Problem 5: For the system shown below, write the differential equations for small motions of the ...
Problem 6: For the two systems shown below, separately, identify the degrees of freedom and then write the equations of motion, respectively. Also, for each system, determine the natural frequency and damping ratio. The two systems shown are set into motion via initial condition. For the first figure of problem 6 (the circular disc), the disc is performing fixed axis rotation about its center of mass, G. It has a radius of gyration kG about the axis through the center...
Write the differential equation of motion for the system shown in the figure, and find the damped natural frequency and damping ratio of this system.
Q1- For the system shown below, with small mass of value (m) and lever of mass moment of inertia (J). • find equivalent mass, equivalent stiffness, and equivalent damping, all these interms of (x) displacement . Get equation of motion Interms of these equivalent quantities. • Find natural frequency (Wn) and damping ratio (zeta). • Find X(t) when the system condition is critically damping ,,X(0)=M and v(0)=0. tinfring
Model for Evaluation The model used for evaluation is the single degree of freedom lumped mass model defined by second order differential equation with constant coefficients. This model is shown in Figure 1. x(t)m m f(t) Figure 1 - Single Degree of Freedom Model The equation of motion describing this system can easily be shown to be md-x + cdx + kx = f(t) dt dt where m is the mass, c is the damping and k is the stiffness...
M[kg] CN) ( 1. Derive the differential equations of motion for the system (two degrees of freedom). Let the angle 9 be small. A (4] 6 [Na] e mikg]
For the system shown in Figure 6, a. How many degrees of freedom is this system and why? b. Write the equations of motion. For the remainder parts, assume alll the dampers are removed: c. If Ki=K3 and mim3, set the necessary matrix to find the natural frequencies and mode shapes d. For part c above, determine and explain how to get the natural frequencies. m1 Ty Absorber тз k1 С1 k3 m2 C2 For the system shown in Figure...
7. 150 points) A one-degree-of-freedom system is shown below. (a) (50 points) Derive the differential equation governing the motion of the system usingq, the (b) (25 points) what are the natural frequency and damping ratto of the system? c) (25 points) Mc)-0 (d) (25 points) (e) (25 points) If M(t) =1.2 sin m N clockwise angular displacement of the disk from equilibrium as the generalized coordinate. 10° and the system is given an initial angulan released from rest what is...
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended from springs with spring constants ki and k, respectively. Assume that there is no damping in the system. a) Show that the displacements z1 and 2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above resuit to show that the spring-mass system satisfies the following fourth order differential equation. and ) Find the general solution...
Problem 5 (20%) For the system shown in Figure 5, a. How many degrees of freedom is this system and why? (5) b. If x3 0 (the upper end is fixed and K1 and K2=K Write the equations of motion. Set the necessary matrix to find the natural frequencies and mode shapes (5) (5) (5) 1. 2. 3. Determine and explain how to get the natural frequencies. m2 Figure 5 www Problem 5 (20%) For the system shown in Figure...
3. Consider the spring - mass system shown below, consisting of two masses mi and m2 sus- pended from springs with spring constants ki and k2, respectively. Assume that there is no damping in the system. a) Show that the displacements ai and r2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above result to show that the spring-mass system satisfies the following fourth order differential equation and c) Find the general solution...