This exam contains 6 pages (including this cover page) and 3 problems. Check to see if any pages are missing
Write your name and student number
Use a pen with black or blue ink. In any case, write neatly
For this exam you can get between 0-100 points
You are not allowed to use any electronic devices and calculators
You are not allowed to use any books or notes
Answers must be provided in the answer sheet that is distributed among you
This exam consists of 3 main parts: part 1 that includes true/false, part 2 includes multiple choice questions, and part 3 that is a comprehensive question in which you shall follow the steps in the question in order to find the final answers
Unless stated otherwise, exclusively the final answer should be provided
Take your time and try to judge the intermediate results in order to spot mistakes early enough.

1. Indicate in the answer form whether each of the statements below is true or false

(2 points) When using D’Alambert principle I can always safely discard reaction forces.

True / False

(2 points) In scleronomic systems, if we use kinematically admissible coordinates we do not need to take into account Lagrange multipliers in the equations of motion.

True / False

(2 points) In scleronomic systems, if we use kinematically admissible coordinates we do not need to take into account Lagrange multipliers in the equations of motion.

True / False

(2 points) The pfaffian form of a scleronomic system is ydx + xdy = 0, this system is non-holonomic.

True / False

(2 points) In scleronomic systems, checking the positive definiteness of the linear stiffness matrix is enough for determining if the system is stable or unstable.

True / False

(2 points) I can always find an equilibrium point for rheonomic systems.

True / False

(2 points) In rheonomic systems the product q˙^T Gq˙ is always greater than zero if G is the gyroscopic matrix and q˙ is the vector of generalized velocities.

True / False

(2 points) A dynamic system has as many equilibrium positions as the number of generalized coordinates used to describe the system.

True / False

(2 points) In the frequency response curve of a multi-degree-of-freedom system, anti-resonance is the point where the system remains stationary.

True / False

(2 points) If a system is damped, the work produced by linear damping forces (of mode r) on a displace- ment described by mode s is not zero.

True / False

(2 points) The value of modal mass for a certain generalized coordinate in a multi-degree-of-freedom system can change depending on how the eigenmodes of the system are scaled.

True / False

2. Multiple choice questions

(4 points) A bead of mass m slides on a smooth circular wire of radius a which is constrained to rotate about vertical diameter with constant angular velocity ω as shown in Figure 1. What is the kinetic energy of this system?

Figure 1: bead on a rotating circle.

T = ¹ ma²θ˙2

B. T = ¹ ma²(θ˙ + ω)²

C. T = ¹ ma²(θ˙2 + ω²a² sin² θ)

D. T = ¹ ma²(θ˙2 + ω²a² cos² θ)

(4 points) For the same bead on rotating circle problem ( Figure 1), what is the equation determining the equilibrium points?
- −ω²a sin² θ + g sin θ = 0
- −ω²a sin θ cos θ + g sin θ = 0
- sin θ = 0
- −ω²a cos θ + g sin θ = 0
- (4 points) If θ = 0 is one of the equilibrium points of the system in Figure 1, then what is the linearized stiffness?
  - mω²a² + mga
  - mga
  - −mω²a² + mga
  - −mω²a² − mga
- (3 points) Determine the number of equilibrium points for the system of Figure 1, if ^aω2 = 2?

1
1. 2
1. 3
1. 4
1. (3 points) For the system of Figure 1 and under the condition where ^aω2

= 2, how many equilibrium

points are stable?

1
1. 2
1. 3
1. 4
1. (3 points) Figure 2 shows a quarter car model with a driver which is modelled by a mass m_d over a linear cushion above the sprung mass m_s. Assuming y = 0, find the mass matrix associated with the acceleration vector [x¨_u, x¨_s, x¨_d]^T ?

A. M = 

B. M = 

C. M = 

Figure 2: quarter car model





_d 0 0 

0 m_u + m_s + m_d

D. M =  

(3 points) For the dynamical system of Figure 2, find the stiffness matrix?

 k_u 0 0 

K = 

0 k_s 0

0 0 k_d

 k_u −k_s 0 

K = −k_s k_s −k_d

0 −k_d k_d

 k_u + k_s k_s 0 

K = 

k_s k_s + k_d k_d

0 k_d k_d

 k_u + k_s −k_s 0 

K = 

−k_s k_s + k_d −k_d

0 −k_d k_d

(3 points) For the dynamical system of Figure 2 if the car passes a bumpy road with profile y = Y sin ωt where Y is the amplitude of the bumps and ω their frequency, which expression best describes the external forces acting on the car from the road bumps?

 k_uY sin ωt c_sY ω sin ωt c_dY ω cos ωt 

c_sY ω sin ωt k_sY cos ωt c_dY ω sin ωt

c_dY ω cos ωt c_dY ω sin ωt k_dY sin ωt

 k_uY sin ωt + c_uY ω cos ωt 0 0 

0 k_sY sin ωt + c_sY ω sin ωt 0

0 0 k_dY sin ωt + c_dY ω cos ωt

 k_uY sin ωt + c_uY ω cos ωt 

k_sY sin ωt + c_sY ω cos ωt k_dY sin ωt + c_dY ω cos ωt

 k_uY sin ωt + c_uY ω cos ωt 

Comprehensive question

A spring pendulum as shown in Figure 3 has a mass m suspended by an elastic spring of stiffness k and natural length L. The system lays on the vertical plane and gravity g is acting in y direction. Choose R and θ as the generalized coordinates, and

Figure 3: The spring pendulum

(8 points) write the kinetic energy and potential energy of the system?
- (15 points) Obtain the governing equations using Hamilton’s principle. Include your derivation in the answer sheet

(8 points) Assuming 3kL/2 = mg, first derive the equations that lead to equilibrium positions and then find the equilibrium positions.
- (8 points) Discuss the stability of equilibrium points found earlier.
- (4 points) write the linearized equations about the stable equilibrium point(s). (f) (10 points) Compute the eigenfrequencies and eigenmodes of the system.