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Question: Non Simple Harmonic Pendulum Motion. Differential Equation contains "d__a" as a derivative operator.

Question:Non Simple Harmonic Pendulum Motion. Differential Equation contains "d__a" as a derivative operator.

Ronan 1207 Maple 2016
differential-equation
January 23 2017
0

The first half of this work sheet deals with SHM of pendulum. In the second half of the work sheet I attempt to solve for the general case of a swinging pendulum. Maple introduces a place holder (correct me if I have used the incorrect termonology) " __a" which I do not understand. What variable(s) should I replace it with and is there an automatic way of doing so?
 

restart

NULL

``

``

Simple*Harmonic*Motion*of*a*Pendulum

NULL

Eq1 := diff(Theta(t), t, t) = -omega^2*Theta(t)

diff(diff(Theta(t), t), t) = -omega^2*Theta(t)

 (1)

ics := Theta(0) = 0, (D(Theta))(0) = Vmax

Theta(0) = 0, (D(Theta))(0) = Vmax

 (2)

SHM := dsolve({Eq1, ics})

Theta(t) = Vmax*sin(omega*t)/omega

 (3)

diffSHM := diff(SHM, t)

diff(Theta(t), t) = Vmax*cos(omega*t)

 (4)

convert(diffSHM, D)

(D(Theta))(t) = Vmax*cos(omega*t)

 (5)

eval[recurse](%, {t = 0, (D(Theta))(0) = Vmax})

Vmax = Vmax

 (6)

solve(%, {_C1})

{_C1 = _C1}

 (7)

assign(%); _C1

_C1

 (8)

``

SHM

Theta(t) = Vmax*sin(omega*t)/omega

 (9)

``

(General*Equation*of*Motion*of)*a*Pendulum

restart

diff(Theta(t), t, t) = -omega^2*sin(Theta(t))

diff(diff(Theta(t), t), t) = -omega^2*sin(Theta(t))

 (10)

ics := Theta(0) = 0, (D(Theta))(0) = Vmax

Theta(0) = 0, (D(Theta))(0) = Vmax

 (11)

Sol := dsolve(diff(Theta(t), t, t) = -omega^2*sin(Theta(t)))

Intat(1/(2*omega^2*cos(_a)+_C1)^(1/2), _a = Theta(t))-t-_C2 = 0, Intat(-1/(2*omega^2*cos(_a)+_C1)^(1/2), _a = Theta(t))-t-_C2 = 0

 (12)

Sol[1]

Intat(1/(2*omega^2*cos(_a)+_C1)^(1/2), _a = Theta(t))-t-_C2 = 0

 (13)

_C2 := 0

0

 (14)

Sol[1]

Intat(1/(2*omega^2*cos(_a)+_C1)^(1/2), _a = Theta(t))-t = 0

 (15)

``

dffSol[1] := diff(Sol[1], t)

(diff(Theta(t), t))/(2*omega^2*cos(Theta(t))+_C1)^(1/2)-1 = 0

 (16)

``convert(dffSol[1], D)

(D(Theta))(t)/(2*omega^2*cos(Theta(t))+_C1)^(1/2)-1 = 0

 (17)

 

eval[recurse](%, {t = 0, Theta(0) = 0, (D(Theta))(0) = Vmax})

Vmax/(2*omega^2+_C1)^(1/2)-1 = 0

 (18)

solve(%, {_C1})

{_C1 = Vmax^2-2*omega^2}

 (19)

assign(%); 1; _C1

Vmax^2-2*omega^2

 (20)

dffSol[1]

(diff(Theta(t), t))/(2*omega^2*cos(Theta(t))+Vmax^2-2*omega^2)^(1/2)-1 = 0

 (21)

``

dsolve(dffSol[1]); 1; SOL1 := int((diff(Theta(t), t))/sqrt(2*omega^2*cos(Theta(t))+Vmax^2-2*omega^2)-1, t = 0 .. Theta(t)) = 0

int((diff(Theta(t), t))/(2*omega^2*cos(Theta(t))+Vmax^2-2*omega^2)^(1/2)-1, t = 0 .. Theta(t)) = 0

 (22)

Sol[1]

Intat(1/(2*omega^2*cos(_a)+Vmax^2-2*omega^2)^(1/2), _a = Theta(t))-t = 0

 (23)

"Using ( 1-cos(theta))/(2)=sin(theta/(2))^(2)and substituting by hand"

NotsoSHM := Intat(1/(Vmax*sqrt(1-2*omega^2*sin((1/2)*_a)^2/Vmax^2)), _a = (1/2)*Theta(t))-t = 0

Intat(1/(Vmax*(1-2*omega^2*sin((1/2)*_a)^2/Vmax^2)^(1/2)), _a = (1/2)*Theta(t))-t = 0

 (24)

``

``


 

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