Solution;
Given;
w = 16 l b w=16lb w = 16 l b
F ( t ) = 24 s i n ( 10 t ) F(t)=24sin(10t) F ( t ) = 24 s in ( 10 t ) k = 5 l b f t k=5lbft k = 5 l b f t
c = 4 c=4 c = 4
The differential equation;
w g x ¨ = − c x ˙ − k x + F ( t ) \frac{w}{g}\ddot{x}=-c\dot{x}-kx+F(t) g w x ¨ = − c x ˙ − k x + F ( t )
By direct substitution;
32 16 x ¨ + 4 x ˙ + 5 x = 24 s i n ( 10 t ) \frac{32}{16}\ddot{x}+4\dot{x}+5x=24sin(10t) 16 32 x ¨ + 4 x ˙ + 5 x = 24 s in ( 10 t )
Rewrite as;
d 2 x d t 2 + 8 d x d t + 10 x = 48 s i n 10 t \frac{d^2x}{dt^2}+8\frac{dx}{dt}+10x=48sin10t d t 2 d 2 x + 8 d t d x + 10 x = 48 s in 10 t
Which is a second order Linear differential equation;
The auxiliary equation is;
r 2 + 8 r + 10 = 0 r^2+8r+10=0 r 2 + 8 r + 10 = 0
The roots;
r = − 8 − + 64 − 40 2 r=\frac{-8_-^+\sqrt{64-40}}{2} r = 2 − 8 − + 64 − 40
r 1 = − 4 − 6 r_1=-4-\sqrt{6} r 1 = − 4 − 6
r 2 = − 4 + 6 r_2=-4+\sqrt{6} r 2 = − 4 + 6
The complementary solution is;
x c ( t ) = c 1 e r 1 t + c 2 e r 2 t x_c(t)=c_1e^{r_1t}+c_2e^{r_2t} x c ( t ) = c 1 e r 1 t + c 2 e r 2 t
The complementary solution is;
x p ( t ) = 1 D 2 + 8 D + 10 48 s i n 10 t x_p(t)=\frac{1}{D^2+8D+10}48sin10t x p ( t ) = D 2 + 8 D + 10 1 48 s in 10 t
x p ( t ) = 48 − 100 + 8 D + 10 s i n 10 t = 48 ( 8 D + 90 ) ( 8 D + 90 ) ( 8 D − 90 ) s i n 10 t = 48 ( 8 D + 90 ) 64 D 2 − 8100 s i n 10 t x_p(t)=\frac{48}{-100+8D+10}sin10t=\frac{48(8D+90)}{(8D+90)(8D-90)}sin10t=\frac{48(8D+90)}{64D^2-8100}sin10t x p ( t ) = − 100 + 8 D + 10 48 s in 10 t = ( 8 D + 90 ) ( 8 D − 90 ) 48 ( 8 D + 90 ) s in 10 t = 64 D 2 − 8100 48 ( 8 D + 90 ) s in 10 t
Therefore we can write;
x p ( t ) = 48 ( 80 c o s 10 t + 90 s i n 10 t ) 64 ( − 100 ) − 8100 x_p(t)=\frac{48(80cos10t+90sin10t)}{64(-100)-8100} x p ( t ) = 64 ( − 100 ) − 8100 48 ( 80 cos 10 t + 90 s in 10 t )
x p ( t ) = − 48 1450 ( 8 c o s 10 t + 9 s i n 10 t ) x_p(t)=\frac{-48}{1450}(8cos10t+9sin10t) x p ( t ) = 1450 − 48 ( 8 cos 10 t + 9 s in 10 t )
The position of the weight at any time t;
x ( t ) = x c ( t ) + x p ( t ) x(t)=x_c(t)+x_p(t) x ( t ) = x c ( t ) + x p ( t )
x ( t ) = c 1 e r 1 t + c 2 e r 2 t − 48 1450 ( 8 c o s 10 t + 9 s i n 10 t x(t)=c_1e^{r_1t}+c_2e^{r_2t}-\frac{48}{1450}(8cos10t+9sin10t x ( t ) = c 1 e r 1 t + c 2 e r 2 t − 1450 48 ( 8 cos 10 t + 9 s in 10 t
Using the initial conditions;
x = 0 , t = 0 x=0,t=0 x = 0 , t = 0
We have;
0 = c 1 + c 2 − 48 1450 × 8 0=c_1+c_2-\frac{48}{1450}×8 0 = c 1 + c 2 − 1450 48 × 8
c 1 + c 2 = 384 1450 c_1+c_2=\frac{384}{1450} c 1 + c 2 = 1450 384 ...(a)
Differentiating x(t) w.r.t 't' we get;
x ˙ ( t ) = r 1 c 1 e r 1 t + r 2 c 2 e r 2 t − 48 1450 ( − 8 s i n 10 t + 9 c o s 10 t ) \dot{x}(t)=r_1c_1e^{r_1t}+r_2c_2e^{r_2t}-\frac{48}{1450}(-8sin10t+9cos10t) x ˙ ( t ) = r 1 c 1 e r 1 t + r 2 c 2 e r 2 t − 1450 48 ( − 8 s in 10 t + 9 cos 10 t )
Applying the initial conditions;
x ˙ = 0 , t = 0 \dot{x}=0,t=0 x ˙ = 0 , t = 0
0 = r 1 c 1 + r 2 c 2 − 432 1450 0=r_1c_1+r_2c_2-\frac{432}{1450} 0 = r 1 c 1 + r 2 c 2 − 1450 432 ...(b)
Solving (a) and (b) gives;
x ( t ) = 0.96 e − 1.55 t − 0.69 e − 6.45 t − 0.298 s i n 10 t − 0.265 c o s 10 t x(t)=0.96e^{-1.55t}-0.69e^{-6.45t}-0.298sin10t-0.265cos10t x ( t ) = 0.96 e − 1.55 t − 0.69 e − 6.45 t − 0.298 s in 10 t − 0.265 cos 10 t
The transient solution is;
0.96 e − 1.55 − 0.695 e − 6.45 t 0.96e^{-1.55}-0.695e^{-6.45t} 0.96 e − 1.55 − 0.695 e − 6.45 t
The steady state solution is;
− 0.298 s i n 10 t − 0.265 c o s 10 t -0.298sin10t-0.265cos10t − 0.298 s in 10 t − 0.265 cos 10 t
= 0.397 s i n ( 10 t + 3.87 ) =0.397sin(10t+3.87) = 0.397 s in ( 10 t + 3.87 )
From the steady state part;
A m p l i t u d e = 0.397 f t Amplitude=0.397ft A m pl i t u d e = 0.397 f t
P e r i o d = 2 π w = 2 π 10 = 0.6283 s e c Period=\frac{2π}{w}=\frac{2π}{10}=0.6283sec P er i o d = w 2 π = 10 2 π = 0.6283 sec
F r e q u e n c y = 1 T = 1.591 Frequency=\frac{1}{T}=1.591 F re q u e n cy = T 1 = 1.591
Velocity of the weight at any time (t);
x ˙ ( t ) = − 1.5 e 1.55 t + 4.48 e − 6.45 t − 2.98 c o s 10 t + 2.65 s i n 10 t \dot{x}(t)=-1.5e^{1.55t}+4.48e^{-6.45t}-2.98cos10t+2.65sin10t x ˙ ( t ) = − 1.5 e 1.55 t + 4.48 e − 6.45 t − 2.98 cos 10 t + 2.65 s in 10 t
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