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What is Calculus? Calculus involves mathematics that deals with rates of change that are not constant. In Algebra, you work with constant rates of change. In the formula (rate)(time) = (distance), the rate is a constant rate. There is only one problem...

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A few things move at a constant velocity, but many don’t...

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Here is a case of a sharp decrease in velocity.

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Here, this bungee jumper experiences acceleration, followed by deceleration

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Differential Equations: The Study of Rates of Change In the study of differential equations, we are able to take a rate equation and “solve it”. To “solve” a differential equation means to write the equation in the form that does not contain rates. Have you ever heard of “exponential growth”?

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Exponential Growth Exponential Growth occurs when the rate of growth of some “thing”is directly proportional to the amount of that “thing” present. Example: Plants in a garden grow exponentially. A possible equation describing this growth could be dy/dt = 0.10y where y = the mass of the plant after “t” days and the plant increases in mass approximately 10% (0.10) each day. What is dy/dt?

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dy/dt is the “Rate of Growth” of the plant measured in mass units per time. For example if y = 20grams, dy/dt = 0.1(20) = 2 grams of new growth per day. But the next day, the mass of the plant is about 22 grams so, dy/dt = 0.1(22) = 2.2 grams of growth and so on... The growth rate keeps growing! What is the solution?

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The Solution Using “techniques” from Calculus, we may “solve” the differential equation dy/dt=0.10y to get the equation y = 10e 0.095t where we use y=10 for day 0. A “picture” is worth a lot here!

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Mass of a Garden Plant Graph of y = 10e 0.095t

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How Bad Shocks Affects Car Ride Quality We can use Calculus and Differential Equations to actually simulate the ride of a car with bad shock absorbers! In the suspension system of a car, there are two major components: 1) Springs to cushion the ride. 2) Shocks to “dampen” the bounce.

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Vertical Acceleration in a Car Ride When Hitting a Bump When you hit a bump while driving a car, there is a lot of “up and down” change in position and acceleration and deceleration occurring. We use dy/dt to represent the change in position with respect to time and we use d 2 y/dt 2 to represent acceleration.

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Equation For This System The second order differential equation is m* d 2 y/dt 2 + k d *dy/dt + k s *y = 0 where m = mass of rear end of vehicle k d = the damping coefficient due to the shocks k s = the spring coefficient Note that “damping” is primarily achieved by the shock absorber but additional damping occurs due to frictional heat losses. After looking up a value for “m” and experimentally determining k d and k s, the equation obtained is 24.2*d 2 y/dt 2 + 400*dy/dt + 2400*y = 0

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Comparison of Solutions Graphed Good Shocks 24.2*d 2 y/dt 2 + 400*dy/dt + 2400*y = 0 Bad Shocks 24.2*d 2 y/dt 2 + 200*dy/dt + 2400*y = 0 Very Bad Shocks 24.2*d 2 y/dt 2 + 100*dy/dt + 2400*y = 0 Note: For this problem, the “solutions” are obtained “graphically”.

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1986 Toyota Celica Suspension System Rear Shocks in Good Condition Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds

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0.25 Seconds

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0.50 Seconds

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0.75 Seconds

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1.00 Seconds

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1.25 Seconds

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1986 Toyota Celica Suspension System Rear Shocks in BAD Condition Viscosity is 1/2 as Much Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds

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0.25 Seconds

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0.50 Seconds

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0.75 Seconds

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1.00 Seconds

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1.25 Seconds

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1986 Toyota Celica Suspension System Rear Shocks in VERY BAD Condition Viscosity is 1/4 as Much Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds

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0.2 Seconds

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1.0 Seconds

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1.2 Seconds

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1.4 Seconds

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1.8 Seconds

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2.0 Seconds

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2.2 Seconds

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Mass spring damper system Tutorial Test 1. Derive the differential equation and find the transfer function for the following mass spring damper system.

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