Question

3. (15 pts) Consider a second-order dynamic system composed of the classical spring, mass, and damper, with spring constant k , mass m , and damping constant c .

a) Generate expressions for the undamped natural frequency ω n and the damping ratio ζ in
terms of k , m , and c .

b) If the spring constant k is increased, but m and c are unchanged, predict whether ω n will increase, decrease, or remain the same. Justify.

c) If the spring constant k is increased, but m and c are unchanged, predict whether ζ will increase, decrease, or remain the same. Justify.

Answer 1

WHEN A = 1/2 KSIMPLE HRMONIC SYSTEM OSCILLATES WITH A CONSTANT AMPLITUDE WHICH DOES NOT CHANGE WITH TIME, ITS OSCILLATIONS ARE CALLED UNDAMPED SIMPLE HARMONIC OSCILLATIONS.

HERE ITS ENERGY IS GIVEN BY

E = 1 /2Kx0^2

where K is called force constant and x0 is called the amplitude of oscillations

it shows that the dissipative forces are not present in the system executing undamped simple harmonic oscillations.

WHEN A VIBRATING BODY VIBRATES IN AIR OR IN ANY OTHER RESISITING MEDIUM, THE AMPLITUDE OF VIBRATION DOES NOT REMAIN CONSTANT BUT DECREASES GRADUALLY AND ULTIMATELY THE BODY COMES TO REST. THE RESISTANCE OFFERED BY A DAMPING FORCE IS KNOWN AS DAMPING.

WHEN THE DAMPING IS SMALL, THE  DAMPING FORCE IS PROPORTIONAL TO THE VELOCITY OF THE VIBRATING BODY.A BODY STOPS  VIBRATING AS SOME ENERGY IS INEVITABLY LOST DUE TO RESISITIVE OR VISCOUS FORCES.

LET US CONSIDER THE CASE OF A SPRING MASS SYSTEM

CONSIDER THE DAMPING FORCE Fs = -bv

and the spring force Fs = -kx

here b is called damping co-efficient

THEREFORE THE NET FORCE

F = -kx - bv = ma

or

m(d^2x/ dt^2) + b(dx / dt) + kx = 0

THE SOLUTION OF THIS IS OF THE FORM

x(t) = Ae^-bt/2mCos('t + )

here A is the amplitude and ' is the angular frequency of the damped oscillator given by

' = k /m - b^2/4m^2----------(1)

the factor Ae^-bt/2m represents effective amplitude ie variation of amplitude with respect to time

since the energy decreases with time it is directley proportional to A^2

E(t) = 1/2 kA^2e^-bt/m

hence '' =

where is the angular frequency of the undamped oscillator.

RELAXATION TIME OR THE TIME CONSTANT IS DEFINED ASTHE TIME () REQUIRED FOR THE AMPLITUDE TO DECREASE TO 1 / e OF THE INITIAL VALUE.

THEREFORE

= 2m /b

ie Ae^-bt/2m = A / e for t =

for energy

E = m / b

2) if the spring constant K is increased then will increase

3) if the spring constant K is increased then the relaxation time will decrease.