Some typical choices of(f(dot u) ), (s(u)), and (F(t)) in (90) are listed below: Linear...

 

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Some typical choices of(f(\dot u) ), (s(u)), and (F(t)) in (90) are listed below: • Linear friction force (low velocities): (f(dot u) = 6 pi\mu R \dot u) (Stokes drag), where (R) is the radius of a spherical approximation to the body's geometry, and \(\mu) is the viscosity of the surrounding fluid. • Quadratic friction force (high velocities): (f(\dot u) = \frac {1}{2} C_D \varrho A\dot u\dot u). Here, (C_D) is a drag coefficient, \(\varrho) is the density of the fluid, and (A) is the cross section area of the body perpendicular to the flow. • Linear spring force: \(s(u)-ku), where (k) is a spring constant. • Sinusoidal spring force: (s(u)-k\sin u), where (k) is a constant. • Cubic spring force: (s(u)-k(u- \frac{1}{6}u^3) \), where (k) is a spring constant. Sinusoidal external force: \(F(t)=F_0+ A\sin omega t \), where (F_0\) is the mean value of the force, (A) is the amplitude, and \(\omega) is the frequency. Bump force: \(F(t)= H(t-t_1)(1-H(t-t_2))F_0 \), where (H(t)\) is the Heaviside function ((H=0\) for (x<0 ) and \(H=1 ) for (x\geq 0)). (t_1\) and (t_2\) are two given time points, and (F_0\) is the size of the force. This (F(t)\) is zero for \(t<t_1\) and (t>t_2\), and (F_0\) for \(t\in [t_1.t_2] \). Random force 1: (F(t)-F_0+A\cdot U(t;B)), where (F_0\) and (A) are constants, and (U(t;B) \) denotes a function whose value at time (t) is random and uniformly distributed in the interval ([-B.B]\). • Random force 2: (F(t)=F_0+A\cdot N(t; mu, sigma)), where (F_0\) and (A) are constants, and (N(t; mu, \sigma) \) denotes a function whose value at time (t) is random, Gaussian distributed number with mean ( \mu) and standard deviation \(\sigma). Make a module functions where each of the choices above are implemented as a class with a _call_ special method. Also add a class Zero for a function whose value is always zero. It is natural that the parameters in a function are set as arguments to the constructor. The different classes for spring functions can all have a common base class holding the (k) parameter as data attribute. Filename: functions. Some typical choices of(f(\dot u) ), (s(u)), and (F(t)) in (90) are listed below: • Linear friction force (low velocities): (f(dot u) = 6 pi\mu R \dot u) (Stokes drag), where (R) is the radius of a spherical approximation to the body's geometry, and \(\mu) is the viscosity of the surrounding fluid. • Quadratic friction force (high velocities): (f(\dot u) = \frac {1}{2} C_D \varrho A\dot u\dot u). Here, (C_D) is a drag coefficient, \(\varrho) is the density of the fluid, and (A) is the cross section area of the body perpendicular to the flow. • Linear spring force: \(s(u)-ku), where (k) is a spring constant. • Sinusoidal spring force: (s(u)-k\sin u), where (k) is a constant. • Cubic spring force: (s(u)-k(u- \frac{1}{6}u^3) \), where (k) is a spring constant. Sinusoidal external force: \(F(t)=F_0+ A\sin omega t \), where (F_0\) is the mean value of the force, (A) is the amplitude, and \(\omega) is the frequency. Bump force: \(F(t)= H(t-t_1)(1-H(t-t_2))F_0 \), where (H(t)\) is the Heaviside function ((H=0\) for (x<0 ) and \(H=1 ) for (x\geq 0)). (t_1\) and (t_2\) are two given time points, and (F_0\) is the size of the force. This (F(t)\) is zero for \(t<t_1\) and (t>t_2\), and (F_0\) for \(t\in [t_1.t_2] \). Random force 1: (F(t)-F_0+A\cdot U(t;B)), where (F_0\) and (A) are constants, and (U(t;B) \) denotes a function whose value at time (t) is random and uniformly distributed in the interval ([-B.B]\). • Random force 2: (F(t)=F_0+A\cdot N(t; mu, sigma)), where (F_0\) and (A) are constants, and (N(t; mu, \sigma) \) denotes a function whose value at time (t) is random, Gaussian distributed number with mean ( \mu) and standard deviation \(\sigma). Make a module functions where each of the choices above are implemented as a class with a _call_ special method. Also add a class Zero for a function whose value is always zero. It is natural that the parameters in a function are set as arguments to the constructor. The different classes for spring functions can all have a common base class holding the (k) parameter as data attribute. Filename: functions.

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