{\omega }_n}\end{array}$

The expression for the amplitude of the fluctuating force transmitted $\left(F_m\right)$ to the foundation from the above equation as follows:

$F_m=x_m\sqrt{k^2+{\left(c{\omega }_f\right)}^2}$ (1)

The expression for the amplitude of motion to the foundation $\left(x_m\right)$ as follows:

$x_m=\frac{\frac{P_m}{k}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}$

Substitute $\frac{\frac{P_m}{k}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}$ for $x_m$ in equation (1).

$\begin{array}{c}{F}_m=\frac{\frac{P_m}{k}\sqrt{k^2+{\left(c{\omega }_f\right)}^2}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\\ =\frac{\frac{P_m}{k}\left(k\right)\sqrt{1+{\left(\frac{c{\omega }_f}{k}\right)}^2}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\\ =\frac{P_m\sqrt{1+{\left(\frac{c{\omega }_f}{k}\right)}^2}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\end{array}$

Substitute ${\omega }_n^{2}m$ for k and $\frac{c_c}{2{\omega }_n}$ for m.

$\begin{array}{c}{F}_m=\frac{P_m\sqrt{1+{\left(\frac{c{\omega }_f}{{\omega }_n^2\left(\frac{c_c}{2{\omega }_n}\right)}\right)}^2}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\\ =\frac{P_m\sqrt{1+{\left(\frac{c{\omega }_f}{{\omega }_n\left(\frac{c_c}{2}\right)}\right)}^2}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\\ =\frac{P_m\sqrt{1+\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}}{\sqrt{{\left(1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right)}^2+{\left(2\left(\frac{c}{c_c}\right)\frac{{\omega }_f}{{\omega }_n}\right)}^2}}\end{array}$

Thus, the amplitude of the fluctuating force transmitted to the foundation is $F_m=P_m\sqrt{\frac{1+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}{{\left[1-{\left(\frac{{\omega }_f}{{\omega }_n}\right)}^2\right]}^2+{\left[2\left(\frac{c}{c_c}\right)\left(\frac{{\omega }_f}{{\omega }_n}\right)\right]}^2}}$ and hence it is proved.