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Korolek [52]
3 years ago
15

Problem A. Consider the following initial value problem for a damped driven linear oscillator: m 2 + b** + kx} = f sin(St); x(0)

= a, x'(0) = C, where a,b,c, m, k, f, 12 are constants, and [m] = M, [t] = T, [2] = L. Find the dimensions of a,b,c, k, ſ, and 12.
Mathematics
1 answer:
skad [1K]3 years ago
7 0

Answer:

a = L

b = MT^(-1)

c = LT^(-1)

k = MT^(-2)

f = MLT^(-2)

S = T^(-1)

Step-by-step explanation:

x (0) = a

x is denoted by displacement in vibration analysis hence attains units of x.

Hence, a = L

b is the damping coefficient:

b = \frac{F}{\frac{dx}{dt} } \\= MLT^(-2) / LT^(-1)\\= MT^(-1)

x'(0) = c

dx/dt = velocity hence c attains the units of velocity

c = LT^(-1)

Coefficient k is the stiffness:

k = \frac{F}{x}  = \frac{MLT^(-2)}{L} = MT^(-2)

Coefficient f is the magnitude of the exciting force

F = m*acceleration = MLT^(-2)

Coefficient S is the angular frequency

angular frequency is displacement in radians per seconds; hence,

S = T^(-1)

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Answer:

General equation of line : y = mx+c   --1

Where m is the slope or unit rate

Table 1)

p    d

1      3

2     6

4     12

d = Number of dollars (i.e.y axis)

p = number of pound(i.e. x axis)

First find the slope

First calculate the slope of given points

m = \frac{y_2-y_1}{x_2-x_1}   ---A

(x_1,y_1)=(1,3)

(x_2,y_2)=(2,6)

Substitute values in A

m = \frac{6-3}{2-1}

m = 3

Thus the unit rate is 3 dollars per pound.

So, It matches the box 1 (Refer the attached figure)

Equation 1 : p=3d

\frac{p}{3}=d

Since p is the x coordinate and d is the y coordinate

On Comparing with 1

m = \frac{1}{3}

Thus the unit rate is \frac{1}{3} dollars per pound

So, It matches the box 2 (Refer the attached figure)

Equation 2 : \frac{1}{3}d=3p

d=9p

Since p is the x coordinate and d is the y coordinate

On Comparing with 1

m =9

Thus the unit rate is 9 dollars per pound

So, It matches the box 3 (Refer the attached figure)

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p = number of pound(i.e. x axis)

(x_1,y_1)=(\frac{1}{9},1)

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Substitute values in A

m = \frac{9-1}{1-\frac{1}{9}}

m = \frac{8}{frac{8}{9}}

m = 9

Thus the unit rate is 9 dollars per pound

So, It matches the box 3 (Refer the attached figure)

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