A mass is subject to a nonlinear spring force as it moves ho

A mass is subject to a nonlinear spring force as it moves horizontally along a rigid constraint. The rest length of each spring is L_0. Parameters m, c, k, L_0 and H are all positive constants. The equation of motion is: m x^dot dot (1 - L_0/Squareroot H^2 + x^2) x= 0. Find all equilibria (fixed points) of the system. Under what conditions does more than one fixed point exist? Determine the stability of each fixed point (if necessary, express stability in terms of parameters).What type of bifurcation occurs as L_0 is increased (with all other parameters held constant). Plot the bifurcation diagram.

Solution

solution:

here given mass spring system has equation of motion as

mx\'\'+2*k*(1-Lo/(H2+x2)^.5)x=0

where length after elongation=L=(H2+x2)^.5

above equation become

mx\'\'+2*k*(1-Lo/L)x=0

at any instant L is changing and because of which system is oscillating with variable natural frequency as distance from mean position increased.

hence system will be highly unstable for horizontal oscillation as long x has non zero value as it affect the length of spring.

2) hence system would be in equillibria at single point when L is not depend on x ,and it is posible at only central point for first case and another two cases stated follow

for equillibria of system x=0

3) system in laplace form can be reduced as

(ms^2+2*k*(1-Lo/L))x(s)=0

hence root of this eqution as

s=+-(2*(k/m)*(1-Lo/L))^.5*i

as root are imaginary hence on bode plot sytem has root on imaginary axis and it is indicating that

relative stability which is measured as distance from imaginary axis and in this case distance from imaginary axis is zero.

relative stability=0 unit

, which indicate that sytem is highly unstable. and

sytem would be stable for L as constant for

s=+-(2*(k/m)*(1-Lo/L))^.5*i

4)if value of Lo is increased then roots of equation of motion has some real value,it is on negative side means in LHP and it is indication of stable system,here bifurcation as

1) here either Lo can be increased

2)or spring can be compressed

5) hence for stability of system we must increased Lo length of spring or let it to operate in compression then system would have stability.