The one-dimensional harmonic oscillator has the Lagrangian L = mx˙2 − kx2/2. Suppose you did not know the solution of the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form x(t) =∑j=0 aj cos jωt, (taking t = 0 at a turning point) where ω is the (unknown) angular frequency

diagram, F, is the input force exerted on the cart by the voltage applied to the motor, m, is the mass of the cart. The encoder is used to keep track of the position of the cart on the track. Cart Encoder Mater March 7, 2023 Track X Figure 5: Free body diagram of the cart (ignoring friction) Using Figure 5 and basic Newtonian dynamics you can derive the equations governing the system. 3.3 Motor Dynamics The input to your system is actually a voltage to the cart's motor. Thus, you need to derive the dynamics of the system that converts the input voltage to the foren exertal on the cart. These are the dynamics of the motor. Figure 6 shows a diagram of the electrical components of the motor. Figure 6: Clasic armature circuit of a standard DC motor For this derivation of the motor dynamics we assume the following: • We disregard the motor inductance: IR, so we can use the approximation 0. • Perfect efficiency of the motor and gearbox: -- 1. The torque generated by the motor is proportional…

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3. Calculate and solve the characteristic equation for the following system: (t) -1 | (t) 0 [ 8 ] *) + 10 [ 17 ] x ) = ( [1] -1

4) Consider the system below. Write the equation of motion and calculate the response assuming that the system is initially at rest for the values k₁ 100[N/m], k2= 100[N/m], k3 = 200[N/m], 100[N/m], k5 = 50[N/m], k6: 50[N/m], and m = = [10kg]. Solve the equation of motion k4 = as a differential equation = F(t) = 80 cos (20t) K5 Κι www K3 ΚΑ m www K6 K2

3. Consider the vibrating system described below (t) 4 -2 -2 1 [ói]x+ [ $ ? ]x= 11 * 0 Compute the mass - normalized stiffness matrix, the eigenvalues, the normalized eigenvectors, the matrix P, and show that PTP I and PTKP is the diagonal matrix of eigenvalues A.

(b) An object is vibrating with simple harmonic motion (SHM), x(t) = A sinøt of amplitude, A = 15 cm and frequency, f=2 Hz. [Sebuah objek yang bergetar dengan gerakan harmonik mudah, x(t) = A sinot yang amplitud A = 15 cm dan frekuensi, f = 2 Hz.] (i) Determine maximum acceleration and maximum velocity of the vibrating object. [Tentukan pecutan maksimum and halaju maksimum untuk objek bergetar tersebut.] (ii) Calculate the acceleration and velocity for the displacement vibrating object, x(1) = 12 cm. [Kirakan pecutan dan halaju bagi sesaran objek bergetar, x(t) = 12 cm.]

please solve this question. Question 4: For the mechanical system shown, find the input-output differential equation relating xo to the displacement input xa (t). K₁ 000 000 1,(1)

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The problem is in the photo