B$.6$ ANALYZE SPRING$-$MASS$-$DAMPER FORCED VIBRATION

Consider a mass$-$spring$-$damper vibrating system excited by a time$-$

dependent force $F(t)$ as shown in Figure $8.$

at rest

excited by $F(t)$

free$-$body diagram

Figure $8$ spring$-$mass$-$damper system excited by force $F(t)$

Do the following:

$(1)$ State your assumptions in performing this analysis

$(2)$ Draw free$-$body diagram $($FBD$)$ and deduce the sum of forces

in the $x$ direction.

$(3)$ Write NLM equation

$(4)$ Deduce the equation of motion $($EOM$)$ as an ODE in terms of

$m,c,k,F(t)$

$(5)$ Assume harmonic excitation $F(t)=$hat$(F)e^{\text{i}\text{o}\text{t}},$ substitute in $(4)$

and divide by $m$

$(6)$ Define the normalized excitation amplitude hat$(f)=$hat$\frac{F}{m}$ and

recall the definitions of ${}_n$ and $$ to obtain a nonhomogeneous

linear ODE in ${}_n,,$hat$(f)$

$(7)$ Write the general solution $u(t)$ as a sum of a complementary

solution $u_c(t)$ and a particular solution $u_p(t)$

$(8)$ Write the general form of the complementary solution $u_c(t)$

$(12)$ Develop the expression for the frequency response function

$FRF()$ in terms of ${}_n,,$

$(13)$ Discuss the variation with frequency of FRF magnitude and

phase

$(14)$ Give the definition of the magnification factor $M()$

$(15)$ Give the expression for the magnification factor $M()$ in terms

of ${}_n$ and $$

$(16)$ Give the definition of the resonance frequency ${}_r$

$(17)$ Give the expression for the resonance frequency ${}_r$ in terms of

${}_n$ and $$

$(18)$ Give the expression of the peak magnification $M_r$

$(19)$ Give the expression of the of the FRF phase $(p)$ where

$p=\frac{}{{}_n}$

$(20)$ Give the definition of phase resonance and deduce the FRF

expression at phase resonance

$(21)$ Calculate the magnification factor at phase resonance

$(22)$ Calculate the magnitude of the physical response at phase

resonance

$(23)$ Discuss the difference between true resonance and phase

resonance

$(24)$ Write and compare the expressions for the resonance

frequency ${}_r$ and damped frequency ${}_d$

$(25)$ Establish a formula to calculate the RPM corresponding to a

$ra\frac{d}{sec}$ frequency $$