Control Systems Example Problems PDF

# Mechanical System Components: ## **Spring** - The equation for the force generated by a spring is *F(t) = K x(t)*, where: - *K* represents the stiffness factor - *x(t)* represents the displacement - The equation relating force in the Laplace domain is *F(s) = K X(s)*. ## **Friction*...

# Mechanical System Components: ## **Spring** - The equation for the force generated by a spring is *F(t) = K x(t)*, where: - *K* represents the stiffness factor - *x(t)* represents the displacement - The equation relating force in the Laplace domain is *F(s) = K X(s)*. ## **Friction** - The equation for the force generated by the friction force is *F(t) = B x(t)*, where: - *B* represents the friction coefficient - *x(t)* represents the velocity - The equation relating force in the Laplace domain is *F(s) = Bs x(s)*. ## **Damper** - The force generated by the damper is represented by *F(t) = Bx(t)*, where: - *B* represents the damping constant - *x(t)* represents the velocity - The equation relating force in the Laplace domain is *F(s) = Bs x(s)*. ## **Transitional System** The goal of this system is to find the relationship between input and output. **Transfer Function** (T.F): *T.F. = x(s)/F(s)* **Newton's Second Law** *ΣF = m. a* **Acceleration:** *x(t)* **Example 1: Find the Transfer Function (T.F.)** * **Free body diagram:** This diagram shows the forces acting on the mass *m*. The forces include *F(t)*, *Kx(t)*, and *Bx(t).* * **Applying Newton's Second Law:** *ΣF= m.a* * **Simplifying the equation:** *F(t) - Kx(t) - Bx(t) = m. x(t)* * **Applying Laplace Transform:** *F(s) - KX(s) - BSX(s) = ms² X(s)* * **Rearranging to find the Transfer Function:** *F(s) = x(s) [ms² + BS + K]* * **The Transfer Function:** *T.F = x(s)/F(s) = 1/(ms² + BS + K)* ## **Example 2: Find the Transfer Function (T.F.)** * **Free body diagram:** * **Applying Newton's Second Law:** *ΣF= m.a* * **Simplifying the equation:** *F(t)-KX(t) -Bx(t) = m. x(t)* * **Applying Laplace Transform:** *F(s)=x(s) [ms² +BS+K]* * **The Transfer Function:** *T.F = x(s)/F(s) = 1/(ms² + BS + K)* ## **Example 3: Find the Transfer Function (T.F.)** * **System:** This system consists of two masses, *m1* and *m2*, connected by springs and dampers. The system is subjected to an input force *F(t)*, and the output is the displacement of the second mass *x(t)*. * **For Mass m1:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m1*:** *K1(x-y) -K2(y-z) - B1y = m1.ÿ* * **Applying Laplace Transform:** *K1(x-y) - K2(y-z) - B1ys = m1s²y* * **Simplifying and rearranging:** *K1x(s) - K2y(s) = y(s) [m1s² + B1S + K2 + K1]* * **For Mass m2:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m2*:** *K2(y-z) - K3z - B2ż = m2. ż* * **Applying Laplace Transform:** *K2(y-z)- K3z - B2sz = m2s²z* * **Simplifying and rearranging:** *K2y(s) = z(s) [m2s² + B2S + K3 + K2]* * **Finding the relationship between x(s), y(s), and z(s):** * **From the equation for mass m1:** *K1x(s) + K2z(s) = y(s) [m1s² + B1S + K2 + K1]* * **From the equation for mass m2:** *K2y(s) = z(s) [m2s² + B2S + K3 + K2]* * **Solving for the Transfer Function:** * **The transfer function from x(s) to y(s):** *G1 = K1 / [m1s² + B1S + K2 + K1]* * **The transfer function from y(s) to z(s):** *G2 = K2 / [m2s² + B2S + K3 + K2]* * **The transfer function from z(s) to x(s):** *H = K2 / [m2s² + B2S + K3 + K2]* * **The final transfer function:** *T.F = G1G2 / (1-G1G2H)* ## **Example 4: Find the Transfer Function (T.F.)** * **System:** This system is similar to the system in example 3, but with a different arrangement of components. The two masses are connected by springs and dampers, with the input force being applied to the first mass, *m1*, and the output being the displacement of the second mass *m2*, *y(t)*. * **For Mass m1:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m1*:** *F(t) - K1(x-y) - B1x = m1.ÿ* * **Applying Laplace Transform:** *F(s) - K1(x-y) - B1sx = m1s²x* * **Simplifying and rearranging:** *F(s) + K1y(s) = x(s) [m1s² + B1S + K1]* * **For Mass m2:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m2*:** *K1(x-y) - K2y - B2ÿ = m2. ÿ* * **Applying Laplace Transform:** *K1(x-y) - K2y - B2sy = m2s²y* * **Simplifying and rearranging:***K1x(s) = y(s) [m2s² + B2S + K1 + K2]* * **Finding the relationship between x(s) and y(s):** * **From the equation for mass m1:** *F(s) + K1y(s) = x(s) [m1s² + B1S + K1]* * **From the equation for mass m2:** *K1x(s) = y(s) [m2s² + B2S + K1 + K2]* * **Solving for the Transfer Function:** * **The transfer function from x(s) to y(s):** *G1 = K1 / [m1s² + B1S + K1]* * **The transfer function from y(s) to x(s):** *G2 = K1 / [m2s² + B2S + K1 + K2]* * **The final transfer function:** *T.F = G1G2 / (1-G1G2H)* ## **Example 5: Find the Transfer Function (T.F.)** * **System:** This system consists of two masses, *m1* and *m2*, connected by springs. There is no friction in this system. The input is the force *u(t)* applied to the first mass, *m1*, and the output is displacement *y(t)*. * **For Mass m1:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m1*:** *u(t) - K1x1 - K2(x1-x2) = m1. x1* * **Applying Laplace Transform:** *u(s) - K1x1(s) - K2(x1(s)-x2(s)) = m1s²x1(s)* * **Simplifying and rearranging:** *u(s) + x2(s)[K2 + bs] = x1(s) [m1s² + bs + K1 + K2]* * **For Mass m2:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m2*:** *K2(x1-x2) + b(x1-x2) - K3x2 = m2. x2* * **Applying Laplace Transform:** *K2(x1(s)-x2(s)) + b(x1(s)-x2(s)) - K3x2(s) = m2s²x2(s)* * **Simplifying and rearranging:** *x1(s)[K2 + bs] = x2(s) [m2s² + bs + K2 + K3]* * **Finding the relationship between x1(s) and x2(s):** * **Equation for mass m1:** *u(s) + x2(s)[K2 + bs] = x1(s) [m1s² + bs + K1 + K2]* * **Equation for mass m2:** *x1(s)[K2 + bs] = x2(s) [m2s² + bs + K2 + K3]* * **Solving for the Transfer Function:** * **The transfer function from u(s) to x1(s):** *G1 = 1 / [m1s² + bs + K1 + K2]* * **The transfer function from x1(s) to x2(s):** *G2 = [K2 + bs] / [m2s² + bs + K2 + K3]* * **The final transfer function:** *T.F = G1G2 / (1-G1G2H)* * **Simplifying the final transfer function:** *T.F = (K2 + bs) / [(m1s² + bs + K1 + K2)(m2s² + bs + K2 + K3) - (K2 + bs)²]* ## **Example 6: Find the Transfer Function (T.F.)** * **System:** This system consists of two masses, *m1* and *m2*, connected by springs and dampers. The input is the force *F(t)* applied to the first mass, *m1*, and the output is the displacement of the second mass *m2*, *y(t)*. * **For Mass m1:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m1*:** *F(t) - K1(x-y) - b1(x-y) = m1. x1* * **Applying Laplace Transform:** *F(s) - K1(x(s)-y(s)) - b1s(x(s)-y(s)) = m1 s²x(s)* * **Simplifying and rearranging:** *F(s) + y(s)[K1 + b1s] = x(s) [m1s² + b1s + K1]* * **For Mass m2:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m2*:** *K1(x-y) - K2y - b2y = m2. ÿ* * **Applying Laplace Transform:** *K1(x(s)-y(s)) - K2y(s) - b2sy(s) = m2s²y(s)* * **Simplifying and rearranging:** *x(s)[K1 + b1s] = y(s) [m2s² + b2S + b1s + K1 + K2]* * **Finding the relationship between x(s) and y(s):** * **Equation for mass m1:** *F(s) + y(s)[K1 + b1s] = x(s) [m1s² + b1s + K1]* * **Equation for mass m2:** *x(s)[K1 + b1s] = y(s) [m2s² + b2S + b1s + K1 + K2]* * **Solving for the Transfer Function:** * **The transfer function from x(s) to y(s):** *G1 = 1 / [m1s² + b1s + K1]* * **The transfer function from y(s) to x(s):** *G2 = [K1 + b1s] / [m2s² + b2S + b1s + K1 + K2]* * **The final transfer function:** *T.F = G1G2 / (1-G1G2H)* * **Simplifying the final transfer function:** *T.F = (K1 + b1s) / [(m1s² + b1s + K1)(m2s² + b2S + b1s + K1 + K2) - (K1 + b1s)²]* ## **Example 7: Find the Transfer Function (T.F.)** * **System:** This system consists of two masses, *m1* and *m2*, connected by springs and dampers. The input is a force *u1(t)* applied to the first mass, *m1*, and a force *u2(t)* applied to the second mass, *m2*. The output is the displacement of the second mass, *m2*, *y2(t)*. * **For Mass m1:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m1*:** *u1 - K1y1 - b1(y1-y2) = m1. ÿ1* * **Applying Laplace Transform:** *u1(s) - K1y1(s) - b1s(y1(s)-y2(s)) = m1s²y1(s)* * **Simplifying and rearranging:** *u1(s) + b1sy2(s) = y1(s)[m1s² + b1s + K1]* * **Rearranging:** *u1(s)y1(s)[m1s² +b1s + K1] = b1sy2(s)* * **For Mass m2:** * **Newton's second law:** *ΣF = ma* * **Forces acting on *m2*:** *u2 + b(y1-y2) - K2y2 = m2. ÿ2* * **Applying Laplace Transform:** *u2(s) + b(y1(s)-y2(s)) - K2y2(s) = m2s²y2(s)* * **Simplifying and rearranging:** *u2(s) + y1(s)bis = y2(s) [m2s² + b1s + K2]* * **Finding the transfer function from u1(s) to y2(s):** * **Set u2(s) equal to 0:** *u2(s) = 0* * **Solving for y2(s):** *y2(s) = (u1(s)G1G2G3) / (1-G1G2G3H). * **Where:** * *G1 = 1 / [m1s² + b1s + K1]* * *G2 = b1s / [m2s² + b1s + K2]* * *G3 = 1/ b1s* * *H = [m1s² + b1s + K1]/ [m2s² + b1s + K2]* * **Finding the transfer function from u2(s) to y2(s):** * **Set u1(s) equal to 0:** *u1(s) = 0* * **Solving for y2(s):** *y2(s) = (u2(s)G1G2G3) / (1-G1G2G3H). * **Where:** * *G1 = [m1s² + b1s + K1] / b1s* * *G2 = 1 / [m2s² + b1s + K2]* * *G3 = b1s* * *H = [m1s² + b1s + K1]/ [m2s² + b1s + K2]* ## **Example 8: Electromechanical System** * **System:** This system consists of a spring, damper, mass, resistor, inductor, and voltage supply. The input is voltage *Vin(t)* and the output is the displacement of the mass *x(t)*. * **Applying Kirchhoff's Voltage Law (KVL) to the electrical circuit:** *Uin(t)= I(t).R+LdI(t)/dt + eb(t)* * **Applying Laplace Transform:** *Uin(s) = I(s) [R+Ls]+eb(s)* * **Relating eb(t) to x(t):** *eb(t) = Kb x(t)* * **Relating eb(s) to x(s):** *eb(s)= Kb sx(s)* * **Applying Newton's Second Law to the mechanical system:** *ΣF=ma* * **Forces acting on the mass:** *F(t) - Kx - Bx = m. x* * **Applying Laplace Transform:** *F(s) = x(s)[ms² + Bs + K]* * **Relating F(t) to I(t):** *F(t) = K1 I(t)* * **Relating F(s) to I(s):** *F(s) = K1 I(s)* * **Solving for the Transfer Function:** * **The transfer function from u1(s) to x(s):** *T.F = x(s) / U1(s) = (G1G2G3H) / (1 + G1G2G3H) * * **Where:** * *G1 = 1 / [R+Ls]* * *G2 = K1* * *G3 = 1 / [ms² + Bs + K]* * *H = Kb / {Kb s)*